{"_id": "1735490", "title": "", "text": "$d(x, z) ≤ d(x, y) + d(y, z)$"}
{"_id": "8319375", "title": "", "text": "$f(x)=\\sum_{n=1}^{N}b_{n}\\chi_{B_{n}}(x)$"}
{"_id": "1711213", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac{1+2\\sqrt2+3\\sqrt3+\\ldots+n\\sqrt n}{n^2 \\sqrt{n}}= \\text{?}$"}
{"_id": "5606866", "title": "", "text": "$|x^2 + y| + |y^2 + z| + |z^2 + x| < |x| + |y| + |z|$"}
{"_id": "1156514", "title": "", "text": "$ \\dfrac 1{\\sqrt1} + \\dfrac{1}{\\sqrt2} +....+\\dfrac 1{\\sqrt{k}} $"}
{"_id": "6561561", "title": "", "text": "$\\forall\\epsilon\\gt0,\\exists\\,\\delta\\gt0:\\forall x,y\\in\\mathbb{R},|x-y|\\le\\delta\\implies|f(x)-f(y)|\\le\\epsilon\\tag{1}$"}
{"_id": "957651", "title": "", "text": "$1.2.20, \\operatorname{ord(a)} = n \\ldots$"}
{"_id": "3824936", "title": "", "text": "$x^3-3x^2+3x+r, \\  \\ r\\in\\mathbb R.$"}
{"_id": "1475020", "title": "", "text": "$y_n = \\frac{(x_1 + x_2 + ... + x_n)}{n}$"}
{"_id": "273587", "title": "", "text": "$E_i=Span\\{e_1,e_2, ...., e_i\\}$"}
{"_id": "5907538", "title": "", "text": "$\\left(\\frac{1}{k}\\right)^{n - 2}$"}
{"_id": "6504411", "title": "", "text": "$4^x+9^x+25^x=6^x+10^x+15^x$"}
{"_id": "7720566", "title": "", "text": "$x = \\cos(\\varphi), y = \\sin(\\varphi) \\cos(\\varphi)$"}
{"_id": "7553356", "title": "", "text": "$\\{g, g+N, g+2N, \\dots \\}$"}
{"_id": "2344634", "title": "", "text": "$d=\\gcd(a+b,a^2+b^2)$"}
{"_id": "3706197", "title": "", "text": "$A_1\\subseteq A_2\\subseteq A_3\\subseteq\\cdots\\text{ and }A:=\\bigcup_{n=1}^{\\infty}A_n$"}
{"_id": "2086418", "title": "", "text": "$\\tan(\\angle\\text{A})=\\frac{\\text{a}}{\\text{b}}\\space\\space\\wedge\\space\\space\\tan(\\angle\\text{B})=\\frac{\\text{b}}{\\text{a}}$"}
{"_id": "6678700", "title": "", "text": "$f(K)⊂[f(a),f(b)]$"}
{"_id": "3687012", "title": "", "text": "$f_3(n)=nlog_2n$"}
{"_id": "7391499", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\sum_{i=1}^\\infty |x^{(n)}_i| = 0$"}
{"_id": "5940301", "title": "", "text": "$1/\\frac{dy}{dx}\\mid_{y=-1}$"}
{"_id": "8527655", "title": "", "text": "$\n \\sum_{n = 1}^{\\infty} e^{-n^2 \\pi x} = \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} e^{-n^2 \\pi x } - \\frac{1}{2} := \\frac{1}{2} f(x) - \\frac{1}{2}.\n $"}
{"_id": "3314964", "title": "", "text": "$e_i=c_1e_1'+....+c_ie_i'+........c_ne_n' \\ $"}
{"_id": "3347669", "title": "", "text": "$d=ax+by,$"}
{"_id": "837966", "title": "", "text": "$2x \\sqrt{1-x^2} = 2\\sin(t) \\cos(t) = \\sin(2t)$"}
{"_id": "6246504", "title": "", "text": "$c^{log_b(a)}=p$"}
{"_id": "271308", "title": "", "text": "$\\mathbb{Z}/(2)$"}
{"_id": "7307530", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{n^{11}}{e^{n\\pi}-1}-16^3\\sum_{n=1}^{\\infty}\\frac{n^{11}}{e^{4n\\pi}-1}=\\frac{691}{16}$"}
{"_id": "2496362", "title": "", "text": "$f(n) = \\frac{(n+3)(n+2)(n+1)(n)(n-1)!}{24(n-1)!} = \\frac{(n+3)(n+2)(n+1)(n)}{24}$"}
{"_id": "569761", "title": "", "text": "$\\forall \\epsilon > 0, \\ \\exists \\ \\delta > 0: \\ 0 < |x - a| < \\delta \\implies |f(x) - l| < \\epsilon$"}
{"_id": "179543", "title": "", "text": "$\\frac{40}{\\sqrt{69}}\\approx4.82\\text{ ft/s}$"}
{"_id": "33244", "title": "", "text": "$p^n - p^{n-1}$"}
{"_id": "2346901", "title": "", "text": "$\\int_{-\\infty}^\\infty \\psi(t)\\cos(t)\\; dt = \\int_{-\\infty}^\\infty \\psi(t) \\sin(t)\\; dt = 0$"}
{"_id": "1057285", "title": "", "text": "$Y,Y,Y,Y,Y,Y$"}
{"_id": "4496053", "title": "", "text": "$P(a) \\rightarrow P(a+1)$"}
{"_id": "929057", "title": "", "text": "$ F(x):=\\int_a^xf(t)dt $"}
{"_id": "5426553", "title": "", "text": "$\\lfloor{\\frac x m}\\rfloor=\\lfloor\\frac n m\\rfloor$"}
{"_id": "247823", "title": "", "text": "$(55,2)=1$"}
{"_id": "4315264", "title": "", "text": "$f(n)=n^2-2$"}
{"_id": "9287350", "title": "", "text": "$\\vartheta(z) = e^{p(z)}$"}
{"_id": "1700354", "title": "", "text": "$0=\\lim_{x\\to \\infty}f'(x)=\\lim_{x\\to \\infty}f'(c_x) =\\lim_{x\\to \\infty}[f(x)-f(x+1)] $"}
{"_id": "1413434", "title": "", "text": "$P\\oplus Q=F$"}
{"_id": "7433358", "title": "", "text": "$f'(x) = \\frac{5^x}{5^x + 1} ~~~~~~~~~~~ \\to ~~~~~ f(x) = \\frac{\\ln(5^x + 1)}{\\ln(5)}$"}
{"_id": "6553936", "title": "", "text": "$f(x)=\\frac{e^{-x}(x+e^{-x-1})}{(1-e^{-x-1})^2}$"}
{"_id": "9024646", "title": "", "text": "$|f(z)|<K|\\phi (z)|$"}
{"_id": "502577", "title": "", "text": "$\\chi_{\\mathcal{A}}=(x-1)(x-(n-1))^{n-1}$"}
{"_id": "8826016", "title": "", "text": "$I_n=\\int_0^{\\frac {\\pi}{2}} \\frac {\\sin ((2n+1)x)dx}{\\sin x}$"}
{"_id": "7100584", "title": "", "text": "$p_1\\times p_2=0$"}
{"_id": "5964133", "title": "", "text": "$ \\sum_{n=-\\infty}^\\infty e^{i n x} = 2 \\pi \\sum_{n=-\\infty}^\\infty \\delta(x - 2 \\pi n) $"}
{"_id": "1578712", "title": "", "text": "$f(x) = \\frac{4^x}{x^3}$"}
{"_id": "137736", "title": "", "text": "$\\{e_1,e_2,\\dots,e_n\\}$"}
{"_id": "2623858", "title": "", "text": "$a\\,b=|BD|$"}
{"_id": "4694763", "title": "", "text": "$\\,\\displaystyle \\left(1+\\frac{x}{s}\\right)^s\\approx e^x$"}
{"_id": "7074917", "title": "", "text": "$(1/6^w)(1/4^{26+w})$"}
{"_id": "4024330", "title": "", "text": "$\\tilde{\\mathbb{R}}_{\\infty} \\subset \\tilde{\\mathbb{R}}_{\\infty}^{\\tilde{\\mathbb{R}}_{\\infty}}$"}
{"_id": "5545499", "title": "", "text": "$F(x) = \\int_a^{x^2} f(t) \\, dt - \\int_a^{x} f(t) \\, dt$"}
{"_id": "205764", "title": "", "text": "$\\gcd(a,\\,a')$"}
{"_id": "1503376", "title": "", "text": "$x = \\{ \\{ x \\} \\}$"}
{"_id": "3961962", "title": "", "text": "$\\int(2\\pi y ds)$"}
{"_id": "4912850", "title": "", "text": "$x R y \\Rightarrow yRx$"}
{"_id": "4816435", "title": "", "text": "$f_X(x)=\\int_{0}^{x} 8xy \\, dy$"}
{"_id": "5362907", "title": "", "text": "$p\\in V\\subseteq\\overline V\\subseteq U$"}
{"_id": "5183079", "title": "", "text": "$u(x)=v(x)=1/|x|$"}
{"_id": "8395720", "title": "", "text": "$I_n = \\int_{0}^{\\pi} \\frac{\\sin nx}{(1+2^x)\\sin x}dx + \\int_{0}^{\\pi} \\frac{\\sin nx}{(1+2^{-x})\\sin x}dx =\\int_{0}^{\\pi} \\frac{\\sin nx}{\\sin x}dx$"}
{"_id": "5678895", "title": "", "text": "$f^{-1}(f(x) + f(y)) = \\sqrt{x^2 + y^2}$"}
{"_id": "5830613", "title": "", "text": "$\\mathcal{N}((1-2p)n,2(1-p)\\sqrt{n})$"}
{"_id": "4256432", "title": "", "text": "$ \\implies A_0 = \\frac{1}{2\\pi} \\int_{0}^{2\\pi} (2\\pi x -x^{2}) dx $"}
{"_id": "9162940", "title": "", "text": "$P(14)\\implies P(18)$"}
{"_id": "4819286", "title": "", "text": "$\\sum_{k=1}^n\\frac1{k^p}=\\frac1{\\Gamma(p)}\\int_0^\\infty\\frac{1-e^{-nx}}{e^x-1}x^{p-1}~{\\rm d}x$"}
{"_id": "7860631", "title": "", "text": "$[-R,R] \\cup C_R$"}
{"_id": "5017663", "title": "", "text": "$ 1 - P(E^c \\cup F^c)  = 1 -[P(E^c) + P(F^c)].$"}
{"_id": "6773414", "title": "", "text": "$N = P \\oplus Q$"}
{"_id": "1633851", "title": "", "text": "$d(x,y)=||x||+||y||$"}
{"_id": "8490546", "title": "", "text": "$a_{2n}=\\frac{1}{2^{n-1}}$"}
{"_id": "3155298", "title": "", "text": "$G = \\gamma_1(G) \\ge \\gamma_2(G) \\ge \\cdots,$"}
{"_id": "692064", "title": "", "text": "$ \\frac{x_1 + x_2 + \\dots + x_n}{n} \\ge \\sqrt[n]{x_1 x_2 \\dots x_n} $"}
{"_id": "7889090", "title": "", "text": "$1< \\left(\\frac{1}{n-2}+1+\\frac{1}{(n-1)^2} \\right)^{\\frac{1}{n-2}}<1+\\frac{1}{(n-2)^2}+\\frac{1}{(n-2)(n-1)^2}$"}
{"_id": "4415256", "title": "", "text": "$\\int_a^xf(t)dt=F(x)$"}
{"_id": "8548930", "title": "", "text": "$k^2\\left\\{ \\begin{array}{c}n\\\\k\\end{array} \\right\\}^2 +2k\\left\\{ \\begin{array}{c}n\\\\k\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n\\\\k-1\\end{array} \\right\\}+\\left\\{ \\begin{array}{c}n\\\\k-1\\end{array} \\right\\}^2-(k-1)(k+1)\\left\\{ \\begin{array}{c}n\\\\k-1\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n\\\\k+1\\end{array} \\right\\}-(k-1)\\left\\{ \\begin{array}{c}n\\\\k-1\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n\\\\k\\end{array} \\right\\}-(k+1)\\left\\{ \\begin{array}{c}n\\\\k-2\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n\\\\k+1\\end{array} \\right\\}-\\left\\{ \\begin{array}{c}n\\\\k-2\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n\\\\k\\end{array} \\right\\}>0$"}
{"_id": "8135256", "title": "", "text": "$F_{ 2n-1 }={ F }_{ n-1 }^{ 2 }+{ F }_{ n }^{ 2 }>{ F }_{ n }^{ 2 }>{ F }_{ n }^{ 2 }-{ F }_{ n-2 }^{ 2 }=F_{ 2n-2 }$"}
{"_id": "1365096", "title": "", "text": "$\\|A\\|_2=\\sqrt{\\rho(A^TA)}=\\rho\\left(\\begin{bmatrix}0&A\\\\A^T&0\\end{bmatrix}\\right)$"}
{"_id": "9059150", "title": "", "text": "$f\\left ( f(x)^{2} \\right )=x^{3}f(x).$"}
{"_id": "8853614", "title": "", "text": "$\\Rightarrow P(\\text{B wins})=P(B \\text{ picks a red ball}|\\text{A picks a white ball})P(\\text{A picks a white ball})=\\frac3{11}\\frac{9}{12}$"}
{"_id": "4572247", "title": "", "text": "$P(m) - P(m-1)$"}
{"_id": "6504412", "title": "", "text": "$f(x)= 4^x+9^x+25^x$"}
{"_id": "7309371", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta_{g,\\epsilon} >0, |x| \\leq \\delta \\implies |g(x)-g(0)| \\leq \\epsilon$"}
{"_id": "7899309", "title": "", "text": "$F'(x) = \\frac{x + 8}{x^3 - 9}$"}
{"_id": "3568235", "title": "", "text": "$=\\left\\lfloor \\{a\\}\\lfloor b\\rfloor+\\{b\\}\\lfloor a\\rfloor+\\{a\\}\\{b\\}\\right\\rfloor\\text{ (1)}$"}
{"_id": "4842875", "title": "", "text": "$s_1 \\supseteq s_2 \\supseteq \\dots \\supseteq s_n \\supseteq s_{n+1} \\supseteq\\ \\dots$"}
{"_id": "4670616", "title": "", "text": "$\\int_0^{\\infty}f_1(x) \\, dx,$"}
{"_id": "2026432", "title": "", "text": "$\\hspace{.2 in} e^{a/(a-1)}<\\frac{a}{a-1}e^{1/(a-1)}\\iff \\frac{a-1}{a}<e^{-1}\\iff 1-\\frac{1}{a}<\\frac{1}{e}\\iff a<\\frac{e}{e-1},\\;$"}
{"_id": "5993504", "title": "", "text": "$c = \\frac{a+b}{a b-1}$"}
{"_id": "4421322", "title": "", "text": "$E=\\dfrac{(1+x)^{n+1}-1}{x}$"}
{"_id": "5015406", "title": "", "text": "$\\large{\\int_0^\\infty \\frac{1}{(x^2+1)(x^a+1)}\\ dx=\\frac{\\pi}{4}}$"}
{"_id": "8376628", "title": "", "text": "$d(x,y) = d(A,B)$"}
{"_id": "7791016", "title": "", "text": "$2^x+9^x=5^x+6^x$"}
{"_id": "1506153", "title": "", "text": "$(1, 1, 1, z)$"}
{"_id": "1697758", "title": "", "text": "$\\Delta(\\gamma\\alpha, \\gamma\\beta) = \\gamma\\alpha\\gamma'\\beta' - \\gamma'\\alpha'\\gamma\\beta = \\gamma\\gamma'(\\alpha\\beta' - \\alpha'\\beta) = N(\\gamma)\\Delta(\\alpha, \\beta)$"}
{"_id": "2026128", "title": "", "text": "$[a,c - \\gamma]\\subset [a,x] \\subset X$"}
{"_id": "2916687", "title": "", "text": "$|f(z^2)| \\le |f(z)|$"}
{"_id": "4355671", "title": "", "text": "$X=c_1e_1+\\dots c_ne_n$"}
{"_id": "3181318", "title": "", "text": "$cov(X,Y)>0$"}
{"_id": "2578629", "title": "", "text": "$xa+yb=d$"}
{"_id": "3506176", "title": "", "text": "$|aba^{-1}|=|b|$"}
{"_id": "527893", "title": "", "text": "$TF\\{\\sum_{n=-\\infty}^{\\infty} \\delta (t-nT) = \\sum_{n=-\\infty}^{\\infty} e^{-jnT\\omega} = \\dfrac{2\\pi}{T} \\sum_{n=-\\infty}^{\\infty} \\delta (\\omega-n\\dfrac{2 \\pi n}{T})$"}
{"_id": "4301027", "title": "", "text": "$f_1(n)=(n-1)!$"}
{"_id": "5220787", "title": "", "text": "$x^3,x^5,x^6$"}
{"_id": "276632", "title": "", "text": "$\\Sigma^+$"}
{"_id": "6524786", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac {1 + \\sqrt[3]{2} + \\sqrt[3]{3} +\\cdots+ \\sqrt[3]{n}}{n^{4/3}}$"}
{"_id": "2457567", "title": "", "text": "$\\displaystyle \\sum_{n=-\\infty}^{\\infty}c_ne^{inx}=\\sum_{n=-\\infty}^{\\infty}c_n\\cos(nx)+\\sum_{n=-\\infty}^{\\infty}ic_n\\sin(nx)$"}
{"_id": "8333390", "title": "", "text": "$a_1\\cdot \\dots \\cdot a_n$"}
{"_id": "1264969", "title": "", "text": "$\\delta (a - x) = \\delta(-(x - a)) = \\delta(x - a)$"}
{"_id": "5441108", "title": "", "text": "$kx\\frac{d^2y}{dx^2}=\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}$"}
{"_id": "929049", "title": "", "text": "$g(x)=\\int_a^{x}f(t)\\,dt,$"}
{"_id": "4217580", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}\\sqrt{2+\\sqrt{2+\\sqrt{...+\\sqrt{2}}}}$"}
{"_id": "5395337", "title": "", "text": "$z^n =  re^{i\\varphi} \\implies z_k = r^{\\frac{1}{n}}e^{i\\left( \\frac{\\varphi}{n} + \\frac{2\\pi k}{n}\\right)} \\qquad k=0,\\dots,n-1$"}
{"_id": "3671093", "title": "", "text": "$\\left(y+\\frac{dy}{dx}\\right)^2+\\frac{dy}{dx}=0$"}
{"_id": "6242503", "title": "", "text": "$\\tan\\left(\\frac{2\\tan^{-1}\\left(\\frac{1}{2}\\right)}{2}\\right)=\\tan\\left(\\tan^{-1}\\left(\\frac{1}{2}\\right)\\right)=\\frac{1}{2}$"}
{"_id": "1853274", "title": "", "text": "$\\int_{-\\pi/2}^{\\pi/2} \\frac{\\sin^{2012}{x}}{\\left(1+ \\alpha^x\\right)\\left(\\sin^{2012} {x}+\\cos^{2012}{x}\\right)}\\;{dx} $"}
{"_id": "2685526", "title": "", "text": "$\\mathbb{E}([X + Y]^3) = \\mathbb{E}(X^3) + \\mathbb{E}(Y^3)$"}
{"_id": "5810804", "title": "", "text": "${(1-\\frac{1}{2^n})}^{2^n}$"}
{"_id": "6985284", "title": "", "text": "$Cov(X,Y)=Cov(X,g(Y))$"}
{"_id": "7293073", "title": "", "text": "$\\int_{-1}^{1}\\frac{1}{(x^2+1)^3}\\,dx$"}
{"_id": "904542", "title": "", "text": "$\\|x_n-x_{n+1}\\|$"}
{"_id": "2960020", "title": "", "text": "$\\alpha < \\alpha^{\\omega}$"}
{"_id": "1744118", "title": "", "text": "$\\lim_{n \\to \\infty}{\\displaystyle\\sum_{k=1}^{n} \\frac{1}{k^2}} $"}
{"_id": "1795764", "title": "", "text": "$n|m_1m_2$"}
{"_id": "7499146", "title": "", "text": "$y-1/|x| = 8 +1/3.$"}
{"_id": "4975136", "title": "", "text": "$(\\Bbb{R}[X])^*$"}
{"_id": "6058413", "title": "", "text": "$E[1_{X\\ge x}]=P[X\\ge x]$"}
{"_id": "3822883", "title": "", "text": "$\\diamond ^+$"}
{"_id": "4085847", "title": "", "text": "$\\int_0^\\infty f(t)h(t)dt = 0$"}
{"_id": "5084123", "title": "", "text": "$ u = \\frac{\\gamma(b) - \\gamma(a)}{\\|\\gamma(b) - \\gamma(a)\\|} $"}
{"_id": "5942001", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} f(x_n)=\\liminf\\limits_{x\\to 0} f(x)$"}
{"_id": "2963682", "title": "", "text": "$-\\frac{945\\ 2^{n+6} \\tau ^5 \\,    _{12}F_{11}\\left(\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}    {2},n;\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1    ,\\frac{n}{2}+1;-1\\right)}{\\pi ^5 n^{11}}+\\frac{105\\ 2^{n+5} \\tau ^4 \\,    _{10}F_9\\left(\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},n;\\frac{n}{2}+1,\\frac{n    }{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1;-1\\right)}{\\pi ^4 n^9}-\\frac{15\\ 2^{n+4}    \\tau ^3 \\,    _8F_7\\left(\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},n;\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{    n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1;-1\\right)}{\\pi ^3 n^7}+\\frac{3\\ 2^{n+3} \\tau ^2 \\,    _6F_5\\left(\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},n;\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1;-1    \\right)}{\\pi ^2 n^5}-\\frac{2^{n+2} \\tau  \\,    _4F_3\\left(\\frac{n}{2},\\frac{n}{2},\\frac{n}{2},n;\\frac{n}{2}+1,\\frac{n}{2}+1,\\frac{n}{2}+1;-1\\right)}{\\pi  n^3}+\\frac{2^{n+1} \\,    _2F_1\\left(\\frac{n}{2},n;\\frac{n+2}{2};-1\\right)}{n}$"}
{"_id": "5427223", "title": "", "text": "$\\mathscr{A}=\\big\\{\\{x\\}:x\\in Y\\big\\}$"}
{"_id": "1227170", "title": "", "text": "$\\implies\\tan\\theta=\\dfrac x5$"}
{"_id": "9019166", "title": "", "text": "$f(x)=\\frac{2}{4^{x}+2}$"}
{"_id": "7345393", "title": "", "text": "$\\lim_{x\\to5^+}f'(x)= \\lim_{x\\to5^-}f'(x)=1$"}
{"_id": "8281049", "title": "", "text": "$\\int_{1}^{10}\\frac{1}{(x^{2}+1)^{10}}dx\\approx .00009843725636$"}
{"_id": "303547", "title": "", "text": "$f(a + b) = f(a) \\cdot f(b)$"}
{"_id": "2056910", "title": "", "text": "$S^n\\times\\mathbb{R}^{n+1}$"}
{"_id": "2639030", "title": "", "text": "$\\operatorname{cov}(X,Y)>0$"}
{"_id": "6655049", "title": "", "text": "$\\mathbb{R}^{n+1}-M$"}
{"_id": "3649762", "title": "", "text": "$P : \\overline{\\mathbb{R}} \\times \\overline{\\mathbb{R}} \\to \\overline{\\mathbb{R}}$"}
{"_id": "6829126", "title": "", "text": "$\\left \\| A \\right \\| _2=\\sqrt{\\lambda_{\\text{max}}(A^{^*}A)}$"}
{"_id": "8114810", "title": "", "text": "$\\underset{1 \\times 2}{x^+}$"}
{"_id": "6273036", "title": "", "text": "$\\alpha (\\gamma) = (\\gamma - \\gamma , \\gamma -\\gamma, \\dots) = (0,0, \\dots ) = 0 \\gamma$"}
{"_id": "1314643", "title": "", "text": "$y'''+y'=y'+y$"}
{"_id": "5351614", "title": "", "text": "$ |f(z)| \\le (1+|z|) |g(z)|$"}
{"_id": "814171", "title": "", "text": "$\\forall\\varepsilon\\gt 0,\\,\\exists \\delta\\gt 0 :|h(x)-h(y)|\\lt\\varepsilon$"}
{"_id": "2172487", "title": "", "text": "$\\pi(f)=f(\\pi(X))=f(x)=0$"}
{"_id": "552587", "title": "", "text": "$\\dfrac{\\sqrt{1-x^2}}{\\sqrt{1-x^2}+\\sqrt{1 - 2x^2}} + \\dfrac{2\\sqrt{x}}{1+\\sqrt{1 - 2x^2}}> 0$"}
{"_id": "3333213", "title": "", "text": "$\\mathbb{R}^{n+m}=\\mathbb{R}^{n}\\oplus \\mathbb{R}^{m}$"}
{"_id": "2955280", "title": "", "text": "$\\int_0^\\infty f'(x)\\,dx$"}
{"_id": "5980328", "title": "", "text": "$e^x - (1+\\frac{x}{n})^n \\rightarrow 0$"}
{"_id": "4823578", "title": "", "text": "$\\begin{align} \\left|e^x-\\left(1+\\frac xn\\right)^n\\right|&=\\left|e^x-e^{n\\log\\left(1+\\frac xn\\right)}\\right|\\\\\\\\ &\\le \\left|e^x-e^{\\frac{x}{1+x/n}}\\right|\\\\\\\\ &=e^x\\,\\left|1-e^{-x^2/(x+n)}\\right|\\\\\\\\ &\\le e^x\\,\\left|\\frac{x^2}{x+n}\\right|\\\\\\\\ &\\le e^{b}\\frac{|\\max^2(a,b)|}{n+a}\\\\\\\\ &<\\epsilon \\end{align}$"}
{"_id": "3414867", "title": "", "text": "$\\int_0^{+\\infty}f(x) = \\int_0^{1}f(x) + \\int_1^{+\\infty}f(x)$"}
{"_id": "525575", "title": "", "text": "$T(n) = \\dfrac{n(n+1)}{2}$"}
{"_id": "4131960", "title": "", "text": "$\\left\\lfloor{x\\over 2^n}\\right\\rfloor =\\left\\lfloor{\\lfloor x\\rfloor\\over 2^n}\\right\\rfloor\\ .\\tag{1}$"}
{"_id": "213893", "title": "", "text": "$\\sum_{k=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{n^2k^2(n+k)^2}=  \\frac{1}{3}\\zeta(6)$"}
{"_id": "4335927", "title": "", "text": "$N! + 2, ..., N! + N$"}
{"_id": "8721542", "title": "", "text": "$ \\sum_{n \\geq 1} \\frac{n}{n^3 + 1} < \\sum_{n \\geq 1} \\frac{n}{n^3} = \\sum_{n \\geq 1} \\frac{1}{n^2}.$"}
{"_id": "6097094", "title": "", "text": "$f(n) = n^2\\!+3n+2\\,$"}
{"_id": "2789040", "title": "", "text": "$4/x+3/y+d/z=1$"}
{"_id": "1569701", "title": "", "text": "$\\sum_{k}|\\langle f,\\phi_p\\rangle |^2= A||f||^2$"}
{"_id": "483934", "title": "", "text": "$X=\\overline{X}={\\mathbb R}^2$"}
{"_id": "3182083", "title": "", "text": "$\\frac{1}{2^{N-2}} < \\varepsilon$"}
{"_id": "99925", "title": "", "text": "$ \\int \\frac{\\sin(x)}{1+\\sin^2(x)}\\, dx $"}
{"_id": "4546956", "title": "", "text": "$\\int_1^\\infty\\frac{\\sin^2(x)}{x}\\mathrm d x$"}
{"_id": "7602463", "title": "", "text": "$ax+by = d. (\\dagger)$"}
{"_id": "3322971", "title": "", "text": "$\\frac{ax + b}{cx} = n$"}
{"_id": "7734360", "title": "", "text": "$\\implies\\sum_{k=0}^n\\cos\\dfrac{2\\pi k}n=0+\\cos\\dfrac{2\\pi n}n=?$"}
{"_id": "3180443", "title": "", "text": "$ 6^{x+1}-6^x = 3^{x+4}-3^x $"}
{"_id": "9136625", "title": "", "text": "$\\displaystyle J_n=\\int_0^{\\pi}\\left(\\frac{\\sin nx}{\\sin x}\\right)^2\\mathrm{d}x$"}
{"_id": "5687657", "title": "", "text": "$|\\theta-\\sin \\theta| < |2\\tan\\frac\\theta 2 - \\sin\\theta|$"}
{"_id": "5476098", "title": "", "text": "$\\frac{1}{a} \\biggl( 1 - \\frac{x-a}{a} + \\frac{(x-a)^2}{a^2} - \\frac{(x-a)^3}{a^3} \\pm \\cdots \\biggr) $"}
{"_id": "7140313", "title": "", "text": "$\\wedge^{1,0}F=(T_a^*X)^{1,0}$"}
{"_id": "4875133", "title": "", "text": "$\\mathbb{E}|X-Y| \\leq 1/2$"}
{"_id": "7099806", "title": "", "text": "$E(X^2-2XE(X)+E(X)^2)= E(X^2)-2E(X)^2+E(X)^2= E(X^2)-E(X)^2.$"}
{"_id": "1125395", "title": "", "text": "$\"1+2+3+4+5+...=-{1\\over 12}.\"$"}
{"_id": "3295158", "title": "", "text": "$\\frac{X_1+X_2+ \\ldots +X_n}{n} \\to 0$"}
{"_id": "7892644", "title": "", "text": "$ \\int_{\\gamma -\\ic\\infty}^{\\gamma + \\ic\\infty} {s\\expo{st} \\over \\root{\\pars{s + a}^{3}}}\\,{\\dd s \\over 2\\pi\\ic} = \\pars{1 + 2a\\,\\partiald{}{a}}{\\expo{-at} \\over \\root{\\pi t}} = {1 \\over \\root{\\pi}}\\,{\\pars{1 - 2at}\\expo{-at} \\over \\root{t}} $"}
{"_id": "7916237", "title": "", "text": "$\\lim_{n\\to\\infty} \\inf(x_n) \\leq \\lim_{k\\to\\infty} \\inf(x_{n_k})$"}
{"_id": "1368460", "title": "", "text": "$\\begin{cases} A = 1 \\\\ B = -6 \\\\ C = -30 \\\\ D = -48 \\end{cases}$"}
{"_id": "8260266", "title": "", "text": "$\\sum_{k=0}^n \\cos\\left(\\frac{k\\pi}{n}\\right) = 0$"}
{"_id": "7457858", "title": "", "text": "$\\sum\\limits_{k=1}^n\\frac k{(1+r)^k}$"}
{"_id": "4041263", "title": "", "text": "$\\cdot^+$"}
{"_id": "2391658", "title": "", "text": "$T_U^{0,1}$"}
{"_id": "6069744", "title": "", "text": "$\\rho(A)< \\|A\\|_{\\infty}$"}
{"_id": "5553389", "title": "", "text": "$6^x+3^x=10$"}
{"_id": "5550184", "title": "", "text": "$x^3, x^7, x^9$"}
{"_id": "8067854", "title": "", "text": "$\\forall \\epsilon \\gt 0 \\exists \\delta \\gt 0  : |x-a| \\lt \\delta \\implies |f(x)-f(a)|\\lt \\epsilon$"}
{"_id": "2129991", "title": "", "text": "$\\infty\\d a=\\dfrac 1 a$"}
{"_id": "7672249", "title": "", "text": "$[\\gamma_1,...,\\gamma_{\\ell}] \\in \\Pi(\\gamma,\\ell)$"}
{"_id": "5348636", "title": "", "text": "$ \\displaystyle \\int_{-\\infty}^{+\\infty} \\delta'(x-3)e^{x^2}dx $"}
{"_id": "7544043", "title": "", "text": "$\\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{m^2+n^2}.$"}
{"_id": "5711340", "title": "", "text": "$5 \\mid n^5-n$"}
{"_id": "749961", "title": "", "text": "$ 1+2\\sum_{n=1}^\\infty e^{-\\pi n^2t} =\\frac1{\\sqrt{t}}+\\frac2{\\sqrt{t}}\\sum_{n=1}^\\infty e^{-\\pi n^2/t}\\tag{14} $"}
{"_id": "5544789", "title": "", "text": "$\\int_0^{\\infty}x dF(x)$"}
{"_id": "3488360", "title": "", "text": "$c = {a \\over {a + b} }$"}
{"_id": "7184480", "title": "", "text": "$\\alpha \\Omega + \\beta \\Omega \\subset (\\alpha+ \\beta) \\Omega$"}
{"_id": "310031", "title": "", "text": "$|ab|=1$"}
{"_id": "2284455", "title": "", "text": "$9^x+6^x< 2^{(x+1)}$"}
{"_id": "3586763", "title": "", "text": "$A_n=\\{a_1,a_2,\\cdots, a_n\\}$"}
{"_id": "4623950", "title": "", "text": "$P_X(x) = \\frac{1}{a}e^{\\frac{-x}{a}}u(x)$"}
{"_id": "1373745", "title": "", "text": "$|f^{(n)}(x)| \\leq \\alpha^{n-1}|f(x)|$"}
{"_id": "782887", "title": "", "text": "$(P(s)\\implies P(s+1))$"}
{"_id": "3121787", "title": "", "text": "$f_X(x) = F_X'(x) = \\theta x^{-(\\theta+1)}, \\quad x > 1, \\; \\theta > 0$"}
{"_id": "4664792", "title": "", "text": "$T^{1,0}_e M \\subset T_e M \\otimes \\mathbb{C}$"}
{"_id": "2027562", "title": "", "text": "$d = o( \\sqrt{p})$"}
{"_id": "7132928", "title": "", "text": "$\\beta b - \\alpha a = (\\beta + \\gamma + \\delta) a' - (\\alpha + \\gamma + \\delta) b'\\\\ \\beta b + (\\alpha + \\gamma + \\delta) b' = \\alpha a + (\\beta + \\gamma + \\delta) a'\\\\ \\frac{\\beta b + (\\alpha + \\gamma + \\delta) b'}{\\alpha + \\beta + \\gamma + \\delta} = \\frac{\\alpha a + (\\beta + \\gamma + \\delta) a'}{\\alpha + \\beta + \\gamma + \\delta} = e$"}
{"_id": "575460", "title": "", "text": "$(2a+b)^2-(2a-b)^2= (a+2b)^2-(a-2b)^2$"}
{"_id": "9271826", "title": "", "text": "$S_1 = 1+2+3+4+5+\\cdots = -\\frac{1}{12}$"}
{"_id": "4839669", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}\\sum_{i=1}^{\\infty}\\mu_n(E_i)=\\sum_{i=1}^{\\infty}\\lim_{n \\rightarrow \\infty}\\mu_n(E_i)$"}
{"_id": "1787388", "title": "", "text": "$\\overline{V} \\subset W \\subset \\overline{W}$"}
{"_id": "845820", "title": "", "text": "$c = ax + by$"}
{"_id": "7238645", "title": "", "text": "$f(m) = f(n-1) = {n-1 \\choose 2}$"}
{"_id": "5397776", "title": "", "text": "$\\frac{H(p)}{2\\sqrt{p(1-p)}}$"}
{"_id": "7722858", "title": "", "text": "$O(nlog_2 (b))$"}
{"_id": "859381", "title": "", "text": "$ L = \\lim_{x \\to 0^-} f(x) = \\lim_{x \\to 0^+} f(-x) = \\lim_{x \\to 0^+} f(x) = L. $"}
{"_id": "772143", "title": "", "text": "$R_2(n) = n^2 - n + 2$"}
{"_id": "2722701", "title": "", "text": "$\\lambda \\begin {vmatrix} x & y & 1\\\\ x_1 & y_1 & 1\\\\ x_2 & y_2 & 1\\end {vmatrix} $"}
{"_id": "2833733", "title": "", "text": "$f'(x)= 9/2 \\sqrt{x} (x^{3/2}+1)^2 ->f'(1)=18$"}
{"_id": "7019653", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{m^2+n^2}  = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{(2m)^2+(2n)^2} + \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{(2m-1)^2+(2n)^2} $"}
{"_id": "4689755", "title": "", "text": "$\\Delta f=A\\Delta x+o(\\Delta x)$"}
{"_id": "3161427", "title": "", "text": "$ \\forall \\epsilon>0 , \\exists \\delta > 0, if |x -a| < \\delta \\implies |f(x) - c| < \\epsilon $"}
{"_id": "189023", "title": "", "text": "$x \\in V, \\overline{V} \\subseteq U$"}
{"_id": "4984779", "title": "", "text": "$1/\\left|x\\right| < 1$"}
{"_id": "4628617", "title": "", "text": "$    \\lVert A\\rVert_{(1,\\infty)} \\text{ and } \\lVert A\\rVert_{(2,\\infty)} \\text{ and sketch the proof} \\text{ ( A is a Matrix Norm)} $"}
{"_id": "3514477", "title": "", "text": "$a=br=ars$"}
{"_id": "4570598", "title": "", "text": "$|{\\frac{1}{\\gamma_2!} x^{\\gamma_2+\\alpha} e^{\\langle{s,x}\\rangle}}| \\overset{(1)}{\\leq} \\frac{1}{\\gamma_2!} (\\gamma_2+\\alpha)! \\delta^{-|{\\gamma}_2| - |{\\alpha}|} e^C = \\frac{(\\gamma_2+\\alpha)!}{\\gamma_2!} \\delta^{-|{\\gamma_2}|} \\cdot \\delta^{-|{\\alpha}|} e^C$"}
{"_id": "9062433", "title": "", "text": "$=\\left[\\begin{array}{ccc|c}2&-4&4&0\\\\2&k&-1&3\\\\0&-1&-1&5\\end{array}\\right]$"}
{"_id": "2705189", "title": "", "text": "$\\frac{(1+n)^{k+1} - n -1}{n}C$"}
{"_id": "7924938", "title": "", "text": "$F_2[x]/(x^4+1)$"}
{"_id": "5700379", "title": "", "text": "$log_{a} y = b$"}
{"_id": "163657", "title": "", "text": "$cov(X,Y) = 0)$"}
{"_id": "8661361", "title": "", "text": "$x \\to x^3-x-3$"}
{"_id": "768185", "title": "", "text": "$\\zeta(2):=\\sum_{k=1}^\\infty{1\\over k^2}=\\sum_{k\\geq 1\\ {\\rm even}}^\\infty{1\\over k^2}+\\sum_{k\\geq 1\\ {\\rm odd}}^\\infty{1\\over k^2}={1\\over4}\\zeta(2)+{\\pi^2\\over 8}\\ ,$"}
{"_id": "6198324", "title": "", "text": "$f(x) =\\frac{ae^{-\\frac{a}{x}}+be^{-\\frac{b}{x}}}{e^{-\\frac{a}{x}}+e^{-\\frac{b}{x}}} =\\frac{ae^{-\\frac{a}{x}}+ae^{-\\frac{b}{x}}+(b-a)e^{-\\frac{b}{x}}}{e^{-\\frac{a}{x}}+e^{-\\frac{b}{x}}}=a+(b-a)\\frac{e^{-\\frac{b}{x}}}{e^{-\\frac{a}{x}}+e^{-\\frac{b}{x}}}=\\\\=a+(b-a)\\frac{1}{e^{-\\frac{a-b}{x}}+1}$"}
{"_id": "2195461", "title": "", "text": "$\\alpha \\text{ and }\\gamma \\text{ injective (surjective) }\\implies \\beta \\text{ injective (surjective)}.$"}
{"_id": "6606956", "title": "", "text": "$P[X_0=x]$"}
{"_id": "3649765", "title": "", "text": "$S : \\overline{\\mathbb{R}} \\times \\overline{\\mathbb{R}} \\to \\overline{\\mathbb{R}}$"}
{"_id": "8506413", "title": "", "text": "$x^* \\in (M^\\perp)^\\perp=M$"}
{"_id": "7825406", "title": "", "text": "$ \\mathcal{X} := \\{ \\{ x \\} : x \\in X \\} $"}
{"_id": "4151614", "title": "", "text": "$\\lfloor{\\frac{a}{b+1}}\\rfloor = \\lfloor \\frac{a}{b} \\rfloor  - 1 $"}
{"_id": "2648887", "title": "", "text": "$rn^{n - r - 1}$"}
{"_id": "2781988", "title": "", "text": "$[X_{p}^{k},X_{p}^{l}] \\subset X_{p}^{max(-1,k+l)}$"}
{"_id": "3846263", "title": "", "text": "$\\displaystyle -\\frac{dx}{dy} = \\frac{\\sqrt{1-y^2}}{y}$"}
{"_id": "9148142", "title": "", "text": "$ y (x' - x) = y x' - y x = y' x - y x = (y' - y) x $"}
{"_id": "8844118", "title": "", "text": "$E|X+Y|^r \\leq E|X|^r+E|Y|^r$"}
{"_id": "274256", "title": "", "text": "$F(x) = \\int_a^x f(x) \\, dx$"}
{"_id": "9267764", "title": "", "text": "$ (\\forall m<n' \\ P(m) \\implies P(n'))$"}
{"_id": "8392576", "title": "", "text": "$Var(X) = \\mathbb{E}(X-\\mathbb{E}(X))^2 = \\mathbb{E}(X^2+\\mathbb{E}(X)^2-2X*\\mathbb{E}(X)) = \\mathbb{E}(X^2)+\\mathbb{E}(X)^2-2\\mathbb{E}(X)^2 = \\mathbb{E}(X^2)-\\mathbb{E}(X)^2$"}
{"_id": "433443", "title": "", "text": "$P[K=k]=\\frac{1}{3}$"}
{"_id": "6722283", "title": "", "text": "$[X, Y] = XY - YX,$"}
{"_id": "1145784", "title": "", "text": "$ f_+'(c) = \\lim_{x\\to c^+}\\frac{f(x) - f(c)}{x - c} = \\lim_{x\\to c^+}f'(c_x) = \\lim_{x\\to c}f'(x). $"}
{"_id": "6404036", "title": "", "text": "$F(y) - F(x) = \\lim_{k\\to \\infty} F_{n_k}(y) - F_{n_k}(x) = \\lim_{k\\to \\infty} \\int_0^1 f_n(t)\\chi_{[x,y]}(t)\\,dt = \\int_0^1 f(t)\\chi_{[x,y]}(t)\\,dt,$"}
{"_id": "4429184", "title": "", "text": "$\\lim_{N\\to \\infty} ||\\sum _{n=0}^{N} \\frac{(if)^{n}}{n!}||\\leq e^{r}$"}
{"_id": "1216363", "title": "", "text": "$\\sum_{a \\in A^+} \\frac{1}{a^2} < \\sum_{n=1}^{\\infty} \\frac{1}{n^2}.$"}
{"_id": "2264726", "title": "", "text": "$L\\ = {\\dfrac{(1+x)^n -1}{x(1+x)^n} }(Monthly\\ payment)$"}
{"_id": "6272562", "title": "", "text": "$A = \\{\\{0\\}\\}$"}
{"_id": "4924120", "title": "", "text": "$R/M \\cong \\left( \\mathbb{Z}[x]/(x^2-2) \\right)/(5)$"}
{"_id": "6651706", "title": "", "text": "$y=x^2-4x-5 \\\\\\frac{dy}{dx}=\\operatorname{slope}(x)=2x-4\\\\\\text{ @ (-2, 7):}\\\\\\frac{dy}{dx}\\Big|_{x=-2,\\;y=7}=-8=m\\\\\\text{\"normal\" = perpendicular (}\\perp\\text{)}\\\\m_{\\perp\\xi}=-\\frac1{m_\\xi}\\\\\\text{(slope of the line }\\xi_\\perp\\text{, the line perpendicular to } \\xi\\text{ is the negative reciprocal of }\\xi\\text{'s slope)}\\\\\\frac{dy_\\perp}{dx}\\Big|_{x=-2,\\;y=7}=\\frac18=m_\\perp\\\\y_\\perp=m_\\perp x+b=\\frac{x}8+b\\\\\\text{If they meet at (-2,7):}\\\\7=\\require{cancel}-\\cancelto{\\frac14}{\\frac28}+b \\\\ \\Rightarrow b=\\frac{29}{4}$"}
{"_id": "503328", "title": "", "text": "$A\\subseteq U,B\\subseteq V$"}
{"_id": "2371054", "title": "", "text": "$S=\\{a_1,a_2,a_3,\\cdots,a_n\\}$"}
{"_id": "3457646", "title": "", "text": "$f(x)=2b\\gamma\\left(a-b\\frac{1}{2}\\left(\\gamma-\\frac{1}{\\gamma}\\right)\\right)=-b^2\\left(\\gamma^2-2\\frac{a}{b}\\gamma-1\\right)$"}
{"_id": "1964053", "title": "", "text": "$ax^n=[anx^{n-1}][an(n-1)x^{n-2}].$"}
{"_id": "6606631", "title": "", "text": "$L = \\left(x + \\dfrac{d}{dx} \\right)$"}
{"_id": "6299490", "title": "", "text": "$\\frac{1}{2^{n^2-1}}$"}
{"_id": "8203038", "title": "", "text": "$\\begin{cases}2y+\\frac 1y=3\\\\y=\\frac 1y\\end{cases}\\iff \\begin{cases}x=2\\\\y=1\\end{cases}$"}
{"_id": "2237681", "title": "", "text": "$e^x =(1+\\frac{x}{n})^n $"}
{"_id": "2557791", "title": "", "text": "$F = E + V$"}
{"_id": "9082067", "title": "", "text": "$\\frac{10-x}{4}$"}
{"_id": "1525934", "title": "", "text": "$(\\mathbb F_{p}[x]/(x^m))^{\\times}$"}
{"_id": "305333", "title": "", "text": "$\\sqrt1+\\sqrt2+\\dots+\\sqrt n$"}
{"_id": "5611075", "title": "", "text": "$ E(U^s)=\\sqrt[3]{\\frac{(1+\\epsilon)^{s+1}-(1-\\epsilon)^{s+1}}{2(s+1)\\epsilon}}. $"}
{"_id": "1978188", "title": "", "text": "$|\\int_\\gamma f | \\leq M l(\\gamma)$"}
{"_id": "1634290", "title": "", "text": "$ \\frac{1}{n^2(n+1)^2} = \\frac{1}{n^2}+\\frac{1}{(n+1)^2}+2\\left(\\frac{1}{n+1}-\\frac{1}{n}\\right) $"}
{"_id": "2282371", "title": "", "text": "$ K = (\\frac{\\gamma + ax}{\\alpha + ax})^{2a} = (1 + \\frac{\\gamma -\\alpha}{\\alpha + ax})^{2a}  $"}
{"_id": "5473904", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx = \\int_{0}^{\\pi}\\frac {(1+2^x)\\sin nx}{(1+2^x)\\sin x}dx = \\int_{0}^{\\pi}\\frac {\\sin nx}{\\sin x}dx$"}
{"_id": "3134386", "title": "", "text": "$n! \\simeq  \\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n$"}
{"_id": "9360811", "title": "", "text": "$\\#_2 A$"}
{"_id": "2181950", "title": "", "text": "$\\sum\\limits_{n=1}^\\infty\\frac{-1}{n^2}\\leq\\sum\\limits_{n=1}^K \\frac{-1}{n^2}\\leq \\sum\\limits_{n=1}^K (-1)^n\\leq \\sum\\limits_{n=1}^K\\frac{1}{n^2}\\leq\\sum\\limits_{n=1}^\\infty \\frac{1}{n^2}$"}
{"_id": "3220924", "title": "", "text": "$ds = \\sqrt{(dx)^2+(dy)^2}$"}
{"_id": "8887429", "title": "", "text": "$ \\{x, y, z\\}, \\{x, z, y\\}, \\{y, x, z\\}, \\{y, z, x\\}, \\{z, x, y\\}, \\{z, y, x\\}. $"}
{"_id": "6909472", "title": "", "text": "$(k(1) \\,\\, k(3) \\, \\, k(5)) \\,\\,\\,\\, (k(2) \\,\\, k(4)) = (2 \\,\\, 4 \\,\\, 6) \\,\\,\\, (1\\,\\, 3)$"}
{"_id": "6141478", "title": "", "text": "$t_x x + (1-t_x)y$"}
{"_id": "2077576", "title": "", "text": "$f_{X,Y}(x,y)= \\frac{e^{\\frac{-x}{\\beta}}}{\\beta}\\frac{e^{\\frac{-y}{\\beta}}}{\\beta}=\\frac1{\\beta^2}e^{\\frac{-x-y}{\\beta}}$"}
{"_id": "2676943", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1) \\cong \\mathbb{R}[x]$"}
{"_id": "6800065", "title": "", "text": "$\\det(a_{ij})_{i,j\\in H}$"}
{"_id": "5028557", "title": "", "text": "$\\lim_{n\\to \\infty } \\mu (A_n)=\\mu (X)$"}
{"_id": "5125492", "title": "", "text": "$ \\begin{cases} \\mu   &= \\rho\\frac{\\sin(\\gamma + \\phi)}{\\sin\\gamma},\\\\ 1-\\mu &= \\rho\\frac{\\sin(\\gamma-\\phi)}{\\sin\\gamma} \\end{cases} \\quad\\implies\\quad 1 = \\frac{\\rho}{\\sin\\gamma}(\\sin(\\gamma + \\phi) + \\sin(\\gamma - \\phi)) = 2\\rho\\cos\\phi$"}
{"_id": "1303636", "title": "", "text": "$k^3*1/3 + k^2 + k + 1/3 + k^2/2 + k + 1/2 + k/6 + 1/6 = $"}
{"_id": "3386165", "title": "", "text": "$P_1(h=k) = \\frac16, \\;\\forall\\ k \\in\\{1,2,3,4,5,6\\}$"}
{"_id": "1468067", "title": "", "text": "$n! = \\sqrt{2\\pi{n}}(\\frac{n}{e})^n + r_1(n)$"}
{"_id": "7812368", "title": "", "text": "$(a-b)(2a+2b+1)=2a^2-2b^2+a-b=b^2.$"}
{"_id": "961244", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor x\\rfloor}{n}\\right\\rfloor\\ge\\left\\lfloor\\frac{x}{n}\\right\\rfloor$"}
{"_id": "8054982", "title": "", "text": "$ F'(F^{-1}(x))(F^{-1}(x))' = 1 \\implies\\\\ f(F^{-1}(x))(F^{-1}(x))'  = 1$"}
{"_id": "2587011", "title": "", "text": "$\\lim_{x\\to-\\infty}f'(x)=\\infty=\\lim_{x\\to\\infty}f'(x)$"}
{"_id": "5513477", "title": "", "text": "$ \\lim_{n\\to\\infty}\\left( \\sum_{r=1}^n \\dfrac 1 {\\sqrt{n^2 + r}} \\right) = 1$"}
{"_id": "3739383", "title": "", "text": "$\\sin(t) = \\frac x2$"}
{"_id": "1330112", "title": "", "text": "$xRy \\implies yRx$"}
{"_id": "7663571", "title": "", "text": "$\\dfrac{b - 2a\\cos{\\gamma}  }{a\\sin{\\gamma}}=\\dfrac{\\sin{\\beta}}{\\sin{\\alpha}\\sin{\\gamma}}-\\dfrac{2\\cos{\\gamma}}{\\sin{\\gamma}}$"}
{"_id": "728445", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^{n}{\\frac{1}{k}}$"}
{"_id": "205964", "title": "", "text": "$\\forall \\epsilon > 0, \\ \\exists \\ \\delta > 0 \\ s.t \\ if\\ |x-a| < \\delta \\Rightarrow  |f(x) - f(a)| < \\epsilon$"}
{"_id": "6803344", "title": "", "text": "$(x+y,x-y) \\le \\|x+y\\|\\|x-y\\| \\le 1$"}
{"_id": "3413349", "title": "", "text": "$\\,\\, \\forall \\epsilon >0\\,\\,\\exists \\delta >0 : \\forall a,x\\in S\\,\\, |x-a|<\\delta \\implies |f(x)-f(a)|<\\epsilon $"}
{"_id": "4623568", "title": "", "text": "$f(f(x))=xf(0)+f(x)$"}
{"_id": "2247954", "title": "", "text": "$3^x+4^x<5^x$"}
{"_id": "9161164", "title": "", "text": "$h(x_n)=\\sum y_m$"}
{"_id": "18724", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n\\left(1+O\\left(\\frac{1}{n}\\right)\\right)$"}
{"_id": "9017210", "title": "", "text": "$\\vartheta(u,-i\\tau)$"}
{"_id": "9302828", "title": "", "text": "$Ax-b, x \\in \\mathbb{R^n}$"}
{"_id": "9136619", "title": "", "text": "$\\displaystyle I_n = \\int_0^\\pi \\frac{\\sin 2k x}{\\sin x} \\mathrm{d}x$"}
{"_id": "3960663", "title": "", "text": "$=3\\lim_{N\\rightarrow \\infty}\\sum_{n=1}^{N} \\frac{1}{N}-3\\lim_{N\\rightarrow \\infty}\\sum_{n=1}^{N}\\frac{1}{N+1} -\\lim_{N\\rightarrow \\infty}\\sum_{n=1}^{N} \\frac{1}{2N+3}+\\lim_{N\\rightarrow \\infty}\\sum_{n=1}^{N}\\frac{1}{2N+5} $"}
{"_id": "8965212", "title": "", "text": "$S(n)=\\sum_{k=2}^{n}k\\cos\\frac{\\pi}{k}$"}
{"_id": "1178928", "title": "", "text": "$ \\bbox[lightyellow] {   \\left\\{ \\matrix{   \\theta  \\in \\left\\{ {\\pi /2 + k\\pi  - \\alpha ,\\;k\\pi } \\right\\} \\hfill \\cr    \\varphi  = \\arcsin \\left( {\\sin \\alpha /\\sin \\left( {\\alpha  + \\theta } \\right)} \\right) \\hfill \\cr    a = A\\cos \\varphi  \\hfill \\cr    b = A\\sin \\varphi  \\hfill \\cr}  \\right. } \\tag{2}$"}
{"_id": "4984177", "title": "", "text": "$ \\begin{vmatrix} 1 & 1 & 1 \\\\ x_0 & x_1 & x_2 \\\\ x_0^2 & x_1^2 & x_2^2 \\end{vmatrix}. $"}
{"_id": "6466669", "title": "", "text": "$d=xa+by$"}
{"_id": "4202493", "title": "", "text": "$x^2-ax,x^3-ax\\in\\mathbb Q$"}
{"_id": "2860813", "title": "", "text": "$\\displaystyle \\frac{n(n-1)}{2}=m$"}
{"_id": "3820555", "title": "", "text": "$A_1 \\wedge A_2 \\wedge \\ldots \\wedge A_n \\implies A_n$"}
{"_id": "7184701", "title": "", "text": "$f(x)= \\dfrac{2 x}{r^2}$"}
{"_id": "1546810", "title": "", "text": "$x_4>10$"}
{"_id": "8909806", "title": "", "text": "$\\frac{\\gamma u}{(1+\\gamma)c} = \\sqrt{\\frac{\\gamma-1}{\\gamma+1}}$"}
{"_id": "5195480", "title": "", "text": "$\\int_\\gamma V = \\int_0^1 \\langle V(\\gamma(s)), \\gamma'(s)\\rangle ds.$"}
{"_id": "7580469", "title": "", "text": "$S=\\big\\{\\langle x,y\\rangle:0\\le x\\le 1\\text{ and }-1\\le y\\le 0\\big\\}$"}
{"_id": "7502750", "title": "", "text": "$\\Bbb R[X]/{\\left<X^2+1\\right>}$"}
{"_id": "2363827", "title": "", "text": "$x^4-x\\in\\mathbb{Z}$"}
{"_id": "2966877", "title": "", "text": "$\\int_0^{2\\pi}f^2(t)dt=\\int_0^{2\\pi}f(t)F'(t)dt$"}
{"_id": "4135166", "title": "", "text": "$Gl(n, F_3)$"}
{"_id": "681469", "title": "", "text": "$ \\begin{pmatrix} 0 & -1 \\\\ 0 & 0 \\end{pmatrix} $"}
{"_id": "6912208", "title": "", "text": "$f(x)=\\frac{9^{x}}{9^{x}+27}$"}
{"_id": "148502", "title": "", "text": "$C_n(X) = 0$"}
{"_id": "1414452", "title": "", "text": "$\\int_0^{\\infty} g(t)\\phi(t)dt\\leq \\int_0^{\\infty} f(t)\\phi(t)dt.$"}
{"_id": "8258215", "title": "", "text": "$ f(n) = 2*f(n-1)-f(n-2) + 2 $"}
{"_id": "3178982", "title": "", "text": "$ \\sum_{n=1}^{\\infty}\\mu(A_n)=\\lim_{N\\to\\infty}\\sum_{n=1}^N\\mu(A_n)\\leq \\mu(A) $"}
{"_id": "4867574", "title": "", "text": "$\\partial, \\bar{\\partial}$"}
{"_id": "4400688", "title": "", "text": "$d(0,r) \\leqslant d(0,y) + d(y,r).$"}
{"_id": "3108144", "title": "", "text": "$\\lim_{\\epsilon \\to 0}\\int_0^{2\\pi}\\lvert\\frac{\\epsilon e^{i \\theta} i d\\theta}{\\sqrt[\\alpha]{\\epsilon} e^{i \\theta/\\alpha}(\\epsilon e^{i \\theta}+1)}\\rvert\\le$"}
{"_id": "8451810", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1&1&k&1\\\\k&1&1&1\\\\0&k-1&1-k&0\\end{array}\\right]$"}
{"_id": "5930124", "title": "", "text": "$F(\\gamma (i))=\\int_i^{\\gamma(i)}f(s)ds=\\int_i^{\\gamma(i)}f(\\gamma_2(s))d\\gamma_2(s)=\\int_{\\gamma_2(i)}^{\\gamma_2(\\gamma(i))}f(s)ds =F(\\gamma_2(\\gamma (i)))-F(\\gamma_2 (i))$"}
{"_id": "6510769", "title": "", "text": "$g: \\begin{cases} x=3s \\\\ y=0 \\\\ z=1-4s \\end{cases}$"}
{"_id": "8370290", "title": "", "text": "$\\sqrt w=\\sqrt re^{it/2}$"}
{"_id": "7696805", "title": "", "text": "$m! \\approx \\sqrt{2 \\pi m} \\left(\\frac{m}{e}\\right)^m,$"}
{"_id": "1473473", "title": "", "text": "$n! = \\sqrt{2 \\pi n}  \\left(\\frac{n}{e}\\right)^n e^{\\lambda_n}$"}
{"_id": "5622111", "title": "", "text": "$ a_n = \\sqrt {1!\\sqrt {2!\\cdots\\sqrt {n!} } }, \\quad n\\in\\mathbb N. $"}
{"_id": "5522926", "title": "", "text": "$d(x,V_1)=\\frac 1 2$"}
{"_id": "1124336", "title": "", "text": "$ \\bbox[lightyellow] {   F(x) = {{a_{\\,0} \\left( {1 - x} \\right)^{\\,4}  + x\\left( {1 + 4x + x^{\\,2} } \\right)} \\over {\\left( {1 - x} \\right)^{\\,5} }}\\quad  \\Leftrightarrow \\quad a_{\\,n}  = a_{\\,0}  + \\left( {{{n\\left( {n + 1} \\right)} \\over 2}} \\right)^{\\,2} \\quad  \\Leftrightarrow \\quad \\left\\{ {\\matrix{    {a_{\\,0}  = a_{\\,0} }  \\cr     {a_{\\,n}  - a_{\\,n - 1}  = n^{\\,3} }  \\cr     {a_{\\,n + 1}  - a_{\\,n}  = \\left( {n + 1} \\right)^{\\,3} }  \\cr   } } \\right.  }$"}
{"_id": "6518040", "title": "", "text": "$\\int_0^\\infty f(x)\\,dx<\\infty,$"}
{"_id": "7224653", "title": "", "text": "$2 \\mid b^2 + b$"}
{"_id": "3808763", "title": "", "text": "$\\frac{3}{a}+ \\frac{2}{b}=1$"}
{"_id": "5445023", "title": "", "text": "$  T_n = 1+2+\\cdots+n=\\frac{n(n+1)}{2} $"}
{"_id": "2336845", "title": "", "text": "$f(x;\\lambda)= \\frac{1}{\\lambda}e^\\frac{-x}{\\lambda}\\chi\\{x>0\\}.$"}
{"_id": "3568233", "title": "", "text": "$=\\left\\lfloor (\\lfloor a\\rfloor+\\{a\\})(\\lfloor b\\rfloor+\\{b\\})\\right\\rfloor-\\lfloor a\\rfloor\\lfloor b\\rfloor$"}
{"_id": "7584929", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty \\frac{1}{an^2+bn+c}=\\frac{1}{a}\\sum_{n=-\\infty}^{\\infty}\\frac{1}{\\left(n-z_{0}\\right)\\left(n-z_{1}\\right)}.$"}
{"_id": "8944", "title": "", "text": "$\\mathbf Z[i]/(q)\\simeq (\\mathbf Z[x]/(x^2+1))/q(\\mathbf Z[x]/(x^2+1))\\simeq  \\mathbf (Z/q\\mathbf Z)[x]/(x^2+1)$"}
{"_id": "1202909", "title": "", "text": "$n! + 2, \\dots, n!+n$"}
{"_id": "6798600", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n-1} \\frac{1}{(1 + (k/n))^2}$"}
{"_id": "438562", "title": "", "text": "$(K[X]/(h))^m$"}
{"_id": "7195885", "title": "", "text": "$x' y z + x y' z + x y z' + x' y z' + x y' z' + x' y' z = x y' + y z' + z x'$"}
{"_id": "2377133", "title": "", "text": "$y_n= \\dfrac{x_1+x_2+\\cdots+x_n}{n}$"}
{"_id": "7912208", "title": "", "text": "$y=4[x^9*x^6]$"}
{"_id": "3358562", "title": "", "text": "$(X_n(\\omega_1))=(1,-1,-1,1,-1,1,-1,\\dots)$"}
{"_id": "5574521", "title": "", "text": "$n^{p+1}-n^{p+1}$"}
{"_id": "743560", "title": "", "text": "$\\le_{\\overline{\\mathbb R}}\\;\\subseteq \\overline{\\mathbb R}\\times\\overline{\\mathbb R}$"}
{"_id": "8490476", "title": "", "text": "$\\int_0^x f(t)\\,dt = -\\int_0^y g(t)\\,dt = u$"}
{"_id": "8831509", "title": "", "text": "$M=\\left(\\begin{smallmatrix} 0 & 1 & -1 \\\\ -1 & 0 & \\alpha \\\\ 2 & -\\alpha & 0 \\end{smallmatrix}\\right)$"}
{"_id": "9174977", "title": "", "text": "$z \\in \\mathbb{R}[x]/(x^2+1)$"}
{"_id": "712666", "title": "", "text": "$\\tan u = \\frac{x}{2},$"}
{"_id": "6341426", "title": "", "text": "$\\frac{\\sqrt{1-x^2}-x+1}{2(x-1)}$"}
{"_id": "45957", "title": "", "text": "$\\int_0^{\\infty}f(x) dx$"}
{"_id": "2583265", "title": "", "text": "$\\tau=\\{\\emptyset,\\mathbb{Z}\\}\\cup\\bigl\\{\\{a,a+1,a+2,\\ldots\\}\\,|\\,a\\in\\mathbb Z\\bigr\\}.$"}
{"_id": "813172", "title": "", "text": "$ds=\\sqrt{1+\\left( \\frac{dy}{dx}\\right) ^{2}}$"}
{"_id": "6971717", "title": "", "text": "$z=\\sqrt[4]{w}e^{\\frac{\\pi ki}{2}}$"}
{"_id": "6907535", "title": "", "text": "$ f(x +f(y) + yf(x)) = y +f(x) + xf(y) ,\\forall x,y \\in \\mathbb R$"}
{"_id": "6050158", "title": "", "text": "$ \\sqrt{x^2+y^2}[1-a(1-2x)(1-2y)]\\to r\\left(1-a(1-2r\\cos \\theta)(1-2r\\sin \\theta)\\right) $"}
{"_id": "5202340", "title": "", "text": "$|f(z)|\\leq 2$"}
{"_id": "5791060", "title": "", "text": "$\\frac{(1-p)^2 p}{1-(1-p)^3}.$"}
{"_id": "5942002", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} f(x_n')=\\limsup\\limits_{x\\to 0} f(x)$"}
{"_id": "9018742", "title": "", "text": "$P_1P_2M'$"}
{"_id": "6982435", "title": "", "text": "$[\\gamma_{n+1}(G),\\gamma_l(G)] = [[G,\\gamma_n(G)],\\gamma_{l}(G)]\\leq [\\gamma_n(G),[G,\\gamma_{l}(G)]][G,[\\gamma_{n}(G),\\gamma_l(G)]]\\leq[\\gamma_n(G),\\gamma_{l+1}(G)][G,\\gamma_{n+l}(G)]\\leq \\gamma_{n+1+l}(G)$"}
{"_id": "1328978", "title": "", "text": "$\\frac{\\zeta'(1-s)}{\\zeta(1-s)}=log(2)+log(\\pi)+\\frac{\\pi}{2}tan(\\frac{\\pi s}{2})-\\frac{\\Gamma'(s)}{\\Gamma(s)}-\\frac{\\zeta'(s)}{\\zeta(s)}$"}
{"_id": "7583538", "title": "", "text": "$\\exists C_0, P_0$"}
{"_id": "2994240", "title": "", "text": "$(1+\\frac{X}{n})^n$"}
{"_id": "5069005", "title": "", "text": "$P[X_n=0]=1-1/n$"}
{"_id": "5946811", "title": "", "text": "$u=\\arcsin(\\sqrt{x})$"}
{"_id": "4817694", "title": "", "text": "$F(x) := \\int_{a}^{x}f(t) dt.$"}
{"_id": "5150146", "title": "", "text": "$\\langle f,\\cos(t) \\rangle = \\int_{0}^{\\pi} f \\cos(t) dt = 0$"}
{"_id": "6203578", "title": "", "text": "$1+2+3+4+...=(-1/12).$"}
{"_id": "6887057", "title": "", "text": "$p(n)=n^2-n+41$"}
{"_id": "3655310", "title": "", "text": "$\\vert PQ \\vert = rsin(\\theta)$"}
{"_id": "6377205", "title": "", "text": "$G = \\{\\ e_1\\ ,\\  e_2\\ ,\\  e_1e_2\\  ,\\ e_2e_1\\  ,\\ e_1e_2e_1\\ \\ , ... \\}$"}
{"_id": "2000525", "title": "", "text": "$A=\\mathbb{R}[x]/(x^2+1)$"}
{"_id": "7330436", "title": "", "text": "$f(x) = \\int_a^x f(t)dt,$"}
{"_id": "1221752", "title": "", "text": "$\\gamma'(t) = \\lim_{t \\to a} \\frac{\\|\\gamma(t) - \\gamma(a)\\|}{\\|t-a\\|}= 1$"}
{"_id": "1217624", "title": "", "text": "$f(a+b)>f(a)+f(b)$"}
{"_id": "850870", "title": "", "text": "$\\forall \\epsilon, \\exists \\delta > 0 s.t. |x-c| < \\delta, x \\in D \\implies |f(x) - f(c)| < \\epsilon$"}
{"_id": "1642159", "title": "", "text": "$\\langle x,z\\rangle\\in R^2,$"}
{"_id": "1550857", "title": "", "text": "$|RS| = r-s$"}
{"_id": "1119304", "title": "", "text": "$f_3.$"}
{"_id": "7074912", "title": "", "text": "$(1/6^{25+l})(1/4^l)$"}
{"_id": "5278366", "title": "", "text": "$n!+2, n!+3, n!+4, ..., n!+n$"}
{"_id": "604446", "title": "", "text": "$1+2+3+\\cdots+n = \\frac{(n+1)n}2 $"}
{"_id": "1203723", "title": "", "text": "$u(x,v)=|\\langle v,\\nu_x\\rangle|$"}
{"_id": "6033759", "title": "", "text": "$\\frac{1}{P(S_2 = j | S_0 = i)}\\sum_{R^{*}_{1}=0}^r\\sum_{l=1}^N P(S_1=l|S_0=i)I(R_{il}=R^*_1)P(S_2=j|S_1=l)I(R_{lj}=r-R^{*}_{1})$"}
{"_id": "3126325", "title": "", "text": "$\\int\\frac{1}{x^2+1}\\,dx$"}
{"_id": "3096810", "title": "", "text": "$\\forall x\\in\\mathbb{R}, \\forall\\varepsilon>0,\\exists r\\in \\mathbb{Q}, |x-r|<\\varepsilon $"}
{"_id": "9241403", "title": "", "text": "$r=[(rs)(r^{2}s)]^{-1}=(r^{2}s)^{-1}(rs)^{-1}=(r^{2}s)(rs)$"}
{"_id": "10751", "title": "", "text": "$f(x+y)=f(x)+f(y)$"}
{"_id": "8032561", "title": "", "text": "$ T^k_l(V) := V^{\\otimes l} \\otimes (V^\\ast)^{\\otimes k} \\cong (V^{\\ast\\ast})^{\\otimes l} \\otimes (V^\\ast)^{\\otimes k} \\cong ((V^\\ast)^{\\otimes l} \\otimes V^{\\otimes k})^\\ast \\cong T^l_k(V)^\\ast. $"}
{"_id": "4452812", "title": "", "text": "$A = \\bigcup_{x\\in A} B(x,r) $"}
{"_id": "630048", "title": "", "text": "$\\left\\lfloor \\frac {\\lfloor px \\rfloor}{p} \\right\\rfloor=\\lfloor x\\rfloor$"}
{"_id": "4705497", "title": "", "text": "$a^{log_b n} = n^{log_b a}$"}
{"_id": "3121008", "title": "", "text": "$f(a+b)=f(a)+f(b)\\in J$"}
{"_id": "3579421", "title": "", "text": "$\\mathop {\\lim }\\limits_{N \\to \\infty }N \\sum\\limits_{m = 1}^N {\\frac{{{a^{N - m}}}}{m}} $"}
{"_id": "1213392", "title": "", "text": "$P[X_2\\leq x_2 |X_1=u_1]$"}
{"_id": "6450168", "title": "", "text": "$ P =  \\begin{bmatrix} 1 & 2 & 1 & 2 & 0 \\\\ 1 & -1 & -1 & 1 & 0 \\\\ 2 & -5 & -4 & 1 & 0 \\\\ 4 & 2 & 0 & 6 & 0 \\end{bmatrix} $"}
{"_id": "2192334", "title": "", "text": "$s = \\{v_1, v_2, v_3, ... , v_n\\}$"}
{"_id": "7463337", "title": "", "text": "$T: p\\mapsto f(p) - f(-p)$"}
{"_id": "1641620", "title": "", "text": "$(1,1,-1,1,1,-1,1,1,-1,\\cdots)$"}
{"_id": "4977353", "title": "", "text": "$ f(x) = 1/|x|, $"}
{"_id": "3838763", "title": "", "text": "$Y=\\{\\langle x,y\\rangle\\in\\Bbb R^2:y=3\\}$"}
{"_id": "8841089", "title": "", "text": "$l=\\overline{p_1p_2}$"}
{"_id": "1401768", "title": "", "text": "$au + bv = d.$"}
{"_id": "4077877", "title": "", "text": "$x \\to x^3-x$"}
{"_id": "8111165", "title": "", "text": "$\\int\\int\\int\\int\\int\\int\\int dx$"}
{"_id": "6646064", "title": "", "text": "$\\lim_{n \\to \\infty} \\inf (s_n) = s_*$"}
{"_id": "8857529", "title": "", "text": "$I = \\displaystyle\\int_{-\\pi}^\\pi\\dfrac{\\sin^2(x)}{1+a^x}dx $"}
{"_id": "7730333", "title": "", "text": "$ ...  \\qquad =  - \\lfloor a \\rfloor f(a)   + \\lfloor a \\rfloor f\\left(\\lfloor a\\rfloor+1\\right)\\\\ +  \\left(\\lfloor a \\rfloor + 1 \\right)  \\left[-f\\left(\\lfloor a\\rfloor+1\\right)+f\\left(\\lfloor a\\rfloor+2 \\right)\\right]\\\\ + \\left(\\lfloor a \\rfloor + 2 \\right)  \\left[-f\\left(\\lfloor a\\rfloor+2\\right)+f\\left(\\lfloor a\\rfloor+3 \\right)\\right]\\\\ + ... \\\\ + \\left(\\lfloor b \\rfloor -1 \\right)  \\left[-f\\left(\\lfloor b\\rfloor-1\\right)+f\\left(\\lfloor b\\rfloor\\right)\\right]\\\\ + \\lfloor b\\rfloor~\\left[f(b)-f\\left( \\lfloor b \\rfloor\\right) \\right]\\\\[5mm]   = \\quad - \\lfloor a \\rfloor f(a)  - f\\left(\\lfloor a\\rfloor+1\\right) - f\\left(\\lfloor a\\rfloor+2 \\right) ... - f\\left(\\lfloor b\\rfloor\\right) + \\lfloor b\\rfloor~f(b) $"}
{"_id": "6565885", "title": "", "text": "$b_n=\\int_{-\\pi}^{\\pi}\\frac{1-a\\cos(x)}{1-2a\\cos(x)+a^2}\\sin(nx)dx$"}
{"_id": "65626", "title": "", "text": "$\\frac{1}{2} \\begin{vmatrix} x_{1} & y_{1} & 1   \\\\  x_{3} & y_{3} & 1  \\\\ x_{2} & y_{2} & 1  \\\\   \\end{vmatrix}$"}
{"_id": "149410", "title": "", "text": "$\\|A\\|_F=\\sqrt{\\text{Trace}(A^TA)}$"}
{"_id": "5210784", "title": "", "text": "$a_{2n}=\\frac{1}{2^{n-2}}$"}
{"_id": "5023774", "title": "", "text": "$Cov(X,Y) < 0$"}
{"_id": "4459083", "title": "", "text": "$\\sum_n |\\langle f, x_n \\rangle |^2 = ||f||^2$"}
{"_id": "6744033", "title": "", "text": "$\\frac75=1+\\frac25=1+\\frac1{\\frac52}=1+\\frac1{2+\\frac12}=1+\\frac1{2+\\frac1{1+1}}$"}
{"_id": "7367232", "title": "", "text": "$\\mathcal{X}(\\mathcal{A})$"}
{"_id": "1348971", "title": "", "text": "$\\sum\\limits_{n=-\\infty }^{\\infty }{{{e}^{i\\pi {{n}^{2}}\\tau +i2nz}}}=\\frac{1}{\\sqrt{-i\\tau }}\\sum\\limits_{n=-\\infty }^{\\infty }{{{e}^{{{\\left( z+n\\pi  \\right)}^{2}}/\\pi i\\tau }}}$"}
{"_id": "4892972", "title": "", "text": "$(\\gamma_1\\wedge\\gamma_2)((\\gamma_2 + \\gamma_3)\\wedge(\\gamma_4 + \\gamma_1))\\\\  =\\gamma_1\\gamma_2((\\gamma_2+\\gamma_3)(\\gamma_4+\\gamma_1)-(\\gamma_2+\\gamma_3)\\cdot(\\gamma_4+\\gamma_1))\\\\ \\gamma_1\\gamma_2(\\gamma_2\\gamma_4+\\gamma_2\\gamma_1+\\gamma_3\\gamma_4+\\gamma_3\\gamma_1-0)\\\\ =\\gamma_1\\gamma_2\\gamma_2\\gamma_4+\\gamma_1\\gamma_2\\gamma_2\\gamma_1+\\gamma_1\\gamma_2\\gamma_3\\gamma_4+\\gamma_1\\gamma_2\\gamma_3\\gamma_1\\\\ =\\gamma_1\\gamma_4+1+\\gamma_1\\gamma_2\\gamma_3\\gamma_4+\\gamma_2\\gamma_3$"}
{"_id": "6313884", "title": "", "text": "$n! = \\sqrt{2\\pi n} (\\frac{n}{e})^n \\big(1 + \\mathcal{O}(\\frac{1}{n})\\big)$"}
{"_id": "4081173", "title": "", "text": "$f(x^2)+f(xy)=f(x)\\,f(y)+y\\,f(x)+x\\,f(x+y)\\,.$"}
{"_id": "8122005", "title": "", "text": "$\\left|\\delta(x^{n-1}+...+a^{n-1})\\right|$"}
{"_id": "351379", "title": "", "text": "$f(n)=f(n-3)+(n-2)/2$"}
{"_id": "5641558", "title": "", "text": "$ \\delta x= -A^{-1}\\delta A (x+\\delta x), $"}
{"_id": "468864", "title": "", "text": "$W'+ (\\Phi^n)$"}
{"_id": "251495", "title": "", "text": "$p_1p_2\\in P$"}
{"_id": "4949284", "title": "", "text": "$\\int_0^1 f(t)h(t)dt = \\int_{0}^1 g(t)h(t)dt.$"}
{"_id": "1092480", "title": "", "text": "$1/|x| = |1/x|$"}
{"_id": "421156", "title": "", "text": "$\\gamma_1 \\geq \\gamma_2 \\geq \\cdots \\geq \\gamma_n$"}
{"_id": "3098978", "title": "", "text": "$k:=\\mathbb{R}[x]/(x^2+1)$"}
{"_id": "4285250", "title": "", "text": "$Q(a,b,c)=Q(b,a,c)=Q(a,c,d)=Q(c,b,a)$"}
{"_id": "1080636", "title": "", "text": "$\\| x_n - x_{n-1} \\|$"}
{"_id": "8995296", "title": "", "text": "$\\sum\\limits_{n=-\\infty}^{\\infty} e^{-\\pi n^2 /t}e^{2\\pi i n x} =t^{1/2}e^{-\\pi t x^2}\\sum\\limits_{n=-\\infty}^{\\infty} e^{-\\pi n^2 t}e^{-2\\pi n xt}. $"}
{"_id": "6472769", "title": "", "text": "$M:=\\begin{bmatrix} A & -B \\\\ B & A\\end{bmatrix}$"}
{"_id": "5339528", "title": "", "text": "$V_1 = \\pi\\int_{0}^{1}\\left(x^{3/2}-1\\right)^2dx = 9/20\\pi$"}
{"_id": "1514532", "title": "", "text": "$T_\\Bbb R = \\begin{pmatrix} A & -B \\\\ B & A \\end{pmatrix}$"}
{"_id": "5159663", "title": "", "text": "$A_{0}\\subseteq A_{1}\\subseteq A_{2}\\subseteq A_{3}\\cdots\\subseteq\\Bbb N$"}
{"_id": "7410215", "title": "", "text": "$\\vec x = \\left[\\matrix{1\\cr -1\\cr 2\\cr 1\\cr 1\\cr}\\right]$"}
{"_id": "695689", "title": "", "text": "$10^{-40}$"}
{"_id": "2908742", "title": "", "text": "$1 = \\int_0^\\infty f(x) \\, dx$"}
{"_id": "4792755", "title": "", "text": "$s(x)=\\displaystyle\\sum_{i=1}^k{c_iI_{A_i}(x)}$"}
{"_id": "6603989", "title": "", "text": "$\\sqrt r e^{\\frac {i \\theta}{2}+i\\pi}$"}
{"_id": "7734359", "title": "", "text": "$\\sum_{k=0}^{n-1}\\cos\\dfrac{2\\pi k}n=0$"}
{"_id": "4401770", "title": "", "text": "$3|d^3-d.$"}
{"_id": "6606957", "title": "", "text": "$1-P[X_0=x]$"}
{"_id": "45045", "title": "", "text": "$\\mathbb{R}^{n+1}$"}
{"_id": "855262", "title": "", "text": "$\\int_{0}^{\\pi} \\frac{x}{1+\\sin^2 x} \\, dx = \\frac{\\pi}{2} \\int_{0}^{\\pi} \\frac{1}{1+\\sin^2 x} \\, dx $"}
{"_id": "2139691", "title": "", "text": "$a_n = \\frac{n(n+1)}{2}$"}
{"_id": "1867782", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta>0: |x-y|<\\delta \\implies |f(x)-f(y)|<\\epsilon, $"}
{"_id": "5316832", "title": "", "text": "$((A^+)^+)^+=(A^+)^+\\cup \\{(A^+)^+\\}=\\{\\emptyset,\\{\\emptyset\\}\\}\\cup\\{\\{\\emptyset,\\{\\emptyset\\}\\}\\}=$"}
{"_id": "5687662", "title": "", "text": "$|2\\tan\\frac\\theta 2 - \\sin\\theta| = |\\sin\\theta \\frac{\\sin^2\\theta}{4\\cos^4\\frac{\\theta}2}|$"}
{"_id": "1818582", "title": "", "text": "$\\liminf\\mu(A_n)\\ge\\mu(A)\\ge\\limsup\\mu(A_n).$"}
{"_id": "2591653", "title": "", "text": "$\\sqrt{\\sqrt2+\\sqrt3}-\\sqrt{\\sqrt2-\\sqrt3}>0$"}
{"_id": "2912309", "title": "", "text": "$\\sum_{n=1}^\\infty e^{-(n+x)^2}=e^{-x^2} \\sum_{n=1}^\\infty e^{-n^2} e^{-2nx}$"}
{"_id": "3974500", "title": "", "text": "$\\mathbb{E}[X^r] - \\mathbb{E}[Y^r] + \\sum_{i=1}^{r-1}c_{r,i}(\\mathbb{E}[X^i] - \\mathbb{E}[Y^i]) = 0$"}
{"_id": "32020", "title": "", "text": "$ \\zeta(s) = \\pi^{- \\frac{1}{2} + s} \\frac{\\Gamma(\\frac{1}{2} - \\frac{s}{2})}{\\Gamma(\\frac{s}{2})} \\zeta(1 - s).$"}
{"_id": "4143", "title": "", "text": "$x^n + x^{n+1}$"}
{"_id": "701219", "title": "", "text": "$\\|A\\|_2  = \\sqrt{\\lambda_\\max(A^*A)}$"}
{"_id": "1637050", "title": "", "text": "$\\exists x \\forall y Rxy$"}
{"_id": "4358488", "title": "", "text": "$\\displaystyle \\sum_{n=1}^\\infty \\frac{1}{n}=\\sum_{n=1}^\\infty \\frac{1}{n^2}+\\sum_{n\\ne\\text{square}}^\\infty \\frac{1}{n}$"}
{"_id": "2104807", "title": "", "text": "$ S_N-T_N=\\sum_{n=1}^N\\frac1{n^2} \\to \\sum_{n=1}^\\infty\\frac1{n^2}=\\frac{\\pi^2}6<\\infty. $"}
{"_id": "177147", "title": "", "text": "$a^{\\log_a b} = b$"}
{"_id": "639013", "title": "", "text": "$\\OR_a:\\MX\\to\\Bbb{Z}\\cup\\{\\infty\\},\\quad \\OR_a(f)=\\begin{cases}0,&\\text{ if } f \\text{ is nonzero and holomorphic at }a\\\\k&,\\text{ if f has a zero of order }k\\text{ at a}\\\\-k,&\\text{ if f has a pole of order k at a}\\\\ \\infty,&\\text{ if f is identically zero at a}\\end{cases}$"}
{"_id": "4932790", "title": "", "text": "${(\\mathbb Z}[x]/(x + 3))/(x^2+1) \\cong {\\mathbb Z}/(10)$"}
{"_id": "5824203", "title": "", "text": "$[40, 10]$"}
{"_id": "4601094", "title": "", "text": "$p_1p_2=16$"}
{"_id": "1077451", "title": "", "text": "$E(X+Y)^p \\le 2^p (E(X^p)+E(Y^p)).$"}
{"_id": "2238804", "title": "", "text": "$\\alpha=(a_1,...,a_n,a'_1,...,,a'_n)$"}
{"_id": "466921", "title": "", "text": "$  k^3 = 6 \\; \\left( \\begin{array}{c} k \\\\ 3 \\end{array} \\right) + 6 \\; \\left( \\begin{array}{c} k \\\\ 2 \\end{array} \\right) + \\left( \\begin{array}{c} k \\\\ 1 \\end{array} \\right),  $"}
{"_id": "7890131", "title": "", "text": "$|x^n-a^n| < \\delta$"}
{"_id": "6325757", "title": "", "text": "$gcd(a, a+2)$"}
{"_id": "1683386", "title": "", "text": "$(1+\\frac{1}{n^2})^{n^2}$"}
{"_id": "5985419", "title": "", "text": "$v_1\\cdot v_2 = \\sum_{k=1}^d\\cos\\frac{(k-1)\\pi}d\\sin\\frac{(k-1)\\pi}d = \\frac12\\sum_{k=1}^d\\sin\\frac{(k+1)2\\pi}{d} = 0$"}
{"_id": "6492429", "title": "", "text": "$f(x)=f(x\\cdot 1)=f(x)\\,f(1)-f(x+1)+1=f(x)-f(x+1)+1$"}
{"_id": "8657121", "title": "", "text": "$\\tag{*}\\{\\,\\ldots, a-2n, a-n,a,a+n,a+2n,\\ldots\\,\\}.$"}
{"_id": "6418774", "title": "", "text": "$a\\equiv e\\mod s$"}
{"_id": "8712967", "title": "", "text": "$\\mathbb{P}(X_{n+1}=X_n/2|\\mathcal{F}_n)=1-X_n$"}
{"_id": "802382", "title": "", "text": "$(1+r)^n>=1+rn$"}
{"_id": "4465024", "title": "", "text": "$\\lvert DC \\rvert = kb$"}
{"_id": "320104", "title": "", "text": "$\\Phi(x)=\\int_a^x f(t)dt$"}
{"_id": "9163761", "title": "", "text": "$a_1,f(a_1),f(f(a_1)),...$"}
{"_id": "8211963", "title": "", "text": "$\\begin{bmatrix}1&-3&0&5\\\\-1&1&5&2\\\\-1&1&5&3\\end{bmatrix}$"}
{"_id": "1809643", "title": "", "text": "$R=\\{\\langle x,y\\rangle:s\\le x\\le t\\text{ and }0\\le y\\le 1\\}\\;;$"}
{"_id": "8743024", "title": "", "text": "$\\chi_A=x^{n-1}(x-\\tr A)$"}
{"_id": "1500805", "title": "", "text": "$m \\times \\mu(A\\times B)=m(A)\\mu(B)$"}
{"_id": "681682", "title": "", "text": "$\\sum_{j=1}^\\infty\\sum_{k=1}^\\infty \\frac{1}{j^2(j+k)^2}$"}
{"_id": "1233784", "title": "", "text": "$f(a^k)=b^k$"}
{"_id": "3522289", "title": "", "text": "$=|x+y-z|+|x-y+z|+|x-y+z|+|-x+y+z|+|-x+y+z|+|x+y-z|$"}
{"_id": "6264311", "title": "", "text": "$Cov(x,y)\\neq0$"}
{"_id": "5247209", "title": "", "text": "$x,y \\in A \\rightarrow xRy \\lor yRx$"}
{"_id": "1882076", "title": "", "text": "$f(z) = z^2-z-2$"}
{"_id": "7875671", "title": "", "text": "$\\left\\lfloor \\frac{1}{b} \\left\\lfloor \\frac{a}{b} \\right\\rfloor \\right\\rfloor = \\left\\lfloor \\frac{a}{b^2} \\right\\rfloor.$"}
{"_id": "4668937", "title": "", "text": "$\\alpha+$"}
{"_id": "7388625", "title": "", "text": "$ \\left\\{ \\begin{array}{c} a=\\frac{1}{2}\\\\ b=1\\\\ c=0\\\\ d=1\\\\ e=-2 \\end{array} \\right.$"}
{"_id": "4749808", "title": "", "text": "$m = m' + m''$"}
{"_id": "1297389", "title": "", "text": "$d(x, A) \\leq d(x, z) + d(z, A). \\tag{1}$"}
{"_id": "1379418", "title": "", "text": "$p_1 \\mid p_2$"}
{"_id": "2116685", "title": "", "text": "$ P(k) : k^3 < 3^k $"}
{"_id": "5028093", "title": "", "text": "$R:=\\mathbb{R}[x]/(x^2+1)$"}
{"_id": "4778996", "title": "", "text": "$\\sum_{n\\geq1}\\frac1{f_nf_{n+2}}$"}
{"_id": "5658668", "title": "", "text": "$0=f(t,\\gamma+\\delta\\gamma,\\dot\\gamma+\\delta\\dot\\gamma,\\ddot\\gamma+\\delta\\ddot\\gamma,T+\\delta T)$"}
{"_id": "3118232", "title": "", "text": "$n = \\left\\lfloor\\frac{M}{\\sqrt{a}}\\right\\rfloor\\left\\lfloor\\frac{N}{\\sqrt{a}}\\right\\rfloor$"}
{"_id": "415543", "title": "", "text": "$x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!$"}
{"_id": "6392263", "title": "", "text": "$[a]=\\{\\dots,a-2b,a-b,a,a+b,a+2b,\\dots\\}.$"}
{"_id": "7788155", "title": "", "text": "$\\{e_n,e_{n-1},e_{n-2},\\ldots,e_m\\}$"}
{"_id": "3508763", "title": "", "text": "$y_0=\\frac{a+b}{b}$"}
{"_id": "567830", "title": "", "text": "$I_n = \\int_{-\\pi}^{\\pi} \\frac{\\sin (nx)}{\\sin x} dx$"}
{"_id": "3948831", "title": "", "text": "$B_n = \\frac {n(n+1)}2$"}
{"_id": "4186079", "title": "", "text": "$z=y^k+x\\,, x\\in \\mathbb N$"}
{"_id": "497143", "title": "", "text": "$\\cos(2x) - \\cos(x) = (1-2\\sin^2(x)) - (1-2\\sin^2(x/2)) = 2\\left(\\sin^2(x/2) - \\sin^2(x)\\right)$"}
{"_id": "3815487", "title": "", "text": "$B=P^{-1}A P=\\left[\\matrix{1&0&0\\cr 0&0&-\\sqrt{2}\\cr 0&\\sqrt{2}&0\\cr}\\right]\\ .$"}
{"_id": "4565378", "title": "", "text": "$f(x)=\\frac {9^x}{x!}=\\frac {9^x}{\\Gamma(x+1)}$"}
{"_id": "2284454", "title": "", "text": "$9^x+6^x=2^{(x+1)}$"}
{"_id": "614756", "title": "", "text": "$f=\\sum f_j=\\sum(-1)^j\\chi_{I_j}.$"}
{"_id": "9257829", "title": "", "text": "$\\lim_{x\\to a}f(x)=f(a)\\Leftrightarrow \\forall\\varepsilon_1>0,\\exists\\delta_1>0,|x-a|<\\delta_1\\Rightarrow|f(x)-f(a)|<\\varepsilon_1$"}
{"_id": "207130", "title": "", "text": "$ \\int_{b}^{a} 2\\pi \\,f(x) \\,dx $"}
{"_id": "1212825", "title": "", "text": "$F=I\\oplus K$"}
{"_id": "1293613", "title": "", "text": "$\\overline{\\mathbb{R}} = \\mathbb{R}$"}
{"_id": "6058965", "title": "", "text": "$\\sum_{n\\geq 1}\\frac{x}{n(x+n)} = \\sum_{n\\geq 1} \\frac{x}{n^2} \\frac{1}{1+\\frac{x}{n}}$"}
{"_id": "9295747", "title": "", "text": "$[F(n+2)-F(n)]^2  =  F(n+2)^2 - 2[F(n+2)F(n)] + F(n)^2$"}
{"_id": "5718041", "title": "", "text": "$\\mathcal A:=\\{\\{x\\},x\\in\\mathbb N\\}$"}
{"_id": "3810257", "title": "", "text": "$\\Phi (x)=\\underset{n\\to \\infty }{\\mathop{\\lim }}\\,{{f}_{n}}(x)=0$"}
{"_id": "4588913", "title": "", "text": "$(x,y,z)\\cdot (1,1,1)<0$"}
{"_id": "50085", "title": "", "text": "$\\sum_{i=1}^n\\cos\\left(i\\pi\\frac kn\\right)=0$"}
{"_id": "2395931", "title": "", "text": "$F(x)=\\int_a^x f(t)\\,dx, $"}
{"_id": "1808765", "title": "", "text": "$\\frac{1}{2^{N-3}}$"}
{"_id": "9057672", "title": "", "text": "$x:=(x,y)$"}
{"_id": "3727771", "title": "", "text": "$\\begin{bmatrix}A& -B^t\\\\B& A^t \\end{bmatrix}$"}
{"_id": "793402", "title": "", "text": "$xRy \\leftrightarrow yRx$"}
{"_id": "7215578", "title": "", "text": "$ P_{0,a}^\\alpha(\\gamma)=P_{\\gamma(\\nu^\\alpha (a)),\\gamma(a)}...P_{\\gamma(\\alpha),\\gamma(2\\alpha)}P_{\\gamma(0),\\gamma(\\alpha)},$"}
{"_id": "6198578", "title": "", "text": "$ \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{i=1}^n 1 =\\lim_{n\\rightarrow \\infty} \\frac{1}{n} n = 1$"}
{"_id": "8416767", "title": "", "text": "$T = \\begin{pmatrix}A & -\\overline{B} \\\\ B & \\overline{A} \\end{pmatrix},$"}
{"_id": "7324286", "title": "", "text": "$f(0)=f(\\pi)+f(-\\pi)$"}
{"_id": "137500", "title": "", "text": "$p(n) \\implies p(n+1)$"}
{"_id": "1814903", "title": "", "text": "$                 f(0) = 0,\\;\\; f(\\pi)=0 $"}
{"_id": "813054", "title": "", "text": "$ \\begin{vmatrix} a &   a+b & a+2b\\\\ a+2b & a  &  a+b\\\\ a+b & a+2b &  a \\end{vmatrix}  =  9b^2 (a+b) $"}
{"_id": "439994", "title": "", "text": "$\n \\begin{align*}\n xy \\cdot\n \\left|\n \\begin{array}{ccc}\n 1 & 1 & 1 \\\\\n x & y & 1 \\\\\n x^2 & y^2 & 1 \\\\\n \\end{array} \\right|\n &= xy \\cdot\n \\left|\n \\begin{array}{ccc}\n 0 & 0 & 1 \\\\\n x - 1 & y -1 & 1 \\\\\n x^2 -1 & y^2 - 1 & 1 \\\\\n \\end{array} \\right| \\\\\n &= xy \\cdot (x-1)(y-1)\n \\left|\n \\begin{array}{ccc}\n 0 & 0 & 1 \\\\\n 1 & 1 & 1 \\\\\n x + 1 & y + 1 & 1\n \\end{array}\n \\right| & \\text{because } x^2 - 1^2 = (x-1)(x+1)\\\\\n &=xy \\cdot (x-1)(y-1) \\cdot 1 \\cdot(1 \\cdot (y+1) - 1 \\cdot (x + 1)) & \\text{Laplace with row 1}\\\\\n &= xy \\cdot (x-1)(x+1)(y-x) \n \\end{align*}\n $"}
{"_id": "780592", "title": "", "text": "$L=\\lim_{x\\to a^-}f(x)=\\lim_{x\\to a^+}f(x)$"}
{"_id": "818097", "title": "", "text": "$b^{log_b(x)}=x$"}
{"_id": "7358475", "title": "", "text": "$f(A+B)=f(A)+f(B)-2$"}
{"_id": "3265027", "title": "", "text": "$ |ab| =ab = |a||b| $"}
{"_id": "356425", "title": "", "text": "$f(x) = \\frac{x(x-1)}{2} = \\binom{x}{2}$"}
{"_id": "148575", "title": "", "text": "$\\lim_{x\\to a} f(x) g(x) =\\lim_{x\\to a} f(x) \\cdot \\lim_{x\\to a} g(x) =L\\lim_{x\\to a} g(x) $"}
{"_id": "7141108", "title": "", "text": "$\\left\\Vert x_{n+1}-x_{n}\\right\\Vert \\leq r_{n}-r_{n+1}$"}
{"_id": "228225", "title": "", "text": "$\\text{Cov}(X,Y)=0$"}
{"_id": "2502829", "title": "", "text": "$\\;a^b\\;$"}
{"_id": "2638428", "title": "", "text": "$\\int\\frac{\\sin^3(x)}{(\\cos^4(x)+3\\cos^2(x)+1)\\cdot\\arctan(\\sec(x)+\\cos(x))}$"}
{"_id": "8931941", "title": "", "text": "$ \\int_{0}^{\\infty}{\\sin^{n}\\left(x\\right)\\over  x}\\,\\mathrm{d}x $"}
{"_id": "3503936", "title": "", "text": "$I_n = n\\pi$"}
{"_id": "7436724", "title": "", "text": "$S = \\{ v,e_1, e_2,\\ldots,e_n\\}$"}
{"_id": "8554584", "title": "", "text": "$\\displaystyle\\lim_{h \\to 0} f'(x) = \\displaystyle\\lim_{x \\to c} f'(x) = d$"}
{"_id": "2500897", "title": "", "text": "$ \\left(1 + \\sqrt{2} + \\sqrt{3} + \\dots + \\sqrt{n} \\right) - \\frac{2}{3} n \\sqrt{n}$"}
{"_id": "8910806", "title": "", "text": "$=\\frac{(x+y+z)(y-z)^2(x-y)(x^2(y+z)^2-y^2(x+z)^2)}{(x+z)^2(y+z)^2}=$"}
{"_id": "6722297", "title": "", "text": "$ K\\subset D \\subset\\overline{D}\\subset A. $"}
{"_id": "5348318", "title": "", "text": "$A=\\{ \\left(x,y \\right)\\in \\mathbb{R}^2: 0 \\le x \\le \\pi, 0 \\le y \\le \\pi\\}$"}
{"_id": "870852", "title": "", "text": "$g[n] = \\delta[n] - A \\delta[n-1]$"}
{"_id": "5350506", "title": "", "text": "$\\lim_{x\\to c^+}f(x)=\\lim_{x\\to c^-} f(x)$"}
{"_id": "7647289", "title": "", "text": "$f(x;3)=\\frac{1}{3} e^{-x/2} x \\left(e^{3 x/2}-2 \\sin \\left(\\frac{1}{6} \\left(3 \\sqrt{3}    x+\\pi \\right)\\right)\\right)$"}
{"_id": "374431", "title": "", "text": "$p(e) = p(e')$"}
{"_id": "6575591", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\sum_{i=1}^\\infty a^{(n)}_i =\\sum_{i=1}^\\infty a_i$"}
{"_id": "7889153", "title": "", "text": "$=3+\\frac1{1+\\frac2{95}}=3+\\frac1{1+\\frac1{\\frac{95}2}}=3+\\frac1{1+\\frac1{47+\\frac12}}$"}
{"_id": "5207512", "title": "", "text": "$\\det\\begin{pmatrix} 1 & s & s^2 \\\\ 1 & t & t^2 \\\\ 1 & u & u^2\\end{pmatrix}$"}
{"_id": "1102586", "title": "", "text": "$y'(x)={dy\\over dx}={dy\\over dt}\\cdot{dt\\over dx}={2a\\cos(2t)\\over 2a\\left({\\cos(t)\\over \\sin(t)}-2\\sin(t)\\cos(t)\\right)}=\\tan(t)$"}
{"_id": "3334258", "title": "", "text": "$g(y)=\\int_y^{\\sqrt y } 15y \\, dx$"}
{"_id": "3502799", "title": "", "text": "$\\Bbb R^n\\times\\Bbb R=\\Bbb R^{n+1}$"}
{"_id": "1247997", "title": "", "text": "$A_1 \\times A_2 \\times ... \\times~ A_k$"}
{"_id": "6628778", "title": "", "text": "$\\mathbb{R}[x]/((x^2 +1)^2 )$"}
{"_id": "8609739", "title": "", "text": "$l = \\begin{cases} x=-2t+1 \\\\  y=3t-2 \\\\  z=t+4 \\end{cases} $"}
{"_id": "198386", "title": "", "text": "$\\text{ } \\text{ } \\text{ } \\text{ } \\text{ } \\text{ } \\text{ } \\color{red}{p_{1}p_{2}}p_{3}$"}
{"_id": "6126244", "title": "", "text": "$ J=\\int_{-\\pi}^{\\pi}\\frac{\\sin^2\\left(x\\right)}{1+a^{x}}\\text{d}x $"}
{"_id": "1999101", "title": "", "text": "$\\mathcal{P}(n) \\implies \\mathcal{P}(n+1)$"}
{"_id": "3427502", "title": "", "text": "$\\eta(s)  = \\frac{1}{\\Gamma(s)}\\int_{0}^{\\infty} \\frac{x^{s-1}}{1+e^{x}} \\, dx , \\quad \\text{Re}(s) >0,\\tag{1}$"}
{"_id": "5943281", "title": "", "text": "$\\left\\lfloor \\frac{y}{b}\\right\\rfloor = \\left\\lfloor \\frac{\\lfloor y \\rfloor}b\\right\\rfloor$"}
{"_id": "1952468", "title": "", "text": "$\\frac{2 + bx}{b + x}$"}
{"_id": "609602", "title": "", "text": "$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos(nx) \\, dx $"}
{"_id": "596349", "title": "", "text": "$\\lim_{r \\to \\infty} \\frac 1 r$"}
{"_id": "1065957", "title": "", "text": "$\\sum _{n=1}^{ \\infty}\\sum _{k=1}^{\\infty }\\frac{1}{k^2\\left(n^2+\\left(\\frac{\\pi }{x}\\right)^2k^2\\right)}$"}
{"_id": "1250099", "title": "", "text": "$g[x0] = x1+ zg$"}
{"_id": "2136416", "title": "", "text": "$40 + 40 = x + 40 + 10$"}
{"_id": "6996451", "title": "", "text": "$(p_1p_2\\cdots p_n)+1$"}
{"_id": "140551", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\frac{n}{\\sqrt[n]{n!}}$"}
{"_id": "5633399", "title": "", "text": "$\\{a + \\delta,\\ a+2\\delta,\\ a+3\\delta,\\ \\dots\\} \\subseteq S$"}
{"_id": "78156", "title": "", "text": "$l: \\begin{cases} x=1 \\\\ y=z \\end{cases}$"}
{"_id": "6574990", "title": "", "text": "$\\left\\lfloor \\frac{r + b}{c}\\right\\rfloor = \\left\\lfloor \\frac{\\lfloor r \\rfloor + b}{c}\\right\\rfloor$"}
{"_id": "6083740", "title": "", "text": "$A=\\{\\langle x,y\\rangle\\in\\Bbb I\\times(\\Bbb R\\setminus\\Bbb N):xy>1\\}$"}
{"_id": "7814457", "title": "", "text": "$|AB|=|AB|$"}
{"_id": "1843654", "title": "", "text": "$f_D(d) = \\dfrac{2d}{R^2}.$"}
{"_id": "4497074", "title": "", "text": "$ \\sum_{n=1}^{\\infty}\\frac{n^n}{(n!)^2}$"}
{"_id": "973300", "title": "", "text": "$\\displaystyle\\phi(x) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\dfrac{1}{2}x^2}$"}
{"_id": "1381568", "title": "", "text": "$\\mathrm{cov}(X,Y)=0$"}
{"_id": "5337170", "title": "", "text": "$\\Sigma = \\{\\emptyset, \\mathbb{N}, \\mathcal{P}(\\mathbb{N}), \\mathbb{R}, \\mathbb{R}-\\mathbb{N}, \\mathbb{R}-\\mathcal{P}(\\mathbb{N}),(\\mathbb{R}-\\mathcal{P}(\\mathbb{N})) \\cup \\mathbb{N}, (\\mathbb{R}-\\mathbb{N}) \\cup (\\mathbb{R}-\\mathcal{P}(\\mathbb{N})) \\}$"}
{"_id": "8988679", "title": "", "text": "$\\displaystyle f(1-x)=\\frac{2}{4^{1-x}+2}=\\frac{2\\cdot 4^x}{4+2\\cdot 4^x}=\\frac{4^x}{2+ 4^x}$"}
{"_id": "60863", "title": "", "text": "$e^x=\\lim_{n\\to\\infty}\\left(1+\\frac xn\\right)^n$"}
{"_id": "6499150", "title": "", "text": "$I_n=[2n \\pi,2n\\pi+\\pi]$"}
{"_id": "6607857", "title": "", "text": "$\\alpha + \\omega > \\alpha$"}
{"_id": "1443051", "title": "", "text": "${x\\over 2}=\\tan\\theta$"}
{"_id": "7823733", "title": "", "text": "$(a, b)+_\\mathbb{C}(c, d)=(a+_\\mathbb{R}c, b+_\\mathbb{R}d)=(c+_\\mathbb{R}a, d+_\\mathbb{R}b)=(c, d)+_\\mathbb{C}(a, b).$"}
{"_id": "7638707", "title": "", "text": "$\\gamma(a) = \\gamma(0) = \\gamma(2\\pi) = \\gamma(b).  \\tag{2}$"}
{"_id": "6886802", "title": "", "text": "$\\left(\\begin{array}{ccc|c}3&1&-1&0\\\\1&-1&2&0\\\\2&2&-3&0\\end{array}\\right)$"}
{"_id": "4382150", "title": "", "text": "$|f(z)^2-1|=|f(z)+1||f(z)-1|$"}
{"_id": "1400743", "title": "", "text": "$f((a,b)+(h,k)) - f((a,b)) = f(a+h, b+k) - f(a,b) = f(a,b)+f(h,k)-f(a,b) = f(h,k)$"}
{"_id": "6264308", "title": "", "text": "$Cov(x,y)=E(xy)-E(x)E(y).$"}
{"_id": "8172884", "title": "", "text": "$\\sum\\limits_{r=0}^{n+1} \\frac{{(n+1)}!}{r!(n+1-r)!} =2^{n+1}.$"}
{"_id": "9019309", "title": "", "text": "$ \\frac1{\\sqrt{N}}\\,\\frac{\\sqrt1+\\sqrt2+\\cdots+\\sqrt{N}}N $"}
{"_id": "464786", "title": "", "text": "$\\sum_{k=1}^\\infty\\frac{\\cos 2\\pi k}{k}$"}
{"_id": "6848948", "title": "", "text": "$ \\det\\begin{bmatrix} 1 & 1 & 1 \\\\ x & y & z \\\\ x^2 & y^2 & z^2 \\end{bmatrix} = \\det\\begin{bmatrix} 1 & 1 & 1 \\\\ 0 & y-x & z-x \\\\ 0 & y^2-x^2 & z^2-x^2 \\end{bmatrix} = \\det\\begin{bmatrix} 1 & 1 & 1 \\\\ 0 & y-x & z-x \\\\ 0 & 0 & z^2-x^2-(z-x)(y+x) \\end{bmatrix} $"}
{"_id": "2254118", "title": "", "text": "$(P\\oplus Q)\\oplus P=P\\oplus Q\\oplus P=P\\oplus P\\oplus Q=0\\oplus Q=Q$"}
{"_id": "6235861", "title": "", "text": "$f_X(x)=\\frac{1}{2}e^{\\frac{-x}{2}}$"}
{"_id": "8929033", "title": "", "text": "$\\langle\\alpha+\\beta,\\gamma\\rangle=\\langle\\alpha,\\gamma\\rangle+\\langle\\beta,\\gamma\\rangle$"}
{"_id": "1708969", "title": "", "text": "$\\mathbb{Q}[x]/(x^2+1)\\cong \\mathbb{Q}[i]$"}
{"_id": "2007940", "title": "", "text": "$f'(x)= \\frac{1}{3} ((x+3)^\\frac{-2}{3} - x^\\frac{-2}{3}) = 0$"}
{"_id": "6338859", "title": "", "text": "$\\mathbb{A}^2/(\\mathbb{Z}/2)$"}
{"_id": "7069375", "title": "", "text": "$\\vartheta^4(z+2) = \\vartheta^4(z), \\; \\; \\vartheta^4(-1/z) = -z^2 \\vartheta^4(z).$"}
{"_id": "5825658", "title": "", "text": "$f_n(x)=\\frac{nx}{(n^2x^2+1)^2}$"}
{"_id": "6119774", "title": "", "text": "$x=a^{log_a x}$"}
{"_id": "9322383", "title": "", "text": "$\\left(\\begin{array}{c}-1\\\\ -1\\\\1\\end{array}\\right)$"}
{"_id": "1461008", "title": "", "text": "$2x! > \\sqrt{4\\pi{x}}(\\frac{2x}{e})^{2x}$"}
{"_id": "6798636", "title": "", "text": "$b=a^{\\log_a(b)}$"}
{"_id": "7322967", "title": "", "text": "$(x+y)^r=\\sum_{k\\ge 0}\\binom{r}kx^{r-k}y^k\\;,$"}
{"_id": "2557239", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{m^2+n^2}=+\\infty$"}
{"_id": "3638904", "title": "", "text": "$\\int_0^x f(t)g(t)\\,dt = \\int_a^{x+a} f(t)g(t)\\,dt$"}
{"_id": "1627068", "title": "", "text": "$ \\begin{bmatrix} 1 & -1 & -a & 1 \\\\ -2 & 2 & -1 & 2\\\\ 2 & 2 & b & -2 \\end{bmatrix} $"}
{"_id": "924508", "title": "", "text": "$|AB| = 4$"}
{"_id": "6419495", "title": "", "text": "$  \\mathbb{P}(-X_n < x)\\leq\\mathbb{P}(-X_n \\leq x) \\leq \\mathbb{P}(-X_n < x+\\delta). $"}
{"_id": "890491", "title": "", "text": "$\\gamma,\\tilde{\\gamma}:[a,b]\\to M:\\gamma(a)=\\tilde{\\gamma}(a),\\gamma(b)=\\tilde{\\gamma}(b)$"}
{"_id": "7270226", "title": "", "text": "$\\frac{r}{2\\pi(1-\\rho^2)}\\exp{\\left({\\frac{\\rho r^2\\sin(2\\theta)}{(1-\\rho^2)}-\\frac{r^2}{2(1 -\\rho^2)}}\\right)}$"}
{"_id": "4895698", "title": "", "text": "$x\\rightarrow P(X\\leq x)$"}
{"_id": "4083324", "title": "", "text": "$A\\subset B\\subset \\bar B\\subset U,$"}
{"_id": "7343386", "title": "", "text": "$\\left(\\frac{1}{2}\\right)^{k+2}$"}
{"_id": "1619972", "title": "", "text": "$N(A) = \\lbrace x\\in \\mathbb R^q : A x = 0\\rbrace$"}
{"_id": "3309577", "title": "", "text": "$F_n(x) = \\int_a^{x}f_n(t)dt, x\\in [a,b]$"}
{"_id": "6463485", "title": "", "text": "$Cov(A,B)=0$"}
{"_id": "9258628", "title": "", "text": "$T=\\big\\{\\{a\\},\\{a,b\\}\\big\\}$"}
{"_id": "8140700", "title": "", "text": "$A=\\{\\{a\\}\\}$"}
{"_id": "7969215", "title": "", "text": "$ \\|x_n-x_m\\|<\\eps $"}
{"_id": "8028202", "title": "", "text": "$\\det(A-xI)=(-x+(n-1))(-x-1)^{n-1}\\color{red }{\\longrightarrow} x=n-1  ,-1$"}
{"_id": "8722492", "title": "", "text": "$3^k+3^{k+1}>k^3+k^{3+1}$"}
{"_id": "4632967", "title": "", "text": "$S \\subseteq A_1 \\times ... \\times A_n$"}
{"_id": "399737", "title": "", "text": "$\\alpha=\\omega^\\alpha$"}
{"_id": "6377161", "title": "", "text": "$A_1 ......A_n$"}
{"_id": "4304204", "title": "", "text": "$\\begin{vmatrix} 1&x&x^{2} \\\\ 1&y&y^{2} \\\\ 1&z&z^{2} \\end{vmatrix} = \\begin{vmatrix} 1& 1 & 1 \\\\ x & y & z \\\\ x ^2 & y^2 & z^2\\end{vmatrix} = (y-x)(z-x)(z-y).$"}
{"_id": "1500132", "title": "", "text": "$\\begin{align}\\left(1+\\frac{x}{n}\\right)^n&\\approx e^x\\\\ \\end{align}$"}
{"_id": "2495391", "title": "", "text": "$1/|x|^2$"}
{"_id": "320065", "title": "", "text": "$F(x)=\\int_a^xf(t)\\,\\mathrm{d}t$"}
{"_id": "1444342", "title": "", "text": "$ S = \\lbrace a_1,a_2,a_3,.............a_n \\rbrace $"}
{"_id": "5779740", "title": "", "text": "$\\sqrt {(-x)^2}=|x|$"}
{"_id": "9090637", "title": "", "text": "$\\begin{bmatrix} \\cos(\\varphi) & -\\sin(\\varphi) \\\\ \\sin(\\varphi) & \\cos(\\varphi) \\end{bmatrix}$"}
{"_id": "9323633", "title": "", "text": "$A_1\\subseteq A_2 \\subseteq \\dots \\subseteq A_n \\subseteq \\dots$"}
{"_id": "4353717", "title": "", "text": "$f(a)+f(b) \\leq f(a+b)$"}
{"_id": "4646932", "title": "", "text": "$b^{log_b{x}} = x$"}
{"_id": "184136", "title": "", "text": "$1+x+x^2+\\cdots=\\frac1{1-x}$"}
{"_id": "5733233", "title": "", "text": "$ \\frac{df}{d\\gamma} = {\\frac {-2-\\gamma\\,s+2\\,\\gamma-\\gamma\\,n+2\\,\\sqrt { \\left( 1+\\gamma\\,n -\\gamma \\right)  \\left( 1+\\gamma\\,s-\\gamma \\right) }}{{\\gamma}^{2} \\sqrt { \\left( 1+\\gamma\\,n-\\gamma \\right)  \\left( 1+\\gamma\\,s-\\gamma  \\right) }}}$"}
{"_id": "2350344", "title": "", "text": "$(A_1, A_2,\\dots, A_r)$"}
{"_id": "6643089", "title": "", "text": "$2I= \\int_{-\\pi}^{\\pi} \\dfrac{\\sin nx}{\\sin x} \\dfrac{\\pi^x+1}{1+\\pi^x}dx$"}
{"_id": "5934604", "title": "", "text": "$ \\lim\\limits_{n\\rightarrow\\infty} \\left\\lfloor  \\frac{1}{n}\\right\\rfloor =?$"}
{"_id": "4203024", "title": "", "text": "$\\sum_{r=1}^n\\cos\\frac{2r\\pi}n=0$"}
{"_id": "5286357", "title": "", "text": "$ (\\frac{(-1)^{n-1} x^{n-1} e^{-x}}{(n-1)!}) $"}
{"_id": "3588308", "title": "", "text": "$U\\subset S \\implies U \\subset X$"}
{"_id": "5360733", "title": "", "text": "$\\binom{2^n}{2^{n-1}}$"}
{"_id": "3061309", "title": "", "text": "$\\text{ Sum = } \\lim_{n\\to\\infty} = \\dfrac{n-1}{n+1} = 1$"}
{"_id": "4214074", "title": "", "text": "$\\gamma(x,y)=|\\langle x,y\\rangle|$"}
{"_id": "4890144", "title": "", "text": "$4!+1,4!+2,4!+3,4!+4$"}
{"_id": "5792073", "title": "", "text": "$\\int_0^1 p'(t)f(t)\\,dt = \\int_0^1 p(t)g(t)\\,dt$"}
{"_id": "239837", "title": "", "text": "$\\{a,a+d,a+2d, \\dots\\}$"}
{"_id": "4484159", "title": "", "text": "$\\bigl([x]+[y]\\bigr)+[z]=[x+y]+[z]=[x+y+z]=[x]+[y+z]=[x]+\\bigl([y]+[z]\\bigr)\\ .$"}
{"_id": "2505445", "title": "", "text": "$1\\leq k\\leq e_i$"}
{"_id": "1633237", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}(\\sqrt2-\\sqrt[3]2)\\cdots(\\sqrt2-\\sqrt[n]2)$"}
{"_id": "3017565", "title": "", "text": "$x \\in \\mathbb{R}^{n+1} - S^n$"}
{"_id": "6631415", "title": "", "text": "$\\displaystyle f(x) = \\frac{x^3}{e^{x}+1}$"}
{"_id": "1213620", "title": "", "text": "$F(x)=\\int_a^x f(t)$"}
{"_id": "7607850", "title": "", "text": "$as+b=r$"}
{"_id": "6253672", "title": "", "text": "$(a+i,a+j)=1$"}
{"_id": "6659552", "title": "", "text": "$ 1-x = 1-\\frac{\\gamma_1}{\\gamma_1 + \\gamma_2} = \\frac{\\gamma_1 + \\gamma_2}{\\gamma_1 + \\gamma_2}-\\frac{\\gamma_1}{\\gamma_1 + \\gamma_2} = \\frac{\\gamma_2}{\\gamma_1 + \\gamma_2} $"}
{"_id": "7970472", "title": "", "text": "$|M|+|M^∗|−|M|=|M^∗|$"}
{"_id": "7957619", "title": "", "text": "$|\\hat{f}(n)|\\leq C/|n|^{k+a}$"}
{"_id": "381410", "title": "", "text": "$\\int \\left(\\frac{\\tan^{-1}x}{x-\\tan^{-1}x}\\right)^{2}dx$"}
{"_id": "3271718", "title": "", "text": "$\\theta_{\\gamma_1\\gamma_2,\\gamma_3}\\theta_{\\gamma_1,\\gamma_2}=\\theta_{\\gamma_1,\\gamma_2\\gamma_3}\\theta_{\\gamma_2,\\gamma_3}.$"}
{"_id": "6601315", "title": "", "text": "$\\omega^{\\gamma'} + \\omega^{\\gamma} = \\omega^{\\gamma} < \\omega^{\\gamma} + \\omega^{\\gamma'} = \\omega^{\\gamma'} \\oplus \\omega^{\\gamma}$"}
{"_id": "4137532", "title": "", "text": "$v = (1/12,1/12,-1/6,1/12,-1/6,1/12)$"}
{"_id": "2386765", "title": "", "text": "$f(yf(x)+x)  = xf(y) +f(x)  $"}
{"_id": "5539070", "title": "", "text": "$\\frac{1}{1-x}\\log\\frac{1}{1-x}=\\sum_{n=1}^{\\infty}\\left(1+\\frac{1}{2}+\\frac{1}{3}+...+\\frac{1}{n}\\right)x^n$"}
{"_id": "7000250", "title": "", "text": "$\\Delta^n(X^n)=\\Delta^n(X^n)(a)=n!$"}
{"_id": "7412257", "title": "", "text": "$A\\cong A_1\\times\\dots\\times A_r$"}
{"_id": "1858226", "title": "", "text": "$d(x,z) + d(y,z) = d(x,y)$"}
{"_id": "2449383", "title": "", "text": "$g(b) \\in A_1 \\times \\dotsc \\times A_n$"}
{"_id": "4218845", "title": "", "text": "$\\frac{a+bx}{c-dx}$"}
{"_id": "3489292", "title": "", "text": "$n!\\approx\\sqrt{2\\pi n}*(\\frac{n}{e})^n$"}
{"_id": "241", "title": "", "text": "$*$"}
{"_id": "5087030", "title": "", "text": "$(2x \\frac{dy}{dx})(y)+x^2$"}
{"_id": "307799", "title": "", "text": "$T_n=n(n+1)/2$"}
{"_id": "6183369", "title": "", "text": "$f(a+_Sb)= f(a)+_R f(b)$"}
{"_id": "2769061", "title": "", "text": "$\\lim_{x\\to c} f'(x) = \\lim_{x\\to c} g'(x) = 0$"}
{"_id": "4811834", "title": "", "text": "$\\lim_{n\\to\\infty}\\int_{E_n}f(x)d\\mu=\\int_E f(x)d\\mu\\,\\,\\,\\,\\,\\,\\text{ where }E=\\cup_{n=1}^\\infty E_n$"}
{"_id": "4039167", "title": "", "text": "$\\{X_1\\otimes X_2|X_1\\in\\mathfrak{g}_1,X_2\\in\\mathfrak{g}_2\\}$"}
{"_id": "205641", "title": "", "text": "$ y=\\frac{a}{x-c}+c = \\frac{a + cx - c^2}{ x - c}$"}
{"_id": "6069857", "title": "", "text": "$(\\cos^2\\varphi + \\sin^2\\varphi) = 1 $"}
{"_id": "2842861", "title": "", "text": "$ M\\leq f(x) \\leq \\lim_{n\\rightarrow\\infty}\\inf f(x_n). $"}
{"_id": "1828656", "title": "", "text": "$\\frac 1 2 < x <1 \\implies \\pi < \\frac{\\pi}{x} < 2 \\pi \\implies 0 < 1+ \\sin \\frac{\\pi}x <1$"}
{"_id": "4023022", "title": "", "text": "$x\\in (a-\\gamma, a+ \\gamma)\\setminus \\{a\\}$"}
{"_id": "9307884", "title": "", "text": "$\\|x^m_n - x_n\\| < \\epsilon/2$"}
{"_id": "3893237", "title": "", "text": "$|f(z)|\\leq M|\\varphi^{-1} (z) |$"}
{"_id": "8930639", "title": "", "text": "$\\sum_{n=-\\infty}^{+\\infty}\\delta(t-nT)=\\frac{1}{T} \\sum_{m=-\\infty}^{+\\infty} e^{\\frac {-i2\\pi m t}{T}}$"}
{"_id": "663723", "title": "", "text": "$\\mathcal{A}^+$"}
{"_id": "1471391", "title": "", "text": "$x R y\\land y R x$"}
{"_id": "4791507", "title": "", "text": "$\\begin{bmatrix} A & I \\\\ A & -I \\end{bmatrix}$"}
{"_id": "6988680", "title": "", "text": "$\\sum_n ||e_n - f_n||^2 < 1$"}
{"_id": "1815924", "title": "", "text": "$\\frac{2b(b-a) -b^2-b+a^2+a +2b-2a}{2b-2a+2}$"}
{"_id": "7591947", "title": "", "text": "$A_1 \\times A_2 \\times \\dots \\times A_n \\times Y \\times Y \\times \\cdots.$"}
{"_id": "5172526", "title": "", "text": "$\\frac{x}{|x|} = -1$"}
{"_id": "5471491", "title": "", "text": "$\\vartheta (x) = \\sum_{n\\in \\mathbb Z} e^{-\\pi {n^2} x} = \\sum_{k\\in \\mathbb Z} {e^{-\\pi {k^2} {1\\over x}}} \\int_{-\\infty}^{\\infty} {e^{-\\pi x \\big( y+ i {k\\over x}\\big)^2}} \\space dy$"}
{"_id": "1861893", "title": "", "text": "$c_{i_{0}}\\in C_{i_{0}} $"}
{"_id": "7286549", "title": "", "text": "$\\int_{-\\infty}^{\\infty}f(t)\\exp(-st)dt=\\int_{0}^{\\infty}f(t)\\exp(-st)dt+\\int_{-\\infty}^{0}f(t)\\exp(-st)dt$"}
{"_id": "2149575", "title": "", "text": "$\\large f(n) = \\frac{n(n+1)}{2}$"}
{"_id": "54293", "title": "", "text": "$ D=\\{(x,y)\\in\\mathbb{R}^2:0\\le x\\le 1, 0\\le y \\le x\\}. $"}
{"_id": "68160", "title": "", "text": "$\\int_0^{\\infty} f(x)$"}
{"_id": "5150148", "title": "", "text": "$\\int_{0}^{\\pi} f^{\\prime} \\sin(t) dt = f \\sin(t)$"}
{"_id": "7996116", "title": "", "text": "$4^x+15^x=9^x+10^x(2^x-3^x)(2^x-3^x-5^x)$"}
{"_id": "5938417", "title": "", "text": "$\\liminf_{n \\rightarrow \\infty} \\mu (A_n) \\leq \\limsup_{n \\rightarrow \\infty} \\mu (A_n)$"}
{"_id": "9250496", "title": "", "text": "$\\sum_{n=-N}^Ne^{-n^2/2}\\le\\sum_{n=-\\infty}^\\infty e^{-n^2/2} \\le\\sum_{n=-N}^Ne^{-n^2/2}+\\frac2Ne^{-N^2/2}.$"}
{"_id": "660609", "title": "", "text": "$\\sum_{k=1}^{\\infty} \\frac 1k = 1 + (\\frac 12 + \\frac 13) + (\\frac 14 + \\frac 15 + \\frac 16 + \\frac 17) + ...... + (\\frac 1{2^n} + \\frac 1{2^n + 1} + .... + \\frac 1{2^n + (2^n -1)}) + .....  > \\frac 12 + (\\frac 14 + \\frac 14) + (\\frac 18 + \\frac 18 + \\frac 18 + \\frac 18) + ...... (\\frac 1{2^{n+1}} + \\frac 1{2^{n+1}} + .... + \\frac 1{2^{n+1}}) + ... = \\frac 12 + \\frac 12 + .... + \\frac 12 + .....$"}
{"_id": "5063793", "title": "", "text": "$(\\mathbb{Q}[x]/(x^2+3))[y]/(y^2-5)$"}
{"_id": "5675041", "title": "", "text": "$ \\left( \\begin{array}_ 0 & 1 & 4 & 9 & ... & (n-1)^2  \\\\ 1 & 0 & 1 & 4 & ... & (n-2)^2  \\\\ ... &... &... & ... & ... & ... \\\\ (n-1)^2  & (n-2)^2  & (n-3)^2  & (n-4)^2  & ... & 0  \\end{array} \\right)$"}
{"_id": "7253088", "title": "", "text": "$z=y \\ \\ \\text{or} \\ \\ x=-2\\lambda-1$"}
{"_id": "2107891", "title": "", "text": "$\\int\\limits_\\gamma f dz = \\int\\limits_\\gamma \\frac{1}{g} dz = \\int\\limits_\\gamma \\frac{1}{(z + 1)(z + 2) \\cdot \\ldots \\cdot (z + r)} dz$"}
{"_id": "4987977", "title": "", "text": "$Var(u|x)=E(u^2|x)-[E(u|x)]^2$"}
{"_id": "4773766", "title": "", "text": "$f(x)<1/|x|$"}
{"_id": "6450467", "title": "", "text": "$P(k+1)\\implies P(k)$"}
{"_id": "7122287", "title": "", "text": "$\\forall x\\in E,\\quad ||x||^2=\\sum_{i=1}^n\\langle e_i,x\\rangle^2$"}
{"_id": "4114721", "title": "", "text": "$ J(a) = \\frac{1}{2} \\int_0^1 t^{(a-1)/2} (1-t)^{-1/2}\\; dt = \\frac{B((a+1)/2,1/2)}{2} = \\frac{\\Gamma(a+1/2) \\Gamma(1/2)}{\\Gamma(a+1)} $"}
{"_id": "6197787", "title": "", "text": "$\\sum_{n\\leq x} \\frac{\\sigma (n)}{n} = \\sum_{n \\leq x} \\frac{1}{n} \\lfloor \\frac{x}{n} \\rfloor $"}
{"_id": "6512995", "title": "", "text": "$f(1-x)=\\dfrac2{2+4^{1-x}}=\\dfrac{2\\cdot4^x}{2\\cdot4^x+4}=\\dfrac{4^x}{4^x+2}=\\dfrac{4^x+2-2}{4^x+2}=1-f(x)$"}
{"_id": "7087612", "title": "", "text": "$[x,y,z]:=(x+y+z)(-x+y+z)(x-y+z)(x+y-z)$"}
{"_id": "3984387", "title": "", "text": "$f(\\perp)=g(\\perp)=\\perp$"}
{"_id": "2955195", "title": "", "text": "$f = (1-x y) (1+x+z-x z) (1-y z) = 1+x-x y-x^2 y+z-x z-y z  \\mathbf{-2}  x y z+x^2 y z+x y^2 z+x^2 y^2 z-y z^2+x y z^2+x y^2 z^2-x^2 y^2 z^2$"}
{"_id": "2887698", "title": "", "text": "$E(n, p) = n^ 2 - n + p$"}
{"_id": "7321367", "title": "", "text": "$ \\begin{array}{llllcrrr} [1/9  -2/27, & 3/9, & 5/9 + 2/27 &]&=&[ 1/27, &9/27,& 17/27]\\\\ [1/9   , & 3/9 + 2/27, & 5/9 - 2/27 &]&=& [3/27, &11/27, &13/27]\\\\ [1/9  +2/27, & 3/9- 2/27, & 5/9  &]&=& [5/27, &7/27,& 15/27]\\\\ [3/9  -2/27, & 5/9 , & 1/9+ 2/27 &]&=& [7/27, &15/27,&5/27]\\\\ [3/9   , & 5/9 + 2/27, & 1/9 - 2/27 &]&=& [9/27, &17/27,& 1/27]\\\\ [3/9  +2/27, & 5/9  - 2/27, & 1/9 &]&=& [11/27, &13/27,& 3/27]\\\\ [5/9  -2/27, & 1/9 , & 3/9 + 2/27&]&=& [13/27, &3/27, &11/27]\\\\ [5/9  , & 1/9 + 2/27, & 3/9 - 2/27& ]&=& [15/27, &5/27,& 7/27]\\\\ [5/9  +2/27, & 1/9 - 2/27, & 3/9 &] &=& [17/27, &1/27,& 9/27]. \\end{array} $"}
{"_id": "4481642", "title": "", "text": "$\\sum_{n=1}^{\\infty}||x_n-e_n||^2<1.$"}
{"_id": "4073902", "title": "", "text": "$|\\gamma(1)-\\gamma(t)+\\gamma(t)-\\gamma(0)|= |\\gamma(1)-\\gamma(t)|+|\\gamma(t)-\\gamma(0)|,  $"}
{"_id": "1159861", "title": "", "text": "$\\lim_{n \\to \\infty} \\left(\\sum_{r=1}^n \\frac{1}{x_r}\\right)$"}
{"_id": "5662974", "title": "", "text": "$ \\left(  \\left(  R^{+}\\right)  ^{\\sigma}\\right)  ^{=}=\\left(  \\left( R^{=}\\right)  ^{+}\\right)  ^{\\sigma}=\\left(  \\left(  R^{+}\\right) ^{=}\\right)  ^{\\sigma}=\\bigcup_{n=-\\infty}^{\\infty}R^{n}% $"}
{"_id": "1053513", "title": "", "text": "$ s(x)=\\sum\\limits_{j=1}^ma_j\\chi_{E_j}(x). $"}
{"_id": "816806", "title": "", "text": "$A\\subset B\\subset C\\subset A$"}
{"_id": "6476009", "title": "", "text": "$P(n) \\implies S(m)$"}
{"_id": "9283703", "title": "", "text": "$\\frac{1}{3^{n-2}}< \\epsilon $"}
{"_id": "2096912", "title": "", "text": "$f(a + b) \\leq f(a) + f(b).$"}
{"_id": "4782900", "title": "", "text": "$ \\mathbb{E}[(X - \\mathbb{E}[X])^2] = \\mathbb{E}[X^2 - 2\\mathbb{E}[X]\\cdot X + \\mathbb{E}[X]^2] = \\mathbb{E}[X^2] - 2\\mathbb{E}[X]\\mathbb{E}[X] + \\mathbb{E}[X]^2 =  \\mathbb{E}[X^2] - \\mathbb{E}[X]^2 $"}
{"_id": "968244", "title": "", "text": "$(c_1e_1,…..,c_ne_n)$"}
{"_id": "5920579", "title": "", "text": "$  \\left\\{ \\begin{array}{l}  x=1+2t\\\\ y=1+7t\\\\ z=3t\\\\ \\end{array} \\right. $"}
{"_id": "6941673", "title": "", "text": "$ \\sum_{n\\le x}\\frac{1}{n^2}=\\frac{\\pi^2}{6}-\\frac{1}{x}+O(x^{-2}), $"}
{"_id": "3697577", "title": "", "text": "$f_X(x,\\theta)$"}
{"_id": "4555358", "title": "", "text": "$\\begin{align}\\int_0^\\infty f(x) \\;dx &= \\int_0^1f(x)\\;dx + \\int_1^\\infty f(x)\\;dx \\\\ &\\le \\ln(2) + 1  \\end{align}$"}
{"_id": "3467303", "title": "", "text": "$|\\mathbb P\\{X_n\\leq x\\}-\\mathbb P\\{X\\leq x\\}|\\leq \\mathbb P\\{X_n\\leq x,X>x\\}+\\mathbb P\\{X_n>x,X\\leq x\\}$"}
{"_id": "7765153", "title": "", "text": "$\\Re\\left(\\text{A}\\cdot\\text{a}\\cdot e^{-\\text{a}\\cdot\\text{T}}\\right)=\\Re\\left\\{\\text{A}\\cdot\\text{a}\\cdot\\frac{\\cos\\left(\\text{a}\\zeta\\right)-\\sin\\left(\\text{a}\\zeta\\right)i}{\\exp\\left[\\text{a}\\sigma\\right]}\\right\\}=\\text{A}\\cdot\\text{a}\\cdot\\frac{\\cos\\left(\\text{a}\\zeta\\right)}{\\exp\\left[\\text{a}\\sigma\\right]}\\tag2$"}
{"_id": "2125076", "title": "", "text": "$Y= \\{a_1,a_2,...,n\\}$"}
{"_id": "300042", "title": "", "text": "$f(f(0) + f(z)) = -f(z)$"}
{"_id": "531141", "title": "", "text": "$\\lim\\limits_{x \\to 0} 2 \\dfrac {(e^{\\frac{x}{2}}-e^{\\frac {-x}{2}})^{\\frac {2} {x^2}}} {x^2}$"}
{"_id": "7806596", "title": "", "text": "$ \\dot x = x \\ , \\quad x \\in \\mathbb R $"}
{"_id": "4151615", "title": "", "text": "$\\lfloor{\\frac{a}{b+1}}\\rfloor = \\lfloor \\frac{a}{b} \\rfloor  $"}
{"_id": "2869845", "title": "", "text": "$\\tag 2 n^2 - n + 2 \\ge 0$"}
{"_id": "4951529", "title": "", "text": "$\\begin{array}{ccc}1&k&1\\\\0&(1-k^2)&0\\\\0&(-k^2-k-1)&-k\\end{array}$"}
{"_id": "9158737", "title": "", "text": "$\\sum_{r=1}^{\\infty} \\frac{1}{2^r}=2$"}
{"_id": "1002347", "title": "", "text": "$x^{-n}e^x>\\frac{x^{-n}x^{n+1}}{(n+1)!}=\\frac{x}{(n+1)!}$"}
{"_id": "913294", "title": "", "text": "$\\alpha=\\omega+\\alpha.$"}
{"_id": "64335", "title": "", "text": "$P(n) \\implies P(n+1)$"}
{"_id": "235915", "title": "", "text": "$a^+$"}
{"_id": "9150550", "title": "", "text": "$\\int_0^\\pi f\\left(\\frac{v+k\\pi}{n}\\right)\\cos(v)dv \\sim \\int_0^\\pi f\\left(\\frac{k\\pi}{n}\\right)\\cos(v)dv =0$"}
{"_id": "7111769", "title": "", "text": "$\\vartheta(z+1, \\tau) = \\vartheta(z, \\tau)$"}
{"_id": "9225935", "title": "", "text": "$\\left[\\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\\\ 0 & 0 &  0  &  0  & -1  & 0  & 1  \\\\ 0 & 0 & -1 & 2  & a^{2}+2a &   2  & 3  \\\\ 0  & 0  & 1  & -2 & 2    & -2  & -4 \\end{array}\\right]$"}
{"_id": "9349851", "title": "", "text": "$\\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix}$"}
{"_id": "4856399", "title": "", "text": "$(\\mathbb{R}^n, \\mathcal{O})$"}
{"_id": "5261471", "title": "", "text": "$x=1/|q|$"}
{"_id": "7893094", "title": "", "text": "$\\frac ax + \\frac by = 1$"}
{"_id": "385279", "title": "", "text": "$|f(z)|\\leq c|g(z)|$"}
{"_id": "7356866", "title": "", "text": "$\\frac{\\pi^2}6=\\sum_{n=1}^\\infty\\frac1{n^2}=\\frac14\\sum_{n=1}^\\infty\\frac1{n^2}+\\sum_{n=1}^\\infty\\frac1{(2n-1)^2}\\implies$"}
{"_id": "947638", "title": "", "text": "$\\sum I_j)$"}
{"_id": "653865", "title": "", "text": "$\\gamma_1 =S_4\\ge \\gamma_2=A_4\\ge\\gamma_3=A_4\\ge \\gamma_4 =A_4\\ge...  $"}
{"_id": "1741496", "title": "", "text": "$\\exists c_0$"}
{"_id": "2608410", "title": "", "text": "$a_k = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)\\cos(nx) dx,$"}
{"_id": "9087377", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{1+\\sqrt[n]{2}+\\sqrt[n]{3}+\\cdots+\\sqrt[n]{n}}{n}$"}
{"_id": "2734633", "title": "", "text": "$rs-r's'=(rs-rs')+(rs'-r's')=r(s-s') + s'(r-r') \\lt 0\\,$"}
{"_id": "6334197", "title": "", "text": "$T\\bigg(\\sum x_n e_n\\bigg) \\mapsto \\sum y_n x_n e_n.$"}
{"_id": "6645426", "title": "", "text": "$\\mathbb{E}|XY|\\le\\sqrt{\\mathbb{E}X^2\\mathbb{E}Y^2}$"}
{"_id": "231167", "title": "", "text": "$\\mathbb{Q}[x]/(x^2+1)$"}
{"_id": "8177110", "title": "", "text": "$\\lim_{n \\to \\infty}\\dfrac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n-1}}{n\\sqrt{n}}=\\lim_{n \\to \\infty}\\dfrac{\\sqrt{\\dfrac{1}{n}}+\\sqrt{\\dfrac{2}{n}}+\\sqrt{\\dfrac{3}{n}}+\\cdots+\\sqrt{\\dfrac{n-1}{n}}}{n}\\ge \\lim_{n \\to \\infty}\\bigg(\\sqrt{\\dfrac{1}{n}}.\\sqrt{\\dfrac{2}{n}}.\\sqrt{\\dfrac{3}{n}}\\cdots\\sqrt{\\dfrac{n-1}{n}}\\bigg )^\\dfrac{1}{n-1}=\\lim_{n \\to \\infty}\\bigg(\\dfrac{(n-1)!}{n^n}\\bigg )^\\dfrac{1}{2(n-1)}$"}
{"_id": "6458230", "title": "", "text": "$ k\\cdot\\gcd(a,b)=\\lambda ka+\\gamma kb=\\lambda 0+\\gamma 0\\in\\mathbb Z_n $"}
{"_id": "8937325", "title": "", "text": "$\\theta_x=\\frac\\pi2-\\arctan\\left(\\frac{\\sqrt{1-x^2}}x\\right)$"}
{"_id": "6087354", "title": "", "text": "$\\text{lcm}(a+c,b+c)=k$"}
{"_id": "4529288", "title": "", "text": "$h = \\left(h_1, h_2, h_3, .... ,h_n \\right)$"}
{"_id": "8633704", "title": "", "text": "$A_1\\times \\dots A_k$"}
{"_id": "4410996", "title": "", "text": "$1+2+3+4+5+..... = -\\frac{1}{12}$"}
{"_id": "2590970", "title": "", "text": "$\\binom{-n}{r} = \\frac{(-n)(-n-1)(-n-2)…(-n-r+1)}{r!} = \\frac{(-1)^rn(n+1)(n+2)…(n+r-1)}{r!} = (-1)^r \\binom{n+r-1}{r}$"}
{"_id": "6199345", "title": "", "text": "$1/2^{n-2}$"}
{"_id": "3741811", "title": "", "text": "$\\begin{array}\\\\ \\sum_{n=1}^∞ \\prod_{j=1}^n(\\frac1{a} + \\frac1{b j})  &= -\\dfrac{(\\frac{a - 1}{a})^{-a/b} (a (\\frac{a - 1}{a})^{a/b} -  (\\frac{a - 1}{a})^{a/b} - a)}{a - 1}\\\\ &= -\\dfrac{a - 1 - a(\\frac{a - 1}{a})^{-a/b}}{a - 1}\\\\ &= \\dfrac{ a(\\frac{a - 1}{a})^{-a/b}}{a - 1}-1\\\\ &= \\dfrac{ a^{1+a/b}}{(a - 1)^{1+a/b}}-1\\\\ &= \\left(\\dfrac{ a}{a - 1}\\right)^{1+a/b}-1\\\\ \\end{array} $"}
{"_id": "3396463", "title": "", "text": "$P(t^*<t|H_0)$"}
{"_id": "1603747", "title": "", "text": "$f(x) = \\dfrac{3x^2}{\\theta^3}$"}
{"_id": "1982608", "title": "", "text": "$V = X \\oplus Y$"}
{"_id": "6589519", "title": "", "text": "$B\\,'(p,q)=\\left\\{\\langle x,y\\rangle:a\\le x<b\\text{ and }c\\le y<d\\right\\}\\;;$"}
{"_id": "3188047", "title": "", "text": "$(f(z))^2=(f(-z))^2$"}
{"_id": "160294", "title": "", "text": "$f(n) = n^2$"}
{"_id": "1823787", "title": "", "text": "$\\lim\\limits_{n\\rightarrow\\infty} \\mu(A_n)=\\mu(A)$"}
{"_id": "4073246", "title": "", "text": "$f(f(x)-x)=f(-1-x)=1=f(-1)=f(f(x))$"}
{"_id": "4483651", "title": "", "text": "$\\int_{0}^\\frac{1}{2}\\frac{\\sin^{-1}x}{\\sqrt{1-x^2}} dx $"}
{"_id": "286", "title": "", "text": "$\\:$"}
{"_id": "4401132", "title": "", "text": "$A = \\{ \\{m\\}, \\{x\\}\\}$"}
{"_id": "543503", "title": "", "text": "$\\left| {TP} \\right| = r\\;\\theta $"}
{"_id": "541507", "title": "", "text": "$\\lim_{n\\rightarrow \\infty} \\sum_{i=1}^n a_i$"}
{"_id": "1144406", "title": "", "text": "$\\frac{1}{2}\\sqrt{\\frac{\\gamma}{mg}}\\int\\frac{1}{v - \\sqrt{\\frac{mg}{\\gamma}}}- \\frac{1}{v + \\sqrt{\\frac{mg}{\\gamma}}}dv=\\int \\frac{\\gamma}{m}dt$"}
{"_id": "1620767", "title": "", "text": "$B = \\{\\{a, b\\}\\}$"}
{"_id": "2116132", "title": "", "text": "$AP\\cap AZ\\cap PZ$"}
{"_id": "761843", "title": "", "text": "$0 \\rightarrow A_1 \\rightarrow A_2 \\rightarrow ... \\rightarrow A_{n-1} \\rightarrow A_n \\rightarrow 0$"}
{"_id": "4228268", "title": "", "text": "$a_n= n^2, b_n = \\frac{-1}{n}, c_n = n$"}
{"_id": "1343220", "title": "", "text": "$5 \\cdot -5 = 1 \\bmod{26}$"}
{"_id": "1204310", "title": "", "text": "$m+n=p+n \\implies m=p$"}
{"_id": "41278", "title": "", "text": "$/$"}
{"_id": "6053791", "title": "", "text": "$\\frac{\\ln(1+x)}{\\arcsin(x)}=\\frac{\\sqrt{1-y^2}}{1+y}$"}
{"_id": "2122281", "title": "", "text": "$ = \\left[     \\begin{array}{ccc|c}       1&-2&3&2\\\\       1&1&1&k\\\\       2&-1&4&k^2     \\end{array} \\right] $"}
{"_id": "3013923", "title": "", "text": "$P(n)=P(m)$"}
{"_id": "742400", "title": "", "text": "$d = xa + yb$"}
{"_id": "6643088", "title": "", "text": "$I=\\int_{-\\pi}^{\\pi}\\dfrac{\\sin nx}{(1+\\pi^{x})\\sin x}dx$"}
{"_id": "8406693", "title": "", "text": "$s(\\omega) = \\sum \\limits_{i = 1}^{n} \\alpha_{i} \\chi_{A_{i}}(\\omega)$"}
{"_id": "6575068", "title": "", "text": "$\\mathbb P(E) = \\frac 14$"}
{"_id": "690514", "title": "", "text": "$\\frac{a^{2}+b}{b^{2}-a}$"}
{"_id": "3537786", "title": "", "text": "$F(x) = \\int_{0}^x f(t) \\sin t dt$"}
{"_id": "1266662", "title": "", "text": "$\\lim _ {n\\rightarrow \\infty }\\inf $"}
{"_id": "5347567", "title": "", "text": "$f(x;θ) =  \\dfrac{2(θ−x)}{θ^2}$"}
{"_id": "2238677", "title": "", "text": "$ R = \\{(x,y) \\in \\mathbb{R}^2: a \\leq x \\leq b, c \\leq y \\leq d\\}. $"}
{"_id": "5269467", "title": "", "text": "$f^\\#f_\\# f^\\# =f^\\#$"}
{"_id": "6643087", "title": "", "text": "$I=\\int_{-\\pi}^{\\pi}\\dfrac{\\pi^x \\sin n(x)}{(\\pi^x+1)\\sin x} dx$"}
{"_id": "974704", "title": "", "text": "$\\dfrac{1 - x^{2}}{1 - 2x\\cos t + x^{2}}$"}
{"_id": "2806504", "title": "", "text": "$-5<\\frac1x\\le1\\stackrel{\\text{mult. by}\\;x^2}\\iff-5x^2<x\\le x^2\\iff\\begin{cases}-5x^2<x\\iff x(5x+1)>0\\\\{}\\\\\\text{and}\\\\{}\\\\x\\le x^2\\iff x(x-1)\\ge 0\\end{cases}$"}
{"_id": "4954152", "title": "", "text": "$[X,Y] = XY - YX?$"}
{"_id": "916098", "title": "", "text": "$P(T_x\\ge N|X_0=x_0)\\le p.$"}
{"_id": "401405", "title": "", "text": "$f_1(n)=\\binom{n+1}{2}$"}
{"_id": "5798989", "title": "", "text": "$\\int_0^{\\infty}\\frac{f(x)}{x}=\\int_0^{\\infty}F(s)~ds$"}
{"_id": "6845307", "title": "", "text": "$ \\frac{9}{27}=\\frac{1}{3}$"}
{"_id": "4391947", "title": "", "text": "$ \\|A\\|_2 = \\sqrt{\\text{largest eigen value of } A^{\\ast}A} $"}
{"_id": "436691", "title": "", "text": "$|f(z)|\\leq |f(z_0)|$"}
{"_id": "6784529", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1 & 2 & 3 & 0\\\\0 & -1 & -2 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right],$"}
{"_id": "5668953", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\mu(X_n) = \\mu(X)$"}
{"_id": "6461772", "title": "", "text": "$(a+b, a^2-ab+1)=1$"}
{"_id": "2557819", "title": "", "text": "$\\det \\begin{pmatrix} A & -B \\\\ B & A \\end{pmatrix}$"}
{"_id": "3292129", "title": "", "text": "$u=\\frac{1-r^2}{1-2r \\cos \\theta + r^2}$"}
{"_id": "1651211", "title": "", "text": "$M' M = M M' + [M, M']$"}
{"_id": "204657", "title": "", "text": "$X^n \\setminus X^{n-1}$"}
{"_id": "8909592", "title": "", "text": "$(N,I_N)$"}
{"_id": "14220", "title": "", "text": "$am+bn=1$"}
{"_id": "1386359", "title": "", "text": "$\\sum_{k=1}^{\\infty}\\frac{1}{k^2+a^2}$"}
{"_id": "8717409", "title": "", "text": "$\\mathbb{E}[W|B \\leq b^*]$"}
{"_id": "6245495", "title": "", "text": "$F(x)=\\int_a^x f(t)\\ \\mathsf dt$"}
{"_id": "8794337", "title": "", "text": "$N\\oplus T$"}
{"_id": "5596172", "title": "", "text": "$A_1 \\times A_2 \\times ...\\times A_p$"}
{"_id": "166049", "title": "", "text": "$|A\\times B|=mn$"}
{"_id": "6888161", "title": "", "text": "$\\alpha^+ < \\beta$"}
{"_id": "8400784", "title": "", "text": "$\\gcd(a, a+2)$"}
{"_id": "2487226", "title": "", "text": "$P(E) = |E| / |S|$"}
{"_id": "7855925", "title": "", "text": "$ \\begin{cases} x = \\alpha \\\\ y = \\beta \\\\ z = - \\alpha - \\beta \\end{cases} $"}
{"_id": "1656777", "title": "", "text": "$ \\det(M)=\\det(U^*U)=\\det(U^*)\\det(U)=\\overline{\\det(U)}\\det(U)=|\\det(U)|^2. $"}
{"_id": "2929808", "title": "", "text": "$P(E) = 1/52$"}
{"_id": "6217792", "title": "", "text": "$aRc\\wedge cRa$"}
{"_id": "5107555", "title": "", "text": "$|f'(z)| \\leq |z^2+1|$"}
{"_id": "5690", "title": "", "text": "$+$"}
{"_id": "311581", "title": "", "text": "$P(k)\\implies P(k+1).$"}
{"_id": "7781121", "title": "", "text": "$ a =   \\begin{cases}     x=-2+s       \\\\     y=2+s     \\\\     z=1-s   \\end{cases} $"}
{"_id": "5285767", "title": "", "text": "$ \\int_\\gamma \\mathrm{d} s = \\int_\\gamma \\sqrt{\\mathrm{d}x ^2 + \\mathrm{d}y^2 }.  $"}
{"_id": "3774498", "title": "", "text": "$|f(z)| \\le |\\exp(z)|$"}
{"_id": "3828979", "title": "", "text": "$\\displaystyle {\\frac{{dx}}{{ax + b}}}$ = $\\displaystyle {\\frac{{1}}{{a}}}$"}
{"_id": "1557541", "title": "", "text": "$v=\\frac{a+bx}{c+dx}$"}
{"_id": "9017205", "title": "", "text": "$\\vartheta(-u,-i\\tau)=\\vartheta(u,-i\\tau)=\\vartheta(u+1,-i\\tau)$"}
{"_id": "1027015", "title": "", "text": "$sin\\frac{(sin\\theta + tan\\theta)}{2}=\\frac{(sin\\theta + tan\\theta)}{2}$"}
{"_id": "4730210", "title": "", "text": "$f(a)=a(1-a)^{n-1}$"}
{"_id": "2539", "title": "", "text": "$\\displaystyle\\sum_{n=1}^\\infty \\frac{1}{n^2}$"}
{"_id": "170127", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} t_n = \\infty$"}
{"_id": "3188998", "title": "", "text": "$T = A \\oplus B$"}
{"_id": "7683033", "title": "", "text": "$f(x) \\, \\delta''(x-a)=f(a)\\delta''(x-a)-f'(a)\\delta'(x-a)+f''(a)\\delta(x-a)$"}
{"_id": "8227382", "title": "", "text": "$\\{x_\\gamma\\}_{\\gamma\\in\\Gamma}\\subseteq A$"}
{"_id": "3533738", "title": "", "text": "$ \\lim \\limits_{k \\rightarrow \\infty } \\int \\limits_{-\\pi}^\\pi f(x)\\cos(kx)dx = 0 \\textit{ and } \\lim \\limits_{k \\rightarrow \\infty } \\int \\limits_{-\\pi}^\\pi f(x)\\sin(kx)dx = 0 $"}
{"_id": "6622817", "title": "", "text": "$f(x)=x^2-x+9$"}
{"_id": "8948883", "title": "", "text": "$\\frac{ a_{n+1}}{a_n} = \\frac{\\left(1+\\frac 1n\\right)\\ln\\left(1+\\frac{2}{2n+1}\\right)-\\frac 1n}{\\ln\\left(1+\\frac{2}{2n-1}\\right)-\\frac 1n}$"}
{"_id": "6238422", "title": "", "text": "$\\sum_{n\\geq N}\\frac{1}{n^{\\log\\left(\\log\\left(n\\right)\\right)}}\\leq\\sum_{n\\geq N}\\frac{1}{n^{2}}<\\infty.  $"}
{"_id": "7749954", "title": "", "text": "$z=\\sqrt[4]{2}e^{\\frac{2\\pi i}{3}}$"}
{"_id": "1189255", "title": "", "text": "$\\mathbb{P}[X_N = 1 | X_0 = 1]$"}
{"_id": "3669524", "title": "", "text": "$\\inf\\{s_\\gamma\\mid \\gamma<\\alpha\\}=\\inf\\{F_\\gamma(0)\\mid \\gamma<\\alpha\\}=0$"}
{"_id": "6167562", "title": "", "text": "$S=\\sum_{n=-\\infty}^{\\infty}\\sum_{m=-\\infty}^{\\infty}e^{-\\sqrt{n^2+m^2}}$"}
{"_id": "4953571", "title": "", "text": "$I_n=\\int\\frac{1}{(x^2+a^2)^n}dx$"}
{"_id": "4258794", "title": "", "text": "$e^{-x}(\\frac{x^{n+1}-x^{n}}{(1+x^{n+1})(1+x^{n})})$"}
{"_id": "1247750", "title": "", "text": "$\\gamma\\geq\\beta,\\gamma\\geq\\alpha$"}
{"_id": "2862259", "title": "", "text": "$\\begin{align*} \\sum_{n=2}^\\infty \\frac{\\zeta(n)}{k^n} &= \\sum_{n=2}^\\infty \\frac{1}{k^n \\Gamma(n)}\\int_0^\\infty \\frac{u^{n-1}}{e^u-1}du\\\\ &=\\int_0^\\infty\\frac{1}{u(e^u-1)}\\left( \\sum_{n=2}^\\infty \\frac{1}{(n-1)!}\\left(\\frac{u}{k}\\right)^n\\right)du \\\\ &= \\frac{1}{k}\\int_0^\\infty \\frac{e^{\\frac{u}{k}}-1}{e^u-1}du \\tag{4} \\end{align*}$"}
{"_id": "649611", "title": "", "text": "$P(2) \\implies P(3)$"}
{"_id": "6331738", "title": "", "text": "$F(x)F(x)^{-1}=1=F(x)^{-1}F(x)$"}
{"_id": "2482190", "title": "", "text": "$f(x)=\\sum_{k=1}^n a_k {\\mathbf 1}_{A_k}(x)$"}
{"_id": "4824249", "title": "", "text": "$\\frac{(1+x)^{r}(1-(1+x)^{2(k+1)})}{1-(1+x)^2}$"}
{"_id": "730340", "title": "", "text": "$|f(z)| \\le \\max |f(z+r e^{it})|$"}
{"_id": "2970439", "title": "", "text": "$ \\left\\{ \\matrix{   f(0) = a \\hfill \\cr    f'(0) = \\,\\,b\\omega  + \\rho a\\quad  \\Rightarrow \\quad b = \\;{1 \\over \\omega }\\left( {f'(0) - \\,\\,\\rho f(0)} \\right) \\hfill \\cr}  \\right. $"}
{"_id": "9012498", "title": "", "text": "$\\frac{1}{n(n+1)^2(n+2)} = \\frac{1}{2} \\left( \\frac{1}{n} - \\frac{1}{n+2} \\right) - \\frac{1}{(n+1)^2},$"}
{"_id": "6077144", "title": "", "text": "$1+2+3+4+5+6+...=-1/12$"}
{"_id": "6705490", "title": "", "text": "$\\displaystyle J = 3\\int\\frac{1}{(x^2+1)^2}dx\\;,$"}
{"_id": "1537756", "title": "", "text": "$f(x)=\\frac{2x}{(2x-1)^2}$"}
{"_id": "761842", "title": "", "text": "$0 \\rightarrow A_1 \\rightarrow A_2 \\rightarrow ... \\rightarrow A_{n-1} \\rightarrow A_n $"}
{"_id": "6036990", "title": "", "text": "$d(x,y)+d(y,s)\\leqslant d(x,s)$"}
{"_id": "846354", "title": "", "text": "$G=(\\{v\\},\\{e\\})$"}
{"_id": "5438850", "title": "", "text": "$n>3, n \\in \\Bbb N \\;\\;\\text{and}\\;\\; A = \\{1,2,3,...,n\\}$"}
{"_id": "8552068", "title": "", "text": "$27 +9 + 3 + (3-1) = 27 + 9 + (9-3) - 1 = 27 + 27 -9 -3 -1$"}
{"_id": "2448356", "title": "", "text": "$f(n)=2f(n-1)+2f(n-2)$"}
{"_id": "4023021", "title": "", "text": "$(a-\\gamma, a+ \\gamma)\\subset I$"}
{"_id": "7621333", "title": "", "text": "$1-\\frac{(1-r^2)(1-\\rho^2)}{1+r^2\\rho^2-2r\\rho\\cos\\theta'}.$"}
{"_id": "5989775", "title": "", "text": "$\\displaystyle \\zeta(s)-\\frac{1}{s-1} = \\sum_{n=1}^{\\infty} \\int_{n}^{n+1} (\\frac{1}{n^s}-\\frac{1}{x^s}) dx$"}
{"_id": "1257824", "title": "", "text": "$\\Bbb R^{n-1}\\to\\Bbb R^n$"}
{"_id": "3492024", "title": "", "text": "$Z(F_3)<F_3$"}
{"_id": "113252", "title": "", "text": "$A^+$"}
{"_id": "2509467", "title": "", "text": "$X_1\\times\\dots\\times X_n$"}
{"_id": "6061383", "title": "", "text": "$A_1 \\times \\dotsc \\times A_n \\to B$"}
{"_id": "2883163", "title": "", "text": "$dL = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2} dx$"}
{"_id": "6310050", "title": "", "text": "$a\\leq x\\leq b, ~~c\\leq y\\leq d$"}
{"_id": "7496780", "title": "", "text": "$ a_m - a_n = (1 + \\frac{1}{2} + \\ldots + \\frac{1}{m}) - (1+\\frac{1}{2} + \\ldots + \\frac{1}{n}) \\\\ = (1 + \\frac{1}{2} + \\ldots + \\frac{1}{n} + \\frac{1}{n+1} + \\ldots + \\frac{1}{2n}) - (1+\\frac{1}{2} + \\ldots + \\frac{1}{n}) \\\\ = (\\frac{1}{n+1} + \\frac{1}{n+2} + \\ldots + \\frac{1}{2n}) \\geq  \\frac{1}{2n} + \\frac{1}{2n} + \\ldots + \\frac{1}{2n} = \\frac{n}{2n} = \\frac{1}{2} = \\epsilon) $"}
{"_id": "754183", "title": "", "text": "$\\sup_{x>a} |(1+\\frac{x}{n})^n-e^x| $"}
{"_id": "4683392", "title": "", "text": "$\\lVert u \\rVert ^2 = \\lVert u_v \\rVert ^2 +\\lVert u_{\\perp v} \\rVert ^2= \\lvert \\frac{\\langle v,u\\rangle}{\\langle v,v \\rangle} \\rvert ^2 \\cdot \\lVert v\\rVert^2+\\lVert u_{\\perp v} \\rVert ^2 \\ge \\frac{\\lvert\\langle v,u\\rangle \\rvert ^2}{\\lVert v\\rVert ^2}$"}
{"_id": "4863766", "title": "", "text": "$\\sum_{k=0}^n\\frac{1}{k!}\\geq \\left(1+\\frac{1}{n}\\right)^n$"}
{"_id": "1400739", "title": "", "text": "$f(a+h,b+k) = f((a,b)+(h,k)) = f((a,b)) + f((h,k))$"}
{"_id": "2893798", "title": "", "text": "$(k+1)!>6(2^k+3^k)$"}
{"_id": "5830979", "title": "", "text": "$f_0(n) = n\\bmod 2$"}
{"_id": "8160545", "title": "", "text": "$f([a]_6 + [b]_6) = f(a) + f(b)$"}
{"_id": "4024710", "title": "", "text": "$\\sum_{n=1}^\\infty(-1)^na_n=\\sum_{n=1}^\\infty-\\frac1n=-\\sum_{n=1}^\\infty\\frac1n=-\\infty$"}
{"_id": "6535834", "title": "", "text": "$Cov(X + Y, X − Y) = Cov(X, X-Y) + Cov(Y, X-Y) = Var(X)- Cov(X,Y) + Cov(Y,X)- Var(Y)$"}
{"_id": "8329938", "title": "", "text": "$ \\| M \\|_2 = \\sqrt{\\lambda _{max} (MM^T)}. $"}
{"_id": "3202928", "title": "", "text": "$\\frac{\\sin\\alpha}{a}=\\frac{\\sin \\gamma}{c} \\Rightarrow c\\sin \\alpha=a\\sin \\gamma \\Rightarrow 18\\cdot \\frac{1}{2}=a\\sin \\gamma \\Rightarrow a\\sin \\gamma=9 \\ \\ \\ \\ (2)$"}
{"_id": "3842703", "title": "", "text": "$f(B)=\\binom{B}{2010}$"}
{"_id": "8473637", "title": "", "text": "$ M= \\frac {Ar(1+r)^n}{(1+r)^n-1}$"}
{"_id": "1201991", "title": "", "text": "$p_j \\mid (p_1p_2\\cdots p_n)$"}
{"_id": "412106", "title": "", "text": "$A_1\\cap A_2\\cap\\dots\\cap A_k\\cap A_{k+1}$"}
{"_id": "6755194", "title": "", "text": "$\\mathbb{E}|X_n|^p \\rightarrow \\mathbb{E}|X|^p.$"}
{"_id": "7417548", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{\\sum\\limits_{r=1}^{n}r^0}{\\sum\\limits_{r=1}^{n}r}$"}
{"_id": "1692313", "title": "", "text": "$f(ab) = f(a) + \\rho_V(a)f(b)$"}
{"_id": "4035015", "title": "", "text": "$f(x)=\\frac{2x}{b^2}$"}
{"_id": "510693", "title": "", "text": "$(y'+y)' = y'' + y' = y + y'$"}
{"_id": "5214495", "title": "", "text": "$|f(z)+f(-z)|\\leq|f(z)|+|f(-z)|\\leq2|z|^2$"}
{"_id": "1799209", "title": "", "text": "$[x,y] = [e_i,0]$"}
{"_id": "9080305", "title": "", "text": "$S=\\sum_{k=1}^{n-1}(n-k)\\cos\\frac{2\\pi k}{n} = \\sum_{k=1}^{n-1}k\\cos\\frac{2\\pi k}{n}$"}
{"_id": "8119504", "title": "", "text": "$ (x', y') = (x, y) + (a, b) \\\\ (x'-x , y' - y) = (a, b) \\\\ (x'-x) + (y'-y) = a + b $"}
{"_id": "6760357", "title": "", "text": "$f(f(y)+f(x))=f(x)f(f(y))+f(f(x))+f(f(y))-xf(y).$"}
{"_id": "9131118", "title": "", "text": "$U_0=\\mathbb R^{N+1}\\times \\mathbb R^N$"}
{"_id": "1107527", "title": "", "text": "$f(x_{2\\space periodic})= \\begin{equation} \\begin{cases} &f(x), x<n\\\\ &f(x-n),n<x<2n\\\\ &f(x-2n), 2n< x < 3n\\\\ &... \\end{cases} \\end{equation} $"}
{"_id": "5080112", "title": "", "text": "$y = \\frac {a+b-x} {ab}$"}
{"_id": "752166", "title": "", "text": "$1 + 2 + 3 + \\cdots = -\\frac{1}{12},$"}
{"_id": "5252269", "title": "", "text": "$||T(f)||_{p}^{p}$"}
{"_id": "260143", "title": "", "text": "$d'(x,z) \\leqslant d'(x,y) + d'(y,z)$"}
{"_id": "5065403", "title": "", "text": "$\\lim_{x\\to 0^+}  f'(x) = \\lim_{x\\to 0^-} f'(x)=0,$"}
{"_id": "5682457", "title": "", "text": "$ \\frac{a-x}{x} = \\frac{x}{b-x}  \\;\\Rightarrow\\; x = \\frac{ab}{a+b}.$"}
{"_id": "8608620", "title": "", "text": "$∫_{0}^\\infty xdf(x)$"}
{"_id": "4031661", "title": "", "text": "$f(a + (-a))=f(a)+f(-a)$"}
{"_id": "4880325", "title": "", "text": "$\\begin{bmatrix} 2 &-m&-3&1 \\\\ 1 &-1&2&2\\\\ -1&3&m+1&1 \\end{bmatrix} $"}
{"_id": "6303806", "title": "", "text": "$\\{\\langle x,y\\rangle\\mid 1 \\leq x \\leq 2, y = 0\\}$"}
{"_id": "401406", "title": "", "text": "$f_2(n)=\\binom{n+2}{3}$"}
{"_id": "2459274", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}\\frac{1}{n} \\sum_{i=1}^{n}\\frac{1}{i}=0$"}
{"_id": "6400693", "title": "", "text": "$\\frac{2}{a}+\\frac{5}{b}+\\frac{3}{c}=1$"}
{"_id": "9165165", "title": "", "text": "$\\frac{|x-27|}{x^{2/3} + 3x^{1/3} + 9} \\leq \\frac{|x-27|}{9}. $"}
{"_id": "437378", "title": "", "text": "$A=\\{a_0,a_1,a_2,\\ldots\\}$"}
{"_id": "7332497", "title": "", "text": "$ \\sum_{m=1}^{+ \\infty} \\sum_{n=1}^{+ \\infty} \\frac{1}{n^{\\alpha}+ m^{\\alpha }} = \\sum_{m=1}^{+ \\infty} O \\left( \\frac{1}{m^{\\alpha -1}} \\right)$"}
{"_id": "8455648", "title": "", "text": "$\\gamma\\implies G\\leq \\frac{\\min\\left(\\lfloor \\frac{R}{\\gamma}\\rfloor,N\\right)}{k}$"}
{"_id": "8563776", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor 2a\\rfloor}{2}\\right\\rfloor = \\lfloor a\\rfloor$"}
{"_id": "4781308", "title": "", "text": "$a^{x\\log_a b}=b^x$"}
{"_id": "7355086", "title": "", "text": "$\\int_0^{\\infty}e^{-x^2}\\frac{\\sin(2nx)}{\\sin x}{\\rm d}x$"}
{"_id": "6876396", "title": "", "text": "$ 2S=2+4+8+\\dots+2^{n+1}+\\dotsb=S-1 $"}
{"_id": "6647750", "title": "", "text": "$e^{\\frac{2ik\\pi}{p}}\\sqrt[p]t\\notin\\mathbb F_p(t)$"}
{"_id": "2805921", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 1 & 2 & -1 & a \\\\ 2 & 1 & 3 & b \\\\ 1 & -4 & 9 & 2b-3a \\\\ \\end{array}\\right].$"}
{"_id": "8718698", "title": "", "text": "$|\\mu * \\lambda|(\\mathbb{R}) \\leq |\\mu \\times \\lambda|(\\mathbb{R}^2)$"}
{"_id": "6146274", "title": "", "text": "$\\mu(A_n)\\xrightarrow{n\\to\\infty}\\mu(A)\\tag1$"}
{"_id": "3061054", "title": "", "text": "$\\gamma < a \\implies \\exists x\\in E_1 < \\gamma$"}
{"_id": "4768360", "title": "", "text": "$=\\sqrt2-\\sqrt1+\\sqrt3-\\sqrt2+\\sqrt4-\\sqrt3+...+\\sqrt9-\\sqrt8=\\sqrt9-\\sqrt1=3-1=2$"}
{"_id": "5837586", "title": "", "text": "$\\sum\\Vert x_n-y_n\\Vert\\le 1/2K$"}
{"_id": "5455911", "title": "", "text": "$\\lim_{n\\to\\infty}t_{n}\\\\=\\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{r=1}^{n}\\frac{1}{\\sqrt{r}}$"}
{"_id": "302173", "title": "", "text": "$ \\int\\frac{1}{\\sqrt{x^2+1}}\\,dx$"}
{"_id": "4756725", "title": "", "text": "$\\gamma^{4}+\\gamma+1=(\\gamma +b)(\\gamma^{3}+d\\gamma^{2}+e\\gamma+\\gamma)$"}
{"_id": "6838184", "title": "", "text": "$ F(x):=\\int_a^xf(t)\\,\\mathrm dt$"}
{"_id": "8290130", "title": "", "text": "$\\dot \\gamma, \\ddot \\gamma, \\dots, \\gamma^{(m)}$"}
{"_id": "7144081", "title": "", "text": "$a_n=\\frac{z^n-\\bar{z}^n}{z-\\bar{z}}.$"}
{"_id": "3709022", "title": "", "text": "$f(xf(x)+f(y))=(f(x))^2+y$"}
{"_id": "551563", "title": "", "text": "$f(a_1)=f(a_1+)$"}
{"_id": "1671356", "title": "", "text": "$y*x=y*(x*y)*x=(y*x)*(y*x)$"}
{"_id": "4023268", "title": "", "text": "$\\Delta{y} = f'(a) \\Delta{x} + \\epsilon \\Delta{x} $"}
{"_id": "3472431", "title": "", "text": "$\\frac{\\pi}{2z^2} = \\sum_{m=0}^\\infty \\sum_{n=0}^\\infty e^{-z\\sqrt{m^2+n^2}}-\\frac{1}{2}\\sum_{n=0}^\\infty e^{-zn}+\\frac{1}{4}$"}
{"_id": "1704817", "title": "", "text": "$A\\times A\\times A$"}
{"_id": "5259604", "title": "", "text": "$ \\sum_{n=1}^\\infty \\frac 1 {n^3} = 1 + \\sum_{n=2}^\\infty \\frac 1 {n^3} \\le 1 + \\sum_{n=2}^\\infty \\int_{n-1}^n \\frac 1 {x^3}\\,dx = 1 + \\int_1^\\infty \\frac 1 {x^3} \\, dx = 1 + \\frac 1 2.   $"}
{"_id": "737913", "title": "", "text": "$\\CH_3$"}
{"_id": "5613915", "title": "", "text": "$\\gamma(s)=(\\gamma_1,\\gamma_2,\\gamma_3)$"}
{"_id": "3543710", "title": "", "text": "$\\int_a^xf(t)w(t)\\,dt=F(x)-F(a)$"}
{"_id": "2615207", "title": "", "text": "$\\frac{1}{n}\\sum_{i=1}^n X_i^k = E\\left(x\\right)$"}
{"_id": "4172431", "title": "", "text": "$s:\\left\\{\\begin{array}{ll} x\\mapsto y\\\\ y\\mapsto x\\\\ \\lambda\\mapsto \\lambda \\text{ for } \\lambda\\not\\in\\{x,y\\} \\end{array}\\right.$"}
{"_id": "8958437", "title": "", "text": "$\\frac{\\bigl|\\langle v,u\\rangle\\bigr|}{\\|u\\|}$"}
{"_id": "5263830", "title": "", "text": "$f(n)=\\binom n2$"}
{"_id": "267485", "title": "", "text": "$A_1 \\times A_2$"}
{"_id": "2676941", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1) \\cong \\mathbb{C}[x]$"}
{"_id": "8097729", "title": "", "text": "$\\;d=gcd(x,m)\\iff \\begin{cases}x=x'd\\\\{}\\\\m=m'd\\end{cases}\\implies a=x-mk=(x'-m'k)d\\implies d\\mid a$"}
{"_id": "5392795", "title": "", "text": "$\\left\\{\\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix} \\,\\middle |\\, a, b \\in F\\right\\} \\subset M_2(F)$"}
{"_id": "7528586", "title": "", "text": "$\\int_{0}^{+\\infty}\\frac{\\sin(x)}{x^n}$"}
{"_id": "3564615", "title": "", "text": "$ \\int_a^bf(t)dt = F(a) - F(b) \\\\ \\int_a^xf(t)dt = F(x) - F(a) \\\\ \\int_x^bf(t)dt = F(b) - F(x) \\\\ $"}
{"_id": "6485528", "title": "", "text": "$I_n=\\int_0^{\\frac{\\pi}{2}}\\frac{\\sin(nx)}{\\sin x}dx$"}
{"_id": "3432996", "title": "", "text": "$ f(x)= \\left(\\frac{11}{12}+O(e^{-x})\\right)^{1/3}. $"}
{"_id": "1342742", "title": "", "text": "$P(X ≤ 11) = 1/6$"}
{"_id": "3204950", "title": "", "text": "$\\begin{cases}\\dot\\gamma(t)=\\tilde{X}(\\gamma(t))=a-\\langle{a,\\gamma(t)}\\rangle\\gamma(t),\\\\\\gamma(0)=p.\\end{cases}$"}
{"_id": "3212622", "title": "", "text": "$\\frac{ds}{dx} = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}$"}
{"_id": "7655274", "title": "", "text": "$a_n = \\dfrac{(n+2)(n+1)}{2}$"}
{"_id": "8122008", "title": "", "text": "$\\delta|x^{n-1}+...+a^{n-1})|<\\delta M<\\epsilon$"}
{"_id": "3277619", "title": "", "text": "$\\mathbb{R}^{n+1}-A$"}
{"_id": "2377007", "title": "", "text": "$-y''(y'-y'')^3 + y'''(y''-y''') = y''''(y'-y'')$"}
{"_id": "3125827", "title": "", "text": "$P(X_t\\in A|\\mathcal{F}_s)=P(X_t\\in A|X_s)$"}
{"_id": "5287354", "title": "", "text": "$ \\begin{cases} y=22+(-x-2z)\\\\ x=3\\\\ z=7 \\end{cases}\\Longleftrightarrow $"}
{"_id": "3816014", "title": "", "text": "$\\frac{(1+x)^r-1^r}{(1+c)^{r-1}}=xr$"}
{"_id": "5382131", "title": "", "text": "$x^2 +y^2 +ax +by = c$"}
{"_id": "7842311", "title": "", "text": "$\\vec{m} = \\begin{cases} x = 0 \\\\ y = 0 \\\\ z = m \\end{cases}$"}
{"_id": "6154409", "title": "", "text": "$F_i \\subseteq V_i \\subseteq \\overline{V_i} \\subseteq U_i$"}
{"_id": "4605053", "title": "", "text": "$f(x)=\\frac{1}{2^x+1}$"}
{"_id": "430837", "title": "", "text": "$f(x+f(y)) = f(f(x)) + f(y)$"}
{"_id": "94644", "title": "", "text": "$f_3^{(N)}(1)=F_3(-N)$"}
{"_id": "3857726", "title": "", "text": "$\\frac{f}{||f||} = \\frac{g}{||g||},$"}
{"_id": "6070099", "title": "", "text": "$\\int_{0}^{\\pi}f(t) dt=2\\int_{0}^{{\\pi}/{2}} f(t) dt.$"}
{"_id": "8383508", "title": "", "text": "$S_n=\\frac{X_1 + X_2 + . . . + X_n}{n}-n$"}
{"_id": "4324057", "title": "", "text": "$\\int_0^\\pi f(x) \\sin(x) dx= \\int_0^\\pi f(x) \\cos(x) dx=1.$"}
{"_id": "7580533", "title": "", "text": "$\\nabla f(x,y,z)\\equiv(1,1,1)$"}
{"_id": "6959965", "title": "", "text": "$f_X(x)=\\frac{1}{2\\sqrt{2\\pi}}e^{\\frac{-(x-1)^2}{8}} \\text{for all x real numbers}$"}
{"_id": "2977013", "title": "", "text": "$x^2-x\\sin x, x \\in [0,\\pi/2]$"}
{"_id": "787492", "title": "", "text": "$ax'+by'=d$"}
{"_id": "9335963", "title": "", "text": "$1=yz\\lambda$"}
{"_id": "5942199", "title": "", "text": "$ \\lim_{N \\to \\infty} \\sum_{n=1}^N \\frac 1n. $"}
{"_id": "8398768", "title": "", "text": "$ \\text{Card}(A) > 2 \\implies \\text{Card}(A) \\geq \\text{Card}(\\mathbb{R}).$"}
{"_id": "975512", "title": "", "text": "$cov(y,x)=0$"}
{"_id": "1804568", "title": "", "text": "$P(A)=\\frac{1}{6}$"}
{"_id": "7455567", "title": "", "text": "$ \\begin{matrix}a&-b\\\\a&b\\end{matrix}$"}
{"_id": "3609910", "title": "", "text": "$y \\in W, x \\in V \\subseteq \\overline V \\subseteq U$"}
{"_id": "5832819", "title": "", "text": "$\\gamma = \\pm \\delta \\left[\\int \\frac{d p d\\phi_p}{(2\\pi)^2}qp^2(1-\\cos(\\phi_q -\\phi_p))^2 \\pi a^2 e^{- (q^2+ p^2 -2 qp \\cos(\\phi_q -\\phi_q)) a^2/4}\\right]^{1/2}.$"}
{"_id": "9333254", "title": "", "text": "$\\lim_n \\frac{1}{n} \\sum_{k=1}^{[\\frac{n}{2}]} \\cos (\\frac{k \\pi}{n}) $"}
{"_id": "7079194", "title": "", "text": "$ x^3~ \\sin (nx),~~n \\in  (\\mathbb{N}) $"}
{"_id": "3547917", "title": "", "text": "$f(c^*) = f(a) + f(b)$"}
{"_id": "7876271", "title": "", "text": "$ \\lim_{n\\to\\infty}\\frac1n\\sum_{k=1}^n\\frac1k=0 $"}
{"_id": "2343525", "title": "", "text": "$gcd(a+b, a-b)\\geq gcd(a,b)$"}
{"_id": "448041", "title": "", "text": "$[x,y]=y$"}
{"_id": "2381994", "title": "", "text": "$\\mathbb{R}^n\\longrightarrow \\mathbb{R}^n$"}
{"_id": "2690689", "title": "", "text": "$(6/9)^x + (4/9)^x$"}
{"_id": "5550411", "title": "", "text": "$g(x)=1/|x|^{n}$"}
{"_id": "1817912", "title": "", "text": "$\\sum_{n\\geq1}\\frac1{n(n+1)}$"}
{"_id": "2343054", "title": "", "text": "$\\forall \\epsilon > 0, ~ \\exists \\delta > 0 ~ : ~ |x-x_0|<\\delta \\Rightarrow |f(x) - f(x_0)| < \\epsilon$"}
{"_id": "661741", "title": "", "text": "$\\frac{11!}{4!} - \\frac{10!}{4!} = \\frac{11!-10!}{4!}$"}
{"_id": "6098559", "title": "", "text": "$\\gamma[1,0,1]+\\delta[0,1,1]=[\\gamma,\\delta,\\gamma+\\delta]$"}
{"_id": "2623856", "title": "", "text": "$a\\,b=|BE|$"}
{"_id": "1666790", "title": "", "text": "$P[X \\geq x]$"}
{"_id": "2747075", "title": "", "text": "$H(x) = \\int^{x}_{a} f(t)dt$"}
{"_id": "6909324", "title": "", "text": "$x=\\{x,t\\}$"}
{"_id": "5398471", "title": "", "text": "$f_X(x;\\theta)=\\theta\\cdot x$"}
{"_id": "4454851", "title": "", "text": "$ f(x,y,z) = \\text{det} \\begin{bmatrix} x^2&y^2&z^2\\\\ x&y&z\\\\ 1&1&1 \\end{bmatrix},$"}
{"_id": "2733232", "title": "", "text": "$b_n = \\left(\\frac 1 2 \\right)^{n/2}$"}
{"_id": "7346461", "title": "", "text": "$I=\\int_{0}^{\\pi}\\frac{(\\pi-x)\\sin xdx}{1+\\cos^2 x}$"}
{"_id": "4335087", "title": "", "text": "$\\langle x \\rangle ^{\\perp} = \\langle y \\rangle ^{\\perp} \\Rightarrow \\left( \\langle x \\rangle ^{\\perp} \\right)^\\perp = \\langle x \\rangle = \\left( \\langle y \\rangle ^{\\perp} \\right)^\\perp =  \\langle y \\rangle, $"}
{"_id": "7913905", "title": "", "text": "$ (\\gamma^3+1)(\\gamma^2+1)=\\gamma^{14}\\cdot\\gamma^8=\\gamma^{22}=\\gamma^7=\\gamma^3+\\gamma+1. $"}
{"_id": "71619", "title": "", "text": "$\\alpha+1\\mapsto\\begin{cases}\\{\\beta,A\\},&\\beta=\\gamma\\\\\\{\\beta+1,\\gamma\\},&\\beta<\\gamma\\end{cases}$"}
{"_id": "131942", "title": "", "text": "$P\\times T$"}
{"_id": "3741179", "title": "", "text": "$\\epsilon^+$"}
{"_id": "7924469", "title": "", "text": "$   \\gamma(s)^t=\\gamma(s)\\Rightarrow\\gamma(s)^t\\gamma(s)^{-1}=id $"}
{"_id": "6912212", "title": "", "text": "$f(3-x)=\\frac{1}{1+\\frac{9^x}{27}}$"}
{"_id": "3675231", "title": "", "text": "$\\sum (b_n-a_n)<1$"}
{"_id": "3066036", "title": "", "text": "$P(m) := \\forall n \\ (0 < m \\to [(p(m) < p(n)) \\leftrightarrow (m < n)])$"}
{"_id": "9326122", "title": "", "text": "$\\forall \\epsilon \\gt 0, \\exists \\delta_n \\gt 0$"}
{"_id": "6654499", "title": "", "text": "$x = a, x = b, y = a$"}
{"_id": "8320508", "title": "", "text": "$f(z)=f(z/2+z/2)=f(z/2)f(z/2)=f(z/2)^2$"}
{"_id": "7473605", "title": "", "text": "$ \\left \\lfloor{\\frac{\\left \\lfloor{\\frac{n}{2}}\\right \\rfloor}{2}}\\right \\rfloor= \\left \\lfloor{\\frac{n}{2^2}}\\right \\rfloor$"}
{"_id": "514348", "title": "", "text": "$P_1P_2P_x$"}
{"_id": "3742193", "title": "", "text": "$ax+by=N.$"}
{"_id": "7411127", "title": "", "text": "$\\mbox{cov}[X+Y,X-Y] = \\mbox{cov}[X,X]+\\mbox{cov}[Y,X]-\\mbox{cov}[X,Y]-\\mbox{cov}[Y,Y].$"}
{"_id": "8144914", "title": "", "text": "$\\vert f(t)\\phi(\\theta)\\vert\\le M\\vert f(t)\\vert$"}
{"_id": "414696", "title": "", "text": "$[x,y]=$"}
{"_id": "5625375", "title": "", "text": "$G(t,s)=\\begin{cases} s(1-t)~ 0\\leq s\\leq t\\leq 1\\\\ t(1-s)~ 0\\leq t\\leq s\\leq 1\\end{cases}$"}
{"_id": "1401649", "title": "", "text": "$d(x,A) \\leq d(x,y) + d(y,z)$"}
{"_id": "8012346", "title": "", "text": "$2^x+2^x=2^{x+1}$"}
{"_id": "467441", "title": "", "text": "$\\mathbb{R}^n \\times \\mathbb{R}^n$"}
{"_id": "6238596", "title": "", "text": "$γ˜=M(\\gamma)=P\\gamma+a$"}
{"_id": "1537135", "title": "", "text": "$ \\sum_{n=1}^\\infty ||e_n - f_n||^2 < 1 $"}
{"_id": "5293586", "title": "", "text": "$\\sum_{n=1}^{\\infty}(-1)^n < \\infty$"}
{"_id": "6171210", "title": "", "text": "$x^2-x\\in I$"}
{"_id": "5192644", "title": "", "text": "$d(x,z)\\preceq d(x,y)+d(y,z)$"}
{"_id": "1289022", "title": "", "text": "$\\mathbb{R}[x]/(x^4-1)$"}
{"_id": "8889061", "title": "", "text": "$L=\\lim_{x \\rightarrow c^{+}} f'(x)=\\lim_{\\epsilon \\rightarrow 0} f'(c_2(\\delta(\\epsilon), f))=\\lim_{\\epsilon \\rightarrow 0} \\frac{f(c+\\delta(\\epsilon))-f(c)}{\\delta}=\\lim_{x \\rightarrow c^{-}} \\frac{f(x) - f(c)}{x-c}$"}
{"_id": "9080308", "title": "", "text": "$=n\\sum_{k=1}^{n-1}\\cos\\frac{2k\\pi }n-\\sum_{k=1}^{n-1}k\\cos\\frac{2k\\pi }n$"}
{"_id": "5997709", "title": "", "text": "$f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$"}
{"_id": "831817", "title": "", "text": "$F(x) = \\int_a^x g(t) dt $"}
{"_id": "4171487", "title": "", "text": "$ \\sum_{n=1}^{\\infty}\\frac{{(-1)}^{n-1}}{{n}^{x}}=\\frac{1}{\\Gamma(x)}\\int_0^\\infty\\frac{u^{x-1}}{e^u+1} \\mathrm{d}u, \\quad x>1. $"}
{"_id": "2715998", "title": "", "text": "$X = \\left\\{x_1,x_2,\\cdots,x_n\\right\\}$"}
{"_id": "1675113", "title": "", "text": "$10^{40}$"}
{"_id": "4656744", "title": "", "text": "$z = \\sqrt[n] r e^{{(i\\theta+2\\pi m)}/n}$"}
{"_id": "9246351", "title": "", "text": "$\\sqrt[4]{2}e^{\\frac{i\\pi}8}$"}
{"_id": "242310", "title": "", "text": "$K^+$"}
{"_id": "1160655", "title": "", "text": "$y^{y^{y^{y^{.^{.^.}}}}}$"}
{"_id": "3348270", "title": "", "text": "$\\forall x\\in X \\forall y\\in X :(xRy \\vee yRx)$"}
{"_id": "1953967", "title": "", "text": "$xRy \\rightarrow yRx$"}
{"_id": "608502", "title": "", "text": "$ \\left\\|\\pmatrix{A_1x\\\\ A_2x} \\right\\| \\geq \\max \\{\\|A_1x\\|,\\|A_2 x\\|\\} $"}
{"_id": "110095", "title": "", "text": "$\\prod_{k=1}^{n-1}{\\sin \\left(\\frac{k\\pi}{n}\\right)}=\\frac{2n}{2^n}$"}
{"_id": "7690975", "title": "", "text": "$(xRy \\vee x=y)$"}
{"_id": "760094", "title": "", "text": "$\\mathsf{AD}^+$"}
{"_id": "2220773", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\,\\left(\\frac{n}{e}\\right)^n\\!\\left[1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right],\\quad n\\rightarrow\\infty.$"}
{"_id": "6216164", "title": "", "text": "$ \\begin{split} \\sin(e^u)&=(D+b)^{-1}\\bigl[e^u\\cos(e^u)+b\\sin(e^u)\\bigr]\\qquad(**)\\\\ &=e^u(D+b+1)^{-1}\\cos(e^u)+b(D+b)^{-1}\\sin(e^u). \\end{split} $"}
{"_id": "2120880", "title": "", "text": "$dv = \\int_a^x f(t)\\,dt$"}
{"_id": "6560531", "title": "", "text": "$ \\lim_{n\\rightarrow\\infty}\\|x_n\\|_{\\ell^1}=\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^\\infty |x_{i,n}|=0\\qquad ? $"}
{"_id": "5516072", "title": "", "text": "$f(x)= \\dfrac{e^x}{1+9e^x}$"}
{"_id": "8752651", "title": "", "text": "$|AB|=35$"}
{"_id": "2231391", "title": "", "text": "$\\gamma(s) = (\\gamma^0(s), \\gamma^1(s), \\gamma^2(s), \\gamma^3(s))$"}
{"_id": "1407299", "title": "", "text": "$h \\mid g_1 g_2$"}
{"_id": "447665", "title": "", "text": "$T_{\\mathbb{C}}M = T^{1,0}M \\oplus T^{0,1}M$"}
{"_id": "7624899", "title": "", "text": "$\\mathbf{T} = ({dx \\over ds}; {dy \\over ds})$"}
{"_id": "3021954", "title": "", "text": "$ \\mathbb{P}\\{X \\geq 2\\} \\leq \\left(\\frac{e^\\gamma}{(1+\\gamma)^{1+\\gamma}}\\right)^{\\frac{2}{1+\\gamma}} = \\left(\\frac{e^{\\frac{\\gamma}{1+\\gamma}}}{1+\\gamma}\\right)^{2} = \\Theta\\!\\left(\\frac{1}{\\gamma^2}\\right) = O\\!\\left(\\frac{1}{C^2}\\right) $"}
{"_id": "3919622", "title": "", "text": "$Y = \\mathbb{R}[x]/(x^2+1)$"}
{"_id": "3629778", "title": "", "text": "$S = \\{a_1,a_2,...., a_n\\}$"}
{"_id": "7425380", "title": "", "text": "$\\sum_{n=-\\infty}^{-1}\\frac{1}{n^2}=\\sum_{n=1}^{\\infty}\\frac{1}{n^2}$"}
{"_id": "7162683", "title": "", "text": "$Cov(x, y) = E(XY)$"}
{"_id": "142889", "title": "", "text": "$f(n) = n^2 + n$"}
{"_id": "5793882", "title": "", "text": "$ Y := \\left\\{ (x,y) \\in \\mathbb{R}^2 \\; : \\; a \\leq x \\leq b, \\; R \\leq y \\leq f(x) \\right\\} $"}
{"_id": "2249380", "title": "", "text": "$p_1q_1 \\mid m$"}
{"_id": "2479456", "title": "", "text": "$T_a = \\{(x,y)\\in\\Bbb R^2\\mid 0\\leq x\\leq a, 0\\leq y\\leq a-x\\}$"}
{"_id": "6725976", "title": "", "text": "$\\{\\emptyset,\\{a\\}\\}=\\{\\{a\\}\\}$"}
{"_id": "8540531", "title": "", "text": "$Cov(X+Y,Y+Z) = Cov(X,Y)+Cov(X,Z)+Cov(Y,Y)+Cov(Y,Z)$"}
{"_id": "9073031", "title": "", "text": "$\\phi(n) = n^{-s}$"}
{"_id": "4371077", "title": "", "text": "$s(x)=\\sum_{k=1}^na_n\\chi_{A_n}(x).$"}
{"_id": "8982131", "title": "", "text": "$ x\\mapsto F(x) =\\int_a^x f(t)dt ~~~~ $"}
{"_id": "4427664", "title": "", "text": "$ 1-\\cos (xy) \\le\\int_0^xf(t) \\sin {(tf(t))}dt + \\int_0^y f^{-1}(t) \\sin{(tf^{-1}(t))} dt .$"}
{"_id": "3205130", "title": "", "text": "$(-sin(\\varphi), cos(\\varphi))$"}
{"_id": "5331816", "title": "", "text": "$\\left(\\frac12\\right)^{2N-K}$"}
{"_id": "3823962", "title": "", "text": "$\\frac{a}{b}=\\frac{a+3}{b-8}$"}
{"_id": "7584943", "title": "", "text": "$\\sum_{n=0}^\\infty \\frac{1}{an^2+bn+c}= \\frac{1}{a}\\sum_{n=0}^\\infty \\frac{1}{(n-z_0)(n-z_1)} $"}
{"_id": "4967515", "title": "", "text": "$Â(z) = z\\sum_{j\\ge0}(e^z - e^{(1 - 2^{-j})z})$"}
{"_id": "6599336", "title": "", "text": "$(1 - \\cos A)^2 + \\sin^2 A \\over \\sin A(1 - \\cos A)$"}
{"_id": "535042", "title": "", "text": "$xRy \\land yRz$"}
{"_id": "5114319", "title": "", "text": "$f(f(y))+f(x-y)=f(xf(y)-x)$"}
{"_id": "3494871", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} \\int_{-\\pi}^\\pi x^2 \\frac{\\sin(2nx)}{\\sin x} dx$"}
{"_id": "8307266", "title": "", "text": "$ \\sum_{n\\geq 1}\\sum_{k\\geq 1}\\frac{1}{n^2+k^2}=\\infty. $"}
{"_id": "3879034", "title": "", "text": "$Cov_2(x,y)>0$"}
{"_id": "5614299", "title": "", "text": "$\\sum_{k=1}^{N} \\frac1{k^2} = \\frac{\\pi^2}{6} - \\sum_{k=1}^{K} \\frac{a_k}{N^k} +  O\\left (\\frac1{N^{K+1}}\\right )$"}
{"_id": "1960620", "title": "", "text": "$\\int_0^1 x^5 (\\sin^{-1}x) \\, dx$"}
{"_id": "2137690", "title": "", "text": "$f_{Y_{(5)}}(y) = 5y^4/\\theta^5, \\quad 0 \\le y \\le \\theta.$"}
{"_id": "1878146", "title": "", "text": "$d(x,A) - d(x,y) \\leq d(y,A).$"}
{"_id": "8988331", "title": "", "text": "$I=\\int_{-\\pi}^{\\pi}\\frac{\\cos x}{1+2^x}dx$"}
{"_id": "4079173", "title": "", "text": "$\\lim_{k \\to \\infty}|\\frac{{(k+1)^3}(x-3)^{(k+1)!}3^{k!}}{k^3(x-3)^{k!}3^{(k+1)!}}|$"}
{"_id": "1067630", "title": "", "text": "$\\frac{m}{d} \\in D^{-1}M$"}
{"_id": "2979824", "title": "", "text": "$f(x) = \\sum_{j=1}^k c_j 1_{A_j}(x)$"}
{"_id": "4680529", "title": "", "text": "$  = \\frac {log_{c}x}{nlog_{c} b}$"}
{"_id": "4843094", "title": "", "text": "$\\int\\frac{1}{x^2+1}dx.$"}
{"_id": "818986", "title": "", "text": "$\\begin{eqnarray}   (1)\\quad& F(m,k,b) &=& 0 \\quad \\text{if } b < k-1 \\text{ or } m < \\tfrac{k(k-1)}{2} \\text{ or } m > \\tfrac{b(b+1) - (b-k)(b-k+1)}{2} \\\\   (2)\\quad& F(m,k,b) &=& 1 \\quad \\text{if } b \\geq m \\text{ and } k=1 \\\\   (3)\\quad& F(m,k,b) &=& 1 \\quad \\text{if } b \\geq k-1 \\text{ and } m = \\tfrac{k(k-1)}{2} \\text{.} \\end{eqnarray} $"}
{"_id": "316457", "title": "", "text": "$\\{e_1,\\, e_2,\\, e_3,\\, e_1e_2,\\, e_2e_3,\\, e_1e_3,\\, e_1e_2e_3\\}$"}
{"_id": "6593347", "title": "", "text": "$f(x)=\\frac{2x}{\\theta^2}, 0\\leq x \\leq \\theta$"}
{"_id": "5145392", "title": "", "text": "$(*) \\,\\,\\,\\,\\,\\,(\\gamma - a) \\cdot (\\gamma - a) = r^{2} $"}
{"_id": "1112860", "title": "", "text": "$d(x,z)\\leq d(x,y) + d(y,x)$"}
{"_id": "7949684", "title": "", "text": "$x-\\frac{x^3}{6},x\\to0$"}
{"_id": "7136044", "title": "", "text": "$F(x)=\\int_{a}^{x}f(t)dt\\int_{a}^{b}f(t)dt$"}
{"_id": "5207406", "title": "", "text": "$[T]_D=\\begin{bmatrix}T_{1,1} & T_{1,2} \\\\ T_{2,1} & T_{2,2}\\end{bmatrix}$"}
{"_id": "5938425", "title": "", "text": "$ \\lim\\limits_{n\\to\\infty}\\mu(B_n)\\leq  \\lim\\limits_{n\\to\\infty}\\inf\\limits_{k\\geq n}\\mu(A_k)=\\liminf\\limits_{n\\to\\infty}\\mu(A_n)\\tag{1} $"}
{"_id": "1991627", "title": "", "text": "$\\operatorname{frac}(x+y)=\\operatorname{frac}\\big(\\operatorname{frac}(x)+\\operatorname{frac}(y)\\big)$"}
{"_id": "2824822", "title": "", "text": "$ax+by=d,$"}
{"_id": "9088668", "title": "", "text": "$\\text{im}(\\gamma)\\subseteq A$"}
{"_id": "5058745", "title": "", "text": "$y=(40, -60, 40, -60, -10, -10, -10,...)$"}
{"_id": "4950774", "title": "", "text": "$\\left(1+\\dfrac{dy}{dx}\\right)\\neq0$"}
{"_id": "1947074", "title": "", "text": "$e^x\\ge \\left(1+\\frac xn\\right)^n\\ge 1+x$"}
{"_id": "5802324", "title": "", "text": "$(a_n)^2 = \\left(\\dfrac{(-1)^n}{n^{1/2}}\\right)^2 = \\dfrac{1}{n}$"}
{"_id": "7864934", "title": "", "text": "$\\lim_{n\\to\\infty}n\\sum_{i=1}^n\\frac{1}{(n+i)^2}=\\frac{1}{2}.$"}
{"_id": "774224", "title": "", "text": "$f(a+f(b))=f(a+b)+f(b)$"}
{"_id": "4903314", "title": "", "text": "$\\bigg\\lfloor\\frac{[x]}{m}\\bigg\\rfloor=\\bigg\\lfloor\\frac{x}{m}\\bigg\\rfloor$"}
{"_id": "1305233", "title": "", "text": "$\\sum_{r=1}^{n} \\frac{r}{2^r}$"}
{"_id": "5254218", "title": "", "text": "$\\tan\\theta=x/2$"}
{"_id": "4678010", "title": "", "text": "$f(x) = \\frac{2^x}{2^x - 8} - 2$"}
{"_id": "9242193", "title": "", "text": "$Cov(X,a)=0$"}
{"_id": "6347645", "title": "", "text": "$(x^2-3,\\;\\;x<0)$"}
{"_id": "5810920", "title": "", "text": "$ X''(x)Y(y)=X(x)[Y(y)-Y''(y)] $"}
{"_id": "2681634", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\|x_n -y_n\\| < 1$"}
{"_id": "6151678", "title": "", "text": "$\\sum_{m=1}^k\\cos{4m\\pi\\over k}=\\sum_{m=1}^k\\mathfrak{R}(e^{4im\\pi\\over k})=\\mathfrak{R}\\left(\\sum_{m=1}^ke^{4im\\pi\\over k}\\right)$"}
{"_id": "1450718", "title": "", "text": "$E[X_{(1)}]=\\frac{\\alpha}{\\alpha+\\beta}=\\frac{1}{6} $"}
{"_id": "7878102", "title": "", "text": "$|f(x_0)|\\le 2x_0|f(x_2)|$"}
{"_id": "6322416", "title": "", "text": "$\\displaystyle\\sum_1^n\\sin\\frac{2k\\pi}n=0$"}
{"_id": "5397188", "title": "", "text": "$=\\frac{1}{2\\sqrt 2}\\lim_{n\\to \\infty}\\sum_{r=1}^{n}\\frac{1}{\\sqrt {nr}}$"}
{"_id": "7578783", "title": "", "text": "$f(x)=\\frac{2^x+1}{2^x-1}$"}
{"_id": "8263943", "title": "", "text": "$\\int_0^\\infty f(x) dx=\\int_0^\\infty f(x)^2 dx$"}
{"_id": "8790629", "title": "", "text": "$\\big([x,y,z], (x^2+y^2+z^2)\\big)$"}
{"_id": "8374371", "title": "", "text": "$P^\\mu[X_{S+t}\\in\\Gamma|\\mathcal F_{S+}]=0$"}
{"_id": "8983445", "title": "", "text": "$\\frac1{2^{r+1}}$"}
{"_id": "3956265", "title": "", "text": "$P[X_n=x] \\leq P[X_n\\leq x] - P[X_n \\leq y_i] $"}
{"_id": "2999021", "title": "", "text": "$K_1 = \\frac{\\gamma + \\beta - r}{2\\gamma}\\\\K_2 = \\frac{\\gamma - \\beta + r}{2\\gamma}\\\\w(i) = \\frac{(\\beta + \\gamma - r)(\\beta + \\gamma)^i - (\\beta - \\gamma - r)(\\beta - \\gamma)^i}{2\\gamma}$"}
{"_id": "8848397", "title": "", "text": "$f(-\\pi)=f'(-\\pi)=0$"}
{"_id": "3464230", "title": "", "text": "$4 ^{x} +6 ^{x ^{2}} =5 ^{x} +5 ^{x ^{2}}$"}
{"_id": "2023575", "title": "", "text": "$|G|=rs$"}
{"_id": "483291", "title": "", "text": "$\\int{1\\over{(1+x^2)^n}}dx, n\\in \\mathbb{Z}^+$"}
{"_id": "1178712", "title": "", "text": "$M=N\\oplus T$"}
{"_id": "943930", "title": "", "text": "$f_Y(y) = \\frac{2}{y^3}$"}
{"_id": "8187610", "title": "", "text": "$I_n = \\pi^{-1}A_n(\\pi^{-2})   $"}
{"_id": "80658", "title": "", "text": "$y=x^3-x,$"}
{"_id": "1697715", "title": "", "text": "$ A^T = A^* = (UDU^*)^* = UD^*U^* = UDU^* = A$"}
{"_id": "3704924", "title": "", "text": "$x = \\pmatrix{1\\cr 1\\cr-2\\cr}$"}
{"_id": "7532613", "title": "", "text": "$f(x^2+y)=xf(x)+f(y).$"}
{"_id": "8875556", "title": "", "text": "$I_r = \\{s\\in R\\mid rs = 0\\}$"}
{"_id": "5928185", "title": "", "text": "$\\frac{\\sin{(\\tan{\\theta})}-\\sin{(\\sin{\\theta})}}{\\tan{(\\tan{\\theta})}-\\tan{(\\sin{\\theta})}}= \\frac{\\sin{(\\tan{\\theta})}-\\sin{(\\sin{\\theta})}}{\\frac{\\sin{(\\tan{\\theta})}\\cos{(\\sin{\\theta})-\\cos{(\\tan{\\theta})\\sin{(\\sin{\\theta})}}}}{\\cos{(\\tan{\\theta})\\cos{(\\sin{\\theta})}}}}.$"}
{"_id": "4512655", "title": "", "text": "$b=\\{\\{\\varnothing\\}\\}$"}
{"_id": "2527841", "title": "", "text": "$d_1 \\mid a$"}
{"_id": "2433364", "title": "", "text": "$A_1 \\ldots A_p$"}
{"_id": "8174013", "title": "", "text": "$\\int_{-x}^{1-x}f(x+t)Q_n(t)dt = \\int_{0}^{1}f(t)Q_n(t-x)dt $"}
{"_id": "233569", "title": "", "text": "$\\mu(A) = \\lim_{n \\to \\infty} \\mu(A_n)$"}
{"_id": "4429961", "title": "", "text": "$\\sum_{n\\in \\mathbb{Z}}e^{-n^2\\pi a}=\\frac{1}{\\sqrt{a}}\\sum_{k\\in \\mathbb{Z}}e^{-k^2 \\pi / a}.$"}
{"_id": "73592", "title": "", "text": "$z^+$"}
{"_id": "490523", "title": "", "text": "$\\int _0 ^\\infty f(x) dx=0$"}
{"_id": "792774", "title": "", "text": "$P(n - 1) \\implies P(n)$"}
{"_id": "862887", "title": "", "text": "$\\det(A)=P((a_{i,j})_{i,j})$"}
{"_id": "5936833", "title": "", "text": "$B = \\{B_1, B_2, B_3,\\cdots, B_n\\}$"}
{"_id": "796091", "title": "", "text": "$p(V)\\subseteq p(U)$"}
{"_id": "2564309", "title": "", "text": "$\\sum_{k\\geq 0}\\frac{1}{(k+a)^s} \\stackrel{\\mathcal{L}^{-1}}{=}\\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{e^{(1-a)u}u^{s-1}}{e^u-1}\\,du $"}
{"_id": "8527520", "title": "", "text": "$(b^n-b^{n-1})$"}
{"_id": "4350348", "title": "", "text": "$S=E-P$"}
{"_id": "422396", "title": "", "text": "$\n a_n=-3+{n(n+1)\\over 2},\n $"}
{"_id": "4514167", "title": "", "text": "$f(x)=\\frac{2^x-1}{2^{x-1}}$"}
{"_id": "1332441", "title": "", "text": "$x\\mathcal{R}y \\land y\\mathcal{R}x\\implies x=y$"}
{"_id": "1703223", "title": "", "text": "$a\\sin(\\gamma)+b\\cos(\\gamma)=\\sqrt{a^2+b^2}\\left(\\frac{a}{\\sqrt{a^2+b^2}}\\sin(\\gamma)+\\frac{b}{\\sqrt{a^2+b^2}}\\cos(\\gamma)\\right)\\\\=\\sqrt{a^2+b^2}\\left(\\cos(\\theta)\\sin(\\gamma)+\\sin(\\theta)\\cos(\\gamma)\\right)\\\\= \\sqrt{a^2+b^2}\\sin(\\gamma+\\theta).$"}
{"_id": "1233794", "title": "", "text": "$\\alpha^+\\cdot\\beta^+=\\alpha^+\\cdot\\beta+\\alpha^+$"}
{"_id": "6721089", "title": "", "text": "$-\\frac{1}{2^{n-1}}$"}
{"_id": "8950591", "title": "", "text": "$K: \\begin{cases} x=t+1  \\\\ y=-t+2s-1 \\\\ z=s-1 \\end{cases}$"}
{"_id": "107707", "title": "", "text": "$L^+$"}
{"_id": "2039819", "title": "", "text": "$\\lim_{N\\to\\infty}\\sum_{k=1}^N\\dfrac{k^r}{N^{r+1}}=\\lim_{N\\to\\infty}\\dfrac1N\\sum_{k=1}^N\\left(\\dfrac kN\\right)^r$"}
{"_id": "311588", "title": "", "text": "$P(1)\\implies P(2)$"}
{"_id": "7515653", "title": "", "text": "$xRy ∧ yRx$"}
{"_id": "7679112", "title": "", "text": "$y'/y = y'''/y''$"}
{"_id": "5197084", "title": "", "text": "$\\int_0^x f_1(t) dt = \\int_0^{Y(x)} f_2(t) \\sin(t) dt$"}
{"_id": "384526", "title": "", "text": "$g(n)=n^2-n$"}
{"_id": "6154910", "title": "", "text": "$A=\\begin{bmatrix}1&1&1\\\\1&\\omega^2&\\omega\\\\1&\\omega&\\omega^2\\end{bmatrix},$"}
{"_id": "4812214", "title": "", "text": "$ab = |ab|$"}
{"_id": "4363177", "title": "", "text": "$|f(z^2)|\\le|f(z)|^2$"}
{"_id": "2777309", "title": "", "text": "$W^{1,1}\\cap W^{1,\\infty}$"}
{"_id": "2506446", "title": "", "text": "$\\mathbb{R}^{n+2}\\rightarrow \\mathbb{R}^n$"}
{"_id": "1464899", "title": "", "text": "$\\rm\\:p_1 p_2.$"}
{"_id": "3669545", "title": "", "text": "$\\inf\\{F_\\gamma(x)\\mid\\gamma\\le x<\\alpha\\}=\\inf(\\{t_\\gamma\\}\\cup\\{s_\\gamma\\mid\\gamma<\\alpha\\})=0$"}
{"_id": "8014746", "title": "", "text": "$\\frac{ax+b}{x+c}$"}
{"_id": "7031670", "title": "", "text": "$\\begin{align}xRy, \\;yRz&\\implies x-y,y-z\\in\\mathbb{Q} \\\\ &\\implies (x-y)-(y-z)\\in\\mathbb{Q} \\\\ &\\implies x-z\\in \\mathbb{Q} \\\\ &\\implies xRz.\\end{align}$"}
{"_id": "1050609", "title": "", "text": "$0 \\to F_{1,n-1} \\to F_{0,n}  \\to E_{\\infty}^{0,n} \\subset E_2^{0,n}$"}
{"_id": "1261465", "title": "", "text": "$P(E_2) = 1/6$"}
{"_id": "5898385", "title": "", "text": "$ \\begin{cases} x=t\\\\ y=0\\\\ z=0 \\end{cases} $"}
{"_id": "7660016", "title": "", "text": "$s(x)=\\sum_{i=1}^{n}a_i\\chi_{E_i}(x),$"}
{"_id": "6373686", "title": "", "text": "$ B=\\{A_1,A_2,...,A_{n-1},Ab\\}\\subset\\mathbb R^{n-1}, $"}
{"_id": "2133423", "title": "", "text": "$a_n =2\\left(\\frac{\\binom{2^{n-1}}{2^{n-2}}}{\\binom{2^n}{2^{n-1}}}\\right)^{1/2^{n-1}}$"}
{"_id": "8879195", "title": "", "text": "$\\vartheta, \\vartheta'$"}
{"_id": "5267381", "title": "", "text": "$X=\\frac{\\frac{c(1+r)^{T+1}+(1-c)r^2}{r^2(1+r)^{T+1}}}{\\frac{(c-r)(1+r)^{T}+(1-c)r}{r(1+r)^{T}}}$"}
{"_id": "8162012", "title": "", "text": "$\\begin{pmatrix} {1}, {0}, {0}  \\\\ {0}, {0}, {1}\\end{pmatrix}$"}
{"_id": "1587952", "title": "", "text": "$\\displaystyle\\sum _{k=1}^{n-1} (n-k)\\cos\\frac{2k\\pi}{n}= $"}
{"_id": "6700199", "title": "", "text": "$ \\lim_{n \\to \\infty} \\frac {\\sqrt{n+1}+\\sqrt{n+2}+...+\\sqrt{2n}}{n^{3/2}}$"}
{"_id": "9357605", "title": "", "text": "$P(S_t\\gt x\\mid\\mathcal F_0)=M_0/x$"}
{"_id": "4408669", "title": "", "text": "$\\lim_{n\\to\\infty}n\\sum_{r=2}^{n+1}\\frac1{(n+r)^2}$"}
{"_id": "1780327", "title": "", "text": "$\\sum_{k=1}^{\\frac{n-1}2} \\frac{1}{\\cos^2 \\frac{k\\pi}{n}}$"}
{"_id": "8911919", "title": "", "text": "$a_n=\\frac{n^2}{2}+\\frac{n}{2}=\\frac{n(n+1)}{2}$"}
{"_id": "2756932", "title": "", "text": "$R_3 = \\left(\\dfrac{1}{k^2-1}\\right)R_3$"}
{"_id": "1859947", "title": "", "text": "$9^x - 6^x - 2\\cdot 4^x = 0 $"}
{"_id": "4905930", "title": "", "text": "$\\lim_{n\\rightarrow \\omega}\\ |x_n-p|=\\lim_{n\\rightarrow \\omega}\\ |x_n-q|=0$"}
{"_id": "1898383", "title": "", "text": "$ 1+2+3+4+5+\\cdots=-\\frac{1}{12}.$"}
{"_id": "7693978", "title": "", "text": "$\\sum_{n\\in \\Bbb N}=1+2+3+4+\\dots =-\\frac 1{12}$"}
{"_id": "246312", "title": "", "text": "$ \\begin{align} \\sum_{j=1}^\\infty\\sum_{k=1}^\\infty\\frac1{j^2+k^2} &\\ge\\sum_{j=1}^\\infty\\sum_{k=1}^\\infty\\frac1{(j+k)^2-(j+k)}\\\\ &=\\sum_{j=1}^\\infty\\sum_{k=1}^\\infty\\left(\\frac1{j+k-1}-\\frac1{j+k}\\right)\\\\ &=\\sum_{j=1}^\\infty\\frac1j\\tag1 \\end{align} $"}
{"_id": "9111913", "title": "", "text": "$ a \\le x \\le b \\\\ c \\le y \\le d $"}
{"_id": "4064148", "title": "", "text": "$\\displaystyle F(x+k)=\\int^{x}_{a}f(t)dt+\\int^{x+k}_{x}f(t)dt$"}
{"_id": "8594174", "title": "", "text": "$a_n=\\frac{vn(n+1)}{2}$"}
{"_id": "4985323", "title": "", "text": "$(a+b,a-b).$"}
{"_id": "6366015", "title": "", "text": "$\\lim_{x\\rightarrow c^{+}} f(x)=\\lim_{x\\rightarrow c^{-}} f(x)=f(c)$"}
{"_id": "6052363", "title": "", "text": "$a^k\\equiv a_s^k \\equiv a_s^{k\\ \\bmod\\ S}\\not\\equiv 1\\pmod s$"}
{"_id": "6958711", "title": "", "text": "$\\int_0^\\pi f(t) \\sin t dt=-f(t)\\cos t + \\int_0^\\pi f'(t) \\cos t dt\\\\ =-f(t)\\cos t + f'(t)\\sin t - \\int_0^\\pi f''(t) \\sin t dt $"}
{"_id": "4756726", "title": "", "text": "$\\gamma^{4} + \\gamma+1=(\\gamma)(\\gamma^{3}+d\\gamma^{2}+e\\gamma+\\gamma)$"}
{"_id": "4684111", "title": "", "text": "$V_1\\subseteq V_2^c\\subseteq U$"}
{"_id": "8876787", "title": "", "text": "$f_R(r)=F'(r)={2r\\over a^2}\\ ,$"}
{"_id": "3395043", "title": "", "text": "$f(xy)=xf(y)+f(x).$"}
{"_id": "4032079", "title": "", "text": "$P(X_{16}=2 \\mid X_0=0)$"}
{"_id": "660444", "title": "", "text": "$\\mathbb{R}^n \\rightarrow \\mathbb{R}^{n+1}$"}
{"_id": "5970114", "title": "", "text": "$p_1 p_2 = p_3 p_4$"}
{"_id": "3812344", "title": "", "text": "$ \\limsup_{n\\to\\infty}  \\left|\\int_0^\\pi f(x+t)F_n(t)\\,dt - \\int_0^\\delta f(x+t)F_n(t)\\,dt\\right| \\leq \\limsup_{n\\to\\infty}\\int_\\delta^\\pi|f(x+t)|F_n(t)\\,dt = 0 $"}
{"_id": "3276026", "title": "", "text": "$p_1 p_2 p_3 \\cdots p_{n-1}$"}
{"_id": "1928205", "title": "", "text": "$ \\left[\\begin{array}{ccc|c}  1 & 2 & 1 &  3\\\\ 1 & 1 & 1 &  1/5\\\\ 0 & 0 & 1 &  -k+42/5 \\end{array}\\right] $"}
{"_id": "6197788", "title": "", "text": "$\\sum_{n \\leq x} \\frac{1}{n} \\lfloor \\frac{x}{n} \\rfloor = \\sum_{n \\leq x} \\frac{x}{n^2}  + O(1) = x \\sum_{n =1}^{\\infty} \\frac{1}{n^2}  + \\sum_{n \\leq x} O(1) - \\sum_{x}^{\\infty} \\frac {1}{x} =  x \\space \\zeta(2)  + \\sum_{n \\leq x} O(1) - O(\\log (x))$"}
{"_id": "5995685", "title": "", "text": "$\\mathcal{B}= \\{(x,y): a<x\\leq b, c<y\\leq d\\}$"}
{"_id": "6989100", "title": "", "text": "$=\\int_0^\\pi(\\pi-t)f(\\sin t)dt$"}
{"_id": "8775041", "title": "", "text": "$\\sum_{n=1}^{+\\infty}\\|f_n-e_n\\|<1,$"}
{"_id": "1629858", "title": "", "text": "$\\gamma \\wedge \\gamma = - \\gamma \\wedge \\gamma$"}
{"_id": "6660056", "title": "", "text": "$p\\mid k_1k_2$"}
{"_id": "163674", "title": "", "text": "$cov(X,Y)=0$"}
{"_id": "1661811", "title": "", "text": "$ \\begin{align} \\int_0^1\\frac{\\tan^{-1}(x)}{1+x}\\,\\mathrm{d}x &=\\int_0^1\\frac{\\frac\\pi4-\\tan^{-1}(y)}{\\frac2{1+y}}\\frac{2\\mathrm{d}y}{(1+y)^2}\\\\ &=\\int_0^1\\frac{\\frac\\pi4-\\tan^{-1}(y)}{1+y}\\,\\mathrm{d}y\\\\[4pt] &=\\frac\\pi4\\log(2)-\\int_0^1\\frac{\\tan^{-1}(y)}{1+y}\\,\\mathrm{d}y\\\\ \\end{align} $"}
{"_id": "1393042", "title": "", "text": "$\\sum\\limits_{r=1}^n \\cot \\frac{r\\pi}{n+1}=0$"}
{"_id": "3956150", "title": "", "text": "$X = \\{ \\{a \\} \\} = \\{ \\{c\\} \\}$"}
{"_id": "608274", "title": "", "text": "$4^x=7^x=8^x=9^x$"}
{"_id": "3379442", "title": "", "text": "$P(X_t\\in A\\mid\\mathcal{F}_s)=P(X_t\\in A\\mid X_s)$"}
{"_id": "4120095", "title": "", "text": "$x+y+z=(x-y)^{2}+(y-z)^{2}+(z-x)^{2}$"}
{"_id": "6612570", "title": "", "text": "$.2, .4, .4 = \\frac{2}{10}, \\frac{4}{10}, \\frac{4}{10}$"}
{"_id": "2448921", "title": "", "text": "$\\nabla_{\\dot{\\gamma}} \\dot{\\gamma} - \\frac{g(\\nabla_{\\dot{\\gamma}}\\dot{\\gamma}, \\dot{\\gamma} )}{g(\\dot{\\gamma}, \\dot{\\gamma})}\\dot{\\gamma} = 0.$"}
{"_id": "287864", "title": "", "text": "$C_R \\subseteq C_R'$"}
{"_id": "2923173", "title": "", "text": "$F(x)=\\int_{\\infty}^x f(t) dt$"}
{"_id": "1100278", "title": "", "text": "$\\lambda = \\frac{\\alpha}{||F||}$"}
{"_id": "4764257", "title": "", "text": "$i=\\lbrace a+b,a+2b,...,a+nb\\rbrace$"}
{"_id": "7819375", "title": "", "text": "$\\lim_{n\\to\\infty} \\frac{\\sqrt{n+1}+\\sqrt{2(n+2)}+...+\\sqrt{n(n+n)}}{n^2}$"}
{"_id": "946461", "title": "", "text": "$\\alpha^{n+1}-\\beta^{n+1}$"}
{"_id": "8668217", "title": "", "text": "$f(t)=(t^p-x^p)(t^p-y^p)$"}
{"_id": "1579611", "title": "", "text": "$2^x+3^x+4^x=5^x$"}
{"_id": "2852468", "title": "", "text": "$ \\det \\begin{pmatrix} A & B\\\\B& A\\end{pmatrix}=\\det \\begin{pmatrix} A & B\\\\B-A&  A-B\\end{pmatrix}\\\\ =\\det \\begin{pmatrix} A+B & B\\\\0&  A-B\\end{pmatrix} =\\det(A-B)\\det(A+B) $"}
{"_id": "541498", "title": "", "text": "$1+2+3+4+\\cdots=\\dfrac{-1}{12}$"}
{"_id": "8222834", "title": "", "text": "$f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n+1}$"}
{"_id": "3129980", "title": "", "text": "$ r(t)= \\begin{cases} x=t\\\\ y=1-t\\\\ z=2t^2-2t+1 \\end{cases}\\quad \\text{where }t\\in[-1,1] $"}
{"_id": "543644", "title": "", "text": "$f(x-y+f(y))=f(x)+f(y)$"}
{"_id": "5157514", "title": "", "text": "$\\mathcal{C}^{+}$"}
{"_id": "5676716", "title": "", "text": "$\\frac 1{2^{i+2}}$"}
{"_id": "7017425", "title": "", "text": "$\\frac{1}{a+(a^{2}-1)^{1/2}}=\\frac{(a-(a^{2}-1)^{1/2})}{(a-(a^{2}-1)^{1/2})(a+(a^{2}-1)^{1/2})}=\\frac{a-(a^{2}-1)^{1/2}}{a^{2}-(a^{2}-1)}=a-(a^{2}-1)^{1/2}$"}
{"_id": "3245734", "title": "", "text": "$x^4+ \\frac{1}{x^4} +y^4+ \\frac{1}{y^4}$"}
{"_id": "5783103", "title": "", "text": "$a_{k_n}=(2+1)\\frac{2n}{2n+1}=6\\frac{1}{2+\\frac1n}\\to\\frac{6}{2}=3$"}
{"_id": "8099977", "title": "", "text": "$f'(x)=1/√|x|$"}
{"_id": "1046959", "title": "", "text": "$[x, y]^3=[x, y]$"}
{"_id": "7080612", "title": "", "text": "$(M^{s})^{a}\\equiv M^{sa}\\equiv M^{sa}\\times 1\\equiv M^{sa}M^{x(p-1)}\\equiv M^{sa+x(p-1)}\\equiv M^{1}\\equiv M(\\mod p)$"}
{"_id": "8043153", "title": "", "text": "$ \\left( \\begin{array}{cc} AB & 0\\\\ B & 0\\\\ \\end{array} \\right) $"}
{"_id": "1471147", "title": "", "text": "$ f(2f(x)+f(y)) = 2x+f(y) $"}
{"_id": "9289398", "title": "", "text": "$\\Gamma(x) = \\sqrt{2\\pi(x-1)}\\left(\\frac{x-1}{e}\\right)^{x-1}$"}
{"_id": "215144", "title": "", "text": "$\\frac{3}{x} +\\frac{4}{y} +\\frac{5}{z} =6$"}
{"_id": "7817680", "title": "", "text": "$g(x)=\\frac{1}{2} e^{-2}(x-e^2+1)+1.$"}
{"_id": "3711594", "title": "", "text": "$M^{n}=M*M*M*...*M$"}
{"_id": "7521203", "title": "", "text": "$\\langle f,g\\rangle = \\int_{-1}^{1}f(t)g(t)dt = \\int_{-1}^{1}f(t)\\cdot\\left(\\delta(t-t_0)+\\delta(t+t_o)\\right)dt$"}
{"_id": "125356", "title": "", "text": "$y' = -y+\\frac{10}{4}\\frac{xy}{1+x}$"}
{"_id": "3027862", "title": "", "text": "$g\\in W_0^{\\alpha,1}(0,T)$"}
{"_id": "8147587", "title": "", "text": "$k^{log_kB}=B$"}
{"_id": "4535787", "title": "", "text": "$ax+b\\color{red}{y}=d$"}
{"_id": "1155430", "title": "", "text": "$\\exists f_1,f_2\\in M \\forall d\\in M$"}
{"_id": "7865538", "title": "", "text": "$aX+bY = d$"}
{"_id": "3278736", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} X_n(\\omega)\\rightarrow X(\\omega)$"}
{"_id": "5781767", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y(x)}=1$"}
{"_id": "6399457", "title": "", "text": "$ P(E\\cap E) = P(E)P(E)$"}
{"_id": "4122564", "title": "", "text": "$\\sum_{p}\\frac{1}{p^{2s}}\\leq \\sum_{n}\\frac{1}{n^2}$"}
{"_id": "4983029", "title": "", "text": "$ 1 + \\sum_{n \\ge 1} \\frac{1}{n^2}. $"}
{"_id": "280708", "title": "", "text": "$S^n \\subset \\mathbb R^{n+1}$"}
{"_id": "117801", "title": "", "text": "$\\int_{a}^{\\infty}f(x)dx$"}
{"_id": "3482166", "title": "", "text": "$\\Vert e_n-c_n\\Vert^2=r_n^2$"}
{"_id": "6303067", "title": "", "text": "$\\sec\\theta = h/2$"}
{"_id": "3554489", "title": "", "text": "$m<n,f^{m}\\in \\langle x \\rangle$"}
{"_id": "749990", "title": "", "text": "$\\Gamma(x)\\zeta(x) = \\sum_{n=1}^\\infty\\frac{\\Gamma(x)}{n^x} = \\sum_{n=1}^\\infty\\int_0^\\infty \\frac{t^{x-1}}{e^{nt}}dt = \\int_0^\\infty t^{x-1}\\sum_{n=1}^\\infty\\frac{1}{(e^{t})^n}dt = \\int_0^\\infty\\frac{t^{x-1}}{e^{t} - 1}dt$"}
{"_id": "4386398", "title": "", "text": "$f_3(n) = n^3$"}
{"_id": "5269466", "title": "", "text": "$f_\\#f^\\#f_\\#=f_\\#$"}
{"_id": "2855589", "title": "", "text": "$\\zeta (s) = \\frac{\\Pi(- s)}{2 \\pi i} \\int_{+ \\infty}^{+ \\infty} \\frac{(- x)^s}{e^x - 1}\\frac{dx}{x}$"}
{"_id": "1677591", "title": "", "text": "$\\mathbb{R}^{N+1}/\\mathbb{R}^N$"}
{"_id": "5006233", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{ab}$"}
{"_id": "6520014", "title": "", "text": "$\\int_\\gamma\\omega=\\int_{\\partial\\gamma} f=f(\\gamma(1))-f(\\gamma(0))$"}
{"_id": "525660", "title": "", "text": "$\\int \\frac{1}{(1+x^2)^2}dx$"}
{"_id": "2805354", "title": "", "text": "$\\alpha=\\{\\langle{x},y\\rangle\\in{\\mathbb{R}\\times\\mathbb{R}}: x+2y=1, 0\\lt{x}\\le{1},0\\le{y}\\lt{1/2}\\}$"}
{"_id": "4212471", "title": "", "text": "$a^{\\log_a(x)} = x$"}
{"_id": "7598294", "title": "", "text": "$\\gamma(a \\wedge b) = \\gamma(a) \\vee \\gamma(b)$"}
{"_id": "4823576", "title": "", "text": "$\\displaystyle \\left|e^x-\\left(1+\\frac xn\\right)^n\\right|<\\epsilon$"}
{"_id": "4951938", "title": "", "text": "$b^{\\log_b(a)} = x$"}
{"_id": "2962263", "title": "", "text": "$\\{a, a+d, a+2d, a+3d\\}$"}
{"_id": "1443023", "title": "", "text": "$\\tan\\theta = \\frac{x}{2} => x = 2\\tan\\theta => dx = 2\\sec^2\\theta d\\theta$"}
{"_id": "9295072", "title": "", "text": "$\\begin{align} [X, Y] = XY − YX. \\end{align}$"}
{"_id": "2514495", "title": "", "text": "$\\sin(\\theta)=\\frac{\\sin^2(\\theta)}{\\sin(\\theta)}=\\sin(\\theta)$"}
{"_id": "6391569", "title": "", "text": "$ \\lim_{n \\to \\infty} \\frac{\\sqrt{1} + \\sqrt{2} + ... + \\sqrt{n}}{\\sqrt{n}}\\frac1n= \\lim_{n \\to \\infty} \\left(\\sqrt{\\frac1n} + \\sqrt{\\frac2n} + \\sqrt{\\frac3n} +\\cdots+\\sqrt{\\frac{n}{ n}} \\right) \\frac1n $"}
{"_id": "7501132", "title": "", "text": "$\\mathcal{A}(x)=\\mathcal{A}\\cap[1,x]$"}
{"_id": "1037035", "title": "", "text": "$A = V \\oplus W$"}
{"_id": "8874122", "title": "", "text": "$GF(3)[x]/(x^2+1)=GF(3)[w]$"}
{"_id": "1733278", "title": "", "text": "$x = c_1e_1 + \\ldots + c_ne_n$"}
{"_id": "4175348", "title": "", "text": "$\\partial_i\\gamma\\partial_i\\gamma=\\partial_i\\partial_i\\gamma\\gamma$"}
{"_id": "2878940", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\\\ 1 & -1 & 1 & 1 \\\\ -2 & 1 & -2 & k \\\\ \\end{array}\\right]$"}
{"_id": "3400983", "title": "", "text": "$Cov(X,Y) = 3$"}
{"_id": "3613391", "title": "", "text": "$A \\cap B = \\emptyset: \\mu(A \\cup B) = \\mu(A) + \\mu(B)$"}
{"_id": "601302", "title": "", "text": "$\\alpha_{m}\\le\\dfrac{(n-1)^{m-1}+(-1)^m}{n(n-1)^{(m/2)-1}}$"}
{"_id": "6257813", "title": "", "text": "$r_3 > 2^{k-1} $"}
{"_id": "1622995", "title": "", "text": "$f(x) = \\dfrac{(1+x)^3}{x \\, (1+x^2)}$"}
{"_id": "7320978", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       1&-3&0&6\\\\       1&0&3&-3\\\\       2&k&3-k&1     \\end{array} \\right] $"}
{"_id": "6935225", "title": "", "text": "$ \\begin{array}{ccccc} \\text{order}    & 1& 3& 5& 15 \\\\ \\# \\text{elems} & 1& 2& 4& 8  \\end{array}$"}
{"_id": "3988616", "title": "", "text": "$\\int_0^{+\\infty}f(x)$"}
{"_id": "6007186", "title": "", "text": "$cI = \\gamma'\\gamma I \\subset \\gamma'R \\subset R$"}
{"_id": "5827650", "title": "", "text": "$f(f(x)) = x f(x) - F(x)$"}
{"_id": "5818205", "title": "", "text": "$S_n=\\sqrt1+\\sqrt2+\\cdots+\\sqrt n$"}
{"_id": "8055446", "title": "", "text": "$\\int\\tan^nx\\ dx=\\frac{\\tan^{n-1}x}{n-1}-\\int\\tan^{n-2}x\\ dx$"}
{"_id": "1374512", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} \\frac{1}{\\sqrt{n}}\\sum_{k=1}^n\\frac{1}{\\sqrt{n+k}}$"}
{"_id": "890131", "title": "", "text": "$\\sum\\limits_{n = 1}^\\infty \\sum\\limits_{m = 1}^\\infty \\frac{m^2}{(n^2 + m^2)^4}$"}
{"_id": "3396462", "title": "", "text": "$ P(t^*>t|H_0) $"}
{"_id": "213380", "title": "", "text": "$\\lim_{x\\to c}f'(x)=L$"}
{"_id": "8448348", "title": "", "text": "$(z-x,z+x)=d>1$"}
{"_id": "30135", "title": "", "text": "$(1+a(n-1))(1+a) \\ge 1+a(n)$"}
{"_id": "541502", "title": "", "text": "$1+2+3+4+\\cdots=-1/12$"}
{"_id": "8046608", "title": "", "text": "$\\mathbb P\\left[X_n\\to x \\text{ as } n\\to\\infty\\right]=1$"}
{"_id": "6760341", "title": "", "text": "$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$"}
{"_id": "5450364", "title": "", "text": "$|f(z)| \\le M |T(z)|$"}
{"_id": "5465844", "title": "", "text": "$\\begin{align}    \\binom{5 + 5}{5, 5} = \\binom{10}{5} = 252 \\end{align}$"}
{"_id": "451449", "title": "", "text": "$f(n)={n\\choose{\\log{n}}}$"}
{"_id": "6902976", "title": "", "text": "$S_n = \\frac{(n+1)n}{2}$"}
{"_id": "7009451", "title": "", "text": "$f(a*b) = f(a) + f(b)$"}
{"_id": "7256842", "title": "", "text": "$E(X)=N[1-(\\frac{N-1}{N})^n]$"}
{"_id": "6631294", "title": "", "text": "$\\textbf{G}(w_1,x_{\\gamma},y_{\\gamma},z_{\\gamma})=0\\\\ F_2(x_{\\gamma},y_{\\gamma},z_{\\gamma})=0\\\\ F_3(x_{\\gamma},y_{\\gamma},z_{\\gamma})=0$"}
{"_id": "2146869", "title": "", "text": "$\\begin{array}\\\\ t(m) &=1+2a+3a^2+...+(m-1)a^{m-2}\\\\ &=((m-1) a^{m+1}-m a^m+a)/((a-1)^2 a)\\\\ &=(ma^m(a-1)-a^{m+1}+a)/((a-1)^2 a)\\\\ \\end{array} $"}
{"_id": "96384", "title": "", "text": "$\\mathbb{R}^{n+m} \\to \\mathbb{R}^n$"}
{"_id": "5909516", "title": "", "text": "$T^{1,0}M\\cap  T^{0,1}M\\not=(0)$"}
{"_id": "7906523", "title": "", "text": "$d(x,\\alpha)\\leqslant d(x,y)+d(y,\\alpha)<\\frac{2}{m}$"}
{"_id": "4427185", "title": "", "text": "$B \\subseteq A \\subseteq N$"}
{"_id": "3760616", "title": "", "text": "$p(E) = 1/2$"}
{"_id": "7620836", "title": "", "text": "$e^{\\sqrt{\\frac{m}{2}}(1+i)}-e^{-\\sqrt{\\frac{m}{2}}(1+i)}=e^{\\sqrt{\\frac{m}{2}}}‌​[\\cos \\sqrt{\\frac{m}{2}}+i \\sin\\sqrt{\\frac{m}{2}}] -e^{-\\sqrt{\\frac{m}{2}}}[\\cos \\sqrt{\\frac{m}{2}}-i \\sin\\sqrt{\\frac{m}{2}}]$"}
{"_id": "8763495", "title": "", "text": "$(a_n, b_n) \\subset U_n$"}
{"_id": "7599353", "title": "", "text": "$\\det ((a-b)I + b e e^T) = (a-b)^n \\times \\left( 1 + n \\dfrac{b}{a-b} \\right) = (a-b)^{n-1} (a+(n-1)b)$"}
{"_id": "4388300", "title": "", "text": "$\\frac{1}x+\\frac{2}y=\\frac{3}n.$"}
{"_id": "3955311", "title": "", "text": "$   \\mathbb{E}f\\left(\\frac{|X+Y|}{a+b} \\right)  \\le \\frac{a}{a+b}\\mathbb{E}f\\left(\\frac{|X|}{a}\\right) + \\frac{b}{a+b}\\mathbb{E}f\\left(\\frac{|Y|}{b}\\right). $"}
{"_id": "8245287", "title": "", "text": "$\\rm\\: \\  \\lfloor x/(mn)\\rfloor\\ =\\ \\lfloor{\\lfloor x/m\\rfloor}/n\\rfloor\\ \\  $"}
{"_id": "775675", "title": "", "text": "$|x-a|<\\delta\\implies -\\delta<x-a<\\delta\\implies-\\delta+a<x<\\delta+a$"}
{"_id": "2832598", "title": "", "text": "$L = L^2= \\lim_{x \\to 0} f(x) \\lim_{x \\to 0}f(-x) = \\lim_{x \\to 0} f(x)f(-x) = \\lim_{x \\to 0} f(x-x) = \\lim_{x \\to 0} f(0) = f(0)$"}
{"_id": "544956", "title": "", "text": "$\\frac{1}{\\sin x}$"}
{"_id": "6965713", "title": "", "text": "$\\text{trace}(\\gamma) = \\text{trace}(\\gamma^{-1})$"}
{"_id": "3782735", "title": "", "text": "$1+2+3+4\\cdots=-1/12$"}
{"_id": "241298", "title": "", "text": "$f_{X}(x)= \\frac{1}{x^{2}}$"}
{"_id": "9091868", "title": "", "text": "$(a,\\gamma)\\mapsto \\gamma(1)$"}
{"_id": "6486535", "title": "", "text": "$e^{x} > (1 + x/n)^n $"}
{"_id": "7599896", "title": "", "text": "$f(n) = f(n-1) * f(n-2)\\\\ f(n) = (f(n-2))^2 * (f(n-3))^1\\\\ f(n) = (f(n-3))^3 * (f(n-3))^2\\\\ f(n) = (f(n-3))^5 * (f(n-4))^3$"}
{"_id": "977170", "title": "", "text": "$e^x\\ge \\left(1+\\frac xm\\right)^m$"}
{"_id": "2074909", "title": "", "text": "$\\left\\|A \\right\\| = \\sqrt{\\lambda_{max}(A^TA)}$"}
{"_id": "1652840", "title": "", "text": "$A\\not R B \\land B \\not R A$"}
{"_id": "2530679", "title": "", "text": "$\\frac{n}{2(n+1)(n+2)}=\\frac{1}{2}(\\frac{-1}{n+1}+\\frac{2}{n+2})$"}
{"_id": "5064724", "title": "", "text": "$|x^3-x_0^3|\\le|x-x_0||x+x_0|^2\\le\\delta(2|x_0|+\\delta)^2$"}
{"_id": "3970655", "title": "", "text": "$ax+by=h$"}
{"_id": "2371680", "title": "", "text": "$ T = \\{(x,y)\\in\\mathbb{R}^2: 0\\leq x\\leq 1,0\\leq y\\leq x\\},$"}
{"_id": "3787806", "title": "", "text": "$\\frac{1}{2}\\left(e^{x/20}-e^{-x/20}\\right)$"}
{"_id": "7235485", "title": "", "text": "$E_1^{0,n} = \\frac{\\ker (E_0^{0,n} \\to E_0^{0,n+1})}{\\text{im}(E_0^{0,n-1} \\to E_0^{0,n})}$"}
{"_id": "6007948", "title": "", "text": "$\\sqrt[k]{\\frac{a_{1}^{k}+a_{2}^{k}+\\ldots +a_{n}^{k}}{n}}\\geq \\frac{a_{1}+a_{2}+\\ldots + a_{n}}{n}$"}
{"_id": "8129903", "title": "", "text": "$\\frac{1}{\\pi}\\sum_{n = -\\infty}^{\\infty}\\frac{a}{a^2 + n^2} = \\sum_{n = -\\infty}^{\\infty}e^{-2\\pi a |n|}$"}
{"_id": "1006952", "title": "", "text": "$\\sum_{\\gamma\\in \\Gamma} a_\\gamma X^\\gamma$"}
{"_id": "136838", "title": "", "text": "$x^2-y^2=1$"}
{"_id": "206363", "title": "", "text": "$\\begin{cases}x=-t-z\\\\y=tz\\\\t=-u-q\\\\z=uq \\\\u=-x-y\\\\q=xy\\end{cases}$"}
{"_id": "6247086", "title": "", "text": "$(\\mathbb{Z}/2^r\\mathbb{Z})^*$"}
{"_id": "3278309", "title": "", "text": "$\\mathbb E\\lvert X-Y\\rvert^p= \\mathbb E\\lvert X\\rvert^p+p(p-1)\\int_0^1(1-s)\\mathbb E\\left[Y^2\\lvert X+sY\\rvert^{p-2}      \\right] \\mathrm ds.           $"}
{"_id": "3127643", "title": "", "text": "$reg(\\alpha) = \\alpha^+$"}
{"_id": "3169571", "title": "", "text": "$ \\mathbb{E}\\|X\\|_2^r\\le C_r'\\left(\\mathbb{E}\\|X\\|_2^2\\right)^{r/2}. $"}
{"_id": "426095", "title": "", "text": "$\\zeta(s) = \\displaystyle\\sum_{n=1}^{\\infty} \\displaystyle\\int_n^{n+1} \\frac{1}{n^s} - \\frac{1}{x^s} dx + \\frac{1}{s-1}$"}
{"_id": "7242684", "title": "", "text": "$ \\begin{align} a_n & =\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi}{f(x)\\cos(nx)dx} =\\frac{2}{\\pi}\\int_{0}^{\\pi}{f(x)\\cos(nx)dx} =\\frac{2}{\\pi}\\int_{0}^{\\pi}{(e^x+e^{-x})\\cos(nx)dx}\\\\ & =\\frac{2}{\\pi}\\int_{0}^{\\pi}{e^x\\cos(nx)dx}   +\\frac{2}{\\pi}\\int_{0}^{\\pi}{e^{-x}\\cos(nx)dx} =I+J \\end{align} $"}
{"_id": "2175727", "title": "", "text": "$S^5/(Z/6)$"}
{"_id": "6327888", "title": "", "text": "$1+2+3+4+5.... = -\\frac{1}{12}$"}
{"_id": "7806303", "title": "", "text": "$f_X(x)=\\theta e^{-\\theta x}$"}
{"_id": "6081534", "title": "", "text": "$d(x,y) \\le d(x,x_{n_k}) + d(x_{n_k}, y_{n_k}) + d( y_{n_k}, y)$"}
{"_id": "7197542", "title": "", "text": "$V(Y|x)=E(Y^2|x)-[E(Y|x)]^2$"}
{"_id": "3083634", "title": "", "text": "$\\int_0^\\infty \\frac{\\sin^3(x)}{x^2}\\,dx$"}
{"_id": "2551093", "title": "", "text": "$\\sum_{n=1}^\\infty\\sum_{m=1}^\\infty{a_{nm}}.$"}
{"_id": "6476158", "title": "", "text": "$\\alpha R\\beta\\lor\\beta R\\alpha$"}
{"_id": "5309613", "title": "", "text": "$\\operatorname{True} \\implies P(x).$"}
{"_id": "3524577", "title": "", "text": "$\\tag{1}\\sum_{n=-\\infty}^\\infty e^{-s\\pi(n+x)^2}=s^{-1/2} \\sum_{n=-\\infty}^\\infty e^{-\\pi n^2/s}e^{2\\pi i n x}$"}
{"_id": "1630934", "title": "", "text": "$\\ker \\Phi = T \\oplus F$"}
{"_id": "5540822", "title": "", "text": "$\\frak \\overline{a^+}$"}
{"_id": "6149755", "title": "", "text": "$f(f(m)^2+f(n)^2)=m^2+n^2.$"}
{"_id": "6597014", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty}\\frac{1}{(n\\!-\\!a)^2\\!-\\!b^2} = \\frac{1}{2b}\\sum_{n=-\\infty}^{\\infty}\\frac{a-b}{n\\!-\\!(a\\!-\\!b)^2}\\!+\\!\\frac{a+b}{n\\!-\\!(a\\!+\\!b)^2} =\\frac{\\pi}{2b}\\left[\\cot\\pi (a\\!-\\!b)\\!-\\!\\cot\\pi (a\\!+\\!b)\\right]\\tag{1}$"}
{"_id": "5134201", "title": "", "text": "$r^2 \\equiv s^2 \\equiv 1$"}
{"_id": "5165016", "title": "", "text": "$\\mathrm{d}(a,L)=\\frac{ |\\langle a, \\alpha \\rangle-c|} {\\| \\alpha\\|}$"}
{"_id": "4992664", "title": "", "text": "$\\forall \\epsilon \\gt 0, \\forall n \\ge N, |c_n-x| \\lt \\epsilon$"}
{"_id": "3825225", "title": "", "text": "$\\int \\frac{\\sin^{-1}\\sqrt{x}}{\\sqrt{1-x}}\\,dx$"}
{"_id": "909987", "title": "", "text": "$\\bigg\\lfloor\\dfrac{\\lfloor x \\rfloor }{m}\\bigg\\rfloor = y = \\bigg\\lfloor\\dfrac{x }{m}\\bigg\\rfloor$"}
{"_id": "1238441", "title": "", "text": "$\\mathrm{x} = (x, y)$"}
{"_id": "5497492", "title": "", "text": "$ \\frac14 \\sum_{m=1}^\\infty \\frac1m \\frac{1}{\\sinh^2 \\frac{m\\alpha}2} = - \\sum_{n=1}^\\infty n\\log (1-q^n)$"}
{"_id": "6576913", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\sum\\limits_{r=0}^{n-1}\\frac{1}{n}f(\\frac{r}{n})$"}
{"_id": "3870595", "title": "", "text": "$y_n=\\frac{x_1+x_2+...x_n}{n}$"}
{"_id": "5988638", "title": "", "text": "$E_i \\subset (a_i, b_i)$"}
{"_id": "4434077", "title": "", "text": "$ \\left(1+\\frac xn\\right)^n \\le e^x \\le \\left(1-\\frac xn\\right)^{-n} $"}
{"_id": "4787459", "title": "", "text": "$\\{a,a+b,a+2b,\\cdots\\}$"}
{"_id": "2150986", "title": "", "text": "$|m'-m|=|m''-m'|=|m-m''|$"}
{"_id": "3198526", "title": "", "text": "$g'(x, y)=\\left(\\dfrac{\\sqrt{\\gamma (1-\\gamma)}y}{\\left(\\sqrt{\\gamma}x+\\sqrt{1-\\gamma}y\\right)^2}, -\\dfrac{\\sqrt{\\gamma (1-\\gamma)}x}{\\left(\\sqrt{\\gamma}x+\\sqrt{1-\\gamma}y\\right)^2} \\right)^t$"}
{"_id": "4676813", "title": "", "text": "$F_n=\\int_0^1\\frac{\\sin(nx)}{2+\\cos(nx)}f_n(x)dx,\\:\\:\\:n\\geq 1.$"}
{"_id": "5612109", "title": "", "text": "$I = \\int_0^\\infty f(x) ~ dx$"}
{"_id": "7737843", "title": "", "text": "$0 = V_0 \\subseteq V_1 \\subseteq V_2 \\subseteq ... \\subseteq V_r \\subseteq V$"}
{"_id": "5853768", "title": "", "text": "$ F(\\alpha) = a_2 (\\alpha-\\beta) (\\alpha-\\gamma) \\\\  F(\\beta) = a_3 (\\beta-\\gamma) (\\beta-\\alpha) \\\\ F(\\gamma) = a_1 (\\gamma-\\alpha) (\\gamma-\\beta)$"}
{"_id": "809103", "title": "", "text": "$A = A_{1} \\times \\dots \\times A_{n}$"}
{"_id": "3568434", "title": "", "text": "$a_n = \\frac{n(n+1)}{2} * (\\frac{(2n+4)}{6})$"}
{"_id": "4164398", "title": "", "text": "$\\alpha(f)=\\frac1{1/T}\\sum^\\infty_{m=-\\infty}c_ne^{-i2\\pi nf/(1/T)}=\\sum^\\infty_{m=-\\infty}e^{-i2\\pi nfT}$"}
{"_id": "1562354", "title": "", "text": "$\\sum\\limits_{i = 1}^{n} r^{i - 1} = \\sum\\limits_{i = 0}^{n - 1} r^{i} = \\frac{1 - r^n}{1 - r}$"}
{"_id": "309011", "title": "", "text": "$\\Phi(s \\otimes n) = s\\varphi(n)$"}
{"_id": "5110714", "title": "", "text": "$xRy\\Rightarrow \\neg yRx$"}
{"_id": "147345", "title": "", "text": "$H(z,\\alpha)\\restriction \\beta=\\{(\\gamma,H(z,\\alpha)(\\gamma))\\mid \\gamma<\\beta\\}=\\{(\\gamma,H(z,\\gamma+1)(\\gamma))\\mid \\gamma<\\beta\\}=$"}
{"_id": "1683834", "title": "", "text": "$\\mathbb{E}|X_n|^p \\to \\mathbb{E}|X|^p$"}
{"_id": "7828712", "title": "", "text": "$a^{(\\log_ab)^2}=a^{\\log_a(b)\\times \\log_a(b)}=b^{\\log_a(b)}$"}
{"_id": "940505", "title": "", "text": "$ \\begin{matrix}   & 1 & (12) & (123)\\\\ \\hline  \\chi_{S^2(V)} & 3 & 1 & 0 \\end{matrix}$"}
{"_id": "8475455", "title": "", "text": "$(\\mathbb{R}[t]/(t^2+1))[x]$"}
{"_id": "8273133", "title": "", "text": "$f_X ( x ; \\theta ) = (1/\\theta) x \\exp (-x/\\theta).$"}
{"_id": "705632", "title": "", "text": "$\\left(\\begin{array}{rrr|r} 1 & 1 & 2 & 7\\\\ -2 & -2 & k & -14\\\\ 3 & 3 & 6 & 14 \\end{array}\\right)$"}
{"_id": "1168397", "title": "", "text": "$\\sum_{k=1}^{10} (\\sin \\frac{2 \\pi k}{11}-i \\cos \\frac{2 \\pi k}{11})=?$"}
{"_id": "1734489", "title": "", "text": "$\\gamma=(\\gamma_1,\\gamma_2)$"}
{"_id": "9143029", "title": "", "text": "$\\int_{-\\infty}^{\\infty}\\frac{1}{(x^2+1)(2-2x+x^2)}dx$"}
{"_id": "1518185", "title": "", "text": "$\\left[\\begin{array}{ccc|c}       1&   0&   2&  1\\\\       0&   1&  -1&  2\\\\       1&  -2& k+4&  5     \\end{array}\\right]$"}
{"_id": "4890340", "title": "", "text": "$(x - y)^z + (x - z)^y + (y - x)^z + (y - z)^x + (z - x)^y + (z - y)^x$"}
{"_id": "2066961", "title": "", "text": "$f(x)=\\sum_{j=1}^{n}a_{j}p_{j}(x)$"}
{"_id": "2474349", "title": "", "text": "$[\\forall m,P(m,0)]\\land[\\forall n,P(0,n)]$"}
{"_id": "4736154", "title": "", "text": "$(-)\\cdot(-)=(-)\\cdot(+)=(+)\\cdot(-)=(-)$"}
{"_id": "3420738", "title": "", "text": "$4^x+6^x=9^x$"}
{"_id": "4638956", "title": "", "text": "$A_1\\subseteq A_2\\subseteq \\cdots \\subseteq A_{n+1} $"}
{"_id": "6255733", "title": "", "text": "$|x_i + y_i|^p \\leq (\\alpha+ \\beta)^p \\bigg( \\frac{\\alpha}{\\alpha + \\beta} |\\hat{x}_i|^p + \\frac{\\beta}{\\alpha + \\beta} |\\hat{y}_i|^p \\bigg). $"}
{"_id": "6294588", "title": "", "text": "$\\geq{1\\over 2^{n-2}}$"}
{"_id": "781104", "title": "", "text": "$f(a)\\oplus f(b)=f(a+b)\\ .$"}
{"_id": "8867502", "title": "", "text": "$a, b, c, d ∈ \\mathbb{Q}$"}
{"_id": "3061051", "title": "", "text": "$\\gamma^\\prime = \\gamma + \\frac{a-\\gamma}{2} <a \\implies \\gamma^\\prime \\in E_1$"}
{"_id": "4124630", "title": "", "text": "$2\\pi \\int\\int y\\  dy\\ dx$"}
{"_id": "1165207", "title": "", "text": "$\\frac1{\\sqrt{n}}\\|A\\|_\\infty = \\|A\\|_2 = \\sqrt{m}\\|A\\|_\\infty$"}
{"_id": "7403678", "title": "", "text": "$\\xrightarrow{R_2-R_1\\to R_2} \\left[     \\begin{array}{ccc|c}       1&-3&0&6\\\\       1&3&3&-9\\\\       0&k&24-6k&-5     \\end{array} \\right] $"}
{"_id": "1973250", "title": "", "text": "$(a+b,a+2b)$"}
{"_id": "7040887", "title": "", "text": "$\\forall \\epsilon >0, \\exists \\delta>0,\\text{ s.t. }|x-a|< \\delta \\implies |f(x)-f(a)| <\\epsilon$"}
{"_id": "930185", "title": "", "text": "$\\qquad =(a-b)^{n-1}(a+(n-1)b)$"}
{"_id": "3604360", "title": "", "text": "$A = \\begin{bmatrix} 1 &-1  & 2 & -1\\\\  -1 & 0 & -1 &2 \\\\  2 &-4  &6  & 0 \\end{bmatrix}$"}
{"_id": "6776478", "title": "", "text": "$f(n) = n^2 - 7n + 12$"}
{"_id": "7036985", "title": "", "text": "$\\exists \\epsilon >0, \\forall \\delta >0, \\exists x \\in \\mathbb R, s.t. |f(x)-f(a)|> \\epsilon, |x-a|< \\delta$"}
{"_id": "2906030", "title": "", "text": "$ \\frac{k^3}{3}+k^2<\\frac{(k+1)^3}{3} $"}
{"_id": "6908264", "title": "", "text": "$A(M(\\gamma))=\\frac{1}{2}\\int_0^T  ||(\\gamma+a) \\times \\gamma'||$"}
{"_id": "2924411", "title": "", "text": "$f(a+\\Delta x)-f(a-\\Delta x)$"}
{"_id": "1794728", "title": "", "text": "$(6,1,-6,2) = \\alpha(1,-1,-1,0) + \\beta(-1,0,1,1) +\\gamma(1,1,-1,1).$"}
{"_id": "2153343", "title": "", "text": "$s > 1/|x|$"}
{"_id": "8442279", "title": "", "text": "$\\mathcal C^n-\\mathcal C^{n+1}$"}
{"_id": "3732278", "title": "", "text": "$5^x=2^x+3^x$"}
{"_id": "3168766", "title": "", "text": "$\\exists! h':C_0' \\to C_1'$"}
{"_id": "4382649", "title": "", "text": "$\\displaystyle \\int \\limits _{\\gamma (n)}\\varphi=\\int \\limits _{\\gamma _1(n)}\\varphi +\\int \\limits_{\\gamma _2(n)}\\varphi \\tag {*}$"}
{"_id": "704716", "title": "", "text": "$q!+2,\\ldots, q!+q-1$"}
{"_id": "5976828", "title": "", "text": "$x \\neq y \\rightarrow ( x \\not\\mathrel{R} y \\vee y \\not\\mathrel{R} x )$"}
{"_id": "8788964", "title": "", "text": "$E[Y\\mid X = x_0] = 1 + x_0$"}
{"_id": "3397390", "title": "", "text": "$P_\\theta[\\Theta] = \\frac{2r(\\theta)}{R^2} = \\frac{2d}{R^2 \\tan(\\theta)}$"}
{"_id": "2972518", "title": "", "text": "$\\|AB\\|_2 = \\sqrt{\\rho((AB)^TAB)} = \\sqrt{\\rho((AB)^2)} = \\rho(AB)$"}
{"_id": "4157368", "title": "", "text": "$U \\subseteq V \\subseteq \\operatorname{Spec} A$"}
{"_id": "639274", "title": "", "text": "$\\mathrm{dist}(\\gamma_\\varepsilon(t),\\gamma[0,1]) = \\inf_{s \\in [0,1]}|\\gamma_\\varepsilon(t) - \\gamma(s)| \\leq |\\gamma_\\varepsilon(t) - \\gamma(t)| \\leq \\varepsilon \\|f\\|_\\infty$"}
{"_id": "5476286", "title": "", "text": "$P(H...TT) = 1/6$"}
{"_id": "2068871", "title": "", "text": "$g(x)=\\frac{a+bx}{c+dx}$"}
{"_id": "6658848", "title": "", "text": "$(\\forall n[  \\forall m<n, P(m)\\implies P(n)]) \\implies \\forall n P(n))$"}
{"_id": "7411327", "title": "", "text": "$\\sum_{n\\geq1}\\frac{\\log\\left(n\\right)}{n^{2}}\\leq C + \\sum_{n\\geq N}\\frac{1}{n^{2-\\alpha}}<\\infty.$"}
{"_id": "3892272", "title": "", "text": "$ \\int_{0}^{1}\\frac{\\tan^{-1}(x)}{1+x}dx$"}
{"_id": "7534041", "title": "", "text": "$k_1 = 2+(n)!, k_2 =3+(n)!,....$"}
{"_id": "1393089", "title": "", "text": "$\\sum_{k=1}^{n}\\cot\\left(\\frac{k\\pi}{n+1}\\right)=0$"}
{"_id": "1921677", "title": "", "text": "$|R|=m=rs$"}
{"_id": "4537184", "title": "", "text": "$\\mathbb R^{2n} - \\mathbb R^{n}$"}
{"_id": "550782", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=1.$"}
{"_id": "8880254", "title": "", "text": "$\\int_0^1\\frac{\\sin^{-1}x}x dx.$"}
{"_id": "8728072", "title": "", "text": "$\\mathbb{Q} [x]/(x^{4}+1) \\simeq \\mathbb{Q}(i,\\sqrt{2})$"}
{"_id": "1982543", "title": "", "text": "$1/|b|$"}
{"_id": "1840128", "title": "", "text": "$f(x^2+f(x))=f(x)^2+x$"}
{"_id": "1097361", "title": "", "text": "$aRc \\lor cRa$"}
{"_id": "8484709", "title": "", "text": "$P(m)\\to P(b)$"}
{"_id": "2922887", "title": "", "text": "$|\\omega^{\\alpha}| = |\\alpha|$"}
{"_id": "3266643", "title": "", "text": "$\\lim_{(x,y)\\to(0,0)} f(x,y)=L\\ne \\lim_{x\\to 0} \\lim_{y=0} f(x,y) \\ \\ \\ ?$"}
{"_id": "7630431", "title": "", "text": "$ 2x^2-8x,  x>2 $"}
{"_id": "4517325", "title": "", "text": "$(1,x,y,x^2, xy, y^2, x^3, x^2y,xy^2, y^3)$"}
{"_id": "928966", "title": "", "text": "$\\frac{|\\langle f,x\\rangle|}{\\|f\\|}\\le d(x,M).$"}
{"_id": "3725683", "title": "", "text": "$a,b,c,A,B,C,D\\in\\mathbb{R}$"}
{"_id": "2278298", "title": "", "text": "$a^{\\log_a(y)}=y,$"}
{"_id": "5126057", "title": "", "text": "$\\sum_{n = 0}^\\infty \\sum_{m = 0}^\\infty \\frac{x^n y^m}{n! m!}.$"}
{"_id": "1450669", "title": "", "text": "$5\\mid k^5-k$"}
{"_id": "1020960", "title": "", "text": "$\\lbrace a \\rbrace \\cup\\lbrace b \\rbrace= \\lbrace a,b\\rbrace\\not\\in \\mathcal{F}$"}
{"_id": "5986474", "title": "", "text": "$R^{n+1}_k\\subset\\mathbb{R}^{n+1}$"}
{"_id": "88942", "title": "", "text": "$\\sqrt[]{1 + {\\frac {dy} {dx}}^2}$"}
{"_id": "5477793", "title": "", "text": "$\\lim_{x\\rightarrow1^-}f'(x)\\neq\\lim_{x\\rightarrow1^+}f'(x)$"}
{"_id": "2700862", "title": "", "text": "$\\frac {b}{a} = -\\alpha - \\beta  - \\gamma\\\\ \\frac {c}{a} = \\alpha \\beta + \\beta\\gamma + \\gamma\\alpha\\\\ \\frac {d}{a} = -\\alpha \\beta\\gamma\\\\ -\\frac {b}{a}\\frac {c}{a} + 3\\frac {d}{a} = (\\alpha^2 \\beta + \\beta^2\\gamma + \\gamma^2\\alpha)+(\\alpha \\beta^2 + \\beta\\gamma^2 + \\gamma\\alpha^2)$"}
{"_id": "3026782", "title": "", "text": "$\\lim_{n\\to\\infty}\\sum_{k=1}^n\\sin\\frac{k\\pi}{n}=\\lim_{n\\to\\infty}\\frac{2n}{\\pi}=\\infty.$"}
{"_id": "2703537", "title": "", "text": "$a^{b+2a}\\mid a^{a+2b}=b^{b+2a}$"}
{"_id": "7876203", "title": "", "text": "$=\\frac {Tan^2\\theta - sin^2\\theta}{sin^2\\theta}$"}
{"_id": "7999548", "title": "", "text": "$||x+y|^t-|x|^t-|y|^t|\\leq C(|x|^{t-1}|y|+|x||y|^{t-1})$"}
{"_id": "3809236", "title": "", "text": "$1+\\frac {dy}{dx}y=f(x)$"}
{"_id": "2853398", "title": "", "text": "$0 \\begin{pmatrix} 1 & 1 & -2  \\\\ 3 & 1 &-6 \\\\ -1 & 8 & 7   \\end{pmatrix}$"}
{"_id": "5496703", "title": "", "text": "$\\displaystyle \\sum_{i = 1}^n \\sum_{j = 1}^n \\frac1{i^2+j^2}$"}
{"_id": "225376", "title": "", "text": "$ f(xy) = f(x) + f(y) $"}
{"_id": "7574332", "title": "", "text": "$m(X-a)^{m-1}$"}
{"_id": "119131", "title": "", "text": "$ \\begin{pmatrix} 0 & -1\\\\1 & 0 \\end{pmatrix}$"}
{"_id": "2459737", "title": "", "text": "$\\int_{-\\pi}^\\pi \\overline{f}(x)\\sin(nx) dx =2 \\int_{-\\pi}^\\pi \\overline{f}(x)\\sin(nx)dx=2\\int_0^\\pi f(x)\\sin(nx)d=0$"}
{"_id": "8568086", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{\\sqrt{a(4^0n)}+\\sqrt{a(4^1n)}+\\cdots+\\sqrt{a(4^{10}n)}}{\\sqrt{a(2^0n)}+\\sqrt{a(2^1n)}+\\cdots+\\sqrt{a(2^{10}n)}}$"}
{"_id": "5984518", "title": "", "text": "$ \\int^{\\infty}_{-\\infty}\\frac{1}{\\sqrt{x^{2}+1}} dx $"}
{"_id": "2004046", "title": "", "text": "$f(x,y,z)=\\frac{1}{3}e^{\\frac{-x}{3}}$"}
{"_id": "3517094", "title": "", "text": "$\\left\\lfloor\\sum_{k = 1}^{n}{\\varphi^{3k}}\\right\\rfloor$"}
{"_id": "2375998", "title": "", "text": "$\\tan \\theta = \\frac y{100}$"}
{"_id": "3462652", "title": "", "text": "$\\dfrac{1}{\\sqrt{2\\pi ep(1-p)}}$"}
{"_id": "8310030", "title": "", "text": "$\\gamma \\implies N(\\gamma) \\mid 21 \\implies N(\\gamma) = 1,3$"}
{"_id": "647786", "title": "", "text": "$ \\sin(x)=x^2-x, \\quad x\\in (1,2).  $"}
{"_id": "9355133", "title": "", "text": "$\\;\\;\\fbox{X}\\fbox{X}\\fbox{2}\\fbox{X}\\fbox{1}\\fbox{X}\\fbox{1}\\fbox{2}\\fbox{X}\\fbox{X}\\;$"}
{"_id": "8949371", "title": "", "text": "$f_X(x \\mid \\gamma) = \\begin{cases} \\frac{1}{2\\log \\gamma} \\frac{1}{x} \\mathbb 1(1/\\gamma \\le x \\le \\gamma), & \\gamma \\ge 1 \\\\ \\frac{1}{-2 \\log \\gamma} \\frac{1}{x} \\mathbb 1(\\gamma \\le x \\le 1/\\gamma), & \\gamma < 1. \\end{cases}$"}
{"_id": "1993679", "title": "", "text": "$ \\begin{align} F(x)=\\int_a^xf(t)dt, &&x\\in[a,b] \\end{align} $"}
{"_id": "3961480", "title": "", "text": "$\\bigl|f\\bigl(\\gamma(t)\\bigr)\\gamma'(t)\\bigr|\\leqslant\\sup|f|.\\bigl|\\gamma'(t)\\bigr|,$"}
{"_id": "2133005", "title": "", "text": "$\\mathbb{B}=\\left\\{ \\left\\{ a\\right\\} ,\\left\\{ b\\right\\} \\right\\} $"}
{"_id": "2630149", "title": "", "text": "$f(x) = 1/(1+x), \\\\ f(f(x)) = 1/(1 + (1/(1+x)), \\\\ f(f(f(x))) = 1/(1+1/(1 + (1/(1+x)))  , ...$"}
{"_id": "6547681", "title": "", "text": "$\\Leftrightarrow \\sin \\gamma\\frac{\\tan a +\\tan \\gamma}{1-\\tan a \\tan \\gamma}+\\cos \\gamma = \\frac{b}{c}$"}
{"_id": "3045928", "title": "", "text": "$\\mu(C_r)<r$"}
{"_id": "7923371", "title": "", "text": "$ \\begin{align} \\frac{(1+x)^{n+1}-(1+y)^{n+1}}{x-y} &=\\frac{(1+x)^{n+1}-(1+y)^{n+1}}{(1+x)-(1+y)}\\\\ &=\\sum_{k=0}^n(1+x)^k(1+y)^{n-k}\\\\ &=\\sum_{k=0}^n\\sum_{i=0}^k\\binom{k}{i}x^i\\sum_{j=0}^{n-k}\\binom{n-k}{j}y^j\\\\ &=\\sum_{i=0}^k\\sum_{j=0}^{n-k}x^iy^j\\sum_{k=0}^n\\binom{k}{i}\\binom{n-k}{j}\\tag{6} \\end{align} $"}
{"_id": "8213374", "title": "", "text": "$\\int_{-\\infty}^{x} f(x) dx=1+\\frac{-1}2xe^{\\frac {-x}2}-e^{\\frac {-x}2}=\\frac 12$"}
{"_id": "8177928", "title": "", "text": "$\\gamma_s(a) = \\gamma(a)$"}
{"_id": "6032293", "title": "", "text": "$ J_0\\supset J_1\\supset\\cdots\\supset J_n\\supset J_{n+1}\\supset\\cdots. $"}
{"_id": "4261205", "title": "", "text": "$R=P\\oplus P$"}
{"_id": "2434262", "title": "", "text": "$\\lim_{x \\to 0} f(x) = L \\ \\Longleftrightarrow \\ \\lim_{x \\to \\infty} f(1/x) = \\lim_{x \\to -\\infty} f(1/x) = L$"}
{"_id": "4120312", "title": "", "text": "$d(a,X)=\\inf|x-a|$"}
{"_id": "1794724", "title": "", "text": "$\\{(1,-1,-1,0), (-1,0,1,1), (1,1,-1,1)\\}$"}
{"_id": "5940603", "title": "", "text": "$\\int_{0}^{\\infty}\\frac{\\sin^{2n+1}(x)}{x} \\mathrm {d}x$"}
{"_id": "4952805", "title": "", "text": "$\\lim_{y\\to\\infty}\\sum_{x=1}^y\\frac{1}{2^x},$"}
{"_id": "7075989", "title": "", "text": "$\\sum_{k=1}^\\infty |\\langle x,x_k\\rangle|^2$"}
{"_id": "2742276", "title": "", "text": "$-f_3$"}
{"_id": "2766207", "title": "", "text": "$\\gamma^{-1}(\\gamma(x * y)) = \\gamma^{-1}(\\gamma(x) \\circ \\gamma(y)) \\implies (x * y) = \\gamma^{-1}(\\gamma(x) \\circ \\gamma(y))$"}
{"_id": "4990615", "title": "", "text": "$\\displaystyle\\int_0^{\\pi} f'(t)^2dt\\geq\\int_0^{\\pi}f(t)^2dt$"}
{"_id": "4480287", "title": "", "text": "$ar + bs = 1$"}
{"_id": "5269457", "title": "", "text": "$(f_\\#,f^\\#)$"}
{"_id": "3928133", "title": "", "text": "$\\dfrac 1x+\\dfrac 2y-\\dfrac3z=1 $"}
{"_id": "8896163", "title": "", "text": "$\\frac{\\left|\\langle f,v \\rangle \\right|}{\\|v\\|_{H^1_0}}=\\|u_f\\|_{L^2}$"}
{"_id": "2522320", "title": "", "text": "$ x\\delta^{(n)} = -n \\delta^{(n-1)}. $"}
{"_id": "204534", "title": "", "text": "$d(x,y) + d(y,z) \\geq d(x,z)$"}
{"_id": "6963061", "title": "", "text": "$\\frac{ax+b}{bx+c}=\\frac{b-cx}{xb-a}$"}
{"_id": "2527948", "title": "", "text": "$P(E) = 1/2$"}
{"_id": "1381469", "title": "", "text": "$(2a+b)-(a+b) = 2a+b-a-b = a$"}
{"_id": "5824490", "title": "", "text": "$|\\tilde \\gamma'|\\le |\\gamma'|$"}
{"_id": "5639870", "title": "", "text": "$M×M →M$"}
{"_id": "7893702", "title": "", "text": "$C\\subset V\\subset \\overline{V} \\subset U$"}
{"_id": "94195", "title": "", "text": "$ J_1 =\\int_0^\\infty f(x)dx $"}
{"_id": "9029941", "title": "", "text": "$V(x)=1/|x|$"}
{"_id": "7860041", "title": "", "text": "$(x_1, y_2, z_3)=(1, 1, 1)$"}
{"_id": "4667968", "title": "", "text": "$\\{(x,y)\\mid 0\\leq x \\leq 2, 1\\leq y\\leq 3\\}$"}
{"_id": "2414728", "title": "", "text": "$f(z) = z^2-z+1$"}
{"_id": "4352063", "title": "", "text": "$f(n)=\\binom n2=\\frac{n(n-1)}2$"}
{"_id": "8632831", "title": "", "text": "$\\mathbb{A} = \\mathbb{A}_{\\mathbb{C}/\\mathbb{Q}}$"}
{"_id": "7630953", "title": "", "text": "$\\mathcal{H}=V^\\perp \\oplus (V^\\perp)^\\perp = \\{0\\} \\oplus \\overline{V}=\\overline{V}$"}
{"_id": "2117966", "title": "", "text": "$\\gamma(1)=1,\\gamma(2)=\\gamma(3)=2,\\gamma(4)=\\gamma(5)=\\gamma(6)=3$"}
{"_id": "3714806", "title": "", "text": "$y=e^{\\gamma x}\\implies\\left(e^{\\gamma x}\\right)''-e^{\\gamma x}=e^{\\gamma x}(\\gamma^2-1)=0\\implies\\gamma=\\pm1$"}
{"_id": "3466077", "title": "", "text": "$[x,y] = e$"}
{"_id": "4205178", "title": "", "text": "$10y=x-4 \\Rightarrow y=\\frac{x}{10}-\\frac{4}{10}$"}
{"_id": "5848194", "title": "", "text": "$(e_1,e_2,..)$"}
{"_id": "6922990", "title": "", "text": "$\\mathbb{Z}_2[X]/(X^4+X^2+1)$"}
{"_id": "170985", "title": "", "text": "$|f(y)| \\leq \\frac{1}{2}|f(x)|$"}
{"_id": "8389681", "title": "", "text": "$\\sin\\gamma=\\sqrt{\\frac{a+b}{a+b+c}}.$"}
{"_id": "7217380", "title": "", "text": "$\\rho(2) = \\frac{\\gamma_{2}}{\\gamma_{0}} = \\frac{\\varphi_{1}\\gamma(1) + \\varphi_{2}\\gamma(0)}{\\gamma(0)} = \\frac{\\varphi_{1}\\gamma(1)}{\\gamma(0)} + \\frac{\\varphi_{2}\\gamma(0)}{\\gamma(0)}$"}
{"_id": "374051", "title": "", "text": "$ \\mu\\left(\\lim_{n\\rightarrow \\infty} \\inf E_n\\right) \\leq \\lim_{n\\rightarrow \\infty} \\inf ~\\mu\\left(E_n\\right).$"}
{"_id": "6680554", "title": "", "text": "$ \\begin{vmatrix} x&y&z\\\\ x^2&y^2&z^2\\\\ x^3&y^3&z^3\\\\ \\end{vmatrix}$"}
{"_id": "4136811", "title": "", "text": "$a+b=2a+(b-a)=2b+(a-b)$"}
{"_id": "3760420", "title": "", "text": "$O(f_3(n))$"}
{"_id": "5549465", "title": "", "text": "$Cov(\\mathbf{x},\\mathbf{u})=0$"}
{"_id": "2501648", "title": "", "text": "$2^\\gamma=2^{\\gamma^2}=(2^\\gamma)^\\gamma \\ge \\gamma^\\gamma \\ge \\kappa^\\gamma$"}
{"_id": "7733717", "title": "", "text": "$U \\subseteq P(V)$"}
{"_id": "8113986", "title": "", "text": "$\\displaystyle \\frac{(1+x)^{n+1}-(1+x)^r}{x}$"}
{"_id": "224573", "title": "", "text": "$ 1+x+x^2+x^3 + \\cdots = \\frac{1}{1-x} $"}
{"_id": "8402424", "title": "", "text": "$(\\mathbb{Z}[x] / (x^2-x))_U\\simeq\\mathbb Z_U\\times\\mathbb Z_U$"}
{"_id": "7298413", "title": "", "text": "$\\lim_{x\\to0^-}f'(x)=\\lim_{x\\to0^+}f'(x)=1$"}
{"_id": "8045437", "title": "", "text": "$ f(c)=\\lim_{k\\to \\infty}f(c_k)=m=\\inf f, $"}
{"_id": "87546", "title": "", "text": "$p_1p_2$"}
{"_id": "2513869", "title": "", "text": "$\\lim_{x\\to c^+} = L = \\lim_{x \\to c^-}$"}
{"_id": "1499665", "title": "", "text": "$(n!)+2, (n!)+3, ..., (n!)+n.$"}
{"_id": "6712065", "title": "", "text": "$(a+b)^r\\leqslant 2^{r-1}(a^r+b^r)\\mbox{ if }r\\gt1.$"}
{"_id": "440613", "title": "", "text": "$|ab| = 2.$"}
{"_id": "2741214", "title": "", "text": "$\\lim_{n\\rightarrow \\infty} \\frac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n}}{n^{3/2}}=\\lim_{n\\rightarrow \\infty} \\frac{\\sqrt{n}}{n^{3/2}-(n-1)^{\\frac{3}{2}}}=\\lim_{n\\rightarrow \\infty} \\frac{\\sqrt{n}\\left(n^{3/2}+(n-1)^{\\frac{3}{2}}\\right)}{n^3-(n-1)^3}=$"}
{"_id": "3542488", "title": "", "text": "$( (N-m+Nr)(1+r)^x -m((1+r)^x-1)/r )r $"}
{"_id": "8554163", "title": "", "text": "$\\gcd(a,a^2+2)=\\gcd(a,2)$"}
{"_id": "1745922", "title": "", "text": "$(a+b,a^2-ab+b^2) = 1$"}
{"_id": "2229389", "title": "", "text": "$ \\frac{-a_2}{x} + \\frac{b_2}{y} = 1 $"}
{"_id": "3540151", "title": "", "text": "$ \\displaystyle s(x)=\\int_a^x f(t) \\, dt $"}
{"_id": "1233795", "title": "", "text": "$\\alpha^+\\cdot\\beta+\\alpha$"}
{"_id": "5242786", "title": "", "text": "$B = \\{B_1,B_2,\\cdots,B_n\\}$"}
{"_id": "1265773", "title": "", "text": "$\\left[         \\begin{array}{ccc|c}          4 & -4 & 2 & 20\\\\         -3 & -1 & 3 &-18\\\\         1 & -5 & 5 &k\\\\         \\end{array} \\right] $"}
{"_id": "625833", "title": "", "text": "$    \\det(A) = (x-2)^{n-1} (x + 2(n-1)). $"}
{"_id": "8655283", "title": "", "text": "$x_1 + x_2 + \\cdots + x_n < m$"}
{"_id": "1225639", "title": "", "text": "$\\left( 1+ \\frac{x}{p} \\right) ^{p}  < \\left( 1+ \\frac{x}{q} \\right)  ^{q} $"}
{"_id": "8149848", "title": "", "text": "$P(X_1 = 1 | X_0 = 0) = 1$"}
{"_id": "9073166", "title": "", "text": "$f_X(x)=\\frac{2x}{\\theta^2}$"}
{"_id": "9219142", "title": "", "text": "$\\frac{10^4}{x} - \\frac{1}{x}$"}
{"_id": "1209610", "title": "", "text": "$\\tilde{A} = \\cases{a \\text{ if } \\hat A \\leq a\\\\\\hat A \\text{ if } a< \\hat A <b\\\\{b \\text{ if } \\hat A \\geq b}}{}$"}
{"_id": "798283", "title": "", "text": "$x = \\pm 1, y = 0, z = 0, \\lambda = 0$"}
{"_id": "8147016", "title": "", "text": "$\\overline{\\mathbb{R}}^X$"}
{"_id": "5566181", "title": "", "text": "$\\ f(n)=n^3-n+1$"}
{"_id": "6031160", "title": "", "text": "$F(x)=\\int_{a}^{x}f(t)dt,x\\in[a,b]$"}
{"_id": "4796813", "title": "", "text": "$(x-a,y-b) = (x-a)$"}
{"_id": "1673767", "title": "", "text": "$\\lim_{x \\to c^{+}}f'(x) = A, \\lim_{x \\to c^{-}}f'(x) = B$"}
{"_id": "153381", "title": "", "text": "$\\sqrt z=\\sqrt r e^{i\\theta/2}$"}
{"_id": "843459", "title": "", "text": "$d(y,a) \\leq d(x,a) + d(y,x)$"}
{"_id": "4199940", "title": "", "text": "$\\left\\|A\\right\\|_{2} \\le \\sqrt{\\rho(A^{*}A)}$"}
{"_id": "9200106", "title": "", "text": "$ x = f(f^{-1}(x)) \\Rightarrow \\\\ 1 = f'(f^{-1}(x)) (f^{-1})'(x) \\Rightarrow \\\\ (f^{-1})'(x) = 1 / f'(f^{-1}(x)) $"}
{"_id": "9346933", "title": "", "text": "$\\displaystyle J_{1}=\\int_{0}^{\\infty}f(x) dx$"}
{"_id": "2778473", "title": "", "text": "$\\sum_{-\\infty}^\\infty |J_n(x)|^2$"}
{"_id": "4324798", "title": "", "text": "$|(\\omega(z))^{\\varepsilon}f(z)|\\leq M$"}
{"_id": "1284055", "title": "", "text": "$f(x) = f(a+b) = f(a) + f(b) = f(a) + 0 = f(a)$"}
{"_id": "2391659", "title": "", "text": "$[E_C, E_C] \\subseteq E_C$"}
{"_id": "8025679", "title": "", "text": "$\\sum_{k=1}^{\\infty}\\sum_{n=1}^{\\infty}\\frac{1}{k^{2}+n^{1/\\gamma}}$"}
{"_id": "8590570", "title": "", "text": "$3^x + 10^x = 4^x + 9^x.$"}
{"_id": "550950", "title": "", "text": "$\\{\\langle{x , y}\\rangle: 0\\leq x\\wedge 0\\leq y\\wedge x+y\\leq z\\}$"}
{"_id": "6162968", "title": "", "text": "$\\cos^2 x=\\frac{1+\\cos 2x}{2}\\Rightarrow2\\sum_{k=1}^N\\cos^2\\frac{k\\pi}{N+1} =N+\\sum_{k=1}^N\\cos\\frac{2k\\pi}{N+1}=N-1,$"}
{"_id": "8734295", "title": "", "text": "$F = P  \\oplus T$"}
{"_id": "5227829", "title": "", "text": "$E(p) =  \\frac{1-a}{a (n-2)}\\left(1-(1-a)^{n-2}\\right)$"}
{"_id": "447295", "title": "", "text": "$f(x) + f(y) = f(x+y)$"}
{"_id": "4904185", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} > \\frac{1}{10}$"}
{"_id": "8986976", "title": "", "text": "$ I_n=\\int_{0}^{\\infty} \\frac{\\sin(\\frac{x}{n})}{x(1+x^2)}dx\\ ? $"}
{"_id": "1673789", "title": "", "text": "$\\displaystyle\\lim_{x\\to c^+}f'(x)=A\\not= B= \\displaystyle\\lim_{x\\to c^-}f'(x)$"}
{"_id": "13872", "title": "", "text": "$f_n=\\sum a_j\\chi_{I_j}$"}
{"_id": "843851", "title": "", "text": "$\\dfrac{1}{2\\pi} \\cdot \\dfrac{1-r^2}{1-2r \\cos(\\varphi) + r^2 } $"}
{"_id": "5202875", "title": "", "text": "$\\int \\frac{\\sin (\\pi x)}{|x|^a + 1} dx$"}
{"_id": "9238305", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\sum_{k=1}^{\\infty}\\frac{1}{n^2k^2(n+k)^2}$"}
{"_id": "2146732", "title": "", "text": "$f(a + b) \\ne f(a) + f(b)$"}
{"_id": "6599323", "title": "", "text": "$\\begin{align}(1 - \\cos A)^2 + \\sin^2 A \\over \\sin A(1 - \\cos A)  & = \\dfrac{1 - 2 \\cos A + \\cos^2 A + \\sin^2 A}{\\sin A(1 - \\cos A)} \\\\ \\\\ & = {1 \\color{blue}{\\bf + \\cos^2 A} -2\\cos A + 1 \\color{blue}{\\bf - \\cos^2A} \\over \\sin A(1-\\cos A)} \\\\ \\\\ & = \\dfrac{2 - 2\\cos A}{\\sin A(1 - \\cos A)}\\\\ \\\\ & = \\dfrac{2\\color{red}{\\bf (1-\\cos A)}}{\\sin A\\color{red}{\\bf (1 - \\cos A)}}\\\\ \\\\ & = \\frac{2}{\\sin A} \\\\ \\\\ & = 2 \\csc A \\end{align}$"}
{"_id": "6015325", "title": "", "text": "$r:\\begin{cases}x=t \\\\ y=2t+1\\\\z=t-2 \\end {cases}$"}
{"_id": "8121536", "title": "", "text": "$\\nabla(x,y,z) = (1,1,1)$"}
{"_id": "8512000", "title": "", "text": "$\\int_{0}^{\\infty}f^2(x)dx< \\infty$"}
{"_id": "1637823", "title": "", "text": "$\\frac{x_1+x_2+\\cdots +x_n}{n}\\in X$"}
{"_id": "8872297", "title": "", "text": "$10-40$"}
{"_id": "4002158", "title": "", "text": "$ \\left( \\matrix{   x \\cr    m \\cr}  \\right) = \\left\\{ {\\matrix{    {{{x^{\\,\\underline {\\,m\\,} } } \\over {m!}} = {1 \\over {m!}}\\prod\\limits_{0\\, \\le \\,k\\, \\le \\,m - 1} {\\left( {x - k} \\right)} } & {0 \\le m \\in Z}  \\cr     0 & {m < 0\\; \\vee \\;m \\notin Z}  \\cr   } } \\right. $"}
{"_id": "6278028", "title": "", "text": "$|ab|=4$"}
{"_id": "2249216", "title": "", "text": "$\\begin{cases} t\\equiv y \\\\ u\\equiv x\\end{cases}$"}
{"_id": "1449435", "title": "", "text": "$\\sum_{j=1}^n c_j X_{t_j}$"}
{"_id": "6033832", "title": "", "text": "$f(x)[f'(x-1)+f'(x+1)]-f'(x)[f(x-1)+f(x+1)]=0$"}
{"_id": "2445936", "title": "", "text": "$\\hat f (n) = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(x)e^{-inx}\\,dx = \\frac{1}{\\pi}\\int_{0}^\\pi f(x)\\cos (nx)\\,dx = 0$"}
{"_id": "5159840", "title": "", "text": "$\\gamma(s)\\gamma(t)=\\gamma(s+t)=\\gamma(t)\\gamma(s)$"}
{"_id": "3308202", "title": "", "text": "$A = (\\{a\\}, \\{(a, a)\\})$"}
{"_id": "8105504", "title": "", "text": "$ xy=xz^*z=(xz^*)z=z(xz^*)=(zx)z^*=z^*zx=yx. $"}
{"_id": "80447", "title": "", "text": "$\\lbrace$"}
{"_id": "69629", "title": "", "text": "$ LHS = \\frac{n}{n-1} e^{\\frac{-2}{n}}(\\frac{1 - e^{\\frac{2k}{n}}}{1 - e^{\\frac{-2}{n}}}) $"}
{"_id": "5174322", "title": "", "text": "$\\displaystyle\\frac1{2^{n-1}}$"}
{"_id": "7009642", "title": "", "text": "$ p\\mid r_1r_2 $"}
{"_id": "900533", "title": "", "text": "$G=\\{A_1,A_2,A_3,\\dots,A_n\\}$"}
{"_id": "3318746", "title": "", "text": "$P(X_n \\leq x) \\to P(X \\leq x)$"}
{"_id": "2005863", "title": "", "text": "$\\sum_p \\frac1{p^s} = \\log \\frac1{s - 1} + O(1) \\tag2$"}
{"_id": "2134597", "title": "", "text": "$\\pi^+$"}
{"_id": "3331596", "title": "", "text": "$N=\\{x_1,\\dotsc,x_n,\\dotsc\\}$"}
{"_id": "2306263", "title": "", "text": "$f(A_1) = A_1$"}
{"_id": "2255791", "title": "", "text": "$\\bar  X=\\frac{X_1+X_2+...+X_n}{n}$"}
{"_id": "6970820", "title": "", "text": "$x_n^*=e_n$"}
{"_id": "6357144", "title": "", "text": "$f(z) = e^{i \\theta} \\prod_{j=1}^{k} {{z-a_j} \\over {1- \\overline a_jz}}$"}
{"_id": "2669263", "title": "", "text": "$  x^n+1 \\geq (1+\\frac1n \\frac{x^n}{x^n+1})^n $"}
{"_id": "5048969", "title": "", "text": "$\\frac{x!}{2^x}={\\sqrt {2\\pi x}}\\left({\\frac {x}{2e}}\\right)^{x}.$"}
{"_id": "7106096", "title": "", "text": "$\\hat{f}(k)={n\\choose k}.$"}
{"_id": "5631146", "title": "", "text": "$A:= \\frac{x_1+\\ldots+x_n}{n}$"}
{"_id": "6628775", "title": "", "text": "$\\mathbb R[X]/((X^2+1)^2)$"}
{"_id": "5997743", "title": "", "text": "$f(x)^2+f(y)^2=f(x+y)(f(y)+f(x) + a)$"}
{"_id": "7810254", "title": "", "text": "$\\begin{cases}{x^2 +y^2 −z(x+y)=2\\\\ y^2 +z^2 −x(y+z)=4\\\\ z^2 +x^2 −y(z+x)=8}\\end{cases}$"}
{"_id": "3789561", "title": "", "text": "$Cov(X,Y) = 0.0007$"}
{"_id": "92686", "title": "", "text": "$ a_0=\\color{lightgrey}{\\binom 02}+\\binom 20=0+1=1\\\\ a_1=\\color{lightgrey}{\\binom 12}+\\binom 21=0+2=2\\\\ a_2=\\binom 22+\\binom 22=1+1=2\\\\ a_3=\\binom 32+\\color{lightgrey}{\\binom 23}=3+0=3\\\\ a_4=\\binom 42+\\color{lightgrey}{\\binom 24}=6+0=6\\\\ \\vdots$"}
{"_id": "2287576", "title": "", "text": "$x*(y*y)=y*(y*y)=y*y=y$"}
{"_id": "8792139", "title": "", "text": "$\\begin{pmatrix}c&-s\\\\s&c\\end{pmatrix}$"}
{"_id": "6095081", "title": "", "text": "$ O_X(F) \\to A_X^{0,0,r}(F) \\to A_X^{0,1,r-1}(F)\\to \\dots \\to A_X^{0,n,r-n}(F) \\to 0 $"}
{"_id": "1749270", "title": "", "text": "$\\mathrm ds = \\sqrt{r^2 + \\left(\\frac{\\mathrm dr}{\\mathrm dθ}\\right)^2}$"}
{"_id": "5256180", "title": "", "text": "$|f(x_0)|\\le x_0|f(y_0)|$"}
{"_id": "1463504", "title": "", "text": "$ \\zeta(-1) =  \\sum \\limits_{n=1}^{\\infty} \\frac{1}{n^{-1}}= 1+2+3+4 + ... = -\\frac{1}{12}$"}
{"_id": "9025784", "title": "", "text": "$ \\lim\\limits_{x\\to \\infty  } \\lim\\limits_{n\\to \\infty} \\sum\\limits_{r=1}^n \\frac{r^2(\\sin x)^x}{n^3}$"}
{"_id": "668456", "title": "", "text": "$t=0 \\implies \\frac x 2=k\\pi\\implies x=2k\\pi$"}
{"_id": "316973", "title": "", "text": "$\\log_a b$"}
{"_id": "3183852", "title": "", "text": "$b_0\\in V\\subseteq \\overline{V}\\subseteq p(U)$"}
{"_id": "5050583", "title": "", "text": "$\\mathbb{R}^{n+1} \\subset \\mathbb{R}^{n+2}$"}
{"_id": "9316384", "title": "", "text": "$\\sum_{k=2}^{n}\\frac{\\cos\\frac{\\pi}{n+1}}{\\cos\\frac{\\pi}{n+1}-\\cos\\frac{k\\pi}{n+1}}$"}
{"_id": "6840308", "title": "", "text": "$A\\subset U \\subset \\overline U\\subset O$"}
{"_id": "8764271", "title": "", "text": "$y=\\frac{c+d}{b-a}$"}
{"_id": "1091205", "title": "", "text": "$\\tan \\theta=\\frac12$"}
{"_id": "629292", "title": "", "text": "$f(x,\\gamma)=\\frac{2\\gamma+1}{2\\gamma}x^{\\frac{1}{2\\gamma}},\\quad 0\\leq x \\leq 1,\\gamma>0,$"}
{"_id": "769918", "title": "", "text": "$a_n = \\frac{1+\\sqrt2+\\sqrt 3+ \\cdots +\\sqrt n}{n\\sqrt n}$"}
{"_id": "5084901", "title": "", "text": "$\\cos(k\\pi)=\\frac{2k\\sin(\\pi k)}{2\\pi k}+\\sum_{n=1}^{\\infty}\\frac{2k\\sin(\\pi k)(-1)^n}{k^2-n^2}\\cos(n\\pi)$"}
{"_id": "5237776", "title": "", "text": "$P(X\\leq x) = P(X_1\\leq x,X_2\\leq x,\\ldots,X_n\\leq x).$"}
{"_id": "8153567", "title": "", "text": "$40 - x_1 - x_2 -x_4$"}
{"_id": "6016366", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}\\inf J(u_n)\\geq J(u)$"}
{"_id": "5279667", "title": "", "text": "$T(n)=n^2-n+1$"}
{"_id": "7904773", "title": "", "text": "$\\lim_{n\\to\\infty}f_{t_n}(\\omega)$"}
{"_id": "6065247", "title": "", "text": "$f(n)=n^2-n+41$"}
{"_id": "4987736", "title": "", "text": "$2(a,b)=(a+b,a-b)$"}
{"_id": "55145", "title": "", "text": "$x^{3}, x^{2}, x, x^{0}$"}
{"_id": "5301610", "title": "", "text": "$ E|X|^p\\le E|X+y-y|^p\\le E(|X+y|+|y|)^p\\le C_p(E|X+y|^p+|y|^p)<\\infty. $"}
{"_id": "3516267", "title": "", "text": "$b\\cdot c = \\cfrac {2A} {\\sin Â}\\\\(b+c)^2 = b^2 +c^2 +2bc = k^2 \\implies b^2+c^2=k^2-2bc\\\\ a^2=k^2-2bc(1+\\cosÂ)=k^2- 4A \\cfrac {1+\\cos Â} {\\sin Â}=k^2- 4A \\cdot\\cot \\cfrac Â2 \\implies \\tan \\cfrac Â2=\\cfrac {4A} {k^2-a^2}$"}
{"_id": "7549965", "title": "", "text": "$ = \\frac{P(A=t|B=t,F=t) P(B=t) P(F=t) +P(A=t|B=f,F=t)P(B=f)P(F=t)} {P(F=t)}$"}
{"_id": "6736361", "title": "", "text": "$\\forall \\epsilon>0,\\ \\exists\\delta>0,\\ \\forall|x-y|<\\delta,\\ |f(x)-f(y)|<\\epsilon$"}
{"_id": "113217", "title": "", "text": "$t_n = n(n+1)/2$"}
{"_id": "3864482", "title": "", "text": "$\\det\\begin{pmatrix} A & B \\\\ B & A\\end{pmatrix}=\\det(A-B)\\det(A+B)$"}
{"_id": "7076078", "title": "", "text": "$I_n=\\int_{0}^{\\pi/2}\\frac{\\sin \\left ( ax \\right )}{\\sin x+\\cos x}\\, {\\rm d}x$"}
{"_id": "6768", "title": "", "text": "$\\sum_k |\\langle x, e_k \\rangle |^2 = ||x||^2$"}
{"_id": "1563907", "title": "", "text": "$\\frac{\\frac{(n+2)^2}{(n+1)^2}}{\\frac{((n+1)+2)^2}{((n+1)+1)^2}}<1$"}
{"_id": "6824952", "title": "", "text": "$\\alpha(x)=\\int_a^xf(t)dt$"}
{"_id": "6256420", "title": "", "text": "$f(z^2)=(f(z))^2$"}
{"_id": "3493340", "title": "", "text": "$\\mathscr{D}=\\{\\{a\\}:a\\in A\\}$"}
{"_id": "7713905", "title": "", "text": "$\\lim_{n\\to\\infty}{\\sqrt{5+\\sqrt{5+...+\\sqrt 5}}}$"}
{"_id": "1470177", "title": "", "text": "$\\begin{align*} &(1)\\quad \\left\\lceil \\frac{\\left\\lceil \\frac{x}{a} \\right\\rceil} {b}\\right\\rceil = \\left\\lceil {\\frac{x}{ab}}\\right\\rceil\\\\\\\\ &(2)\\quad\\left\\lfloor \\frac{\\left\\lfloor \\frac{x}{a} \\right\\rfloor} {b}\\right\\rfloor = \\left\\lfloor {\\frac{x}{ab}}\\right\\rfloor\\\\\\\\ &(3)\\quad \\left\\lceil {\\frac{a}{b}} \\right\\rceil \\le \\frac{a + b - 1}{b}\\\\\\\\ &(4)\\quad \\left\\lfloor {\\frac{a}{b}} \\right\\rfloor \\le \\frac{a - b + 1}{b} \\end{align*}$"}
{"_id": "6510388", "title": "", "text": "$\\begin{cases} x=s\\\\ y=s\\\\ z=0 \\end{cases}$"}
{"_id": "2787186", "title": "", "text": "$P\\circ T$"}
{"_id": "559146", "title": "", "text": "$E(n) = (\\frac{n}{2})^{(n-1-\\lfloor\\frac{n}{2}\\rfloor)}$"}
{"_id": "9322556", "title": "", "text": "$\\{a+b,a+2b,\\dotsb,a+kb\\}$"}
{"_id": "74586", "title": "", "text": "$q^n-q^{n-1}$"}
{"_id": "8824321", "title": "", "text": "$X = \\bigcup \\mathcal{A}$"}
{"_id": "7792572", "title": "", "text": "$h+x = (-x_n)+(x_n) = (-x_n+x_n) = (0) = 0$"}
{"_id": "7592177", "title": "", "text": "$Cov(X,Y) = Cov(X,Z) = 0$"}
{"_id": "6725454", "title": "", "text": "$2^\\gamma\\le\\alpha^\\gamma\\le(2^\\gamma)^\\gamma=2^{\\gamma^2}=2^\\gamma$"}
{"_id": "1677128", "title": "", "text": "$\\ell^+$"}
{"_id": "7347670", "title": "", "text": "$|OX| = ab$"}
{"_id": "278259", "title": "", "text": "$f(0) = f(\\pi) = 0$"}
{"_id": "9017206", "title": "", "text": "$\\vartheta(u,-i\\tau)=\\frac 1{\\sqrt{-i\\tau}}e^{i\\pi\\tau u^2}\\vartheta(\\tau u,\\frac i\\tau)$"}
{"_id": "2800031", "title": "", "text": "$\\gamma+\\bar\\gamma=\\gamma\\bar\\gamma=1, {\\gamma}^2=\\gamma-1, |\\gamma|=1$"}
{"_id": "3523729", "title": "", "text": "$\\mathbb{E}|X_1|^r < \\infty$"}
{"_id": "2188338", "title": "", "text": "$U\\subseteq B\\subseteq \\overline{A}\\subseteq\\overline{V}$"}
{"_id": "3172522", "title": "", "text": "${ax^k\\over (1+r)^{4k-3}} = {a(1+r)^3 ({x\\over(1+r)^4})^k}$"}
{"_id": "2716735", "title": "", "text": "$\\{e_1,e_2,e_3,e_4,e_5\\}$"}
{"_id": "1986372", "title": "", "text": "$\\Bbb R^n-\\Bbb R^m$"}
{"_id": "2866097", "title": "", "text": "$\\left(  1+x/n\\right)^n\\leq e^x.$"}
{"_id": "4798566", "title": "", "text": "$ \\int_0^\\pi f (x) \\sin x=\\int_0^\\pi f (x) \\cos x=0$"}
{"_id": "5105804", "title": "", "text": "$l_2: ax+by=d$"}
{"_id": "432014", "title": "", "text": "$[x,y]=x$"}
{"_id": "1929169", "title": "", "text": "$ \\sum_{n=-\\infty}^\\infty |f(n)|^2 = \\sum_{n=-\\infty}^\\infty e^{-n^2} =\\vartheta_3(0,e^{-1}) \\approx 1.772637 $"}
{"_id": "6688579", "title": "", "text": "$a=\\frac{l-bc}{b+c}=\\frac{m}{bc}$"}
{"_id": "763743", "title": "", "text": "$\\hat{\\gamma}=\\{ \\gamma(t):0\\leq t\\leq 1 \\} .$"}
{"_id": "4207228", "title": "", "text": "$U_1 \\subset U_2 \\subset ... \\subset U_i \\subset U_{i+1} \\subset ...$"}
{"_id": "4637915", "title": "", "text": "$I=\\int_0^1\\int_0^1 \\frac{\\sin^{-1}(xy)}{xy} \\,\\mathrm dx\\,\\mathrm dy.$"}
{"_id": "1769031", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^{n}\\left(1+\\frac{1}{12n}+O\\left(\\frac{1}{n^{2}}\\right)\\right).$"}
{"_id": "4217674", "title": "", "text": "$\\sum_{k=1}^{n-1}{\\sin{\\frac{2\\pi k}{n}}} = 0$"}
{"_id": "132411", "title": "", "text": "$E|X-a|^{r} \\leq 2^{r} (E|X|^{r}+|a|^{r})$"}
{"_id": "2948340", "title": "", "text": "$\\sum\\limits_{k=1}^n r^k = \\frac {r - r^{n+1}}{1-r} \\implies \\sum\\limits_{k=1}^{n+1} r^k = \\frac {r-r^{n+1}}{1-r}$"}
{"_id": "9116000", "title": "", "text": "$2\\,{\\frac {\\sin \\left( x \\right) \\cos \\left( x \\right)  \\left( \\cos  \\left( x \\right) -\\sin \\left( x \\right)  \\right) }{ \\left( 2\\,  \\left( \\cos \\left( x \\right)  \\right) ^{2}-1 \\right)  \\left( \\cos  \\left( x \\right) +\\sin \\left( x \\right)  \\right) }} $"}
{"_id": "4199943", "title": "", "text": "$\\left\\|A\\right\\|_{2} \\ge \\sqrt{\\rho(A^{*}A)}$"}
{"_id": "856354", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac1{2^n}\\sum_{k=1}^n \\frac1{\\sqrt k}\\binom nk$"}
{"_id": "3652508", "title": "", "text": "$\\mu=\\frac{x_1+x_2+\\cdots+x_n}{n}$"}
{"_id": "8659624", "title": "", "text": "$ \\mathbb{E}|X|^r\\le\\bigl(\\mathbb{E}|X|^s\\bigr)^{r/s} $"}
{"_id": "4170635", "title": "", "text": "$ T[\\gamma] = \\lambda\\gamma \\quad \\Rightarrow \\quad (\\gamma,\\beta)\\gamma = \\lambda\\gamma. $"}
{"_id": "1075816", "title": "", "text": "$\\begin{cases} x=t \\\\ y=6-t \\\\ z= 1\\end{cases}$"}
{"_id": "213936", "title": "", "text": "$ \\begin{align} \\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\frac1{mn(m+n)^2} &=\\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\left(\\color{#C00}{\\frac1{nm^2(n+m)}}-\\color{#090}{\\frac1{m^2(n+m)^2}}\\right)\\tag{3a}\\\\ &=\\color{#C00}{\\frac{\\pi^4}{72}}-\\color{#090}{\\frac{\\pi^4}{120}}\\tag{3b}\\\\[3pt] &=\\frac{\\pi^4}{180}\\tag{3c} \\end{align} $"}
{"_id": "1780523", "title": "", "text": "$c_n = \\frac{1}{3}(-1)^n + \\frac{2}{3}2^n$"}
{"_id": "8053334", "title": "", "text": "$ \\gamma ~:=~\\det\\gamma_{\\mu\\nu}, \\qquad  \\gamma^{-1} ~=~\\det\\gamma^{\\mu\\nu}.\\tag{E} $"}
{"_id": "3604999", "title": "", "text": "$(-1)^n{2\\pi p^n\\over1-p^2}$"}
{"_id": "500996", "title": "", "text": "$a=1/|x|$"}
{"_id": "6450579", "title": "", "text": "$\\lim_{x\\to c}f(x)=L\\implies\\lim_{x\\to c^+}f(x)=L$"}
{"_id": "2293067", "title": "", "text": "$\\frac{1}{\\pi + \\cos(z)}  = \\frac{2\\gamma}{1+\\gamma^2 + 2\\gamma\\cos(z)} = \\frac{2\\gamma}{(\\gamma + e^{iz})(\\gamma + e^{-iz})} = \\frac{\\gamma}{\\gamma^2-1}\\left[\\frac{\\gamma - e^{iz}}{\\gamma + e^{iz}} + \\frac{\\gamma-e^{-iz}}{\\gamma + e^{-iz}}\\right] $"}
{"_id": "8867863", "title": "", "text": "$f'(x)=(x+1)^{m-1} (x-)^{n-1} (m(x-1)+n(x+1))$"}
{"_id": "6098594", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1 & 3 & -1 & 4\\\\4 & -1 & 2 & 8\\\\2 & -7 & 4 & -3\\end{array}\\right].$"}
{"_id": "2681055", "title": "", "text": "$  \\lim_{x\\to 0^+} f'(x) \\ne \\lim_{x\\to0^-} f'(x) $"}
{"_id": "4519819", "title": "", "text": "$F \\subseteq V \\subseteq \\overline{V} \\subseteq U$"}
{"_id": "4482782", "title": "", "text": "$\\log_a {\\frac{b}{c}} = \\log_a b -\\log_a c$"}
{"_id": "5480291", "title": "", "text": "$ \\left\\lfloor\\frac{\\left\\lfloor\\frac ab\\right\\rfloor}c\\right\\rfloor=  \\left\\lfloor\\frac{qc+q'}c\\right\\rfloor=q$"}
{"_id": "4263812", "title": "", "text": "$|S \\times T| = mn$"}
{"_id": "6182199", "title": "", "text": "$ \\left[ \\begin{array}{ccc|c}   1&2&-1&-3\\\\   3&5&k&-4\\\\   9&k+13&6&9\\\\ \\end{array} \\right] $"}
{"_id": "4820176", "title": "", "text": "$I_n=\\int^{\\frac{\\pi}{2}}_0 \\frac{\\sin(2nx)}{\\sin(x)}dx$"}
{"_id": "1363585", "title": "", "text": "$\\begin{pmatrix} A & B\\\\ C & D\\\\ \\end{pmatrix}$"}
{"_id": "877824", "title": "", "text": "$Cov(X,Y)=Cov(Y,X)$"}
{"_id": "3063478", "title": "", "text": "$\\{\\,\\sqrt1, \\sqrt2, \\ldots, \\sqrt n\\,\\}$"}
{"_id": "7328115", "title": "", "text": "$ \\sum_{n=1}^{\\infty}\\frac{\\eta(2n+1)}{2^{2n+1}}  \\quad = \\frac{1}{2}\\int_{0}^{\\infty}\\frac{\\cosh(x/2)-1}{e^{x}+1}\\,dx = \\frac{1}{2}\\left(1-\\log2\\right) \\tag{1} \\\\[6mm] $"}
{"_id": "617396", "title": "", "text": "$f(y)=y^2+y-2$"}
{"_id": "9166626", "title": "", "text": "$(p_1-1)(p_2-1)\\mid p_1p_2$"}
{"_id": "3686997", "title": "", "text": "$f_3(n) = n \\log_2 n$"}
{"_id": "4454192", "title": "", "text": "$T^{0,1}_pM\\times T^{0,1}_pM$"}
{"_id": "4213131", "title": "", "text": "$A_n=\\frac{n(n-1)}4$"}
{"_id": "2651669", "title": "", "text": "$\\forall \\epsilon>0, \\exists\\delta>0 : |x-r|<\\delta \\Rightarrow  |f(x) - f(r)| < \\epsilon$"}
{"_id": "5934603", "title": "", "text": "$ \\lim\\limits_{n\\rightarrow\\infty} \\left\\lfloor  \\frac{-1}{n}\\right\\rfloor =?$"}
{"_id": "4209684", "title": "", "text": "$T_n=\\frac{n(n+1)}{2}.$"}
{"_id": "1443456", "title": "", "text": "$=2+\\frac1{1+\\frac1{\\frac{26}9}}=2+\\frac1{1+\\frac1{2+\\frac89}}=2+\\frac1{1+\\frac1{2+\\frac1{\\frac98}}}$"}
{"_id": "2932133", "title": "", "text": "$P[X_0=x]=1$"}
{"_id": "2967328", "title": "", "text": "$\\mbox{Tr}_{F/K}(a) = \\sigma_1(a) + \\ldots + \\sigma_n(a)$"}
{"_id": "8949884", "title": "", "text": "$[\\mathbb{F}_5[X]/(X^2−2)]^*$"}
{"_id": "5036482", "title": "", "text": "$c_1, c_0$"}
{"_id": "54876", "title": "", "text": "$(1-r^{n+1})/(1-r)$"}
{"_id": "5376134", "title": "", "text": "$\\sin(\\arccos(p))$"}
{"_id": "1029986", "title": "", "text": "$\\beta(a,b)=\\gamma(a+b)-\\gamma a -\\gamma b, a,b\\in A$"}
{"_id": "5960469", "title": "", "text": "$|f_n(z)|\\leq M|f_{n-1}(z)|$"}
{"_id": "4334861", "title": "", "text": "$ \\frac{Z^{m/2}-Z^{-m/2}}{Z^{1/2}-Z^{-1/2}} = \\frac{(Z^{m/2}-Z^{-m/2})(Z^{1/2}+Z^{-1/2})}{(Z^{1/2}-Z^{-1/2})(Z^{1/2}+Z^{-1/2})} = \\frac{Z^{(m+1)/2}+Z^{(m-1)/2}-Z^{(1-m)/2}-Z^{-1-m)/2}}{Z-Z^{-1}} = U_{(m-1)/2}\\left(\\frac{X}2\\right) + U_{(m-3)/2}\\left(\\frac{X}2\\right),$"}
{"_id": "3484114", "title": "", "text": "$\\int_0^\\infty f(x)\\,dx=1.$"}
{"_id": "7841276", "title": "", "text": "$V\\oplus N=F$"}
{"_id": "1662122", "title": "", "text": "$\\frac 1r[(1+r)^n -1]$"}
{"_id": "8264911", "title": "", "text": "$\\frac{(1+t)^{n+1}-1}{n+1}$"}
{"_id": "368032", "title": "", "text": "$ \\eqalign{ \\alpha^0&=&&=1,\\\\ \\alpha^1&=&&=\\alpha,\\\\ \\alpha^2&=&&=\\alpha^2,\\\\ \\alpha^3&=&&=1+\\alpha,\\\\ \\alpha^4&=&\\alpha\\cdot\\alpha^3=\\alpha(1+\\alpha)&=\\alpha+\\alpha^2,\\\\ \\alpha^5&=&\\alpha\\cdot\\alpha^4=\\alpha(\\alpha+\\alpha^2)=\\alpha^2+\\alpha^3=\\alpha^2+(1+\\alpha)&=1+\\alpha+\\alpha^2,\\\\ \\alpha^6&=&\\alpha\\cdot\\alpha^5=\\alpha(1+\\alpha+\\alpha^2)=\\alpha+\\alpha^2+\\alpha^3= \\alpha+\\alpha^2+(1+\\alpha)&=1+\\alpha^2,\\\\ \\alpha^7&=&\\alpha\\cdot\\alpha^6=\\alpha(1+\\alpha^2)=\\alpha+\\alpha^3=\\alpha+(1+\\alpha)&=1. }$"}
{"_id": "2997443", "title": "", "text": "$\\langle a, \\alpha a + \\gamma c\\rangle = \\alpha\\langle a, a\\rangle + \\gamma\\langle a, c\\rangle=0$"}
{"_id": "2988585", "title": "", "text": "$\\frac{x}{2} + \\frac{x}{y} - \\frac{3}{2} = \\frac{10}{y}$"}
{"_id": "710862", "title": "", "text": "$\\gamma = (\\gamma^1, \\gamma^2)$"}
{"_id": "855264", "title": "", "text": "$\\frac{1}{2} \\int_{0}^{\\pi} \\frac{1}{1+\\sin^2 x} \\, dx = \\frac{1}{2} \\int_{0}^{\\pi} r^2 \\, d\\theta = \\frac{\\pi}{2\\sqrt{2}} \\implies \\int_{0}^{\\pi} \\frac{1}{1+\\sin^2 x} \\, dx= \\frac{\\pi}{\\sqrt{2}}$"}
{"_id": "1567487", "title": "", "text": "$\\tau=\\dfrac{\\det(\\dot\\gamma,\\ddot\\gamma,\\dddot\\gamma)}{\\|\\dot\\gamma\\times\\ddot\\gamma\\|^2}$"}
{"_id": "3061989", "title": "", "text": "$\\lim_{N \\to \\infty} \\sum_{k=1}^N \\frac{1}{N+k} = \\ln(2)$"}
{"_id": "3066379", "title": "", "text": "$ax+by=2$"}
{"_id": "3685409", "title": "", "text": "$ B = \\{x_1,x_2,x_3,\\dots\\}. $"}
{"_id": "364963", "title": "", "text": "$\\gcd(a+b, a-b) = d$"}
{"_id": "5243456", "title": "", "text": "$a^{d}a^{s-d}\\equiv a^{s}\\equiv -1\\pmod{m}.$"}
{"_id": "7932496", "title": "", "text": "$\\mu( \\lim \\sup A_{n}) \\geq \\lim \\inf _{n} \\mu (A_{n})$"}
{"_id": "5082725", "title": "", "text": "$(A\\otimes B)^{\\bot\\bot}\\vdash A^{\\bot\\bot}\\otimes B^{\\bot\\bot}$"}
{"_id": "9259125", "title": "", "text": "$x-m\\in M^{\\perp\\perp}$"}
{"_id": "1022541", "title": "", "text": "$ f(n) = n^2 - n + 2 $"}
{"_id": "1805016", "title": "", "text": "$M = P\\oplus Q$"}
{"_id": "3686703", "title": "", "text": "$f(x^2 + yf(z)) = xf(x) + zf(y)$"}
{"_id": "1745921", "title": "", "text": "$(a+b,a-b) = 1$"}
{"_id": "532619", "title": "", "text": "$ \\lim_{y\\to 0} \\frac{(1+y)^{P+1}-(1+y)(P+1)+P}{y^2} = \\lim_{y\\to 0} \\frac{(1+y)^{P+1}-1-(P+1)y}{y^2} $"}
{"_id": "360757", "title": "", "text": "$\\lim_{n \\to \\infty} \\sum_{n=0}^\\infty\\frac{\\left \\lfloor \\frac{n}{2} \\right \\rfloor}{n^{2}}$"}
{"_id": "5781625", "title": "", "text": "$\\frac{223}{38}=5+\\frac{33}{38}=5+\\frac1{\\frac{38}{33}}=5+\\frac1{1+\\frac5{33}} =5+\\frac1{1+\\frac1{\\frac{33}5}}=5+\\frac1{1+\\frac1{6+\\frac 35}} =5+\\frac1{1+\\frac1{6+\\frac1{\\frac53}}}=5+\\frac1{1+\\frac1{6+\\frac1{1+\\frac23}}} =5+\\frac1{1+\\frac1{6+\\frac1{1+\\frac1{\\frac32}}}}=5+\\frac1{1+\\frac1{6+\\frac1{1+\\frac1{1+\\frac12}}}}$"}
{"_id": "3514055", "title": "", "text": "$\\frac{1}{2h}\\int_a^b\\mu(A\\cap(x-h,x+h))\\,\\text{d}x\\le \\mu(A).$"}
{"_id": "3828435", "title": "", "text": "$(1/\\sqrt1 + \\sqrt2) + (1/\\sqrt2 + \\sqrt3) +\\cdots + (1/\\sqrt{99} + \\sqrt{100})$"}
{"_id": "7862864", "title": "", "text": "$A=\\dfrac{x_1+x_2+\\dots+x_n}n$"}
{"_id": "6480974", "title": "", "text": "$\\frac{x_1+x_2+...+x_N}{N}$"}
{"_id": "816645", "title": "", "text": "$\\mathcal A[x/a]$"}
{"_id": "5944113", "title": "", "text": "$\\|A\\|=\\sup\\limits_{\\|x\\|=1}\\|Ax\\|=\\sup\\limits_{\\|x\\|=1}\\|Bx\\|=\\|B\\|=\\|A^{-1}\\|$"}
{"_id": "3753682", "title": "", "text": "$f(x) + f(1/x) = f(1/(x+1)) + f(x/(x+1))$"}
{"_id": "369201", "title": "", "text": "$\\left(\\frac{\\pm p\\cos\\,\\varphi}{1\\mp\\varepsilon\\cos\\varphi},\\frac{\\pm p\\sin\\,\\varphi}{1\\mp\\varepsilon\\cos\\,\\varphi}\\right)$"}
{"_id": "4278257", "title": "", "text": "$d=\\inf_{m\\in M}\\|x-m\\|$"}
{"_id": "3370180", "title": "", "text": "$(x,y,z,w)=(1,1,1,w)$"}
{"_id": "1674181", "title": "", "text": "$f_{Z}(z)=\\frac{1}{\\pi}\\frac{\\sqrt{1-\\rho^{2}}}{(1-\\rho^{2})+(z-\\rho)^{2}}$"}
{"_id": "9115231", "title": "", "text": "$\\displaystyle F(x)=\\int_a^x f(t)^2dt$"}
{"_id": "8718941", "title": "", "text": "$f(x) = (x-a)^{n-1}h(x)$"}
{"_id": "1343361", "title": "", "text": "$X \\subset Y \\subset \\overline{X}\\subset \\overline{A}$"}
{"_id": "813743", "title": "", "text": "$g(z)=(z-a)f(z)-(z-a)\\bar{a}$"}
{"_id": "3782967", "title": "", "text": "$E\\tau=|ab|$"}
{"_id": "305461", "title": "", "text": "$|G|=pqr$"}
{"_id": "8132851", "title": "", "text": "$F(x) = \\int_a^x f(t)\\ dt = x^3 - 2x^2 - x - a$"}
{"_id": "524112", "title": "", "text": "$x^3-1= (x-1)(x^2+x+1)=(x-1)[(x-1)^2-3x]$"}
{"_id": "6519681", "title": "", "text": "$\\omega = \\alpha’ = \\alpha$"}
{"_id": "696432", "title": "", "text": "$X_{k}= +/- 1$"}
{"_id": "8054105", "title": "", "text": "$\\lim_{n\\to\\infty}\\inf a_n = -\\lim_{n\\to\\infty}\\sup b_n,$"}
{"_id": "323213", "title": "", "text": "$\\gamma\\le\\gamma_1\\le\\gamma_2\\le...$"}
{"_id": "4698201", "title": "", "text": "$a_1=\\frac{n(n+1)}{2}$"}
{"_id": "367313", "title": "", "text": "$\\mathcal{O}(n,\\mathbb R)$"}
{"_id": "6593358", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\left(1\\over2^{n-1}\\right) $"}
{"_id": "834416", "title": "", "text": "$\\lim_{x \\to \\ c+}f(x)=l$"}
{"_id": "5566567", "title": "", "text": "$\\mathbb{C} := \\mathbb{R}[x]/(x^2+1)$"}
{"_id": "1017010", "title": "", "text": "$   0 = f(0) = f \\left( f(y)^2 - f(y) f(y) \\right) = -f(y) f(y - f(y)) $"}
{"_id": "5242514", "title": "", "text": "$f(f(x)+xf(y))=xf(y+1)$"}
{"_id": "300727", "title": "", "text": "$I_n J_n = J_n I_n$"}
{"_id": "2194174", "title": "", "text": "$\\forall n (n\\geq m\\implies (P(n)\\implies P(n+1))$"}
{"_id": "489756", "title": "", "text": "$|f(z_{0})|\\geq |f(z)|$"}
{"_id": "3080775", "title": "", "text": "$ \\mu(A)= \\lim_{n\\rightarrow \\infty} \\mu(A_n).$"}
{"_id": "782979", "title": "", "text": "$n![z^n](e^z)^s = s^n$"}
{"_id": "5636520", "title": "", "text": "$s(\\gamma)h(s(\\gamma)^{-1}ks(\\gamma))-h(k)=kf(s(\\gamma))-f(s(\\gamma))$"}
{"_id": "934710", "title": "", "text": "$2^{\\alpha} = \\alpha^+$"}
{"_id": "394406", "title": "", "text": "$I_j=(a_j)$"}
{"_id": "8641", "title": "", "text": "$T_{x}(X)$"}
{"_id": "261884", "title": "", "text": "$\\mu(A)=\\lim_{n\\to\\infty} \\mu_n(A\\cap (-n,n))$"}
{"_id": "2789728", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\mu(A_n)\\geq\\mu(A_m)=1$"}
{"_id": "2309279", "title": "", "text": "$\\frac{dx}{dt} = \\mu x - x^3, \\mu \\in \\Bbb R, x \\in \\mathbb R$"}
{"_id": "8333128", "title": "", "text": "$\\lim_{n\\to \\infty} (1 - \\frac{1}{4})(1 - \\frac{1}{9})(1 - \\frac{1}{16}) \\cdots (1 - \\frac{1}{(n+1)^2})$"}
{"_id": "6990020", "title": "", "text": "$1+2+3+4+\\ldots = -1/2$"}
{"_id": "784884", "title": "", "text": "$(x+a)(x-a)$"}
{"_id": "1158123", "title": "", "text": "$\\{A+B,A-B\\}$"}
{"_id": "8239205", "title": "", "text": "$\\widetilde{\\gamma}\\gamma' = \\gamma$"}
{"_id": "4326866", "title": "", "text": "$x^3-x-x-1$"}
{"_id": "5899251", "title": "", "text": "$ \\lambda(v^*v)=v^*(\\lambda v)=v^*Av=v^*A^*v=(Av)^*v=(\\lambda v)^*v=\\lambda^*v^*v $"}
{"_id": "2848684", "title": "", "text": "$\\left(1+\\frac xn\\right)^n\\leqslant e^x$"}
{"_id": "8426775", "title": "", "text": "$ \\lim_{n \\to \\infty} \\sum\\limits_{k=1}^n \\frac{1}{k 2^k}? $"}
{"_id": "671274", "title": "", "text": "$\\int\\limits_0^x f(t)dt=0$"}
{"_id": "2649267", "title": "", "text": "$ \\begin{cases}  3t=0 \\\\ 1-t=0\\\\ 2-2t=s \\end{cases} $"}
{"_id": "3673223", "title": "", "text": "$s=\\sum _{m=1}^{\\infty } \\sum _{n=1}^{\\infty } \\frac{1}{m n \\left(m^2+n^2\\right)}\\tag{1}$"}
{"_id": "8223564", "title": "", "text": "$S_i = (a_i, a_i+d_i, a_i+2d_i, ...)$"}
{"_id": "367865", "title": "", "text": "$|AB|=mn$"}
{"_id": "1225527", "title": "", "text": "$f(-\\pi) = f(\\pi) = -f(-\\pi) = 0$"}
{"_id": "8637149", "title": "", "text": "$y(n) = \\binom{n+2}{2} u(n)$"}
{"_id": "4987737", "title": "", "text": "$D = (a+b, a-b)$"}
{"_id": "1755431", "title": "", "text": "$\\int\\limits_0^\\infty \\frac{x^{s-1}}{e^x-1}dx=\\zeta(s)\\Gamma(s)$"}
{"_id": "5254221", "title": "", "text": "$ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$"}
{"_id": "7957792", "title": "", "text": "$\\|x_n-y_n\\|<\\varepsilon$"}
{"_id": "2337764", "title": "", "text": "$\\mathbb{R} / \\{0\\}$"}
{"_id": "5014430", "title": "", "text": "$k(f(x)+ g(x)) = kf(x) + kg(x)$"}
{"_id": "7914462", "title": "", "text": "$\\|A\\|_2=\\sqrt{\\rho(A^TA)}\\leq\\sqrt{\\|A^TA\\|_{\\infty}}$"}
{"_id": "8514184", "title": "", "text": "$f(x) = 500 (1 - e^{-x/500})$"}
{"_id": "2068922", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}n\\ln(a)=\\ln(a)\\lim_{n\\rightarrow\\infty}n=-\\infty$"}
{"_id": "761936", "title": "", "text": "$\\log_2(0.7-5^x)<\\log_5(0.7-2^x)$"}
{"_id": "5216668", "title": "", "text": "$(x^2 + 5) = (x^2 + 5)$"}
{"_id": "1453788", "title": "", "text": "$\\binom{n}{2}=O(n^2)$"}
{"_id": "1893190", "title": "", "text": "$\\begin{align} \\int\\frac{x}{(x^2+2)^2}dx&=\\frac12\\int  \\frac{1}{(u+2)^2}\\,du\\\\\\\\ &=-\\frac12 \\frac{1}{u+2}\\\\\\\\ &=-\\frac12 \\frac{1}{x^2+2} \\tag 2 \\end{align}$"}
{"_id": "8897375", "title": "", "text": "$(a+b+1,a-b)$"}
{"_id": "7710111", "title": "", "text": "$ \\int \\frac{x^3}{\\left(x^2+1\\right)^2}\\sin^2 x dx $"}
{"_id": "1049734", "title": "", "text": "$f(x+y)(f(x)+f(y))=f(xy)$"}
{"_id": "811374", "title": "", "text": "$T \\oplus F = F \\oplus T = F$"}
{"_id": "60078", "title": "", "text": "$A= \\left[ {\\matrix{\n    x_1 & x_2 \\cr \n    x_3 & x_4 \\cr \n \n  } } \\right]$"}
{"_id": "9136917", "title": "", "text": "$|ab|=$"}
{"_id": "6228751", "title": "", "text": "$P(\\{i\\})=1/6$"}
{"_id": "4968469", "title": "", "text": "$S=\\sum_{k=1}^{n} 4^k\\left(3-\\frac{1}{k+2}\\right) +\\sum_{k=1}^{n}\\frac{(2^{2k+1}-2^{2k})}{(k+1)(k+2)}$"}
{"_id": "1730508", "title": "", "text": "$\\int_0^\\infty x \\frac{M}{c} e^{(\\frac{-x}{c})} (1-e^{\\frac{-x}{c}})^{M-1} \\,dx = c \\sum_{k=1}^M \\frac{1}{k}.$"}
{"_id": "5368489", "title": "", "text": "$\\begin{vmatrix}1&a&a^2\\\\1&b&b^2\\\\1&c&c^2\\end{vmatrix}$"}
{"_id": "5941890", "title": "", "text": "$  \\mathcal{M} := \\left\\{f \\in L^2([0,\\pi]): \\int_0^\\pi f(x)\\cos x dx =  \\int_0^\\pi f(x)\\sin x dx = 1\\right\\}. $"}
{"_id": "442624", "title": "", "text": "$\\zeta(x) = \\sum_{1}^{\\infty}\\frac{1}{r^x}= \\frac{1}{\\Gamma(x)}\\sum_{1}^{\\infty} \\int_{0}^{\\infty}u^{x-1}e^{ru}du$"}
{"_id": "1231492", "title": "", "text": "$\\tan\\theta =\\frac{x}{\\sqrt{1-x^2}}$"}
{"_id": "692859", "title": "", "text": "$\\mathbb H^{+}$"}
{"_id": "2527591", "title": "", "text": "$\\sqrt[!]{x}=\\sqrt{x\\sqrt{(x-1)\\sqrt{(x-2)\\sqrt{\\ddots\\sqrt{2\\sqrt1}}}}}$"}
{"_id": "8000739", "title": "", "text": "$\\frac{1}{24} (n^6 + 6 n^3 + 3 n^4 + 8 n^2 + 6 n^3) = \\frac{1}{24} (n^6 + 3 n^4 + 12 n^3 + 8 n^2) \\\\= \\frac{1}{24} n^6 + \\frac{1}{8} n^4 + \\frac{1}{2} n^3 + \\frac{1}{3} n^2.$"}
{"_id": "6204257", "title": "", "text": "$(t_n^0,x_n^0)\\to(t^0,x^0)$"}
{"_id": "1936821", "title": "", "text": "$x^2 - x, x - 1, x$"}
{"_id": "8280734", "title": "", "text": "$k!+2,k!+3, \\ldots, 2k!-1$"}
{"_id": "3901692", "title": "", "text": "$|p_1p_2|$"}
{"_id": "596161", "title": "", "text": "$f(x,p)= \\frac{1-p^2}{2\\pi(1+p^2)-2p\\cos(x)}$"}
{"_id": "389992", "title": "", "text": "$\\omega^{+}$"}
{"_id": "5528283", "title": "", "text": "$\\int_{-\\infty}^xf_n(t)\\phi(t)dt\\to \\int_{-\\infty}^xf(t)\\phi(t)dt$"}
{"_id": "5540347", "title": "", "text": "$f(1+x) = f(-x-1) = f(1-x) $"}
{"_id": "7283607", "title": "", "text": "$e^{i3\\pi/2(-(1-p/2))}$"}
{"_id": "1476677", "title": "", "text": "$\\log_a(b) + \\log_a(c) = \\log_a(b \\cdot c)$"}
{"_id": "4176408", "title": "", "text": "$\\{a+y, a+2y, a+3y,\\dots\\}$"}
{"_id": "5714722", "title": "", "text": "$[S_n,S_n]\\subseteq A_n$"}
{"_id": "702124", "title": "", "text": "$P(X_n\\leq x) \\to P(X\\leq x)$"}
{"_id": "785882", "title": "", "text": "$x<y,x=y,x>y</y,x=y,x>$"}
{"_id": "1245440", "title": "", "text": "$\\exists\\;\\gamma_1,\\gamma_2\\in \\Bbb R $"}
{"_id": "6554766", "title": "", "text": "$ \\eqalign{   & \\left( \\matrix{   x \\cr    f(x) \\cr}  \\right)\\quad \\left| \\matrix{   \\;{\\rm real }x \\hfill \\cr    \\; - 1 < {\\rm real }f(x) \\hfill \\cr}  \\right.\\;\\quad  = {1 \\over {2\\,\\pi }}\\int_{\\, - \\,\\pi \\;}^{\\,\\pi \\,} {e^{\\, - \\,i\\,\\,t\\,f(x)} \\left( {1 + e^{\\,\\,i\\,\\,t} } \\right)^{\\,x} \\;d\\,t}  =   \\cr    &  = {1 \\over {2\\,\\pi }}\\int_{\\, - \\,\\pi \\;}^{\\,\\pi \\,} {\\sum\\limits_{0\\, \\le \\,j} {\\left( \\matrix{   x \\cr    j \\cr}  \\right)e^{\\,\\,i\\,\\,t\\,\\left( {j - f(x)} \\right)} } \\;d\\,t}  = {1 \\over {2\\,\\pi }}\\sum\\limits_{0\\, \\le \\,j} {{{x^{\\,\\underline {\\,j\\,} } } \\over {j!}}\\int_{\\, - \\,\\pi \\;}^{\\,\\pi \\,} {e^{\\,\\,i\\,\\,t\\,\\left( {j - f(x)} \\right)} \\;d\\,t} }  \\cr}  $"}
{"_id": "2862067", "title": "", "text": "$rb^{r-1} \\leq r(a+b)^{r-1}$"}
{"_id": "5144059", "title": "", "text": "$A\\cong A_1\\times\\ldots\\times A_n?$"}
{"_id": "256263", "title": "", "text": "$\\lim_{n\\to\\infty} (1+\\frac{x}{n}) ^n = e^x $"}
{"_id": "8724372", "title": "", "text": "$\\mathbb{P}(\\sup_{t \\geq 0} M_t > x \\mid \\mathcal{F}_0)= 1 \\wedge \\frac{M_0}{x}$"}
{"_id": "4964496", "title": "", "text": "$Y_n=\\frac{X_1+X_2+... +X_n}{n}$"}
{"_id": "6471438", "title": "", "text": "$\\sum_{i=2}^\\infty\\log(1-x^i)=-\\sum_{n=1}^\\infty\\sum_{i=2}^\\infty\\frac{x^{in}}n=-\\sum_{n=1}^\\infty\\frac{x^n}n\\color{red}{\\frac{x^n}{1-x^n}}\\geqslant-\\sum_{n=1}^\\infty\\frac{x^n}n=\\log(1-x)$"}
{"_id": "7513163", "title": "", "text": "$\\chi(K_n + I) = (-1)^n x^{n-1} (x - n).$"}
{"_id": "7581097", "title": "", "text": "$\\int_0^\\infty f_-(x)\\,dx$"}
{"_id": "5171367", "title": "", "text": "$y = \\ln\\frac{1+x}{1-x}$"}
{"_id": "8821327", "title": "", "text": "$\\mathbb C_R$"}
{"_id": "4138862", "title": "", "text": "$f(n) = n^{2} + 2n + 1$"}
{"_id": "4996265", "title": "", "text": "$x∘(y∘z) = x⊗y⊗z + x⊗z⊗y + y⊗z⊗x + z⊗y⊗x$"}
{"_id": "7167058", "title": "", "text": "$ f_n(a) = \\frac n2 $"}
{"_id": "2763897", "title": "", "text": "$\\forall \\epsilon>0,\\exists \\delta>0:|y-x|<\\delta\\implies |f(x)-f(y)|<\\epsilon$"}
{"_id": "5676576", "title": "", "text": "$1-e^\\frac{-x}{10}$"}
{"_id": "9080903", "title": "", "text": "$rn={n\\choose 2}$"}
{"_id": "6447428", "title": "", "text": "$\\begin{cases}x_1+x_2=k\\\\1\\leq x_1\\\\ 1\\leq x_2\\end{cases}$"}
{"_id": "1598209", "title": "", "text": "$G_1\\subseteq G_2\\subseteq G_3\\subseteq \\dots\\subseteq G_n \\subseteq G_{n+1}\\subseteq \\cdots$"}
{"_id": "852721", "title": "", "text": "$\\int \\frac{x}{(x^2+4)^6}dx$"}
{"_id": "4903965", "title": "", "text": "$ |z| = \\sqrt[n]{|w|}[\\cos (\\frac{2κπ +φ}{n}) + i\\sin (\\frac{2κπ +φ}{n})] = \\sqrt[n]{|w|}e^{\\frac{2κπ +φ}{n}i} , κ \\in \\mathbb Z, κ=0,1,...,n-1$"}
{"_id": "4096655", "title": "", "text": "$z^{\\frac{1}{n}}=\\sqrt[n]{r}\\left(e^{i\\left(\\frac{\\theta}{n}+\\frac{2k\\pi}{n} \\right)}\\right)\\, $"}
{"_id": "3165268", "title": "", "text": "$\\sum_{n\\ge 1}x_n=\\sum_{n\\ge 1}\\left\\lfloor\\frac{x}{2^n}\\right\\rfloor+\\beta(x)\\;.$"}
{"_id": "9218816", "title": "", "text": "$(x-1)^{n-1}(x+n-1).$"}
{"_id": "5816514", "title": "", "text": "$f_3(n)=n \\sqrt{\\log{n}}$"}
{"_id": "4743072", "title": "", "text": "$3 \\mid x^3 - x$"}
{"_id": "3093481", "title": "", "text": "$\\sum_{n\\geq1}\\frac1{n^2}\\sum_{dq=n}\\mu(d) =\\sum_{n\\geq1}\\frac1{n^2}\\sum_{d|n}\\mu(d) =\\sum_{n\\geq1}\\frac1{n^2}I(n)$"}
{"_id": "5057485", "title": "", "text": "$\\mathbb{E}\\sup_{t\\leq T}|X_t|^p\\leq Ce^{CT}(1+\\mathbb{E}|X_0|^p)$"}
{"_id": "1689735", "title": "", "text": "$\\frac{1/2^{k-1}}{1/2^{k-1}}\\;.$"}
{"_id": "7666108", "title": "", "text": "$\\zeta(s) = \\dfrac{1}{\\Gamma(s)} \\int_{0}^{\\infty} \\dfrac{x ^ {s-1}}{{\\mathrm e} ^ x - 1}\\mathrm{d}x,$"}
{"_id": "4669781", "title": "", "text": "$n(\\gamma,a)=-n(-\\gamma,a)$"}
{"_id": "1739274", "title": "", "text": "$\\gcd(a+b, a-b) = 2$"}
{"_id": "339304", "title": "", "text": "$= (X/4+X/6) + (X/4+X/6) + X/6$"}
{"_id": "4601132", "title": "", "text": "$j= \\lfloor N/I \\rfloor \\mod J;\\quad k= \\lfloor\\lfloor N/I \\rfloor/J\\rfloor=\\lfloor N/(IJ)\\rfloor$"}
{"_id": "2451290", "title": "", "text": "$3^x+4^x=7^x$"}
{"_id": "1174009", "title": "", "text": "$f(P)=\\int_0^{2\\pi}\\frac1{\\pi}P\\,dP\\,d\\phi=2P\\,dP$"}
{"_id": "3043129", "title": "", "text": "$\\int \\frac{1}{(x^2-x+1)^2}\\, dx$"}
{"_id": "5225439", "title": "", "text": "$x+\\mathcal a = x' + \\mathcal a$"}
{"_id": "2276409", "title": "", "text": "$f'(x)=n[(x+1)^{n-1}-2x^{n-1}]$"}
{"_id": "2214120", "title": "", "text": "$\\left\\{|\\mathbb{N}|,|\\mathbb{R}|,|\\mathbb{R}^\\mathbb{R}|,\\left|(\\mathbb{R}^\\mathbb{R})^{(\\mathbb{R}^\\mathbb{R})}\\right|,\\dots\\right\\}$"}
{"_id": "709152", "title": "", "text": "$\\displaystyle\\lim_{n\\to\\infty} \\frac{1}{\\sqrt{n}}\\sum_{k=1}^{n}\\frac{1}{\\sqrt{n}+\\sqrt{k}}$"}
{"_id": "1398854", "title": "", "text": "$\\lim_{n\\to\\infty}2\\sin(x_n)=0$"}
{"_id": "2504481", "title": "", "text": "$ \\big( A[x]/(x^2 - d) \\big)^\\times \\simeq A^\\times \\times A^\\times  $"}
{"_id": "8119217", "title": "", "text": "$f(f(x))=xf(x)+a$"}
{"_id": "6453827", "title": "", "text": "$X = (x,y), A = (a,b)$"}
{"_id": "1449481", "title": "", "text": "$h(x) = \\int_a^x{f(t)dt}$"}
{"_id": "699024", "title": "", "text": "$\\alpha_+$"}
{"_id": "7373717", "title": "", "text": "$\\vartheta(\\tau)$"}
{"_id": "971825", "title": "", "text": "$\\mathbb{R}[x]/(x^7+1)$"}
{"_id": "2415247", "title": "", "text": "$\\forall x[P(x) \\vee Q(x)]$"}
{"_id": "242999", "title": "", "text": "$A_1 A_2 A_3$"}
{"_id": "1312503", "title": "", "text": "$\\lim_ {n\\to \\infty} \\mu^*(E_n) < \\mu^*(E) $"}
{"_id": "8861250", "title": "", "text": "$ f(0)=0=f(\\pi)  \\} $"}
{"_id": "46772", "title": "", "text": "$\\int_{0}^\\pi l \\, dφ=\\int_0^\\pi r\\cdot \\sin(φ) \\, dφ=r\\int_0^\\pi \\sin(φ) \\, dφ = r (-\\cos(\\pi) +\\cos(0))=2r$"}
{"_id": "4311249", "title": "", "text": "$B^{\\gamma,\\gamma}_{n-\\mu} + B^{\\gamma,\\gamma+1}_{n-\\mu} + \\ ... + \\, B^{\\gamma,\\mu}_{n-\\mu}$"}
{"_id": "6609826", "title": "", "text": "$\\mathbb{R}^{n+m}\\cong\\mathbb{R}^n\\times\\mathbb{R}^n$"}
{"_id": "1127278", "title": "", "text": "$f(x)=\\dfrac{a^x}{b^{x^c}} $"}
{"_id": "8174922", "title": "", "text": "$(x+y)(x+y) = (x+y)x + (x+y)y=$"}
{"_id": "7648519", "title": "", "text": "$|y-b|+|x-a|=b-y+a-x=(b-x)+(a-y)=(b-x)-(y-a)\\\\=(b-x)-|y-a|\\le|x-b|+|y-a|$"}
{"_id": "3714507", "title": "", "text": "$N! = \\sqrt{2 \\pi N} \\cdot (\\frac{N}{e})^{N}$"}
{"_id": "9254428", "title": "", "text": "$[\\alpha, \\beta, \\gamma, \\delta][\\beta, \\gamma] - [\\alpha, \\beta, \\gamma] \\cdot [\\beta, \\gamma, \\delta] = 1$"}
{"_id": "8214658", "title": "", "text": "$\\gcd(a, -a) = |a|$"}
{"_id": "6420531", "title": "", "text": "$\\eqalign{\\|\\gamma(b)-\\gamma(a)\\|^2&=\\bigl(\\gamma(b)-\\gamma(a)\\bigr)\\cdot\\bigl(\\gamma(b)-\\gamma(a)\\bigr)=\\bigl(\\gamma(b)-\\gamma(a)\\bigr)\\cdot\\int_a^b\\gamma'(t)\\>dt\\cr&=\\int_a^b \\bigl(\\gamma(b)-\\gamma(a)\\bigr)\\cdot\\gamma'(t)\\>dt\\leq\\int_a^b \\bigl|\\gamma(b)-\\gamma(a)\\bigr|\\>\\bigl|\\gamma'(t)\\bigr|\\>dt\\cr&= \\bigl|\\gamma(b)-\\gamma(a)\\bigr|\\int_a^b\\bigl|\\gamma'(t)\\bigr|\\>dt\\ .\\cr}$"}
{"_id": "3534012", "title": "", "text": "$f_R(r) = (n-1)e^{-r}(1 - e^{-r})^{n-2}$"}
{"_id": "8901588", "title": "", "text": "$\\mathcal{A} = \\mathcal{B}(X)$"}
{"_id": "518489", "title": "", "text": "$a, a+b, a +2b, $"}
{"_id": "4403077", "title": "", "text": "$   \\begin{pmatrix}A&B\\\\ B&A\\end{pmatrix} $"}
{"_id": "7688506", "title": "", "text": "$E_i = \\{X \\le 1/i\\}$"}
{"_id": "6353876", "title": "", "text": "$g:\\mathbb{R}^{n+1} \\to \\mathbb{R}^n$"}
{"_id": "7033262", "title": "", "text": "$a_n = z^n + w^n + i \\pi(z^n - w^n) $"}
{"_id": "6444960", "title": "", "text": "$L_1 :\\begin{cases} x = 1 + 2t\\\\ y = 2-2t\\\\ z = 3+4t\\end{cases}$"}
{"_id": "9197070", "title": "", "text": "$[x,y]=z^{p^{n-1}}$"}
{"_id": "3643931", "title": "", "text": "$\\varphi''(x)=-\\frac{d}{dx}\\int_0^1t f(t)\\sin(xt)dt =-\\int_0^1\\frac{\\partial}{\\partial x}t f(t)\\sin(xt)dt =-\\int_0^1t^2 f(t)\\cos(xt)dt$"}
{"_id": "1320410", "title": "", "text": "$9^x-6^x=4^{x+1/2}$"}
{"_id": "7321113", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}X_n(\\omega)=X(\\omega)<\\infty$"}
{"_id": "6267552", "title": "", "text": "$s_n =\\frac {a_1 +a_2 +...+a_n} {n}$"}
{"_id": "686141", "title": "", "text": "$\\Omega^{+}$"}
{"_id": "7394041", "title": "", "text": "$\\forall\\varepsilon\\gt0,\\exists \\delta\\gt0,\\quad\\text{such that}\\ \\space |~x~| \\lt \\delta \\Rightarrow ~|f(x)| \\lt \\varepsilon \\tag{1}$"}
{"_id": "2692535", "title": "", "text": "$|f(y)| \\leq \\frac{1}{2} |f(c)|$"}
{"_id": "5335410", "title": "", "text": "$ I=\\pi^{-s/2}\\Gamma(s/2)\\sum_{n=1}^\\infty\\frac{1}{n^s}=\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) $"}
{"_id": "580158", "title": "", "text": "$ax+by=n$"}
{"_id": "6955901", "title": "", "text": "$B=\\{d,e,..,H,G\\}$"}
{"_id": "4621685", "title": "", "text": "$[x,y]=x\\circ y-y\\circ x$"}
{"_id": "4694978", "title": "", "text": "$\\gamma(t) = (\\gamma_x(t),\\gamma_y(t)), \\; \\gamma(0) = (x_0, y_0), \\; \\gamma(1) = (x, y); \\tag{10}$"}
{"_id": "5065787", "title": "", "text": "$\\int_0^\\infty  {\\frac{x^{2n - 1}}{{\\left( x^2 + 1 \\right)}^{n + 3}}\\,dx}\\quad\\quad n\\geq 1\\;,\\; n\\in\\mathbb{Z}$"}
{"_id": "2285018", "title": "", "text": "$\\color{blue}{\\sqrt z= \\sqrt r e^{i\\varphi/2}}$"}
{"_id": "4445708", "title": "", "text": "$(a+b)f=af+bf$"}
{"_id": "8940267", "title": "", "text": "$\\lim_{x\\to0}\\left(\\frac{\\sin^{-1}(x)}{x} \\right) = 1 $"}
{"_id": "222426", "title": "", "text": "$\\beta+\\gamma\\geq \\alpha,\\ \\gamma+\\alpha\\geq \\beta,\\ \\wedge\\ \\alpha+\\beta\\geq \\gamma.$"}
{"_id": "2234277", "title": "", "text": "$\\mathcal{A}+\\mathcal{X}=\\mathcal{B}$"}
{"_id": "9267762", "title": "", "text": "$ \\mathbf{1.}\\ (\\forall m<n \\ P(m) \\implies P(n))$"}
{"_id": "177268", "title": "", "text": "$\\frac{1}{(1+m)^{1/n}} + \\frac{1}{(1+n)^{1/m}} \\ge \\frac{1}{1+\\frac{m}{n}} + \\frac{1}{1+\\frac{n}{m}} = \\frac{n}{n+m} + \\frac{m}{n+m} = 1$"}
{"_id": "4632969", "title": "", "text": "$A_{l_1} \\times ... \\times A_{l_m} \\subseteq A_1 \\times ... \\times A_n = \\textrm{Soc}(S_1)$"}
{"_id": "9134715", "title": "", "text": "$\\lim \\limits_{n \\to \\infty}\\frac{2}{n}\\left\\lfloor \\frac{n}{3}\\right\\rfloor$"}
{"_id": "351689", "title": "", "text": "${\\vartheta _n} = \\int_{ - \\infty }^{ + \\infty } {\\frac{1}{{{{\\left( {1 + {x^2}} \\right)}^n}}}} \\frac{{dx}}{{1 + {x^2}}}$"}
{"_id": "6790286", "title": "", "text": "$ \\dfrac{2}{x} + \\dfrac {3}{y} = 1 $"}
{"_id": "4498087", "title": "", "text": "$\\widetilde{\\gamma}(\\sum a_s s)=\\sum a_s\\gamma(s)\\quad\\text{and}\\quad\\overline{c}(s)=c(1s).$"}
{"_id": "2068597", "title": "", "text": "$f(n) = [(1^{n-1}) + (-1^{n-1})] u(n-1)$"}
{"_id": "1583102", "title": "", "text": "$P[X_n=0]=1-1/n^2$"}
{"_id": "8048598", "title": "", "text": "$T_x(U) = T_x(X)$"}
{"_id": "438462", "title": "", "text": "$\n d(x, \\{a\\}^{\\perp})=\\frac{\\left|\\langle x,a\\rangle \\right|}{\\left\\|a\\right\\|}.\n $"}
{"_id": "791490", "title": "", "text": "$f_2(n)=2^n$"}
{"_id": "514029", "title": "", "text": "$\\lim_{x\\to 0^{-}} f(x) = \\lim_{x\\to 0^{+}} f(x) = \\lim_{x\\to 0} f(x)$"}
{"_id": "1290882", "title": "", "text": "$\\mathbb E|X|\\mathbb E|Y|=1$"}
{"_id": "8008813", "title": "", "text": "$\\det M=(k-\\lambda)^{n-1}\\left(\\lambda(n-1)+k\\right)$"}
{"_id": "2260037", "title": "", "text": "$[\\gamma]\\le[\\gamma']\\iff\\gamma\\le\\gamma',$"}
{"_id": "6128534", "title": "", "text": "$[\\frac{1}{9}, \\frac{1}{9}+\\frac{1}{27}] \\cup [\\frac{2}{9}-\\frac{1}{27}, \\frac{2}{9}] \\cup [\\frac{1}{3}, \\frac{1}{3}+\\frac{1}{27}]\\cup[\\frac{2}{3}-\\frac{1}{27}, \\frac{2}{3}]\\cup [\\frac{7}{9},\\frac{7}{9}+\\frac{1}{27}]\\cup [\\frac{8}{9}-\\frac{1}{27},\\frac{8}{9}].$"}
{"_id": "3152544", "title": "", "text": "$f(a \\ast b) = f(a) + f(b)$"}
{"_id": "4525355", "title": "", "text": "$\\left[   \\begin{array}{cccc|c}   1 & 2 & -1 & 3  & 3 \\\\   2 & 4 &  4 & 3  & 9 \\\\   3 & 6 & -1 & 8  & 10 \\\\   \\end{array}   \\right]$"}
{"_id": "6407801", "title": "", "text": "$({\\frac {1} {2^{s-1}})}^j$"}
{"_id": "9241404", "title": "", "text": "$s=r^{-1}(rs)=(rs)(r^{2}s)(rs)$"}
{"_id": "9299577", "title": "", "text": "$A_n \\supset A_{n+1} \\supset \\ldots$"}
{"_id": "772003", "title": "", "text": "$ f(a, b, c) = a^\\alpha + b^\\beta + c^\\gamma, g(a, b, c) = abc - 1 = 0 \\\\ \\nabla f = \\left\\langle\\alpha a^{\\alpha - 1}, \\beta b^{\\beta - 1}, \\gamma c^{\\gamma - 1}\\right\\rangle = \\lambda \\nabla g = \\lambda \\langle bc, ac, ab\\rangle \\\\ \\alpha a^\\alpha = \\lambda abc = \\lambda \\rightarrow a = \\left(\\frac{\\lambda}{\\alpha}\\right)^\\frac{1}{\\alpha} \\\\ \\beta b^\\beta = \\lambda abc = \\lambda \\rightarrow b = \\left(\\frac{\\lambda}{\\beta}\\right)^\\frac{1}{\\beta}\\\\ \\gamma c^\\gamma = \\lambda abc = \\lambda \\rightarrow c = \\left(\\frac{\\lambda}{\\gamma}\\right)^\\frac{1}{\\gamma} \\\\ abc = 1 = \\frac{\\lambda^{\\frac{1}{\\alpha} + \\frac{1}{\\beta} + \\frac{1}{\\gamma}}}{\\alpha^\\frac{1}{\\alpha}\\beta^\\frac{1}{\\beta}\\gamma^\\frac{1}{\\gamma}} \\rightarrow \\lambda = \\left(\\alpha^\\frac{1}{\\alpha}\\beta^\\frac{1}{\\beta}\\gamma^\\frac{1}{\\gamma}\\right)^\\frac{\\alpha\\beta\\gamma}{\\alpha\\beta + \\alpha\\gamma + \\beta\\gamma} = \\left(\\alpha^{\\beta\\gamma}\\beta^{\\alpha\\gamma}\\gamma^{\\alpha\\beta}\\right)^\\frac{1}{\\alpha\\beta + \\alpha\\gamma + \\beta\\gamma} $"}
{"_id": "5744261", "title": "", "text": "$\\pmatrix{0\\cr -1\\cr -1\\cr 1\\cr}$"}
{"_id": "5219463", "title": "", "text": "$T_{X,x}^{\\ast0,1}$"}
{"_id": "2112278", "title": "", "text": "$7\\mid a^7-a$"}
{"_id": "9101619", "title": "", "text": "$F := P \\oplus Q$"}
{"_id": "4216598", "title": "", "text": "$ S(x) = \\int_a^x f(t) dt $"}
{"_id": "5676092", "title": "", "text": "$T_X = T_X^1$"}
{"_id": "6873098", "title": "", "text": "$f^n(x)=\\dfrac{ax+b}{cx+d}$"}
{"_id": "504012", "title": "", "text": "$A_{n}\\cdots A_{1}$"}
{"_id": "1215870", "title": "", "text": "$S \\equiv $"}
{"_id": "5276453", "title": "", "text": "$\\tag{1}\\dfrac{(1+r)^k-(1-r)^k}{(1+r)^k+(1-r)^k}=-\\frac {\\sqrt{b}}a$"}
{"_id": "1156226", "title": "", "text": "$a^{log b}$"}
{"_id": "5650068", "title": "", "text": "$\\sum_{i=1}^n \\frac{1}{2^i}$"}
{"_id": "1243735", "title": "", "text": "$ r = |\\sin( \\varphi) \\cos( \\varphi) \\cos( \\psi) \\sin^2 (\\psi)| $"}
{"_id": "5471803", "title": "", "text": "$\\lim_{x\\to a}\\hat{f}(x) = \\lim_{x\\to a}f(x) = L = \\hat{f}(a)$"}
{"_id": "1474644", "title": "", "text": "$\\{\\gamma,-\\gamma\\}=\\{\\beta,-\\beta\\}$"}
{"_id": "2214852", "title": "", "text": "$\\mu(\\lim \\inf A_n) \\leq \\lim \\inf \\mu(A_n)\\leq \\lim \\sup \\mu(A_n) \\leq \\mu(\\lim \\sup A_n)$"}
{"_id": "1531366", "title": "", "text": "$|X\\times Y|=mn$"}
{"_id": "410075", "title": "", "text": "$f'(x)=3f(x)$"}
{"_id": "358500", "title": "", "text": "$x=(x_1.x_2,x_3,...,x_n,...)$"}
{"_id": "5860552", "title": "", "text": "$x = (x_{1},x_{2}....,x_{n})$"}
{"_id": "6444961", "title": "", "text": "$L_2 :\\begin{cases} x = 2+2t\\\\ y = 1\\\\ z = t\\end{cases}$"}
{"_id": "2726994", "title": "", "text": "$\\mathcal{O}_{\\mathbb{R}^{m+n} | \\mathbb{R}^n}$"}
{"_id": "8671247", "title": "", "text": "$\\sum_{n=1}^\\infty\\frac{n+2-2}{(n+2)!}=\\sum_{n=1}^\\infty\\frac1{(n+1)!}-2\\sum_{n=1}^\\infty\\frac1{(n+2)!}$"}
{"_id": "23132", "title": "", "text": "$[x,y]=z$"}
{"_id": "2698444", "title": "", "text": "$\\sum _{i=1}^n r^{i-1}=\\frac{r^n-1}{r-1}$"}
{"_id": "6420525", "title": "", "text": "$ \\| \\gamma(a)-\\gamma(b)\\| \\le  \\int_a^b \\| \\gamma '\\|$"}
{"_id": "2763830", "title": "", "text": "$|2^{\\frac{2^{n}-1}{2^{n}}}-2|=|2^{1-\\frac{1}{2^{n}}}-2|=|2\\times2^{-\\frac{1}{2^{n}}}-2|=|2(2^{-\\frac{1}{2^{n}}}-1)|=2|2^{-\\frac{1}{2^{n}}}-1|=2|\\frac{1}{2^{\\frac{1}{2^{n}}}}-1|=2(1-\\frac{1}{2^{\\frac{1}{2^{n}}}})$"}
{"_id": "6151186", "title": "", "text": "$\\int_{0}^{2\\pi} f(t) \\cos(pt)\\,\\mathrm{d}t=\\int_{0}^{2\\pi} f(t) \\sin(pt)\\,\\mathrm{d}t=0,$"}
{"_id": "4875797", "title": "", "text": "$\\zeta(s)={1\\over \\Gamma(s)}\\int_{0}^{\\infty}{x^{s-1}\\over e^x-1}\\mathrm dx\\tag5$"}
{"_id": "2638465", "title": "", "text": "$\\det_\\ast[\\gamma]=[\\det\\circ\\gamma]$"}
{"_id": "316998", "title": "", "text": "$\\left[\\begin{smallmatrix}A&B\\\\C&D\\end{smallmatrix}\\right]$"}
{"_id": "1903837", "title": "", "text": "$A = \\{a_1,a_2,a_3,..,a_N\\} $"}
{"_id": "501273", "title": "", "text": "$n! = \\sqrt{2\\pi n}(n/e)^n (1+o(1))$"}
{"_id": "2720445", "title": "", "text": "$U \\subset V \\subset X,$"}
{"_id": "2409698", "title": "", "text": "$\\sum x_n e_n$"}
{"_id": "3737116", "title": "", "text": "$d(x,y)=\\frac1{2^L}$"}
{"_id": "679389", "title": "", "text": "$\\gcd(a,a) = |a|$"}
{"_id": "4354032", "title": "", "text": "$ \\frac{e^{-n}-e^{-n-1}}{n+1} \\leq \\int_{n}^{n+1} \\frac{e^{-x}}{x} \\ dx  \\leq \\frac{e^{-n}-e^{-n-1}}{n}, $"}
{"_id": "3492449", "title": "", "text": "$ \\frac{a+b}{b+x}\\in\\mathbb{Z}\\tag{3} $"}
{"_id": "2134113", "title": "", "text": "$ \\dim S = \\dim V-\\dim S^\\perp=\\dim S^\\perp+\\dim S^{\\perp\\perp}-\\dim S^\\perp=\\dim S^{\\perp\\perp}\\tag{1} $"}
{"_id": "6098600", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1 & 3 & -1 & 4\\\\4 & -1 & 2 & 8\\\\2 & -7 & 4 & 0\\end{array}\\right],$"}
{"_id": "408556", "title": "", "text": "$\n  \\lim_{n\\rightarrow\\infty} \\mu(A_n) =\\mu (A)=1.\n $"}
{"_id": "4300827", "title": "", "text": "$A_{1} \\subset A_{2} \\subset A_{3} \\subset \\dots \\subset A_{n} \\subset A_{n + 1} \\subset \\dots$"}
{"_id": "186032", "title": "", "text": "$f_2(x) = \\dfrac{2}{(1-x)^{3}}$"}
{"_id": "6155200", "title": "", "text": "$a|c_1c_2$"}
{"_id": "4491183", "title": "", "text": "$\\tan (-\\theta)=\\frac {sin (-\\theta)}{cos (-\\theta)}\\frac {-sin (\\theta)}{cos (\\theta)}=-tan (\\theta)$"}
{"_id": "5293666", "title": "", "text": "$\\overline{[S]}\\subset\\overline{S^{\\bot\\bot}}=S^{\\bot\\bot}\\Rightarrow \\overline{[S]}^{\\bot}\\supset S^{\\bot\\bot\\bot}\\supset S^{\\bot}$"}
{"_id": "1682448", "title": "", "text": "$ (x-2)[(x-2)^{n-1}-n+1] $"}
{"_id": "6947512", "title": "", "text": "$\\int_0^1 f(t) dt = \\int_0^1 g(t) dt.$"}
{"_id": "7322984", "title": "", "text": "$=\\frac a\\gamma\\int_{0}^x\\frac{\\gamma e^{\\gamma u}du}{e^{\\gamma u}-1+a\\gamma}-a\\int_0^x\\frac{du}{e^{\\gamma u}-1+a\\gamma}$"}
{"_id": "4842522", "title": "", "text": "$\\int_0^\\infty f(y)\\,dy$"}
{"_id": "9054715", "title": "", "text": "$\\int_0^\\infty f^+(x)dx,\\int_0^\\infty f^-(x)dx < \\infty$"}
{"_id": "105190", "title": "", "text": "$\\;\\Bbb Z/(11)\\;$"}
{"_id": "54567", "title": "", "text": "$\\displaystyle{\\lim_{x \\to c}f(x)=L}$"}
{"_id": "5238983", "title": "", "text": "$1+\\frac{2}{n(n+3)} = \\frac{(n+1)(n+2)}{n (n+3)} = \\frac{\\frac{n+2}{n+3}}{\\frac{n}{n+1}} $"}
{"_id": "6569533", "title": "", "text": "$P\\{\\sup_{t\\geq 0} M_t > x \\,| \\,F_0\\} \\leq 1\\wedge \\frac{M_0}{x}$"}
{"_id": "2287577", "title": "", "text": "$(x*y)*y=(y*y)*y=y*y=y.$"}
{"_id": "6461741", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{\\sqrt{n+1}+\\sqrt{n+2}+\\dots+\\sqrt{2n}}{\\sqrt{1}+\\sqrt{2}+\\dots+\\sqrt{n}}$"}
{"_id": "4032089", "title": "", "text": "$P(X_{16}=2|X_0=0)$"}
{"_id": "2782038", "title": "", "text": "$P^*(4) = P^*(5) = P^*(6) = P^*(7)=P^*(8)=P^*(9)= P^*(10) = 0$"}
{"_id": "8061115", "title": "", "text": "$F = x+y$"}
{"_id": "6778554", "title": "", "text": "$\\begin{vmatrix} 1&1 &1 \\\\ x_1& x_2 &x \\\\ y_1& y_2& y \\end{vmatrix}=0$"}
{"_id": "5646271", "title": "", "text": "$D=\\{(x,y)\\in\\mathbb R^2|0\\leq x \\leq m,0 \\leq y \\leq n\\}$"}
{"_id": "6946034", "title": "", "text": "$         \\begin{pmatrix}          A & B^T \\\\          -B & A^\\dagger \\\\         \\end{pmatrix}  $"}
{"_id": "5247080", "title": "", "text": "$f(x) = \\sum_{j=1}^{\\infty} \\frac{1}{j}\\chi_{A_{j}}(x).$"}
{"_id": "4076707", "title": "", "text": "$1/\\vert x \\vert^4$"}
{"_id": "8479233", "title": "", "text": "$e^x > \\bigg(1+ \\frac xn\\bigg)^n $"}
{"_id": "9235296", "title": "", "text": "$\\|\\dot\\gamma\\|^2=\\dot\\gamma\\cdot\\dot\\gamma=1 \\Rightarrow (\\dot\\gamma\\cdot\\dot\\gamma)'=0 \\Rightarrow \\ddot\\gamma\\cdot\\dot\\gamma+\\dot\\gamma\\cdot\\ddot\\gamma=0 \\Rightarrow 2\\ddot\\gamma\\cdot\\dot\\gamma=0 \\Rightarrow \\ddot\\gamma\\cdot\\dot\\gamma=0$"}
{"_id": "799620", "title": "", "text": "$\\sum_{n=N}^\\infty \\frac1{n^2}+\\sum_{n=N}^\\infty\\frac1{n^3}<\\epsilon$"}
{"_id": "6578727", "title": "", "text": "$S=\\{ (x,y) \\in \\mathbb{R^2}\\: | \\:0 \\leq x \\leq 1, 0 \\leq y \\leq 1\\}$"}
{"_id": "6795862", "title": "", "text": "$d(x,a)\\le d(x,b)+d(b,a)$"}
{"_id": "3667018", "title": "", "text": "${Let‌ ‌S = \\{x_1, x_2, x_3, ..., x_n\\}}$"}
{"_id": "4718102", "title": "", "text": "$p\\mid a_1a_2$"}
{"_id": "6344408", "title": "", "text": "$|f(z)|\\leq M |g(z)|$"}
{"_id": "2624031", "title": "", "text": "$ f : \\mathbb{R}^2 \\rightarrow \\mathbb{R}, f((x, y)) = x+y $"}
{"_id": "2196191", "title": "", "text": "$d(x,M)=1$"}
{"_id": "4268456", "title": "", "text": "$2^x+4^x+12=0$"}
{"_id": "3964191", "title": "", "text": "$ \\left[       \\begin{array}{ccc|c}         1&-1&0&k\\\\         3&0&2&5\\\\         2&-2&k^2-k&2\\\\       \\end{array}     \\right]$"}
{"_id": "5037504", "title": "", "text": "$A_1=\\{a_1,f(a_1),f(f(a_1)),...\\}$"}
{"_id": "8249195", "title": "", "text": "$\\forall \\epsilon > 0,  \\exists \\delta > 0 \\text{ such that } \\forall x, y \\in I,  |x-y| < \\delta \\Rightarrow |f(x)-f(y)| < \\epsilon$"}
{"_id": "504324", "title": "", "text": "$f(x) = x^{2}-x+1$"}
{"_id": "8825608", "title": "", "text": "$X_M^c\\subset [X_M^c,X_M^c]$"}
{"_id": "5712174", "title": "", "text": "$\\int_{-\\infty}^\\infty \\frac{1}{(x^2 + 1)(x^4+4)^2} dx$"}
{"_id": "432824", "title": "", "text": "$[x,y]=xy-yx$"}
{"_id": "4593305", "title": "", "text": "$\\frac{\\pi^2}{4} \\frac{\\sin^2(\\pi p/2)}{\\cos(\\pi p /2)}.$"}
{"_id": "4637942", "title": "", "text": "$Cov(X,Z+Y)=Cov(X,Z)+Cov(X,Y)$"}
{"_id": "2058260", "title": "", "text": "$\\{x^2, y^2, z^2, xy, xz, yz\\}$"}
{"_id": "6532898", "title": "", "text": "$  \\left[\\begin{array}{rr|r}7.5 & -5 & 0 \\\\k + 7.5 & 0 & 0 \\\\\\end{array}\\right]$"}
{"_id": "5668460", "title": "", "text": "$A=\\frac {Pr(1+r)^n}{(1+r)^n-1}$"}
{"_id": "6047008", "title": "", "text": "$a,b,c,d\\in\\mathbb{R},a\\leq b,c\\leq d$"}
{"_id": "1787694", "title": "", "text": "$P(\\{\\omega_3\\})=1/6$"}
{"_id": "4538545", "title": "", "text": "$L(x)\\vert E(x)$"}
{"_id": "5983418", "title": "", "text": "$\\mathbb Z\\oplus T$"}
{"_id": "7383455", "title": "", "text": "$\\mathbb R^n\\times\\mathbb R^n\\to\\mathbb R^n\\times\\mathbb R^n$"}
{"_id": "7133883", "title": "", "text": "$\\int_0^\\infty \\sum_{n=-\\infty}^\\infty e^{-\\pi{n^2}/ x}  dx $"}
{"_id": "8236963", "title": "", "text": "$\\Gamma(s) = \\sum_{n=1}^{\\infty} \\frac{\\mu(n)}{n^s} \\int_0^\\infty\\frac{x^{s-1}}{e^x-1} dx .$"}
{"_id": "9200064", "title": "", "text": "$\\det{\\alpha \\gamma} = d^4 = (\\det{\\alpha}) (\\det{\\gamma}) = (\\det{\\alpha})(\\det{\\beta}) = dD.$"}
{"_id": "1858013", "title": "", "text": "$Cov(X+Y, Z+W)=Cov(X,Z)+Cov(X,W)+Cov(Y,Z)+Cov(Y,W)$"}
{"_id": "4043077", "title": "", "text": "$\\det(A) = x (x-a)^{n-1}$"}
{"_id": "9332923", "title": "", "text": "$g(x)=\\frac{a+bx}{b+ax},$"}
{"_id": "3770648", "title": "", "text": "$3^{k+1}2^k-3^k2^{k+1}=6^k(3-2)=6^k$"}
{"_id": "1486889", "title": "", "text": "$\\sum_{j=1}^Nc_j\\chi_{A_j}$"}
{"_id": "5738230", "title": "", "text": "$ \\color{red}{\\sum\\limits_{k=1}^n\\cos\\left(\\frac{2k-1}{2n+1}\\pi\\right)=\\frac12}. $"}
{"_id": "8450161", "title": "", "text": "$f(n)=n^2-n+1$"}
{"_id": "6666741", "title": "", "text": "$\\vert ab\\vert = \\operatorname{lcm}(m,n)$"}
{"_id": "2826988", "title": "", "text": "$\\begin{vmatrix} 1 & a & a^3 \\\\ 1 & b & b^3 \\\\ 1 & c & c^3 \\\\ \\end{vmatrix}$"}
{"_id": "7249661", "title": "", "text": "$\\mu(A)= \\int_{a}^{b} 1~dx$"}
{"_id": "1288323", "title": "", "text": "$\\int \\frac{1}{\\sqrt{x^2+1}} dx$"}
{"_id": "4046053", "title": "", "text": "$\\left\\lfloor{\\frac{x}{n}}\\right\\rfloor=\\left\\lfloor{\\frac{\\lfloor{x}\\rfloor}{n}}\\right\\rfloor,$"}
{"_id": "3919296", "title": "", "text": "$\\mathcal{P}(U)=\\{\\{a_1\\}\\}$"}
{"_id": "913102", "title": "", "text": "$x^3-x^2+x$"}
{"_id": "8265255", "title": "", "text": "$[-m, m] = [-m, m] + [-m, m]$"}
{"_id": "9118231", "title": "", "text": "$\\displaystyle \\tan \\theta = \\frac{x}{20}$"}
{"_id": "8891213", "title": "", "text": "$ \\sqrt {11} = \\frac {a+b}{b}$"}
{"_id": "7667551", "title": "", "text": "$\\int_f \\leq \\lim_{n \\to \\infty}\\inf    \\int f_n\\ d\\mu.$"}
{"_id": "5007746", "title": "", "text": "$\\gcd(a, b) = ax+by = d$"}
{"_id": "3911675", "title": "", "text": "$\\int_\\gamma\\frac{dz}{z+i}=\\int_{\\tilde\\gamma}\\frac{dz}{z+i}-\\int_{\\gamma_0}\\frac{dz}{z+i}=\\int_{\\tilde\\gamma}\\frac{dz}{z+i}+\\ln(\\sqrt2)-\\tfrac14\\pi i,$"}
{"_id": "7558917", "title": "", "text": "$\\alpha=\\omega\\alpha$"}
{"_id": "1694328", "title": "", "text": "$(k+6)^{1/3} > k^{1/3}$"}
{"_id": "86160", "title": "", "text": "$Cov(X,Y) = 0$"}
{"_id": "1528250", "title": "", "text": "$\\sum_{n\\ge 1}\\left(\\frac3{n(n+3)}+\\frac1{2^n}\\right)=\\sum_{n\\ge 1}\\frac3{n(n+3)}+\\sum_{n\\ge 1}\\frac1{2^n}$"}
{"_id": "4626153", "title": "", "text": "$ \\left[ \\begin{array}{cccc|c}   1&2&-1&-5&0\\\\   -1&0&3&5&0\\\\ 1&1&a&b&0 \\end{array} \\right] $"}
{"_id": "5535293", "title": "", "text": "$E((X+Y)^p)\\leq 2^p\\left( E(X^p)+E(Y^p)\\right)$"}
{"_id": "3032614", "title": "", "text": "$b^{log_b(x)}$"}
{"_id": "269992", "title": "", "text": "$\\frac{246}{217}=1+\\frac{29}{217}=1+\\frac1{\\frac{217}{29}}=1+\\frac1{7+\\frac{14}{29}}=1+\\frac1{7+\\frac1{\\frac{29}{14}}}=1+\\frac1{7+\\frac1{2+\\frac1{14}}}$"}
{"_id": "4680522", "title": "", "text": "$log_{a}x=b $"}
{"_id": "2301898", "title": "", "text": "$f(x,y)={{x^3}\\over{y^2}} e^{{-x}\\over{y}}$"}
{"_id": "8009025", "title": "", "text": "$\\lim_{n\\to\\infty}\\sup (-a_n)=-\\lim_{n\\to\\infty}\\inf a_n.$"}
{"_id": "136815", "title": "", "text": "$ f_X(x) = \\frac{1}{\\sqrt{2\\pi}}e^\\frac{-x^2}{2} \\; \\forall x\\in \\mathbb{R} $"}
{"_id": "633884", "title": "", "text": "$ \\int_{\\gamma} \\frac{dz}{z^2+1} = -i\\int_{\\gamma} \\frac{dz}{z-i}  + i\\int_{\\gamma} \\frac{dz}{z+i}.   $"}
{"_id": "3675953", "title": "", "text": "$[X,Y]= X\\wedge Y$"}
{"_id": "1119254", "title": "", "text": "$F(a) + F(b) = F(a+b)$"}
{"_id": "5765201", "title": "", "text": "$\\left\\vert\\begin{matrix}1&1&1\\\\x&y&z\\\\f(x)&f(y)&f(z)\\end{matrix}\\right\\vert\\not=0$"}
{"_id": "5635732", "title": "", "text": "$ \\sum^{\\infty}_{n=1}\\left\\|f_n\\right\\|<\\infty, $"}
{"_id": "4307133", "title": "", "text": "$\\phi(x)=1/|x|$"}
{"_id": "4405997", "title": "", "text": "$\\sum\\Vert x_n-y_n\\Vert <1/(2K)$"}
{"_id": "4454189", "title": "", "text": "$T_pM\\otimes\\mathbb C=T^{1,0}_pM\\oplus T^{0,1}_pM$"}
{"_id": "5506069", "title": "", "text": "$S_3(n+1)=\\binom n2=\\frac{n(n-1)}2=\\sum_{k=1}^{n-1}k.$"}
{"_id": "1102063", "title": "", "text": "$[x, y]=x$"}
{"_id": "2594188", "title": "", "text": "$e^{-x} - (1 - x/n)^n = e^{-x}[ 1 - e^{x}(1- x/n)^n] \\leqslant e^{-x}[ 1 - (1 +x/n)^n(1 - x/n)^n] = e^{-x}[ 1 - (1 -x^2/n^2)^n] \\\\ \\leqslant\\frac{e^{-x}x^2}{n} .$"}
{"_id": "7231440", "title": "", "text": "$U(n)={n\\choose 2}$"}
{"_id": "512736", "title": "", "text": "$\\sum\\limits_{n=0}^{\\infty}xe^{-n^2x} = x\\sum\\limits_{n=0}^{\\infty}e^{-n^2x}$"}
{"_id": "3506018", "title": "", "text": "$\\int_{0}^{1}(\\sin^{-1}x)^ndx$"}
{"_id": "9368218", "title": "", "text": "$\\displaystyle\\;T_n = \\frac{n(n+1)(n+2)}{6}$"}
{"_id": "5648214", "title": "", "text": "$\\bigcup_{\\gamma<\\delta}\\lambda_{\\gamma}\\times{\\{\\gamma\\}}$"}
{"_id": "314157", "title": "", "text": "$w=\\sqrt re^{i\\theta/2}$"}
{"_id": "3740376", "title": "", "text": "$c = b^{\\log_b(c)}$"}
{"_id": "9001329", "title": "", "text": "$[a]+([b]+[c])=[a]+[b+c]=[a+(b+c)]=[(a+b)+c]=([a]+[b])+[c]$"}
{"_id": "4282652", "title": "", "text": "$\\:\\mathrm{d}s=\\sqrt{1+\\frac{\\mathrm{d}^{2}y}{\\mathrm{d}x^{2}}}\\:\\mathrm{d}x$"}
{"_id": "7316381", "title": "", "text": "$A_1\\times\\dots A_n$"}
{"_id": "1505918", "title": "", "text": "$2 + 2 + 2 + 2 + ... = -\\frac12$"}
{"_id": "3415062", "title": "", "text": "$|x^2-a^2|=|x-a||x+a| \\leq \\epsilon |x+a| \\leq \\epsilon (2|a|+\\epsilon)$"}
{"_id": "8292988", "title": "", "text": "$\\text{gcd}(a+b, a+b) = 1$"}
{"_id": "4539559", "title": "", "text": "$[x,y] = \\textrm{lcm}[x,y]$"}
{"_id": "2526436", "title": "", "text": "$\\overline{Z_0} \\subset \\overline{Z_1} \\subset \\dots \\subset \\overline{Z_n}$"}
{"_id": "7331761", "title": "", "text": "$V\\subseteq\\overline{V}\\subseteq V_1\\cap V_2$"}
{"_id": "8720276", "title": "", "text": "$+++−−+−−=1+1+1-\\frac12-\\frac14+\\frac18-\\frac1{16}-\\frac1{32}=2+\\frac14+\\frac1{32}$"}
{"_id": "7749956", "title": "", "text": "$z=\\sqrt[4]{2}e^{\\frac{5\\pi i}{3}}$"}
{"_id": "275059", "title": "", "text": "$\\mathbb{Z}[x]/(x^2+1) \\cong \\mathbb{Z}[i]$"}
{"_id": "5255591", "title": "", "text": "$K[T^2,T^3]\\subset K[T]$"}
{"_id": "3220989", "title": "", "text": "$} \\mapsto \\lbinom{n}{k+1} \\right) = \\left( \\text{int $"}
{"_id": "1112439", "title": "", "text": "$(A_1,A_2,...,A_n)$"}
{"_id": "3348264", "title": "", "text": "$xRx\\lor xRx$"}
{"_id": "1787904", "title": "", "text": "$y_n = (x_1+ x_2+ \\dots + x_n)/n$"}
{"_id": "5766537", "title": "", "text": "$ f(t_x)=0\\quad  \\forall \\;t_x\\in(0,1). $"}
{"_id": "1600569", "title": "", "text": "$\\int_{0}^{\\infty}f(x)\\rm{d}x$"}
{"_id": "6277482", "title": "", "text": "$\\lfloor (a+b+c)/d\\rfloor+\\lfloor (a+b+d)/c\\rfloor+\\lfloor (a+d+c)/b\\rfloor+\\lfloor (d+b+c)/a\\rfloor$"}
{"_id": "7305464", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor x \\rfloor + j}{m}\\right\\rfloor = \\left\\lfloor\\frac{x + j}{m}\\right\\rfloor$"}
{"_id": "5987461", "title": "", "text": "$F(n)F(n+2)-F(n+1)^2=F(n)^2-F(n-1)F(n+1)$"}
{"_id": "6495310", "title": "", "text": "$f(z^2)=f(z)f(z)=z^2$"}
{"_id": "629639", "title": "", "text": "$f(x)=\\sum_{j=1}^n c_j\\lfloor x\\rfloor \\chi_{[j,j+1)}(x)$"}
{"_id": "5566255", "title": "", "text": "$P(E(X\\mid \\mathcal F_0)\\in C)=1$"}
{"_id": "2975805", "title": "", "text": "$A= \\int \\int _D dS = \\int \\int _D 3/2 dA = 3/2(16\\pi ) = 24\\pi $"}
{"_id": "2391437", "title": "", "text": "$|X_n|^p \\leq (|X_n-X|+|X|)^p \\leq |X_n-X|^p + |X|^p$"}
{"_id": "447667", "title": "", "text": "$T^{0,1}M$"}
{"_id": "2834436", "title": "", "text": "$a,a+b,a+2b,\\dots$"}
{"_id": "648179", "title": "", "text": "$2a,2b,a+b,a+b$"}
{"_id": "6501506", "title": "", "text": "$[a]+[n-a]=[a+(n-a)]=[a+(-a+n)]=[(a-a)+n]=[n]=[0]$"}
{"_id": "2430126", "title": "", "text": "$\\|UAZ\\|_2^2=\\rho(Z^*A^*U^*UAZ)=\\rho(Z^*A^*AZ)=\\rho(AZZ^*A^*)=\\rho(AA^*)=\\rho(A^*A)=\\|A\\|_2^2$"}
{"_id": "8495442", "title": "", "text": "$\\int_1^\\infty \\frac{\\sin^2(x)}{x}dx$"}
{"_id": "7038166", "title": "", "text": "$x,y \\in \\mathbb{R} ,f(x+y)=f(x)f(y)$"}
{"_id": "1113157", "title": "", "text": "$ x\\mapsto \\frac{(1+e^{-x-1})^n}{x+1}-\\frac{(1+e^{-x})^n}{n-x}$"}
{"_id": "6100488", "title": "", "text": "$\\int_{-\\pi}^\\pi f^2=\\int_{-\\pi}^\\pi f^2-\\int_{-\\pi}^\\pi fp=\\int_{-\\pi}^\\pi f(f-p)$"}
{"_id": "5788908", "title": "", "text": "$z^{\\frac{1}{n}}=\\sqrt{n}e^{i\\frac{\\theta}{n}}$"}
{"_id": "6691673", "title": "", "text": "$f(xf(y)+f(x))=f(yf(x))+x$"}
{"_id": "2922190", "title": "", "text": "$(x+y)^*(x+y)  = x^*x + x^*y + y^*x + y^*y$"}
{"_id": "3456479", "title": "", "text": "$2^{n/2}/(100 \\sqrt{n})$"}
{"_id": "2362359", "title": "", "text": "$d_h(x,y)=|f(x)-f(y)|$"}
{"_id": "3489638", "title": "", "text": "$x\\in V\\subseteq\\cl_YV\\subseteq U$"}
{"_id": "7829618", "title": "", "text": "$a_n=\\frac{\\frac{n(n+1)}{2}}{n^2}$"}
{"_id": "5710896", "title": "", "text": "$(x^y-y^x) + (y^z -z^y) + (z^x - x^z) = 0$"}
{"_id": "3270964", "title": "", "text": "$x^n+x^{n-1}$"}
{"_id": "3504910", "title": "", "text": "$P_1\\mid P_2 \\mid\\cdots\\mid P_k$"}
{"_id": "805344", "title": "", "text": "$d/dx(2 g(x)/(x^2 + 1))$"}
{"_id": "4550024", "title": "", "text": "$\\displaystyle \\frac{|\\langle u,v\\rangle|}{\\Vert u\\Vert}\\leq\\Vert v\\Vert$"}
{"_id": "1005048", "title": "", "text": "$L(\\gamma)=\\int_\\gamma \\ ds =\\int_a^b \\gamma^*ds=\\int_a^b\\frac{|\\gamma'(t)|}{1+|\\gamma(t)|^2}dt. $"}
{"_id": "71607", "title": "", "text": "$\\begin{bmatrix}\n a & -b \\\\\n b & a\n \\end{bmatrix}$"}
{"_id": "6290332", "title": "", "text": "$P(m)=P(n)$"}
{"_id": "5281368", "title": "", "text": "$x_1=\\sqrt[10]{\\frac43}e^{\\frac{i\\theta}5}$"}
{"_id": "8680067", "title": "", "text": "$A_n\\subseteq...\\subseteq A_2\\subseteq A_1$"}
{"_id": "1021356", "title": "", "text": "$\\sum_{n\\geq1}\\frac{e^{2\\pi inx}}{n}=\\sum_{n\\geq1}\\frac{1}{n}=\\infty  $"}
{"_id": "1915469", "title": "", "text": "$  (\\delta, \\gamma)^{(\\overline k, 1)} = (\\delta^{\\overline k(\\gamma)}, \\gamma^1)   = (\\delta^k, \\gamma) = (\\delta, \\gamma). $"}
{"_id": "5867929", "title": "", "text": "$a_0=0;\\\\a_1=1+a_0=1+0=1;\\\\a_2=2+a_1=2+1=3;\\\\a_3=3+a_2=3+3=6;\\\\a_4=4+a_3=4+6=10; \\\\a_5=5+a_4=5+10=15;\\\\a_6=6+a_5=6+15=21;$"}
{"_id": "443177", "title": "", "text": "$\\frac{a+x}{b+y}=\\frac{a}{b}$"}
{"_id": "1468982", "title": "", "text": "$P[X_n=-n]=1-1/n^2$"}
{"_id": "5979986", "title": "", "text": "$\\mathrm{P}\\left[\\left.X_t\\in A\\space\\right|\\mathcal{F}_s\\right]=\\mathrm{P}\\left[\\left.X_t\\in A\\space\\right|X_s\\right]$"}
{"_id": "223729", "title": "", "text": "$(a+b,a-b)=(c,d)$"}
{"_id": "2632557", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}\\mu(B_n) \\le \\lim_{n \\rightarrow \\infty}(\\inf_{k \\ge n}\\mu(A_k))=\\liminf_{n \\rightarrow \\infty}\\mu(A_n)$"}
{"_id": "1860833", "title": "", "text": "$[x,y]=f$"}
{"_id": "7879148", "title": "", "text": "$R(x) = o((x - a)^{n - 1})$"}
{"_id": "5910979", "title": "", "text": "$l=\\begin{cases} x=t+2\\\\ y=2t-1 & \\\\ z=t\\end{cases} $"}
{"_id": "1505908", "title": "", "text": "$d(x,y)=\\frac{|x-y|}{1+|x-y|}.$"}
{"_id": "4676835", "title": "", "text": "$\\|A\\|_2 \\leq \\sqrt{m} \\|A\\|_{\\infty}$"}
{"_id": "1115185", "title": "", "text": "$P(X_n \\le x)$"}
{"_id": "2557919", "title": "", "text": "$f_X(x;θ)$"}
{"_id": "3672267", "title": "", "text": "$\\gamma(t)=(\\gamma_1(t),\\gamma_2(t),\\gamma_3(t)),0\\le t \\le 1$"}
{"_id": "1396205", "title": "", "text": "$P(\\omega \\in \\Omega : lim_{n \\rightarrow \\infty} X_n(\\omega) = X(\\omega))= 1 $"}
{"_id": "8515029", "title": "", "text": "$f(a_1)=g(a_1)=b_1$"}
{"_id": "1120781", "title": "", "text": "$(\\beta+\\gamma-2), (-\\alpha + 3\\gamma - 5)$"}
{"_id": "4464533", "title": "", "text": "$\\gcd(a,a)=a$"}
{"_id": "8942723", "title": "", "text": "$μ(x,t)=μ·x$"}
{"_id": "4453596", "title": "", "text": "$\\left(1-\\frac{1}{\\frac{n(n+1)}{2}}\\right)^2=(1-\\frac{2}{n+1}+\\frac{2}{n})^2=(1-\\frac{2}{n+1}+\\frac{2}{n})(1+\\frac{2}{n+1}-\\frac{2}{n})$"}
{"_id": "535940", "title": "", "text": "$p=(k+1)^3-k^3$"}
{"_id": "8772540", "title": "", "text": "$F=P \\oplus M$"}
{"_id": "9292569", "title": "", "text": "$f'(x+f(y)+xf(y))\\cdot (f'(y)+xf'(y)) = 1+f(x)$"}
{"_id": "7657180", "title": "", "text": "$\\frac{{(1+\\sqrt{1+4x})}^{n+1}-{(1-\\sqrt{1+4x})}^{n+1}}{2^{n+1}\\sqrt{1+4x}}$"}
{"_id": "7257224", "title": "", "text": "$\\phi_{max}=\\arctan \\frac{\\sqrt{1-k^2}}{k}$"}
{"_id": "2627113", "title": "", "text": "$f(x+f(y)+xf(y)) = y+f(x)+yf(x)$"}
{"_id": "4468829", "title": "", "text": "$\\mbox{Cow} \\approx \\#^g_{i=1}\\mathbb T^2 ,  $"}
{"_id": "3169567", "title": "", "text": "$ \\mathbb{E}\\|X\\|_{\\infty}^r\\le C_{d,r}\\mathbb{E}\\|Y\\|_{\\infty}^r $"}
{"_id": "1413861", "title": "", "text": "$\\sum_{k=1}^{n-1} (n-k) cos\\frac{2πk}{n}=\\frac{-n}{2}$"}
{"_id": "7076493", "title": "", "text": "$|c_n (-2)^n|=|c_n 2^n|=|c_n 4^n|\\cdot \\frac{1}{2^n}\\leq \\frac{1}{2^n} $"}
{"_id": "1468152", "title": "", "text": "$B=\\left\\{\\langle x,y\\rangle\\in\\Bbb R^2:x>0\\text{ and }y=\\frac1x\\right\\}$"}
{"_id": "297919", "title": "", "text": "$f(e_i)=e_i,\\ p+1\\leq i\\leq n$"}
{"_id": "7228404", "title": "", "text": "$\\phi = \\begin{bmatrix} 1 & x & y & z & xy & xz & yz & x^2 & y^2 & z^2 \\end{bmatrix}$"}
{"_id": "1213622", "title": "", "text": "$\\tilde F(x)=\\int_a^x \\tilde f(t) dt$"}
{"_id": "6889966", "title": "", "text": "$\\lim\\limits_{x \\rightarrow x_{0}^{-}} f(x) = \\lim\\limits_{x \\rightarrow x_{0}^{+}} f(x) = \\lim\\limits_{x \\rightarrow x_0}f(x)$"}
{"_id": "2681633", "title": "", "text": "$ \\sum_{n=1}^{\\infty} \\|x_n-y_n\\|^2 \\geq 1 $"}
{"_id": "4515831", "title": "", "text": "$a_1 = g(f(a_1)$"}
{"_id": "1808766", "title": "", "text": "$\\frac{1}{2^{N-1}}$"}
{"_id": "6231036", "title": "", "text": "$(3)^{\\frac{1}{3}} (9)^{\\frac{1}{9}} (27)^{\\frac{1}{27}}$"}
{"_id": "921251", "title": "", "text": "$L_n=\\frac{n(n+1)}{2}$"}
{"_id": "1861346", "title": "", "text": "$\\int_a^x f(t)dt = F(x)-F(a).$"}
{"_id": "5358802", "title": "", "text": "$C_0(X)/C_0(A)$"}
{"_id": "5514635", "title": "", "text": "$\\frac{10/40}{30/40}.$"}
{"_id": "2529650", "title": "", "text": "$\\int_0^\\infty \\frac{\\sin^2(x)}{x}dx$"}
{"_id": "3488114", "title": "", "text": "$q(X)= X^2 + X + 2$"}
{"_id": "8916352", "title": "", "text": "$|x|^{2.1}\\sin(1/|x|)$"}
{"_id": "7829960", "title": "", "text": "$\\sigma(T^0,T^1)\\subset T$"}
{"_id": "8978180", "title": "", "text": "$I(s)=\\displaystyle\\int_{0}^{\\infty}\\dfrac{x^{s-1}}{e^x+1}\\,dx=\\eta(s)\\Gamma(s) \\tag*{} $"}
{"_id": "6885806", "title": "", "text": "$A_{n+1} \\subset A_1 = A_n \\subset A_{n+1}$"}
{"_id": "4668430", "title": "", "text": "$A \\subseteq S \\subseteq \\bar A$"}
{"_id": "7764431", "title": "", "text": "$ \\zeta(2) = \\sum_{n\\geq 1}\\frac{1}{n^2} = 2\\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^2} = 2\\,\\eta(2),$"}
{"_id": "7453142", "title": "", "text": "$\\int_0^{2\\pi} f(t)p(t)\\,dt=0$"}
{"_id": "6197799", "title": "", "text": "$\\sum_{n \\leq x} \\frac{1}{n} \\left\\lfloor\\frac{x}{n}\\right\\rfloor = \\sum_{n \\leq x} \\frac{1}{n} \\left[ \\frac{x}{n} + O(1) \\right] = x \\sum_{n \\leq x} \\frac{1}{n^2} + O(1) \\sum_{n \\leq x} \\frac{1}{n}$"}
{"_id": "3539281", "title": "", "text": "$R=(1/2)(T^*+T)$"}
{"_id": "5512563", "title": "", "text": "$f_Y(y)= \\frac{1}{2}e^{\\frac{-y}{2}}$"}
{"_id": "6561941", "title": "", "text": "$\\left(\\begin{array}{cc} A & B \\\\ B^T & D\\end{array}\\right)$"}
{"_id": "850883", "title": "", "text": "$\\forall\\epsilon\\gt 0,\\exists \\delta\\gt 0,|x-c|\\lt \\delta , x\\in D\\Rightarrow |f(x)-f(c)|\\lt\\epsilon$"}
{"_id": "4470632", "title": "", "text": "$11!+2,\\,11!+3,\\,11!+4,\\,\\ldots,\\,11!+11$"}
{"_id": "9019438", "title": "", "text": "$\\phi = \\sum_{i=0}^{n} a_i \\chi_{A_i}$"}
{"_id": "6519397", "title": "", "text": "$f_3^*$"}
{"_id": "280008", "title": "", "text": "$\\begin{bmatrix}1&1&-2&3\\\\0&1&1&-1\\\\-2&-1&5&-7\\end{bmatrix}$"}
{"_id": "3431257", "title": "", "text": "$P(n) = \\binom{n - 2}{3}$"}
{"_id": "896571", "title": "", "text": "$\\frac{27}{9}$"}
{"_id": "6944336", "title": "", "text": "$\\neg\\neg\\phi\\lor\\neg\\neg\\neg\\phi\\tag{axiom 12}$"}
{"_id": "8265708", "title": "", "text": "$(m, m, m, ...)$"}
{"_id": "9059157", "title": "", "text": "$f\\left ( f(x)^{2}y \\right )=x^{3}f(xy) \\tag{1}.$"}
{"_id": "3785721", "title": "", "text": "$\\Large{a^{\\log_a(b)}=b\\qquad\\qquad a^{b/c}=(a^b)^{1/c}}\\qquad\\qquad a^{b-c}=\\frac{a^b}{a^c}$"}
{"_id": "5607890", "title": "", "text": "$p_1p_2\\mid |G|$"}
{"_id": "1020959", "title": "", "text": "$\\mathcal{F}=\\lbrace  \\lbrace a \\rbrace, \\lbrace b\\rbrace, \\lbrace c\\rbrace , X , \\emptyset\\rbrace.$"}
{"_id": "5934410", "title": "", "text": "$ \\det (a)=a\\quad\\text{and}\\quad\\det\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}=ad-bc $"}
{"_id": "2896570", "title": "", "text": "$ax+by=n.$"}
{"_id": "120862", "title": "", "text": "$\\binom{n}{2} = \\frac{n(n-1)}{2}$"}
{"_id": "3313177", "title": "", "text": "$\\begin{cases}x=4+1t\\\\y=0+0t\\\\z=1-2t\\end{cases}$"}
{"_id": "5759451", "title": "", "text": "$\\gamma + \\gamma \\: < \\: \\Gamma(0,\\gamma) + \\gamma + \\gamma +1 \\: = \\: \\Gamma(0,\\gamma+1) \\: < \\: \\Gamma(0,\\alpha) \\: = \\: \\alpha$"}
{"_id": "7034111", "title": "", "text": "$P[X \\leq x-\\epsilon] - P[|X_n - X| \\geq \\epsilon] \\leq P[X_n \\leq x] \\leq P[X\\leq x+\\epsilon] + P[|X_n - x| \\geq \\epsilon]$"}
{"_id": "6616002", "title": "", "text": "$h(x_n)=e_n$"}
{"_id": "3550214", "title": "", "text": "$d\\in M^\\perp$"}
{"_id": "4909554", "title": "", "text": "$A+U \\subseteq V$"}
{"_id": "2403786", "title": "", "text": "$\\prod\\mathscr{A} = A_1 \\times \\cdots \\times A_n$"}
{"_id": "5727264", "title": "", "text": "${3\\over a}+{5\\over b}+{6\\over c}=1\\ .\\qquad(1)$"}
{"_id": "7327842", "title": "", "text": "$\\mid f(z)\\mid \\leq \\mid f(z^2)\\mid\\leq \\mid z^2 \\mid\\leq \\mid z\\mid$"}
{"_id": "3773328", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac 1{n^2}=\\sum_{n=1}^{\\infty}\\frac 1{(2n)^2}+\\sum_{n=0}^{\\infty}\\frac 1{(2n+1)^2}$"}
{"_id": "5769645", "title": "", "text": "$\\nabla_{\\dot\\gamma}\\dot\\gamma - \\frac{g(\\nabla_{\\dot\\gamma}\\dot\\gamma,\\dot\\gamma)}{g(\\dot\\gamma,\\dot\\gamma)}\\dot\\gamma$"}
{"_id": "1587951", "title": "", "text": "$n\\sum_{k=1}^{n-1}\\cos\\frac{2k\\pi}{n} -[ \\sum_{k=1}^{n-1} cos\\frac{2k\\pi}{n}] -[\\sum_{k=2}^{n-1} cos\\frac{2k\\pi}{n}]-[\\sum_{k=3}^{n-1} cos\\frac{2k\\pi}{n}].........-[\\sum_{k=(n-2)}^{n-1} cos\\frac{2k\\pi}{n}]-cos\\frac{2\\pi}{n}$"}
{"_id": "3411152", "title": "", "text": "$\\int_0^{100} (1-e^{\\frac{-x}{1000}})  $"}
{"_id": "7307531", "title": "", "text": "$F(k)=\\sum_{n=1}^{\\infty}\\frac{n^{4k-1}}{e^{n\\pi}-1}-16^k\\sum_{n=1}^{\\infty}\\frac{n^{4k-1}}{e^{4n\\pi}-1}$"}
{"_id": "8797843", "title": "", "text": "$|\\frac{a_2}{2^{n-2}}|\\leq\\varepsilon$"}
{"_id": "4096924", "title": "", "text": "$\\frac{m(x-\\delta ,x+\\delta )}{2\\delta }=\\frac{f(x+\\delta )-f(x-\\delta )}{2\\delta }$"}
{"_id": "7897314", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty|c_n|^2=\\dfrac{1}{2\\pi}\\int_{-\\pi}^\\pi|\\hat{f}(x)|^2dx,$"}
{"_id": "5494718", "title": "", "text": "$\\frac{1}{\\sqrt{x}}e^{-\\frac{n^2\\pi}{x}} = \\int_{-\\infty}^\\infty e^{-m^2\\pi x -2\\pi i m n} dm.$"}
{"_id": "3026348", "title": "", "text": "$c_{x}/(a/4)$"}
{"_id": "9196259", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor\\frac{a}{b}\\rfloor}{c}\\right\\rfloor = \\left\\lfloor\\frac{\\lfloor\\frac{a}{c}\\rfloor}{b}\\right\\rfloor = \\left\\lfloor\\frac{a}{bc}\\right\\rfloor$"}
{"_id": "9178203", "title": "", "text": "$\\begin{cases}x=u\\\\y=v\\\\ z=w(1-u-v)\\end{cases}$"}
{"_id": "5478925", "title": "", "text": "$\\frac{3}{x}+\\frac{4}{y}=2$"}
{"_id": "6294600", "title": "", "text": "$\\varepsilon=\\frac{1}{2^{n/2-2}}$"}
{"_id": "985560", "title": "", "text": "$a \\pmod 6 \\in \\{0,1,2,4,5\\}$"}
{"_id": "3032232", "title": "", "text": "$ e^{\\ln(a)}=a \\implies b^{\\log_b(a)}=a $"}
{"_id": "3698850", "title": "", "text": "$2(4^x) + 3(9^x)=5(6^x)$"}
{"_id": "9162941", "title": "", "text": "$P(15)\\implies P(19)$"}
{"_id": "4023011", "title": "", "text": "$\\tag 1\\lim_{x\\to \\infty}f(x) = L\\, \\iff\\, \\lim_{x\\to 0^+}f(1/x) = L.$"}
{"_id": "1353758", "title": "", "text": "$\\sum\\left\\|e_{n}-f_{n}\\right\\|^{2}<\\infty$"}
{"_id": "5566261", "title": "", "text": "$P(E(X|{\\mathcal F}_0)=0)=0$"}
{"_id": "1230519", "title": "", "text": "$1 =\\frac{4}{10} * x$"}
{"_id": "6637904", "title": "", "text": "$=8+\\frac1{\\frac{31}{22}}=8+\\frac1{1+\\frac1{\\frac{22}9}}=8+\\frac1{1+\\frac1{2+\\frac49}}=8+\\frac1{1+\\frac1{2+\\frac1{\\frac94}}} =8+\\frac1{1+\\frac1{2+\\frac1{2+\\frac14}}}$"}
{"_id": "2021662", "title": "", "text": "$\\neg\\neg B\\lor \\neg B$"}
{"_id": "3706868", "title": "", "text": "$s'(t)=\\cos{2t}$"}
{"_id": "8177112", "title": "", "text": "$\\lim_{n \\to \\infty}\\dfrac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+...+\\sqrt{n-1}}{n\\sqrt{n}} = \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=0}^{n-1} \\sqrt{\\frac{k}{n}} = \\int_0^1\\sqrt{x}\\mathrm{d}x$"}
{"_id": "7903647", "title": "", "text": "$A = \\begin{pmatrix} 1 & x & x^2 & x^3 \\\\ 1 & y & y^2 & y^3 \\\\ 1 & z & z^2 & z^3\\\\ 1 & w & w^2 & w^3 \\\\ \\end{pmatrix} $"}
{"_id": "222450", "title": "", "text": "$ d(x, y) \\leqslant d(x, z) + d(z, y) $"}
{"_id": "8709720", "title": "", "text": "$(1+r)^{-N} = 1 + r(-N) + r^2\\frac{(-N)(-N-1)}{2!}  ...$"}
{"_id": "8893331", "title": "", "text": "$\\lim_{n\\to\\infty} h(x_n)=\\inf h(x)$"}
{"_id": "5879415", "title": "", "text": "$\\lim_{n\\to\\infty} \\mu(A_n)\\geq 1$"}
{"_id": "1270021", "title": "", "text": "$ f(x) = {2x-1\\over x^2} $"}
{"_id": "9304605", "title": "", "text": "$\\frac{dy}{dx}=(1+\\frac{dy}{dx})(-\\sin(x+y))$"}
{"_id": "3878408", "title": "", "text": "$ \\begin{align} \\int_0^1\\left(\\frac{\\sin^{-1}(x)}{x}\\right)^2\\,\\mathrm{d}x &=\\int_0^{\\pi/2}\\left(\\frac{t}{\\sin(t)}\\right)^2\\,\\mathrm{d}\\sin(t)\\tag{14}\\\\ &=-\\int_0^{\\pi/2}t^2\\,\\mathrm{d}\\frac1{\\sin(t)}\\tag{15}\\\\ &=-\\frac{\\pi^2}4+2\\int_0^{\\pi/2}\\frac{t}{\\sin(t)}\\,\\mathrm{d}t\\tag{16}\\\\ &=-\\frac{\\pi^2}4-2\\int_0^{\\pi/2}t\\,\\mathrm{d}\\log(\\csc(x)+\\cot(x))\\tag{17}\\\\ &=-\\frac{\\pi^2}4+2\\int_0^{\\pi/2}\\log(\\csc(x)+\\cot(x))\\,\\mathrm{d}t\\tag{18}\\\\ &=-\\frac{\\pi^2}4+2\\int_0^{\\pi/2}\\big(\\log(1+\\cos(x))-\\log(\\sin(x))\\big)\\,\\mathrm{d}t\\tag{19}\\\\ &=-\\frac{\\pi^2}4+2\\left(2\\mathrm{G}-\\frac\\pi2\\log(2)+\\frac\\pi2\\log(2)\\right)\\tag{20}\\\\ &=\\bbox[5px,border:2px solid #C0A000]{-\\frac{\\pi^2}4+4\\mathrm{G}}\\tag{21} \\end{align} $"}
{"_id": "2152846", "title": "", "text": "$\\#_a(u)$"}
{"_id": "5341610", "title": "", "text": "$ \\begin{align} \\int \\frac{1}{x^2+1}dx&=\\arctan x +C \\tag3 \\end{align} $"}
{"_id": "2030648", "title": "", "text": "$\\int_a^x f(t)dt = F(x)-F(a)$"}
{"_id": "1780703", "title": "", "text": "$f(f(x) + x) = xf(x^2 + 1)$"}
{"_id": "2692273", "title": "", "text": "$\\{(x,x),(y,y)\\},\\{(x,x),(y,y),(x,y)\\},\\{(x,x),(y,y),(y,x)\\},\\{(x,x),(y,y),(x,y),(y,x)\\}$"}
{"_id": "417788", "title": "", "text": "$\\begin{cases}\n x = 1\\\\\n y = 0\\\\\n z = 0\\\\\n \\end{cases}$"}
{"_id": "8282883", "title": "", "text": "$P(X_1>X_2 | X_2=x) = 1-F_{X}(x)$"}
{"_id": "2396152", "title": "", "text": "$d(x,p)\\leq d(x,y)+d(y,p)  $"}
{"_id": "6707580", "title": "", "text": "$\\dfrac{1}{2\\pi i}\\int_\\gamma \\dfrac{2i}{z^2+1}=\\int_\\gamma \\dfrac{2i}{z^2+1}=\\dfrac{1}{2\\pi i}\\bigg(\\int_\\gamma\\dfrac{1}{z-i}-\\int_\\gamma\\dfrac{1}{z+i}\\bigg)$"}
{"_id": "3791112", "title": "", "text": "$(K[X]/(f))^\\times$"}
{"_id": "5379654", "title": "", "text": "$\\displaystyle \\left[ \\begin{array}{rrr|r} 1 & 2 & 1 & 3 \\\\ 1 & 3 & -1 & 1 \\\\ 1 & 2 & a^2-8 & a \\\\ \\end{array} \\right] $"}
{"_id": "1809450", "title": "", "text": "$x^3 - x^2 - x$"}
{"_id": "5012242", "title": "", "text": "$A_1 \\subseteq A_2 \\subseteq \\ldots \\subseteq A_K \\subseteq S $"}
{"_id": "6083574", "title": "", "text": "$f_X(x)=\\frac{x}{2}$"}
{"_id": "2474357", "title": "", "text": "$\\forall m,[\\forall m'<m,\\forall n,P(m',n)]\\to\\forall n,[\\forall n'<n,P(m,n')]\\to P(m,n)\\implies\\forall xy,P(x,y)$"}
{"_id": "939586", "title": "", "text": "$K \\subseteq M \\subseteq L$"}
{"_id": "6801746", "title": "", "text": "$\\lim_{n \\to \\infty} \\sup_{A \\in \\mathcal F}|\\mu_n (A) - \\mu(A)| \\to 0,$"}
{"_id": "955741", "title": "", "text": "$ s(x)=\\sum_{i=1}^mc_i\\chi_{A_i}(x)\\ . $"}
{"_id": "3311020", "title": "", "text": "$\\int_{-\\infty}^{\\infty}\\frac{1}{(x^2+x+1)^a}\\,\\mathrm dx, a > 0$"}
{"_id": "2200094", "title": "", "text": "$x^{9/4}=(x^{1/4})^9.$"}
{"_id": "4312744", "title": "", "text": "$|f(z_0)| \\leq |f(z)|$"}
{"_id": "4566852", "title": "", "text": "$x^{n-1}-x^n$"}
{"_id": "430215", "title": "", "text": "$\\qquad \\small \\begin{array} {rrrrrrr}  1 & -1 & 1/2 & -1/6 & 1/24 & -1/120 & 1/720 & -1/5040 \\\\  . & 1 & -3/2 & 1 & -5/12 & 1/8 & -7/240 & 1/180 \\\\  . & . & 1 & -11/6 & 35/24 & -17/24 & 35/144 & -23/360 \\\\  . & . & . & 1 & -25/12 & 15/8 & -49/48 & 7/18 \\\\  . & . & . & . & 1 & -137/60 & 203/90 & -967/720 \\\\  . & . & . & . & . & 1 & -49/20 & 469/180 \\\\  . & . & . & . & . & . & 1 & -363/140 \\\\  . & . & . & . & . & . & . & 1  \\end{array} $"}
{"_id": "8142472", "title": "", "text": "$f:=f_{1},...,f_{n}\\in L^{p}(\\mu)$"}
{"_id": "2315550", "title": "", "text": "$1 + 2 + 3 + 4\\ldots=-\\frac{1}{12}$"}
{"_id": "7899350", "title": "", "text": "$f(x)=\\frac{ax^3}{x^3+3}$"}
{"_id": "2096107", "title": "", "text": "$(z-y,z+y)=1$"}
{"_id": "2656462", "title": "", "text": "$\\int_0^\\infty\\int_0^\\infty f(x,y)dxdy$"}
{"_id": "3594922", "title": "", "text": "$\\mathcal{F}: \\mathbb{R}^n\\times\\mathbb{R}^n\\to \\mathbb{R}$"}
{"_id": "3039225", "title": "", "text": "$\\frac{4}{x}+\\frac{5}{y}\\geq 23$"}
{"_id": "3454622", "title": "", "text": "${\\mathcal A}:X = X:{\\mathcal A} = X$"}
{"_id": "2899094", "title": "", "text": "$d(x,y)=\\frac{|x-y|}{|x-y|+1}$"}
{"_id": "5873658", "title": "", "text": "$ds = \\sqrt{\\left(\\frac{dx}{dy}\\right)^2 + 1}$"}
{"_id": "5078125", "title": "", "text": "$\\sum_{n=0}^{\\infty}(2x^{n}-x^{2n}) = \\sum_{n=0}^{\\infty}2x^{n} -\\sum_{n=0}^{\\infty}x^{2n} = 2 \\sum_{n=0}^{\\infty}x^{n} -\\sum_{n=0}^{\\infty}x^{2n} = 2 \\frac{1}{1-x} - \\frac{1}{1-x^2}$"}
{"_id": "539295", "title": "", "text": "$[\\begin{smallmatrix}a & -b\\\\ b & \\hphantom{-}a\\end{smallmatrix}]$"}
{"_id": "712314", "title": "", "text": "$A_{n+1} \\subset A_n \\subset A_{n-1} \\subset \\ldots$"}
{"_id": "8048338", "title": "", "text": "$(1-p)p_1p_2p_3$"}
{"_id": "358433", "title": "", "text": "$d(x,y)=||x-y||$"}
{"_id": "45723", "title": "", "text": "$A_{1}A_{2}$"}
{"_id": "7744313", "title": "", "text": "$\\lim_{ n \\to \\infty }f(a_n) \\ \\ ?$"}
{"_id": "6808479", "title": "", "text": "$ a \\leq x_2 \\leq b, \\qquad c \\leq y_2 \\leq d$"}
{"_id": "1755641", "title": "", "text": "$3|n^3 - n$"}
{"_id": "2850241", "title": "", "text": "$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx dx, \\  b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx dx$"}
{"_id": "1406275", "title": "", "text": "${C_r}: |z| = r$"}
{"_id": "153919", "title": "", "text": "$\\frac{4}{10}\\frac{1}{4}=\\frac{1}{10}$"}
{"_id": "6013214", "title": "", "text": "$\\det\\begin{pmatrix}A& 0\\\\ C& B\\end{pmatrix} = \\det\\begin{pmatrix}A& D\\\\ 0& B\\end{pmatrix} = \\det(A) \\det(B) . $"}
{"_id": "6802197", "title": "", "text": "$\\bigcup_{k=0}^nG(k)=\\bigl\\{\\langle x,y\\rangle\\in\\Bbb N\\times\\Bbb N:x\\leq n,y\\leq n\\bigr\\}$"}
{"_id": "2608217", "title": "", "text": "$P_3P_2 = P_1P_7$"}
{"_id": "5521299", "title": "", "text": "$ \\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}= \\frac{x^2}{2}+\\frac{1}{2x^2}. $"}
{"_id": "5444743", "title": "", "text": "$x_k= \\sqrt[n]{a}e^{\\frac{i(2k+1)\\pi}{n}}$"}
{"_id": "3715123", "title": "", "text": "$\\lfloor\\frac{a+b-1}{b}\\rfloor = \\lfloor\\frac{a}{b} + \\frac{b-1}{b}\\rfloor$"}
{"_id": "388391", "title": "", "text": "$f(z)=z^2+f(z^2)$"}
{"_id": "5410264", "title": "", "text": "$ s(n) =  \\sum_{k=0}^n|\\cos k| - \\frac{2}{\\pi}n  = \\sum_{k=0}^{n^*}|\\cos k| - \\frac{2}{\\pi} n^*  +  \\sum_{k=n^* +1}^n|\\cos k| - \\frac{2}{\\pi}(n-n^*) \\\\ = s(n^*) + \\sum_{k=n^* +1}^n|\\cos k| - \\frac{2}{\\pi}(n-n^*) $"}
{"_id": "8994758", "title": "", "text": "$\\sum_{n\\geq1}\\frac{\\log^{2}\\left(n\\right)}{n^{2}}=\\sum_{n=1}^{N}\\frac{\\log^{2}\\left(n\\right)}{n^{2}}+\\sum_{n\\geq N+1}\\frac{\\log^{2}\\left(n\\right)}{n^{2}}\\leq  $"}
{"_id": "6536475", "title": "", "text": "$d(x,z) \\leq d(x,y)  + d(y,u)  + d(u,z)$"}
{"_id": "6928720", "title": "", "text": "$\\to E[X_{T \\wedge n} | \\mathscr{F_0}] =/ \\le X_{T \\wedge 0} \\ \\text{by choosing m = 0}$"}
{"_id": "3372847", "title": "", "text": "$(1+2^{\\frac n2})^2$"}
{"_id": "23041", "title": "", "text": "$[(n-1)a+b](b-a)^{n-1}$"}
{"_id": "108048", "title": "", "text": "$         \\begin{pmatrix}         a & b \\\\         -\\bar b & \\bar a \\\\         \\end{pmatrix} $"}
{"_id": "7886956", "title": "", "text": "$\\equiv \\; s \\vee r$"}
{"_id": "1759295", "title": "", "text": "$\\left(\\begin{array}{cc}\n A&B\\\\\n C&D\n \\end{array}\\right).$"}
{"_id": "1987432", "title": "", "text": "$b_n=\\frac{a_1+a_2+...+a_n}{n}$"}
{"_id": "7915152", "title": "", "text": "$\\mathrm{Cov}(X,Y)=\\mathrm{Cov}(-X,Y)=-\\mathrm{Cov}(X,Y)$"}
{"_id": "5945165", "title": "", "text": "$I_n = \\pi n$"}
{"_id": "4337683", "title": "", "text": "$u^2 + v^2 = (1+\\cos\\varphi)^2 + \\sin^2\\varphi = 1 + 2\\cos\\varphi + \\cos^2\\varphi + \\sin^2\\varphi = 2(1+\\cos\\varphi) = 2u.$"}
{"_id": "1412395", "title": "", "text": "$\\mu(A_n)\\geq \\mu(A)$"}
{"_id": "8617345", "title": "", "text": "$ \\|M\\|_2 = \\sigma_1(M) = \\sqrt{\\lambda_{max}(M^TM)} $"}
{"_id": "271553", "title": "", "text": "$T_x(X) \\times T_x(X).$"}
{"_id": "4552660", "title": "", "text": "$M_1 = \\begin{bmatrix} AB &  0\\\\ B  &  0\\\\ \\end{bmatrix}$"}
{"_id": "412904", "title": "", "text": "$B_n= { 2^n\\choose 2^{n-1}}$"}
{"_id": "3446079", "title": "", "text": "$\\zeta(3)=\\lim_{N\\to \\infty}{1\\over N}\\sum_{k=1}^{N}{1\\over k^{\\phi^2}\\ln{\\left(1+{k^{\\phi^{-2}}\\over N}\\right)}}$"}
{"_id": "5193715", "title": "", "text": "$\\int\\Pi^2=2$"}
{"_id": "340310", "title": "", "text": "$|\\{\\alpha \\in A: \\alpha < \\omega\\}|<\\omega$"}
{"_id": "2107239", "title": "", "text": "$I=\\log (x) \\sin ^{-1}(x)-\\int \\frac{\\sin ^{-1}(x)}{x}\\,dx$"}
{"_id": "5492085", "title": "", "text": "$ \\begin{align} \\left(1 + \\frac{x}{n}\\right)^n & = 1 + x + \\frac{n(n-1)}{2!n^2}x^2 + \\frac{n(n-1)(n-2)}{3! n^3}x^3 + \\cdots \\\\\\\\ & = \\frac{x^0}{0!} + \\frac{x^1}{1!} + \\left(\\frac{n-1}{n}\\right)\\frac{x^2}{2!} + \\left(\\frac{(n-1)(n-2)}{n^2}\\right)\\frac{x^3}{3!} + \\cdots  \\end{align} $"}
{"_id": "649604", "title": "", "text": "$P(1) \\implies P(2),$"}
{"_id": "3976612", "title": "", "text": "$ \\mu\\left(\\liminf\\limits_{n\\to\\infty}A_n\\right)\\leq \\liminf\\limits_{n\\to\\infty}\\mu(A_n) $"}
{"_id": "6658475", "title": "", "text": "$\\left[\\begin{array}{cc}1&0&0\\\\0&(x^1)^2&0\\\\0&0&(x^1 \\sin x^2)^2\\end{array}\\right]$"}
{"_id": "5817450", "title": "", "text": "$f(t)=t-t^2+2$"}
{"_id": "5904072", "title": "", "text": "$x-\\delta < f(x+\\delta) < x + \\delta$"}
{"_id": "4924271", "title": "", "text": "$a_n = \\|x_{n+1} - x_n\\|$"}
{"_id": "25604", "title": "", "text": "$S=\\{E_1,E_2,E_3\\}$"}
{"_id": "8026137", "title": "", "text": "$y = \\frac{2+kz}{k-1}.$"}
{"_id": "6700200", "title": "", "text": "$ \\lim_{n \\to \\infty} \\frac {\\sqrt{n+1}+\\sqrt{n+2}+...+\\sqrt{2n}}{n^{3/2}}. $"}
{"_id": "4983911", "title": "", "text": "$H(\\omega) = 2\\pi(\\cos(\\omega)-1 - i\\sin(\\omega))\\frac{\\delta(\\omega)}{2-2\\cos(\\omega)}.$"}
{"_id": "8334192", "title": "", "text": "$=2^{k-1}3^{k}(3-1)=6^k.$"}
{"_id": "5690391", "title": "", "text": "$R = \\mathbb Z_4[x]/((x^2+1)\\mathbb Z_4[x])$"}
{"_id": "3896298", "title": "", "text": "$ mn = |S \\times T|$"}
{"_id": "6103228", "title": "", "text": "$(1-r)s=s-rs=1-r^n$"}
{"_id": "9316969", "title": "", "text": "$Y_0=\\{a,a+2,a+4,\\cdots\\}, \\quad Y_1=\\{b,b+2,b+4,\\cdots\\}.$"}
{"_id": "9288034", "title": "", "text": "$\\alpha \\subset \\Omega$"}
{"_id": "2040659", "title": "", "text": "$\\\\n! + 2, \\\\n! + 3, \\\\n! + 4, \\\\n! + 5, \\\\\\dots \\\\n + k$"}
{"_id": "5391875", "title": "", "text": "$\\int_{1}^{-1}dx\\sqrt{1+\\frac{dy}{dx}^2}$"}
{"_id": "3205132", "title": "", "text": "$(sin(\\varphi), -cos(\\varphi))$"}
{"_id": "4350294", "title": "", "text": "$\\int_0^\\infty f(x)dx = 3$"}
{"_id": "2119571", "title": "", "text": "$p_1p_2...p_k \\mid a$"}
{"_id": "9297052", "title": "", "text": "$AX^2+BY^2=d$"}
{"_id": "9086071", "title": "", "text": "$(x+y)^r\\leq x^r+ry^r\\leq x^r+y^r\\mathrm{.}$"}
{"_id": "8043578", "title": "", "text": "$\\forall\\epsilon>0,\\exists\\delta>0:0<|x-a|<\\delta\\implies|f(x)-L|<\\epsilon.$"}
{"_id": "6368667", "title": "", "text": "$\\mathrm{gcd}(a+b,a-b)\\leq 2$"}
{"_id": "454756", "title": "", "text": "$R=S\\oplus T$"}
{"_id": "1465597", "title": "", "text": "$f\\in W_0^{1,2}((0,R),r^{N-1})$"}
{"_id": "4214122", "title": "", "text": "$|AB|=6$"}
{"_id": "4217040", "title": "", "text": "$1-P(A^c\\cup B^c)=1-(P(A^c)+P(B^c))$"}
{"_id": "8408938", "title": "", "text": "$3\\int_0^1 \\frac{\\tan^{-1}(x)}{x}-2\\int_0^{1/2} \\frac{\\tan^{-1}(x)}{x}-\\int_0^{1/3} \\frac{\\tan^{-1}(x)}{x}-\\frac 12 \\int_0^{3/4} \\frac{\\tan^{-1}(x)}{x}=\\frac{\\pi\\log 2}{2}$"}
{"_id": "5104200", "title": "", "text": "$(\\overline {\\mathbb{R}})^2$"}
{"_id": "2014215", "title": "", "text": "$f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{\\frac{-x^2}{2}}$"}
{"_id": "1748380", "title": "", "text": "$\\frac{(1 + r)^{1 - T_1} - (1 + r)^{-T_2}}{r}$"}
{"_id": "3820469", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{1}{n^2}=\\sum_{n\\geq 1}\\frac{1}{(2n)^2}+\\sum_{n\\geq 1}\\frac{1}{(2n-1)^2} $"}
{"_id": "1340769", "title": "", "text": "$n!\\approx \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.$"}
{"_id": "4377556", "title": "", "text": "$ \\sum |x_k|^s \\leq n^{\\frac{r-s}{r}} \\left(\\sum |x_k|^r \\right)^{s/r}. $"}
{"_id": "1646639", "title": "", "text": "$\\frac{-44}{-10}=4+\\frac{4}{10}$"}
{"_id": "1547937", "title": "", "text": "$K_1\\subset K_2\\subset\\dots$"}
{"_id": "7627806", "title": "", "text": "$\\displaystyle\\;f_1(n) = \\binom{m+n}{m}\\;$"}
{"_id": "6933316", "title": "", "text": "$j_1\\mid j_2$"}
{"_id": "9022932", "title": "", "text": "$\\lim_{x\\rightarrow 1^{+}} \\frac{\\sqrt{1-x^2}}{1-x^2}$"}
{"_id": "1531757", "title": "", "text": "$1/x+1/y+1/z = 4$"}
{"_id": "3710880", "title": "", "text": "$r(X+Y)\\subseteq r(r(X)+r(Y))$"}
{"_id": "3881919", "title": "", "text": "$\\{e_1\\}\\subset \\{e_1,e_2\\}\\subset \\dots \\subset \\{e_1,e_2,e_3,\\dots ,e_n\\}$"}
{"_id": "444338", "title": "", "text": "$(*,*,x,y),(*,x,*,y),(*,x,y,*),(x,*,*,y),(x,*,y,*),(x,y,*,*) $"}
{"_id": "3706201", "title": "", "text": "$\\lim_{n\\to\\infty}\\mu(A_n^{\\complement})=\\mu(A^{\\complement})$"}
{"_id": "3336035", "title": "", "text": "$1=\\dfrac{2}{x^2}+\\dfrac{3}{y^2}+\\dfrac{4}{z^2}$"}
{"_id": "1884662", "title": "", "text": "$d \\in \\langle m,n\\rangle$"}
{"_id": "2004048", "title": "", "text": "$\\frac{1}{5}e^{\\frac{-x}{5}}$"}
{"_id": "5818206", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{\\sqrt1+\\cdots+\\sqrt n}{n\\sqrt n}=\\lim_{n\\to\\infty}\\frac{\\sqrt n}{n\\sqrt n-(n-1)\\sqrt{n-1}}=\\lim_{n\\to\\infty}\\frac{n^2\\left(1+\\left(1-\\frac1n\\right)\\sqrt{1-\\frac1n}\\right)}{n^3-(n-1)^3}=\\frac23$"}
{"_id": "4493212", "title": "", "text": "$f(f(x)^{-1}x)=f(x)^{-1}f(x)=1$"}
{"_id": "3735759", "title": "", "text": "$x_1 - x_0, \\ldots, x_k - x_0 \\in \\mathbb{R}^n$"}
{"_id": "7147170", "title": "", "text": "$\\frac{1}{(1+r)}+\\frac{1}{(1+r)^2}+\\dots+\\frac{1}{(1+r)^{n-1}}+\\frac{1}{(1+r)^n}$"}
{"_id": "7878099", "title": "", "text": "$|f'(x)| \\le 2|f(x)|$"}
{"_id": "690366", "title": "", "text": "$\\frac{1}{a}+\\frac{2}{b}+\\frac{3}{c}=1$"}
{"_id": "3182273", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{1}{n^2}=\\sum_{n\\geq 1}\\frac{3}{n^2\\binom{2n}{n}}\\tag{3} $"}
{"_id": "8264545", "title": "", "text": "$F = \\mathbb{R}[x]/(x^2 + 1)$"}
{"_id": "4783374", "title": "", "text": "$f_{x_{(4)}} =  20e^{-2x}(1-e^{-x})^3  $"}
{"_id": "2643379", "title": "", "text": "$E|X_n|^r\\to E|X|^r<\\infty$"}
{"_id": "5915710", "title": "", "text": "$A_1\\subseteq A_2\\subseteq...$"}
{"_id": "5494716", "title": "", "text": "$ \\sum_{n=-\\infty}^\\infty e^{-n^2\\pi x}=\\frac1{\\sqrt{x}}\\sum_{n=-\\infty}^\\infty e^{-\\frac{n^2\\pi}x}, $"}
{"_id": "1089696", "title": "", "text": "$J_n=\\int x^n \\tan^{-1}(x)\\,dx=\\frac{x^{n+1} \\left((n+2) \\tan ^{-1}(x)-x \\,    _2F_1\\left(1,\\frac{n}{2}+1;\\frac{n}{2}+2;-x^2\\right)\\right)}{(n+1) (n+2)}$"}
{"_id": "4210308", "title": "", "text": "$\\int_1^\\infty \\frac{1}{(1+x^2)^\\alpha}\\,dx$"}
{"_id": "2605867", "title": "", "text": "$x> A=N\\delta$"}
{"_id": "3725782", "title": "", "text": "$\\forall \\epsilon>0\\quad\\exists\\delta>0\\quad\\forall y\\quad|x-y|<\\delta\\Rightarrow|f(x)-f(y)|<\\epsilon$"}
{"_id": "7636405", "title": "", "text": "$\\mu\\left(\\bigcup_N \\bigcap_{n=N}^{\\infty} A_n \\right) = \\mu\\left(\\bigcup_N B_N\\right) = \\lim_{N \\to \\infty} \\mu(B_N) \\leq \\lim_{N \\to \\infty} \\inf_{n \\geq N} \\mu(A_n) = \\lim \\inf \\mu(A_n)$"}
{"_id": "2061430", "title": "", "text": "$\\mathcal B = \\{\\; \\{x\\}\\;\\}.$"}
{"_id": "5587550", "title": "", "text": "$ \\displaystyle \\lim \\limits_{n \\rightarrow \\infty}\\hspace{2mm}\\lim \\limits_{h \\rightarrow 0}\\sum \\limits_{i = 1}^n\\frac{1}{n} f(x+\\frac{h}{n}i)]=$"}
{"_id": "2348417", "title": "", "text": "$\\operatorname{cl}V\\subseteq B\\subseteq U$"}
{"_id": "3946168", "title": "", "text": "$\\int_0^{2\\pi} \\left(a\\cos x+\\sqrt{1-a^2\\sin^2(x)}\\right)^{2k} dt=2\\pi\\sum_{m=1}^k \\binom{k}{m} \\binom{k-1}{m-1}a^{2(k-m)}$"}
{"_id": "5119455", "title": "", "text": "$W(\\gamma, 0) = \\text{deg}(\\gamma/|\\gamma|)$"}
{"_id": "2818466", "title": "", "text": "$c=\\{\\{\\varnothing\\}\\}=\\{\\{a\\}\\}=\\{b\\}$"}
{"_id": "508495", "title": "", "text": "$|\\langle x, e_n \\rangle |^2 = |x_i|^2$"}
{"_id": "1517366", "title": "", "text": "$P(S_t\\geqslant x\\mid\\mathcal F_0)=M_0/x$"}
{"_id": "6246095", "title": "", "text": "$\\eqalign{   & S \\equiv y,\\,\\,\\,\\,\\,\\,\\,\\,\\,s \\equiv x  \\cr    & P(s) = {1 \\over {12}}  \\cr    & Q(s) = {k \\over {48}}f(s)  \\cr    & A(s) = \\int_a^s {P(t)dt}  = \\int_a^s {{1 \\over {12}}dt}  = {1 \\over {12}}(s - a) \\cr}\\tag{7} $"}
{"_id": "1310604", "title": "", "text": "$e^x \\leq  \\left(1+\\frac{x}{n}\\right)^{n+1}$"}
{"_id": "5050575", "title": "", "text": "$\\text{add}(\\mathsf{nonstat}_\\gamma)=\\text{cov}(\\mathsf{nonstat}_\\gamma)=\\kappa$"}
{"_id": "4688423", "title": "", "text": "$(a+b, a)$"}
{"_id": "8076601", "title": "", "text": "$\\{(x,y) \\in \\mathbb{R}^n |\\ a \\le x \\le b, \\ g(x) \\le y \\le h(x) \\}$"}
{"_id": "7620063", "title": "", "text": "$ f(r) = \\frac{r^m - \\left(\\frac{r}{1+r}\\right)^m + \\left(\\frac{r^2}{1+r}\\right)^m - r^{m+1}}{r^m} \\implies\\\\ f(r) = \\frac{[r(1+r)]^m - r^m + r^{2m} - r[r(1 + r)]^m}{[r(1+r)]^m} $"}
{"_id": "6644082", "title": "", "text": "$f(x) = \\frac{1 + 2x}{1 - x^2}$"}
{"_id": "6006267", "title": "", "text": "$2^n=(1+1)^n=1+n+\\frac{n(n-1)}{2}+\\frac{n(n-1)(n-2)}{6}+\\frac{n(n-1)(n-2)(n-3)}{24}+\\cdots.$"}
{"_id": "4470857", "title": "", "text": "$\\int_0^{\\pi} f(t)\\sin(nt)~dt=0$"}
{"_id": "1547747", "title": "", "text": "$\\cos^2\\varphi+\\sin^2\\varphi=1$"}
{"_id": "958643", "title": "", "text": "$cov(X,X) - cov(X,Y) cov(Y,Y)^{-1} cov(Y,X)$"}
{"_id": "3023807", "title": "", "text": "$\\therefore \\boxed{f(x)=9^x-3^x}$"}
{"_id": "2171074", "title": "", "text": "$\\Rightarrow \\cdots < A < A\\oplus B<(A\\oplus B)\\oplus [(A\\oplus B )\\oplus \\left\\{(A\\oplus B)\\oplus B\\right\\}] <(A\\oplus B )\\oplus \\left\\{(A\\oplus B)\\oplus B\\right\\}<(A\\oplus B)\\oplus B < B < \\cdots$"}
{"_id": "94427", "title": "", "text": "$(\\sec\\varphi - \\tan\\varphi)^2 = \\left(\\frac{1}{\\cos \\varphi}-\\frac{\\sin \\varphi}{\\cos \\varphi}\\right)^2 = \\frac{(1 - \\sin \\varphi)^2}{\\cos^2 \\varphi}$"}
{"_id": "8295735", "title": "", "text": "$ax+by=dz$"}
{"_id": "542442", "title": "", "text": "$101!+2, 101!+3,...,101!+101$"}
{"_id": "6255419", "title": "", "text": "$ \\sqrt{z}=\\sqrt{r}e^{i\\varphi/2} $"}
{"_id": "1845549", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^n \\frac{1}{k/n+1}.$"}
{"_id": "2169557", "title": "", "text": "$105!+2,105!+3,105!+4,...,105!+105$"}
{"_id": "1368243", "title": "", "text": "$P \\in V \\subseteq U \\subseteq X$"}
{"_id": "5805237", "title": "", "text": "$S^i_j$"}
{"_id": "4668150", "title": "", "text": "$n \\in F_3$"}
{"_id": "8357905", "title": "", "text": "$ \\lim_{n \\to\\infty}\\{x_n\\} = \\lim_{n \\to \\infty} \\inf \\{x_n\\} $"}
{"_id": "4136302", "title": "", "text": "$b_n = \\dfrac{x_1 + x_2 +...+ x_n}{n}$"}
{"_id": "6023374", "title": "", "text": "$1/|x|^{2n}$"}
{"_id": "9242379", "title": "", "text": "$S:=\\begin{cases}  x^{2}+y^{2}=1  \\\\   z=1   \\end{cases}$"}
{"_id": "4565962", "title": "", "text": "$[a]+\\left( [b]+[c]\\right)=[a]+[b+c]=[a+(b+c)]=[(a+b)+c]=[a+b]+[c]=\\left( [a]+[b]\\right)+[c]$"}
{"_id": "407235", "title": "", "text": "$ 1 + 2 + 3 + 4 + \\cdots + n \\; {“ \\;=\\; ”} - \\frac{1}{12} . $"}
{"_id": "8487402", "title": "", "text": "$\\begin{array}{}\n \\frac{1}{i} &= \\frac{1}{i} \\cdot \\frac{-i}{-i} \\\\\n &= \\frac{-i}{i \\cdot (-i)} \\\\\n &= \\frac{-i}{-(i \\cdot i)} \\\\\n &= \\frac{-i}{-(-1)} \\\\\n &= -i.\n \\end{array}$"}
{"_id": "4660589", "title": "", "text": "$\\tan\\theta=\\frac{9}{2}$"}
{"_id": "2298679", "title": "", "text": "$\\forall \\gamma(a),\\gamma(b)\\in G$"}
{"_id": "2519872", "title": "", "text": "$\\forall x,y \\in A: xRy \\wedge yRx \\Rightarrow x=y$"}
{"_id": "363158", "title": "", "text": "$(r,s) = rs$"}
{"_id": "2656371", "title": "", "text": "$ \\left\\vert \\frac{f(x)-f(x_{0})-H(x_{0})\\cdot(x-x_{0})}{\\Vert x-x_{0\\Vert}% }\\right\\vert \\leq4\\varepsilon $"}
{"_id": "8509346", "title": "", "text": "$\n \\begin{align*}\n \\int_0^{2\\pi}\\frac{\\sin x}{x}\\,dx&=\\int_0^{\\pi}\\frac{\\sin x}{x}\\,dx+\\int_\\pi^{2\\pi}\\frac{\\sin x}{x}\\,dx\\\\\n &=\\int_0^{\\pi}\\frac{\\sin x}{x}\\,dx+\\int_0^{\\pi}\\frac{\\sin(x+\\pi)}{x+\\pi}\\,dx\\\\\n &=\\int_0^{\\pi}\\Bigl(\\frac{1}{x}-\\frac{1}{x+\\pi}\\Bigr)\\sin x\\,dx\\\\\n &=\\pi\\int_0^{\\pi}\\frac{\\sin x}{x(x+\\pi)}\\,dx\\\\\n &>0\n \\end{align*}\n $"}
{"_id": "6898987", "title": "", "text": "$P(E)=\\frac{\\operatorname{elements in E}}{\\operatorname{elements in S}}$"}
{"_id": "409532", "title": "", "text": "$\\mathbb{R}^{n+1} - M$"}
{"_id": "2591691", "title": "", "text": "$P(X \\in A_0|H_0)=1$"}
{"_id": "4131729", "title": "", "text": "$y = \\frac{ab+c}{a-b}$"}
{"_id": "1353796", "title": "", "text": "$\\sum_{n=m+1}^\\infty \\|e_n-f_n\\|^2 <1$"}
{"_id": "4283226", "title": "", "text": "$\\{i\\gamma:-1\\le \\gamma \\le 1\\}$"}
{"_id": "3557217", "title": "", "text": "$ T=(T^*)^*=(T^*T)^*=T^*T=T^*. $"}
{"_id": "4898511", "title": "", "text": "$\\gamma + a * \\gamma + \\frac{\\gamma ^ 2 * n}{a} + \\gamma * n \\ge n*\\gamma + \\gamma + a$"}
{"_id": "8205099", "title": "", "text": "$(\\mathbb{Z}[x]/(x^n-1))$"}
{"_id": "81267", "title": "", "text": "$f^+$"}
{"_id": "1272516", "title": "", "text": "$x=b^{log_b(x)}$"}
{"_id": "8630486", "title": "", "text": "$P[X\\geq1]=1$"}
{"_id": "1402309", "title": "", "text": "$d(x,e)\\leqslant d(x,y)+d(y,e)<\\varepsilon.$"}
{"_id": "8308509", "title": "", "text": "$\\sum\\limits_{k=0}^\\infty \\frac{(\\frac{1}{\\gamma})^k}{\\Gamma({1+\\frac{1}{\\gamma}+k})}=\\gamma^{\\frac{1}{\\gamma}}e^{\\frac{1}{\\gamma}} \\frac{\\Gamma_L(\\frac{1}{\\gamma},\\frac{1}{\\gamma})}{\\Gamma(\\frac{1}{\\gamma})}$"}
{"_id": "8356966", "title": "", "text": "$f(\\mathbf{a}+\\mathbf{b})=f(\\mathbf{a})+f(\\mathbf{b})$"}
{"_id": "2432519", "title": "", "text": "$|ab|=k$"}
{"_id": "1869839", "title": "", "text": "$f \\left( x \\right) = \\frac{9^x}{3+9^x}$"}
{"_id": "7354164", "title": "", "text": "$\\alpha=\\alpha^\\omega$"}
{"_id": "356057", "title": "", "text": "$f(x) = \\binom{x}{2} = \\frac{x(x-1)}{2}$"}
{"_id": "3936447", "title": "", "text": "$\\frac{1}{2^{N-1}}.$"}
{"_id": "7803555", "title": "", "text": "$|P_1P_2| = |P_1||P_2|$"}
{"_id": "792773", "title": "", "text": "$P(n) \\implies P(n-1)$"}
{"_id": "4845112", "title": "", "text": "$ x_{2n} = \\frac{(-1)^{2n} + 1}{2n} = \\frac{2}{2n} = \\frac{1}{n} $"}
{"_id": "2650920", "title": "", "text": "$f(x)=\\lim_{n\\to\\infty}[2x+4x^3+\\cdots+2nx^{2n-1}]$"}
{"_id": "6750336", "title": "", "text": "$E(X_1+X_2+X_3+X_4)=E(X_1)+E(X_2)+E(X_3)+E(X_4)=E(40)=40.$"}
{"_id": "5113911", "title": "", "text": "$(sa)b = a(sb)$"}
{"_id": "2162369", "title": "", "text": "$C = \\{(x,y) \\in \\mathbb{R^2} : 0\\leq x\\leq 1, 0\\leq y\\leq 1\\}$"}
{"_id": "6873608", "title": "", "text": "$\\frac{1}{(1 + r)^1}$"}
{"_id": "979212", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}|X_n(\\omega)-E(X_n)|=0,$"}
{"_id": "55722", "title": "", "text": "$\\sum _{n=1}^{\\infty } \\frac{1}{n^2+1}$"}
{"_id": "4218099", "title": "", "text": "$Z=\\frac{\\lambda}a+\\frac1{(a+3)^2}=\\frac{\\lambda}b+\\frac1{(b+3)^2}=\\frac{\\lambda}c+\\frac1{(c+3)^2}=\\frac{\\lambda}d+\\frac1{(d+3)^2}$"}
{"_id": "5821907", "title": "", "text": "$L=\\{\\langle x,y\\rangle\\in\\Bbb R^2:x<y\\}$"}
{"_id": "3580235", "title": "", "text": "$(x_k^{1,2},y_k^{1,2})\\rightarrow(0,0)$"}
{"_id": "5714143", "title": "", "text": "$\\det(A)\\det(C)=\\det(A)\\det(C^T)=\\det(AC^T)=\\det((\\det A)I)=(\\det A)^n$"}
{"_id": "63189", "title": "", "text": "$t^+$"}
{"_id": "6938292", "title": "", "text": "$ \\left[ \\begin{array}{ccccc} 1 & -1 & 5 & -2 & 2 \\\\ 2 & -2 & -2 & 5 & 1 \\\\ 0 &0 &  -12 & 9 & -3 \\\\ -1 & 1 &  7 & -7 & 1 \\\\ \\end{array}\\right] $"}
{"_id": "3133961", "title": "", "text": "$|f(z)|\\,|\\Im(z)|^2\\leq 1$"}
{"_id": "3917650", "title": "", "text": "$f'(a)<k<f'(b)$"}
{"_id": "1790043", "title": "", "text": "$P(max\\space M)-P(max \\space x)$"}
{"_id": "7194957", "title": "", "text": "$\\int_0^1 f(t)\\,dt = \\int_1^2 f(t)\\,dt = 1$"}
{"_id": "6179527", "title": "", "text": "$P(X_{n+N} \\notin S|\\mathcal{F}_n) \\geq \\epsilon 1_{X_n \\in S}$"}
{"_id": "1323786", "title": "", "text": "$=\\int_0^\\infty f(x)\\,dx$"}
{"_id": "4336371", "title": "", "text": "$[x,y]=xy-yx\\in L$"}
{"_id": "6388308", "title": "", "text": "$f(x) = 2e^{\\frac{x}{3}} - e^{-\\frac{x}{2}}$"}
{"_id": "3685313", "title": "", "text": "$d(x,A) =  Inf(d(x,y): y in A)$"}
{"_id": "6656229", "title": "", "text": "$||B||_2=\\sqrt{\\rho(B^TB)}$"}
{"_id": "7978019", "title": "", "text": "$ y = |\\frac{b+cx}{a+x}| $"}
{"_id": "5419333", "title": "", "text": "$f(a+b)< f(a) + f(b)$"}
{"_id": "8140890", "title": "", "text": "$S=(e_1, e_2, e_3)$"}
{"_id": "3586423", "title": "", "text": "$(\\Sigma_{\\gamma\\in \\tau}a_{\\gamma}t^\\gamma) +_{N} (\\Sigma_{\\gamma\\in \\tau}b_{\\gamma}t^\\gamma) =\\Sigma_{\\gamma\\in \\tau}(a_{\\gamma}+b_{\\gamma})t^\\gamma$"}
{"_id": "1063860", "title": "", "text": "$\\frac{a}{x}+\\frac{b}{y}=x$"}
{"_id": "5397184", "title": "", "text": "$\\left(\\frac{1}{2r}\\right)^2$"}
{"_id": "7996117", "title": "", "text": "$4^x+15^x=9^x+10^x(3^x-2^x)(5^x+3^x-2^x)$"}
{"_id": "1916999", "title": "", "text": "$e^x\\ge \\left(1+\\frac x2\\right)^2 \\tag 2$"}
{"_id": "3167750", "title": "", "text": "$X=\\{(c_0,c_1),(c_1,c_0)\\}$"}
{"_id": "4604882", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}(x_n)(1/t_n)=$"}
{"_id": "7128404", "title": "", "text": "$\\int \\frac{\\tan{x}}{1+\\tan{x}}= \\int 1-\\frac{1}{1+\\tan{x}}$"}
{"_id": "6502105", "title": "", "text": "$ a^{log_a u} = \\,u $"}
{"_id": "9011070", "title": "", "text": "$\\Bbb R\\setminus[a,b)=\\bigcup_{x<a}[x,a)\\cup\\bigcup_{x>b}[b,x)$"}
{"_id": "3417257", "title": "", "text": "$f'(0)=f'(\\pi)=0$"}
{"_id": "8061911", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^nE[X_i^+]=\\infty$"}
{"_id": "4470639", "title": "", "text": "$n!, \\dots, n!+n$"}
{"_id": "2218041", "title": "", "text": "$f(\\frac{1}{2^n}) = f(\\frac{2}{2^{n+1}}) = f(\\frac{1}{2^{n+1}})^2$"}
{"_id": "5068324", "title": "", "text": "$a=\\frac{bx}{1-x}$"}
{"_id": "4836117", "title": "", "text": "$\\|x_m - x_n \\| < \\varepsilon$"}
{"_id": "6774046", "title": "", "text": "$m' = P \\frac{r}{1-(1+r)^{-12 y}} - f \\frac{r}{(1+r)^{12}-1}$"}
{"_id": "214142", "title": "", "text": "$\\gamma<\\eta\\implies\\gamma<\\sup X\\implies\\gamma<\\gamma'$"}
{"_id": "7763801", "title": "", "text": "$|f(z)|\\leq {\\rm Max}_{\\theta}|f(\\exp(i\\theta))|$"}
{"_id": "1660703", "title": "", "text": "$ \\int_0^\\infty f(x) < \\int_0^\\infty g(x) $"}
{"_id": "2632517", "title": "", "text": "$\\int_{0}^\\infty {\\sin^3(x)\\over x}dx$"}
{"_id": "6546863", "title": "", "text": "$ T = \\begin{pmatrix} A & B \\\\ B & A \\end{pmatrix} $"}
{"_id": "4343421", "title": "", "text": "$ax+by = A\\tag{1}$"}
{"_id": "7469541", "title": "", "text": "$P(\\{s\\}) = \\frac{1}{6}$"}
{"_id": "7459449", "title": "", "text": "$p_1p_2 \\mid p-1$"}
{"_id": "2246874", "title": "", "text": "$ \\frac a {1-r} = \\frac{(1-p)^n p}{1-(1-p)} = (1-p)^n. $"}
{"_id": "334776", "title": "", "text": "$=\\frac12\\cdot\\frac{(1+r+r^2)-(1-r+r^2)}{(1+r+r^2)(1-r+r^2)}$"}
{"_id": "6801470", "title": "", "text": "$\\int_0^\\infty dx \\ f(x) = \\int_0^\\infty dx \\ g(x) \\rightarrow \\int_0^\\infty dx \\ f(x)h(x) = \\int_0^\\infty dx \\ g(x)h(x) $"}
{"_id": "4552817", "title": "", "text": "$\\psi_n(x)=a^{-1/2}e^{i\\pi nx/a}$"}
{"_id": "7268257", "title": "", "text": "$I(a_1) = a_1$"}
{"_id": "5024619", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} + \\frac{4}{z} = 4$"}
{"_id": "1203721", "title": "", "text": "$\\ell(x,v)=2R|\\langle v,\\nu_x\\rangle|$"}
{"_id": "2918948", "title": "", "text": "$\\psi(x)=\\sum_{j=1}^nx_j\\rho_j(x)$"}
{"_id": "527593", "title": "", "text": "$\\!\\!\\!\\! \\overbrace{\\begin{align}\\color{#0a0}{n^2}\\equiv(\\pm1)^2\\equiv \\color{#0a0}1\\\\  n^{\\color{#c00}{1+2k}}\\!\\equiv n(\\color{#0a0}{n^2})^k\\!\\equiv\\color{#c00} n\\end{align}}^{\\Large\\qquad\\quad n\\ \\ \\equiv\\ \\ \\pm 1\\ \\ {\\rm if}\\ \\  \\color{#a0f}{n\\, \\not \\equiv\\, 0}}\\!\\! $"}
{"_id": "1016996", "title": "", "text": "$f\\left(x^2+xf(y)\\right)=xf(x+y)$"}
{"_id": "769538", "title": "", "text": "$ p_e = \\frac{(1-p)p}{1-(1-p)^2}=\\frac{1-p}{2-p}.$"}
{"_id": "491063", "title": "", "text": "$=\\sigma_w^2*(\\phi^{2(t-1)}+\\phi^{2(t-2)}+...+\\phi^{2*0})=\\sigma_w^2\\sum_{i=0}^{t-1}(\\phi^2)^i=\\sigma_w^2[\\dfrac{1-(\\phi^2)^t}{1-\\phi^2}] $"}
{"_id": "8428514", "title": "", "text": "$xRy \\wedge yRx \\Longrightarrow xRx$"}
{"_id": "440611", "title": "", "text": "$|ab| = 1,$"}
{"_id": "4603284", "title": "", "text": "$[0] + [u] = [0+u] = [u] = [u+0] = [u] + [0]$"}
{"_id": "5376129", "title": "", "text": "$\\cos\\theta = \\dfrac{x}{2}$"}
{"_id": "8744000", "title": "", "text": "$p_1p_2\\ldots p_i2^j$"}
{"_id": "7822426", "title": "", "text": "$A = \\begin{bmatrix} 1 & a & a^2 \\\\ 1 & b & b^2 \\\\ 1 & c & c^2 \\\\ \\end{bmatrix}$"}
{"_id": "1357574", "title": "", "text": "$\\begin{cases}x=0 \\\\ y=0 \\end{cases}$"}
{"_id": "7745668", "title": "", "text": "$|f(z)| \\le k|g(z)|$"}
{"_id": "1893192", "title": "", "text": "$\\begin{align} \\int\\frac{1}{(x^2+2)^2}dx&=\\frac{\\sqrt{2}}{4}\\int  \\frac{\\sec^2 \\theta}{\\sec^4 \\theta}\\,d\\theta\\\\\\\\ &=\\frac{\\sqrt{2}}{4}\\int  \\cos^2 \\theta \\,d\\theta\\\\\\\\ &=\\frac{\\sqrt{2}}{4}\\int  \\left(\\frac{1+\\cos 2x}{2}\\right) \\,d\\theta\\\\\\\\ &=\\frac{\\sqrt{2}}{8} \\left(\\theta +\\frac12 \\sin 2\\theta\\right)\\\\\\\\ &=\\frac{\\sqrt{2}}{8} \\left(\\theta + \\sin \\theta \\cos \\theta \\right)\\\\\\\\ &=\\frac{\\sqrt{2}}{8} \\left(\\arctan(x/\\sqrt{2}) + \\frac{\\sqrt{2}x}{x^2+2} \\right) \\tag 3\\\\\\\\ \\end{align}$"}
{"_id": "7977242", "title": "", "text": "$ \\begin{align} \\left(1+\\frac xn\\right)^n &=\\sum_{k=0}^n\\binom{n}{k}\\frac{x^k}{n^k}\\\\ &=\\sum_{k=0}^n\\left(\\frac{n}{n}\\frac{n-1}{n}\\frac{n-2}{n}\\dots\\frac{n-k+1}{n}\\right)\\frac{x^k}{k!}\\\\ &\\le\\sum_{k=0}^n\\frac{x^k}{k!} \\end{align} $"}
{"_id": "3688364", "title": "", "text": "$Perimeter=2\\int_{-a}^{a}\\sqrt{1+\\frac{dy}{dx}^2}dx$"}
{"_id": "4173829", "title": "", "text": "$p_1p_2\\ldots $"}
{"_id": "5669658", "title": "", "text": "$\\int_{-\\infty}^\\infty \\frac{1}{\\left(x^2+1\\right)^2}\\,dx$"}
{"_id": "4614912", "title": "", "text": "$x^2-x,y^2-y\\in Z(A)$"}
{"_id": "4371005", "title": "", "text": "$\\displaystyle\\sum_{n=1}^{\\infty} b_n=\\sum_{n=1}^{\\infty} b_n^{+}-\\sum_{n=1}^{\\infty} b_n^{-}=\\sum_{n=1}^{\\infty} a_n^{+}-\\sum_{n=1}^{\\infty} a_n^{-}=\\sum_{n=1}^{\\infty} a_n$"}
{"_id": "3652637", "title": "", "text": "$\\sum_{n\\geq0}\\frac{1}{1+n^{2}}=\\frac{1}{2}\\left(1+\\pi\\coth\\left(\\pi\\right)\\right)\\approx2.0767.$"}
{"_id": "4762042", "title": "", "text": "$\\tan\\theta = \\frac{x}{d}$"}
{"_id": "3912552", "title": "", "text": "$|f(z)|\\le\\Re(f(z))+\\Im(f(z))$"}
{"_id": "975273", "title": "", "text": "$\\sum_{i = 1}^k \\frac{1}{10^i}$"}
{"_id": "1189871", "title": "", "text": "$B=\\{b_0,b_1,b_2,\\dots\\}$"}
{"_id": "1493382", "title": "", "text": "$F(x)=\\int_{0}^{x}f(t)dt$"}
{"_id": "8177108", "title": "", "text": "$\\lim_{n \\to \\infty}\\frac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n-1}}{n\\sqrt{n}}$"}
{"_id": "6391018", "title": "", "text": "$\\mathbb{E}|X-Y|^2 \\leq \\frac{1}{2}$"}
{"_id": "1843603", "title": "", "text": "$F=G\\oplus H$"}
{"_id": "8239207", "title": "", "text": "$ (\\widetilde{\\gamma'} \\widetilde{\\gamma})\\gamma' = \\widetilde{\\gamma'}\\gamma = \\gamma' \\ . $"}
{"_id": "6179930", "title": "", "text": "$\\delta_{\\alpha \\gamma}=\\sum_\\beta (-1)^{\\alpha+\\gamma} \\binom{\\alpha}{\\alpha-\\beta, \\beta-\\gamma, \\gamma}=\\pm \\binom{\\alpha}{\\alpha-\\gamma, 0, \\gamma}\\pm \\binom{\\alpha}{\\alpha-\\gamma-1, 1, \\gamma}\\pm \\cdots$"}
{"_id": "1822599", "title": "", "text": "$ \\operatorname{Var}(X)=E[(X-E[X])^2]=E[X^2-2XE[X]+E[X]^2]=E[X^2]-E[X]^2. $"}
{"_id": "6010502", "title": "", "text": "$3(x^2+y^2+z^2)\\ge x(x+y+z)+y(y+z+x)+z(z+x+y) = (x+y+z)^2$"}
{"_id": "2059256", "title": "", "text": "$b=log_a c$"}
{"_id": "2261056", "title": "", "text": "$\\lim_{x\\to -3^-} f(x)=\\lim_{x\\to 4^+} f(x)=+\\infty,$"}
{"_id": "6832421", "title": "", "text": "$\\Bbb P[X=Y|\\mathcal G]=1$"}
{"_id": "1711214", "title": "", "text": "$ \\begin{align} &\\lim_{n\\to \\infty} \\frac{1+2\\sqrt{2}+...+n\\sqrt{n}}{n^2 \\sqrt{n}}\\\\ =&\\lim_{n\\to \\infty} \\frac{(1+2\\sqrt{2}+...+(n+1)\\sqrt{n+1})-(1+2\\sqrt{2}+...+n\\sqrt{n})}{(n+1)^2 \\sqrt{n+1}-n^2\\sqrt{n}}\\\\ =&\\lim_{n\\to \\infty} \\frac{(n+1)\\sqrt{n+1}}{(n+1)^2 \\sqrt{n+1}-n^2\\sqrt{n}}\\\\ =&\\lim_{n\\to \\infty} \\frac{(n+1)\\sqrt{n+1}((n+1)^2 \\sqrt{n+1}+n^2\\sqrt{n})}{((n+1)^2 \\sqrt{n+1}-n^2\\sqrt{n})((n+1)^2 \\sqrt{n+1}+n^2\\sqrt{n})}\\\\ =&\\lim_{n\\to \\infty} \\frac{(n+1)\\sqrt{n+1}((n+1)^2 \\sqrt{n+1}+n^2\\sqrt{n})}{(n+1)^5-n^5}\\\\ =&\\lim_{n\\to \\infty} \\frac{(n+1)^4+(n+1)n^2\\sqrt{n(n+1)}}{5n^4+10n^3+10n^2+5n+1}\\\\ =&\\lim_{n\\to \\infty} \\frac{(1+n^{-1})^4+(1+n^{-1})\\sqrt{1+n^{-1}}}{5+10n^{-1}+10n^{-2}+5n^{-3}+n^{-4}}\\\\ =&\\frac{2}{5} \\end{align} $"}
{"_id": "6098586", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1 & 3 & -1 & -4\\\\0 & -13 & 6 & 19\\\\0 & -7 & -1 & 9\\end{array}\\right]$"}
{"_id": "4693660", "title": "", "text": "$a_0 = (-1)[x^n + a_{n-1}x^{n-1} +...+a_1x]$"}
{"_id": "8626687", "title": "", "text": "$\\lim_{n \\to\\infty}\\inf (\\sin^n(x))$"}
{"_id": "6489614", "title": "", "text": "$f(k)=k^2+k+2$"}
{"_id": "7252607", "title": "", "text": "$p(j) \\implies  p(j+1)$"}
{"_id": "7881766", "title": "", "text": "$X^3, X^2, X, X^0$"}
{"_id": "170994", "title": "", "text": "$f(a) < k < f(b)$"}
{"_id": "3491297", "title": "", "text": "$3\\mid u^3-u.$"}
{"_id": "9126964", "title": "", "text": "$1/|x| < \\delta$"}
{"_id": "5945777", "title": "", "text": "$N_1\\subsetneq N_2\\subsetneq \\cdots\\subsetneq N_m\\subsetneq N_{m+1}\\subsetneq \\cdots.$"}
{"_id": "6322420", "title": "", "text": "$\\sum_{k=1}^n \\sqrt{n^4+k}\\ \\sin\\frac{2k\\pi}{n}=-\\frac{1}{4 \\pi }+\\frac{\\pi }{12 n^2}+\\frac{1}{16 \\pi  n^3}+O\\left(\\left(\\frac{1}{n}\\right)^4\\right)$"}
{"_id": "34161", "title": "", "text": "$P(k) \\implies P(k+1)$"}
{"_id": "1188413", "title": "", "text": "$\\lim_{x\\to c}|f(x)|= |L|$"}
{"_id": "812636", "title": "", "text": "$\\frac{(10+4)!}{4!10!}=\\frac{14!}{10!4!}$"}
{"_id": "7270484", "title": "", "text": "$G=([n],E)$"}
{"_id": "1772552", "title": "", "text": "$(\\widetilde{\\mathcal L}_{x_0},\\widetilde{\\mathcal L}_{x_1})$"}
{"_id": "5909727", "title": "", "text": "$p_1p_2\\dots p_{k-1}p_k$"}
{"_id": "1814396", "title": "", "text": "$i=\\frac{C_1-C_0}{C_0}=\\frac{C_1-C_0}{C_0}\\cdot 100\\% $"}
{"_id": "3030534", "title": "", "text": "$\\begin{pmatrix}0& 1\\\\-1 &1\\\\1 &1\\end{pmatrix}$"}
{"_id": "8612465", "title": "", "text": "$s_n(x) = \\sum_{j=1}^{n} c_j \\chi_{K_j}(x)$"}
{"_id": "5766018", "title": "", "text": "$\\sigma(a)\\,\\theta\\,\\sigma(b)$"}
{"_id": "2643463", "title": "", "text": "$\\mathbb R\\times\\mathbb R^{n-1}=\\mathbb R^n$"}
{"_id": "1500146", "title": "", "text": "$ b^x = \\left(1+\\frac xn\\right)^n. $"}
{"_id": "4728660", "title": "", "text": "$T=\\{\\{x\\},\\emptyset\\}$"}
{"_id": "5015146", "title": "", "text": "$d(x,y) = 1/2^n$"}
{"_id": "404989", "title": "", "text": "$\\det A = \\alpha \\beta^{n-1} = [(n-1)a +b] (b-a)^{n-1}$"}
{"_id": "3091871", "title": "", "text": "$\\big((-1)+1\\big)^n$"}
{"_id": "6562710", "title": "", "text": "$ f(z)= \\frac{1}{z}{e^z}{(1- e^{-z})^{-1}} $"}
{"_id": "7705788", "title": "", "text": "$ \\int \\frac{1}{\\cos^2(x)}\\frac{(1 + \\tan x)^n}{(1 - \\tan x)^{n + 2}} dx,$"}
{"_id": "5706109", "title": "", "text": "$f(x)=\\frac{2x}{x^2+1}\\;.$"}
{"_id": "8210694", "title": "", "text": "$p_G(x) = x(x-1)^{n-1}$"}
{"_id": "2904874", "title": "", "text": "$\\int_{a}^{b} f(t) \\dot{h}(t) dt = \\int_{a}^{b} g(t) h(t) dt$"}
{"_id": "2291088", "title": "", "text": "$117!+2, 117!+3, 117!+4, \\ldots, 117!+117$"}
{"_id": "10689", "title": "", "text": "$[X,Y] = XY - YX$"}
{"_id": "2665691", "title": "", "text": "$1+(1/2^{n-2})$"}
{"_id": "5341533", "title": "", "text": "$\\exists n \\forall m P(m, n) \\implies \\forall m \\exists n P(m, n)$"}
{"_id": "4353880", "title": "", "text": "$P(2,n)=\\frac{2^n}{2^n+2},\\\\ P(3,n)=\\frac{3^n}{3^n+3\\cdot 2^n+3}, \\\\P(4,n)=\\frac{4^n}{4^n+4\\cdot 3^n+6\\cdot 2^n+4}.$"}
{"_id": "7694042", "title": "", "text": "$\\lim_{x\\to c^{+}} ~ f(x) \\neq \\lim_{x\\to c^{-}} ~ f(x)$"}
{"_id": "2528211", "title": "", "text": "$x^r-x^{r-1}$"}
{"_id": "5958639", "title": "", "text": "$\\gamma[\\gamma_1 \\vert \\gamma_2 \\vert \\cdots \\vert \\gamma_n]\\equiv \\gamma \\otimes \\gamma_1 \\otimes \\cdots \\otimes \\gamma_n$"}
{"_id": "6033830", "title": "", "text": "$f'(x-1)+f'(x+1)=[f(x-1)+f(x+1)]'$"}
{"_id": "4423161", "title": "", "text": "$ \\begin{align} \\int_{-\\infty}^\\infty\\frac{\\sin(ax)}{x(1+x^2)}\\,\\mathrm{d}x &=\\color{#C00000}{\\frac1{2i}\\int_{\\gamma_+}\\frac{e^{iaz}}{z(1+z^2)}\\,\\mathrm{d}z} -\\color{#00A000}{\\frac1{2i}\\int_{\\gamma_-}\\frac{e^{-iaz}}{z(1+z^2)}\\,\\mathrm{d}z}\\\\ \\end{align} $"}
{"_id": "5251525", "title": "", "text": "$\\lim_{x\\to y^{+}} f(x) = \\lim_{x\\to y^{-}} f(x), \\qquad \\forall y \\in \\mathbb{R}m $"}
{"_id": "6415963", "title": "", "text": "$ \\left[     \\begin{array}{cccc|c}       1&-1&3&2&b1\\\\       -2&1&5&1&b2\\\\-3&2&2&-1&b3\\\\4&-3&1&3&b4     \\end{array} \\right] $"}
{"_id": "5043821", "title": "", "text": "$\\frac{\\sqrt[4]{x}}{x}=\\frac{x^{\\frac{1}{4}}}{x^1}=x^{\\frac{1}{4}-1}=x^{\\frac{1}{4}-\\frac{4}{4}}=x^{\\frac{1-4}{4}}=x^{-\\frac{3}{4}}=\\frac{1}{x^{\\frac{3}{4}}}$"}
{"_id": "2685972", "title": "", "text": "$A_1 \\supset A_2 \\supset \\ldots \\supset A_n \\supset \\ldots$"}
{"_id": "4399174", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\sum_{k=n}^\\infty 1=\\lim_{n \\rightarrow \\infty} \\infty=\\infty.$"}
{"_id": "4519930", "title": "", "text": "$1/2\\int_{0}^{1/2}\\frac{\\sin^{-1}(t)}{t}dt$"}
{"_id": "4572810", "title": "", "text": "$\\sqrt{2} \\neq \\sqrt{1} + \\sqrt{1} = 2$"}
{"_id": "4765606", "title": "", "text": "$ \\alpha,\\beta=1\\pm\\sqrt{3}\\\\ x_n=\\frac{(1+\\sqrt{3})^{n+1}-(1-\\sqrt{3})^{n+1}}{2\\sqrt{3}} $"}
{"_id": "6109783", "title": "", "text": "$F: \\mathbb{R}^{n+1} \\rightarrow \\mathbb{R}^{n+1}$"}
{"_id": "6025237", "title": "", "text": "$nx+1, nx+2, nx+3, \\ldots, nx+n$"}
{"_id": "4666324", "title": "", "text": "$f_Y(y) = \\frac{1}{2\\pi}e^\\frac{-(y)^2}{2}$"}
{"_id": "7997668", "title": "", "text": "$a=\\{\\{\\varnothing\\}\\}$"}
{"_id": "1451832", "title": "", "text": "$(x, y, z, w)=(1, 1, 1, 0)$"}
{"_id": "5710832", "title": "", "text": "$B=\\{x_{1},x_{2},....,x_{n}\\}$"}
{"_id": "3985534", "title": "", "text": "$           \\int_{0}^{\\infty}f(t)\\cos(st)dt,\\;\\; \\int_{0}^{\\infty}f(t)\\sin(st)dt. $"}
{"_id": "5827707", "title": "", "text": "$\\int\\frac{1}{(x+1)\\sqrt{1+x^2}} dx $"}
{"_id": "7834986", "title": "", "text": "$f(x)=e^{20 x/29} \\left(1-e^{-x}\\right)-y$"}
{"_id": "3801333", "title": "", "text": "$\\{ ( \\gamma , \\alpha ] = \\{ \\xi : \\gamma < \\xi \\leq \\alpha \\} : \\gamma < \\alpha \\}$"}
{"_id": "1522787", "title": "", "text": "$A_1 \\subseteq A_2 \\subseteq \\cdots \\subseteq A_n \\subseteq \\cdots$"}
{"_id": "8205902", "title": "", "text": "$\\sum_{n=0}^∞\\frac{x^n}{(n+1)2^n}$"}
{"_id": "6649194", "title": "", "text": "$\\gcd(A')=\\gcd(A)$"}
{"_id": "925932", "title": "", "text": "$\\Im(f)=-\\frac{\\sin(\\theta)(-2\\cos(\\theta)bcd+abcd+2a\\cos(\\theta)-ab-ac-ad+bc+bd+cd-1)}{(2\\cos(\\theta)b-b^2-1)(2\\cos(\\theta)c-c^2-1)(2\\cos(\\theta)d-d^2-1)}$"}
{"_id": "6110535", "title": "", "text": "$\\theta(x)=f_x$"}
{"_id": "4306628", "title": "", "text": "$\\left\\lfloor\\frac{a+b}{2c}\\right\\rfloor=\\left\\lfloor\\frac{\\left\\lfloor\\frac{a}c\\right\\rfloor+\\left\\lfloor\\frac{b}c\\right\\rfloor}{2}\\right\\rfloor$"}
{"_id": "4534551", "title": "", "text": "$\\lim_{x\\to inf}|r^{n+1}|=\\lim_{n\\to inf}|r^{n}|$"}
{"_id": "4149417", "title": "", "text": "$\\left(1+\\frac{x}{n}\\right)^n = e^{n\\log\\left(1+\\dfrac{x}{n}\\right)}$"}
{"_id": "4950775", "title": "", "text": "$\\left(1+\\dfrac{dy}{dx}\\right)=0\\quad$"}
{"_id": "1673082", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}\\frac{\\pi+\\sqrt{\\pi}+\\cdots+\\sqrt[n]{\\pi}}{n}$"}
{"_id": "1396327", "title": "", "text": "$dS=dx\\sqrt{1+(\\frac{dy}{dx})^2}$"}
{"_id": "521395", "title": "", "text": "$a R b \\land b R a$"}
{"_id": "6323149", "title": "", "text": "$A_1...A_m$"}
{"_id": "9277092", "title": "", "text": "$f(re^{i\\theta};n) = f(\\sqrt[n]{r}e^{i\\theta/n}\\,;1)^n \\qquad (n \\geq 1)$"}
{"_id": "6639158", "title": "", "text": "$ \\begin{align} \\int_0^\\infty f(x)\\,dx &\\leq \\sum_{n = 2}^\\infty \\int_{n^2 - 2}^{n^2 + 2}{dx\\over 1 + x}  \\leq \\sum_{n = 2}^\\infty{4\\over n^2 - 1}. \\end{align} $"}
{"_id": "2171413", "title": "", "text": "$\\int \\frac{x^2 }{(x^2+1)^2} dx$"}
{"_id": "1402171", "title": "", "text": "$\\begin{vmatrix}x_1&y_1&z_1\\\\x_2&y_2&z_2\\\\x_3&y_3&z_3\\end{vmatrix}\\geqslant0$"}
{"_id": "7793079", "title": "", "text": "$\\sin\\frac{\\theta}{2}=\\frac{\\frac{\\sin\\theta}{2}}{\\sqrt{\\sin\\theta}},$"}
{"_id": "6946548", "title": "", "text": "$\\frac{2}{a^2} + \\frac{4}{b^2} = 1.$"}
{"_id": "5471495", "title": "", "text": "$\\left(1+\\frac{x}{n}\\right)^n < e^x.$"}
{"_id": "781749", "title": "", "text": "$\\;\\phi_{\\frac1a,-\\frac ba}\\;$"}
{"_id": "8133354", "title": "", "text": "$\\tan(u) = \\frac{x}{1}$"}
{"_id": "1777289", "title": "", "text": "$f(Z^2) = f(Z)^2.$"}
{"_id": "2507944", "title": "", "text": "$2^x+3^x-4^x+6^x-9^x=1$"}
{"_id": "4570597", "title": "", "text": "$\\sum_{\\gamma_1,\\gamma_2} \\binom{\\beta}{\\gamma_1} s^{\\beta-\\gamma_1} h^{\\gamma_1} \\cdot \\frac{1}{\\gamma_2!} x^{\\gamma_2+\\alpha} e^{\\langle{s,x}\\rangle} h^{\\gamma_2} = \\sum_{\\gamma_2} (s+h)^\\beta \\cdot \\frac{1}{\\gamma_2!} x^{\\gamma_2+\\alpha} e^{\\langle{s,x}\\rangle} h^{\\gamma_2}$"}
{"_id": "871955", "title": "", "text": "$x!\\sim_{+\\infty}\\left(\\frac{x}{e}\\right)^x\\sqrt{2\\pi x}$"}
{"_id": "627766", "title": "", "text": "$cov (\\bar{X},\\bar{Y}) = 0$"}
{"_id": "709345", "title": "", "text": "$n! + 2, n! + 3, \\dots, n! + (n - 1)$"}
{"_id": "6385582", "title": "", "text": "$\\frac1x\\int_0^x\\frac t{e^t-1}\\ dt=\\frac1x\\left(\\frac{\\pi^2}6-\\sum_{n=1}^\\infty\\frac{e^{-nx}}{n^2}\\right)+\\ln(1-e^{-x})$"}
{"_id": "3406648", "title": "", "text": "$\\gamma = (\\gamma_1,\\gamma_2,\\gamma_3)$"}
{"_id": "4058521", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=c_1$"}
{"_id": "404940", "title": "", "text": "$=[a+(n-1)b](a-b)^{n-1} $"}
{"_id": "6917110", "title": "", "text": "$S=A_1 \\times A_2 \\times ...\\times A_n$"}
{"_id": "8728075", "title": "", "text": "$K=\\mathbb{Q} [x]/(x^{4}+1)$"}
{"_id": "476201", "title": "", "text": "$F(x) = \\int_a^x f(t) \\,dt  =\\int_a^x g(t) \\,dt $"}
{"_id": "1910774", "title": "", "text": "$(P ∨ ¬P) = (T ∨ F) = T$"}
{"_id": "5624878", "title": "", "text": "$\\therefore \\frac{dF}{dL} = - \\gamma K x^{\\gamma} L^{-\\gamma - 1} - (1 - \\gamma)x^\\gamma L^{-\\gamma}$"}
{"_id": "3960714", "title": "", "text": "$S=\\{P_1,P_2,P_3,...,P_n\\}$"}
{"_id": "872796", "title": "", "text": "$\\frac{a^{-1}-b^{-1}}{a^{-1/2}-b^{-1/2}}=\\frac{(a^{-1/2}-b^{-1/2})(a^{-1/2}+b^{-1/2})}{a^{-1/2}-b^{-1/2}}=a^{-1/2}+b^{-1/2}=\\frac{\\sqrt a}a+\\frac{\\sqrt b}b$"}
{"_id": "3545142", "title": "", "text": "$|f(x)| \\leq \\frac{1}{3}|f(y)|$"}
{"_id": "5902223", "title": "", "text": "$\\int\\pi (r_1^2 - r_2^2)\\,dx = \\int\\pi r_1^2 \\,dx - \\int\\pi r_2^2\\,dx.$"}
{"_id": "8881685", "title": "", "text": "$\\int\\dfrac{dx}{(x^2+1)^n}=\\int\\dfrac1{2x}\\dfrac{2x}{(x^2+1)^n}dx$"}
{"_id": "2114943", "title": "", "text": "$K\\subset A\\subset \\overline{A}\\subset U$"}
{"_id": "3936210", "title": "", "text": "$\\,p_1 \\mid q_1$"}
{"_id": "805430", "title": "", "text": "$|\\alpha|=|\\alpha+\\omega|$"}
{"_id": "1869833", "title": "", "text": "$\\zeta(s)\\Gamma(\\frac s2)=\\pi^{\\frac12-s} \\frac{\\Gamma(\\frac{1-s}{2})}{\\Gamma(\\frac{s}{2})}\\zeta(1-s)$"}
{"_id": "8656910", "title": "", "text": "$(x,y) \\mapsto [x,y]=xy-yx$"}
{"_id": "6576546", "title": "", "text": "$S(n,d) = \\phi(n)S(n',d')/\\phi(n')$"}
{"_id": "2635114", "title": "", "text": "$\\lvert\\dot{\\gamma}(t)\\rvert \\leqslant \\lvert t\\dot{\\gamma}(t)\\rvert$"}
{"_id": "279551", "title": "", "text": "$x,y\\in\\mathbb{R}^{n+1}-\\{0\\}$"}
{"_id": "1431705", "title": "", "text": "$d(x,A) = \\inf \\{d(x, y) : y ∈ A\\}$"}
{"_id": "3946605", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\frac{3+\\sqrt{3}+\\sqrt[3]{3}+\\dots+\\sqrt[n]{3}-n}{\\ln n}.$"}
{"_id": "5566271", "title": "", "text": "$E(X|{\\mathcal F}_0)$"}
{"_id": "392211", "title": "", "text": "$ \\mu (X_{n} - A_{n}) < {1}/{2^{n}} $"}
{"_id": "2708019", "title": "", "text": "$a_1≤...≤a_n$"}
{"_id": "8359210", "title": "", "text": "$Cov(X,Y)=-1$"}
{"_id": "1413049", "title": "", "text": "$s(x)=\\sum_{i=1}^nc_i1_{A_i}(x)$"}
{"_id": "763587", "title": "", "text": "$\\int_0^\\pi f(\\sin x)\\cos x\\ dx=\\int_0^\\pi(F(\\sin x))'\\ dx=[F(\\sin x)]_0^\\pi=F(0)-F(0)=0$"}
{"_id": "7352776", "title": "", "text": "$ (x-y)^2 (x+y) + (y-z)^2 (y+z) + (z-x)^2 (z+x) \\geq  \\frac{(y-z)^2}{(x+y+z)^2} (x+y+z)^3 $"}
{"_id": "7518859", "title": "", "text": "$\\mathscr O_{X,x}\\subset \\widehat{\\mathscr O_{X,x}}$"}
{"_id": "6318746", "title": "", "text": "$\\gamma, - \\gamma, i \\gamma, -i \\gamma, \\bar{\\gamma}, - \\bar{\\gamma}, i \\bar{\\gamma}, -i \\bar{\\gamma}$"}
{"_id": "1673781", "title": "", "text": "$\\displaystyle\\lim_{x\\to c^+}f'\\big(d(x)\\big)=\\displaystyle\\lim_{x\\to c^+}f'(x)=A$"}
{"_id": "7756501", "title": "", "text": "$f(x,y)= \\frac 1 {2\\pi \\sqrt{1-p^2}} e^{ \\frac {-(x^2 - 2pxy + y^2)}{2(1-p^2)}}$"}
{"_id": "369636", "title": "", "text": "$\\int_0^{2\\pi} g(t) S_n(t) \\ dt \\to \\int_0^{2\\pi} g(t) f(t) \\ dt$"}
{"_id": "718001", "title": "", "text": "$|f'(z)|\\leq|f(z)|$"}
{"_id": "1306579", "title": "", "text": "$\\textbf{x}=(x,y)$"}
{"_id": "2643653", "title": "", "text": "$\\sum_{n\\ge k}\\mu(A_n)\\ge\\mu\\left(\\bigcup_{n\\ge k}A_n\\right)\\ge\\mu\\left(\\limsup A_n\\right)$"}
{"_id": "1863252", "title": "", "text": "$R=\\{(x,y):a\\le{x}\\le{b}, c\\le{y}\\le{d}\\}$"}
{"_id": "6748572", "title": "", "text": "$a_n \\geq (1+\\frac{1}{n})^n$"}
{"_id": "5244268", "title": "", "text": "$\\sqrt{\\gamma^2}=\\begin{cases}\\gamma & \\gamma\\geq 0\\\\-\\gamma & \\gamma<0,\\end{cases}$"}
{"_id": "6626375", "title": "", "text": "$alog(b)=log(b^a)$"}
{"_id": "435695", "title": "", "text": "$P_1 P_2,$"}
{"_id": "5448511", "title": "", "text": "$I_j(a_j)=1$"}
{"_id": "6485521", "title": "", "text": "$I_n=\\int_0^{\\pi/2}\\frac{\\sin(nx)}{\\sin(x)}~\\mathrm dx$"}
{"_id": "5610026", "title": "", "text": "$C_X=\\{\\{a\\}\\}$"}
{"_id": "3666878", "title": "", "text": "$ 1-\\frac{N}{N+RN}=1-\\frac{N}{N(1+R)}=1-\\frac{1}{1+R}=\\frac{1+R-1}{1+R}=\\frac{R}{1+R}. $"}
{"_id": "1489782", "title": "", "text": "$\\frac{x}{|x|}=1$"}
{"_id": "3513204", "title": "", "text": "$Y:=\\sum_{j=1}^Nc_j\\chi_{A_j}$"}
{"_id": "757145", "title": "", "text": "$[x,y]=[y,x]$"}
{"_id": "2975587", "title": "", "text": "$ \\bbox[lightyellow] {   \\left\\{ \\matrix{   q_{\\,k}  = {{n^{\\,\\underline {\\,k\\,} } } \\over {n^{\\,k} }}\\quad \\left| {\\;0 \\le k} \\right. \\hfill \\cr    p_{\\,k}  = q_{\\,k - 1}  - q_{\\,k}  = q_{\\,k - 1} {{k - 1} \\over n} = {{\\left( {k - 1} \\right)\\,n^{\\,\\underline {\\,k - 1\\,} } } \\over {n^{\\,k} }}\\quad \\left| {\\;1 \\le k} \\right. \\hfill \\cr}  \\right. }$"}
{"_id": "52692", "title": "", "text": "$yRx$"}
{"_id": "3787808", "title": "", "text": "$\\frac{1}{4}\\left(e^{x/20}+e^{-x/20}\\right)^2.$"}
{"_id": "3225481", "title": "", "text": "$ X:= \\{x = (x_i)_{i\\in\\mathbb{N}} \\in \\mathbb{R}^\\mathbb{N} \\ | \\ \\exists \\ n \\in \\mathbb{N} \\ \\forall \\ i \\in \\mathbb{N} \\ : \\ i \\ge n \\implies x_i =0 \\}, $"}
{"_id": "858227", "title": "", "text": "$\\lim_{n\\to\\infty}{T_n}$"}
{"_id": "619012", "title": "", "text": "$\\lim_{x \\to 0^-}f'(x) = \\lim_{x  \\to 0^+}f'(x) = f'(0) $"}
{"_id": "7052249", "title": "", "text": "$A\\subset E \\subset tU$"}
{"_id": "5107008", "title": "", "text": "$ a_n = 2^{-n}  \\\\ b_n = \\frac{n^2}{n^3 -10} \\\\ c_n = 1 +(-1)^n \\\\ d_n = 2^{1/n}$"}
{"_id": "1140531", "title": "", "text": "$F(x) = \\int_{c}^{x} f(t)\\; dt$"}
{"_id": "4961154", "title": "", "text": "$m ! = \\frac{n(n+1)}{2}$"}
{"_id": "6907", "title": "", "text": "$($"}
{"_id": "1386306", "title": "", "text": "$ \\gamma\\sigma\\gamma^{-1} = \\gamma(s_{1}\\quad s_{2}\\quad\\ldots\\quad s_{k})\\gamma^{-1} = \\big(\\gamma(s_{1})\\quad \\gamma(s_{2})\\quad\\ldots\\quad \\gamma(s_{k})\\big). $"}
{"_id": "8164721", "title": "", "text": "$\\underline{~b~}~\\underline{~a~}~\\underline{~a~}~\\underline{~b~}$"}
{"_id": "9315128", "title": "", "text": "$S_n = \\frac{X_1 + X_2 + ... + X_n}{n}$"}
{"_id": "2832191", "title": "", "text": "$B = \\{q \\in \\mathbb{Q} \\mid q>0 \\land q^2 >2\\}$"}
{"_id": "4905278", "title": "", "text": "$\\begin{cases} x\\equiv\\theta\\\\x\\equiv\\pi-\\theta \\end{cases}\\mod2\\pi$"}
{"_id": "8061912", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^nE[X_i^-]=\\infty$"}
{"_id": "243030", "title": "", "text": "$(1+\\frac{dy}{dx})$"}
{"_id": "1084371", "title": "", "text": "$\\chi(n)=(\\frac{n}{q})$"}
{"_id": "1396873", "title": "", "text": "$\\frac{\\tan^{n-1}(x)}{n-1}-\\int\\tan^{n-2}dx$"}
{"_id": "7954839", "title": "", "text": "$f(x)=\\frac{1}{e^x+e^{-x}+2}$"}
{"_id": "3062619", "title": "", "text": "$\\int_R \\dfrac{1}{(1+x^2)^t}dx $"}
{"_id": "5767643", "title": "", "text": "$D = \\begin{pmatrix} A & B \\\\ -B & A \\end{pmatrix}$"}
{"_id": "2335268", "title": "", "text": "$\\sum_{n\\ge1}{\\frac{1}{\\sqrt{n}}} = \\sum_{n\\ge1}{\\frac{1}{n^{\\frac12}}}$"}
{"_id": "8018695", "title": "", "text": "$0 \\to A_1 \\xrightarrow{f_1} A_2 \\xrightarrow{f_2} \\ldots \\xrightarrow{f_{n-2}} A_{n-1} \\xrightarrow{f_{n-1}} A_n \\to 0$"}
{"_id": "249869", "title": "", "text": "$|ab|=|a||b|$"}
{"_id": "7718362", "title": "", "text": "$\\operatorname{antilog}_a(y)=x$"}
{"_id": "1364287", "title": "", "text": "$(A[x]/(x-1))_{(x-1)} = k$"}
{"_id": "8618991", "title": "", "text": "$ds = dx \\sqrt{1+\\left( \\frac{dy}{dx} \\right)^2}.$"}
{"_id": "910360", "title": "", "text": "$ D=\\{(x,y)\\in\\mathbb{R}^2|0\\leq x\\leq1,0\\leq y\\leq1\\} $"}
{"_id": "446409", "title": "", "text": "$B = \\{ \\gamma(1) \\mid \\gamma \\text{ is a geodesic}, \\gamma(0)=p, \\text{ and } \\langle \\gamma'(0), \\gamma'(0) \\rangle \\, \\lt \\epsilon \\} .$"}
{"_id": "4502807", "title": "", "text": "$E_1E_2\\mid F$"}
{"_id": "3278312", "title": "", "text": "$\\mathbb E\\lvert X\\rvert^p\\leqslant \\frac 12\\mathbb E\\lvert X-Y\\rvert^p.$"}
{"_id": "7296120", "title": "", "text": "$\\int_0^{2\\pi} f(t) \\sin(nt)dt$"}
{"_id": "1205587", "title": "", "text": "$ \\frac{a^2-b^2-c^2}{bc} = \\frac{c(\\gamma-\\alpha)}{b(\\beta-\\alpha)} + \\frac{b(\\beta-\\alpha)}{c(\\gamma-\\alpha)}\\\\ \\frac{b^2-a^2-c^2}{ac} = \\frac{a(\\alpha-\\beta)}{c(\\gamma-\\beta)} + \\frac{c(\\gamma-\\beta)}{a(\\alpha-\\beta)}\\\\ \\frac{c^2-a^2-b^2}{ba} = \\frac{b(\\beta-\\gamma)}{a(\\alpha - \\gamma)} + \\frac{a(\\alpha - \\gamma)}{b(\\beta-\\gamma)} $"}
{"_id": "5513083", "title": "", "text": "$S=(s_1,s_2,s_3,\\dots,s_n)$"}
{"_id": "8918311", "title": "", "text": "$ax+by=a$"}
{"_id": "6163100", "title": "", "text": "$f_\\theta(x) = \\frac{8x^7}{\\theta^8}$"}
{"_id": "5309612", "title": "", "text": "$P(a) \\implies P(x)$"}
{"_id": "9360602", "title": "", "text": "$m(M) = m^*(M).$"}
{"_id": "5941320", "title": "", "text": "$A(\\gamma)=\\int\\left|\\ddot{\\gamma}\\right|^{2}+K(\\gamma,\\dot{\\gamma})dt=\\int\\left|\\dot{\\gamma}\\right|^{2}+\\left|\\ddot{\\gamma}\\right|^{2}-\\left|\\dot{\\gamma}\\right|^{2}+K(\\gamma,\\dot{\\gamma})=\\int\\left|\\dot{\\gamma}\\right|^{2}+\\left|\\ddot{\\gamma}\\right|^{2}+\\tilde{K}(\\gamma,\\dot{\\gamma})$"}
{"_id": "4585865", "title": "", "text": "$P(E)=P(S\\cup E)-P(S)+P(S\\cap E)$"}
{"_id": "2785007", "title": "", "text": "$\\begin{bmatrix}     A & B \\\\     C & 0 \\\\   \\end{bmatrix} $"}
{"_id": "1194120", "title": "", "text": "$F(x)=\\int_a^{x} f(t)\\,dt \\rightarrow {d\\over dx} F(x)=f(x)$"}
{"_id": "397071", "title": "", "text": "$(p_1p_2\\cdots p_k)+1$"}
{"_id": "6744440", "title": "", "text": "$\\det [a_1, \\cdots, ca_r, \\cdots, a_n] = c \\det A$"}
{"_id": "382709", "title": "", "text": "$[1,1,0,0],[1,2,-1,1],[0,0,1,1],[2,1,2,-1]$"}
{"_id": "6103238", "title": "", "text": "$\\vert f(x)\\vert \\leq  K^n\\vert f(z)\\vert$"}
{"_id": "5263044", "title": "", "text": "$h[n]=(\\frac{1}{2^{n+2}}-\\frac{1}{2^{2n+3}})U[n]$"}
{"_id": "3379444", "title": "", "text": "$E[P(X_t\\leq x|\\mathcal{F}_s);\\{X_k\\leq y\\}]=P[\\{X_t\\leq x\\}\\cap\\{X_k\\leq y\\}]=P[\\{(t-\\tau)^+\\leq x\\}\\cap\\{(k-\\tau)^+\\leq y\\}]$"}
{"_id": "9225934", "title": "", "text": "$\\left[\\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\\\ -1 & -2 &  0  &  0  & -1-a  & -1  & 3  \\\\ -2 & -4 & -1 & 2  & a^{2} &   0  & 7  \\\\ 1  & 2  & 1  & -2 & a+2    & -1  & -6 \\end{array}\\right]$"}
{"_id": "3887275", "title": "", "text": "$ \\int_0^\\infty \\frac{\\sin^2(x)}{x^2}dx=-\\lim\\limits_{x\\to \\infty}\\frac{\\sin^2(x)}{x}+\\lim\\limits_{x\\to 0}\\frac{\\sin^2(x)}{x}+\\int_0^\\infty \\frac{2\\sin(x)\\cos(x)}{x}dx $"}
{"_id": "2795831", "title": "", "text": "$ax+by=ax'+by'=d$"}
{"_id": "3677964", "title": "", "text": "$\\mathsf{Cov}(U,V)=0$"}
{"_id": "6539165", "title": "", "text": "$|f(x)||x|^{\\sigma-1} \\leq 1/|x|^2$"}
{"_id": "6347242", "title": "", "text": "$\\mu^{-1}(0) = \\mathbb{R}^{n+1} \\times S^n$"}
{"_id": "3737412", "title": "", "text": "$\\{u,e_2,e_3,\\cdots,e_n\\}$"}
{"_id": "3131546", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 1 & -17 & 0 & 3\\\\ 0 & 17 & -6 & -2\\end{array}\\right]$"}
{"_id": "482400", "title": "", "text": "$ \\lim_{n \\to \\infty} \\nu(A_n) \\leq c \\lim_{n \\to \\infty}\\mu(A_n) = 0, $"}
{"_id": "8549296", "title": "", "text": "$z=\\left( 1+\\frac {a} {m}\\right) e^{\\frac {2k\\pi i} {m}}$"}
{"_id": "2903089", "title": "", "text": "$a_s\\equiv a(s)$"}
{"_id": "886231", "title": "", "text": "$\\int_0^\\infty x f(x) < \\infty$"}
{"_id": "468447", "title": "", "text": "$f(ab)=f(a)+f(b)$"}
{"_id": "512027", "title": "", "text": "$A^{p+1}(V) \\subseteq A^p(V)$"}
{"_id": "8976141", "title": "", "text": "$p_1p_2\\leq x$"}
{"_id": "4766518", "title": "", "text": "$ \\delta x = A^{-1}\\delta A(x+\\delta x), $"}
{"_id": "1408525", "title": "", "text": "$f(1-x) = \\frac{2}{4^{1-x} + 2} = \\frac{2 \\cdot 4^x}{4 + 2 \\cdot 4^x} = \\frac{4^x}{2 + 4^x} = 1 - f(x).$"}
{"_id": "144075", "title": "", "text": "$H(\\alpha)\\restriction \\beta=\\{(\\gamma,H(\\alpha)(\\gamma))\\mid \\gamma<\\beta\\}=\\{(\\gamma,H(\\gamma+1)(\\gamma))\\mid \\gamma<\\beta\\}=$"}
{"_id": "5900012", "title": "", "text": "$\\lim_{n \\to \\infty} \\left \\lfloor \\frac{an+b}{cn+d}\\right\\rfloor  = \\left \\lfloor{a\\over c} \\right \\rfloor-1 $"}
{"_id": "9129865", "title": "", "text": "$\\zeta(2)=\\sum_{n=1}^\\infty\\frac{1}{n^2}=\\int_0^1\\int_0^1\\frac{1}{1-xy}dA$"}
{"_id": "5997259", "title": "", "text": "$(a, a+1, a+2, \\dots, b)$"}
{"_id": "713049", "title": "", "text": "$\\det  \\begin{pmatrix}    A & C  \\\\     {0} & B  \\end{pmatrix}=\\det A\\cdot \\det B $"}
{"_id": "6791576", "title": "", "text": "$\\displaystyle \\sum_{n=1}^{\\infty}\\frac{1}{n(e^{\\pi n}+1)}-\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{1}{n(e^{2\\pi n}+1)}+\\sum_{n=1}^{\\infty}\\frac{1}{n(e^{\\pi n}-1)}-\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{1}{n(e^{2\\pi n}-1)}=\\frac{log(2)}{8}$"}
{"_id": "1865235", "title": "", "text": "$\\operatorname{Var}[\\hat\\theta] = \\operatorname{E}[\\hat \\theta^2 - 2\\hat\\theta \\operatorname{E}[\\hat \\theta] + \\operatorname{E}[\\hat\\theta]^2] = \\operatorname{E}[\\hat\\theta^2] -  \\operatorname{E}[\\hat\\theta]^2,$"}
{"_id": "3491668", "title": "", "text": "$\\{(-2)^n\\}$"}
{"_id": "1680671", "title": "", "text": "$\\prod\\limits_{k=1}^{n}\\cos\\left(\\frac{k\\pi}{n}\\right)$"}
{"_id": "2580605", "title": "", "text": "$\\lim_{n \\to \\infty} \\int_0^\\pi x^{1/n} \\sin(x) dx = \\int_0^\\pi \\lim_{n \\to \\infty} x^{1/n} \\sin(x) dx = \\int_0^\\pi \\sin(x)dx = 2. $"}
{"_id": "1430332", "title": "", "text": "$\\Bbb{R}^{n+1} - {0}$"}
{"_id": "373901", "title": "", "text": "$\\int_{-\\infty}^{\\infty}-\\frac{2}{\\pi^{2}}\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}\\left(\\frac{1}{u^{2}+n^{2}\\pi^{2}}\\right)du=\\frac{-2}{\\pi^{2}}\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}\\int_{-\\infty}^{\\infty}\\frac{1}{u^{2}+n^{2}\\pi^{2}}du.$"}
{"_id": "2724565", "title": "", "text": "$\\ \\underbrace{b/a = p}_{\\large b\\ =\\ ap}$"}
{"_id": "3715431", "title": "", "text": "$\\lim_{k\\rightarrow\\infty} \\mu(B_k) = \\mu(A)$"}
{"_id": "8203039", "title": "", "text": "$\\begin{cases}-y-\\frac 2y=3\\\\y=\\frac 1y\\end{cases}\\iff \\begin{cases}x=1\\\\y=-1\\end{cases}$"}
{"_id": "2096247", "title": "", "text": "$y=\\frac{c+x}{1-\\ln x}$"}
{"_id": "5160343", "title": "", "text": "$E = \\{e_1,e_2,e_3,e_4,e_5,e_6\\}, X = \\{\\max x, ?, ?, ?, ?, ? \\}$"}
{"_id": "5294016", "title": "", "text": "$\\mathbb{P}[X_\\tau=n\\mid X_0=k]=P_k[X_\\tau=n]=k/n$"}
{"_id": "3380573", "title": "", "text": "$\\lim_{n\\to\\infty}\\sum_{k=1}^n\\frac{k^r}{n^{r+1}}.$"}
{"_id": "5325014", "title": "", "text": "$\\arctan(1/|x|)$"}
{"_id": "7862964", "title": "", "text": "$\\int\\frac{1}{1-\\sin\\left(x\\right)}dx\\stackrel{u=\\tan\\left(\\frac{x}{2}\\right)}{=}2\\int\\frac{1}{\\left(u-1\\right)^{2}}=-\\frac{2}{\\tan\\left(\\frac{x}{2}\\right)-1}  $"}
{"_id": "690155", "title": "", "text": "$\\zeta(s)=\\frac1{(s-1)\\Gamma(s)}+\\frac1{\\Gamma(s)}\\sum_{n=1}^\\infty\\frac{B_n}{n!(n+s-1)}+\\frac1{\\Gamma(s)}\\int_1^\\infty\\frac{t^{s-1}}{\\mathrm e^t-1} \\mathrm dt$"}
{"_id": "4783057", "title": "", "text": "$y = 10 + 8x + x^2 - x^3,\\; x \\geq 0$"}
{"_id": "427236", "title": "", "text": "$|\\mathbb E[\\gamma_k\\mid \\gamma_1,\\dots,\\gamma_{k-1}]|\\leqslant  |\\mathbb E[D_k(\\gamma_k)\\mid \\gamma_1,\\dots,\\gamma_{k-1}]|+|\\mathbb E[D_k(\\gamma_k)-\\gamma_k\\mid \\gamma_1,\\dots,\\gamma_{k-1}]|\\leqslant 2\\cdot 2^{-k}.$"}
{"_id": "5267378", "title": "", "text": "$\\frac{\\frac{c}{r^2}+\\frac{1-c}{(1+r)^{T+1}}}{\\frac{c}{r}+\\frac{1-c}{(1+r)^T}-1}= \\frac{r^2}{r^2}\\frac{\\frac{c}{r^2}+\\frac{1-c}{(1+r)^{T+1}}}{\\frac{c}{r}+\\frac{1-c}{(1+r)^T}-1}= \\frac{{c}+r^2\\frac{1-c}{(1+r)^{T+1}}}{r{c}+r^2\\frac{1-c}{(1+r)^T}-r^2}=\\\\ \\frac{(1+r)^{T+1}}{(1+r)^{T+1}}\\frac{{c}+r^2\\frac{1-c}{(1+r)^{T+1}}}{r{c}+r^2\\frac{1-c}{(1+r)^T}-r^2}= \\frac{{c}(1+r)^{T+1}+r^2({1-c})}{r{c}(1+r)^{T+1}+r^2({1-c}){(1+r)}-r^2(1+r)^{T+1}}= \\frac{{c}(1+r)^{T+1}+r^2({1-c})}{r\\left({c}-r\\right)(1+r)^{T+1}+r^2({1-c}){(1+r)}}= \\frac{{c}(1+r)^{T+1}+r^2({1-c})}{\\left(({c}-r)(1+r)^{T}+r({1-c})\\right){(1+r)r}} $"}
{"_id": "5316121", "title": "", "text": "$y-\\frac{b-c}{2} = ax + \\frac{b+c}{2}$"}
{"_id": "490559", "title": "", "text": "$k=1\\cdot\\left(\\left\\lfloor\\frac{n}{2}\\right\\rfloor-\\left\\lfloor\\frac{n}{4}\\right\\rfloor\\right)+2\\left(\\left\\lfloor\\frac{n}{4}\\right\\rfloor-\\left\\lfloor\\frac{n}{8}\\right\\rfloor\\right)+\\ldots=\\sum_{i=1}^\\infty \\left\\lfloor\\frac{n}{2^i}\\right\\rfloor$"}
{"_id": "3557100", "title": "", "text": "$\\begin{cases} x = 3 + 1 \\\\ y = −2 + 4 \\\\ z = 1 − 2 \\end{cases}$"}
{"_id": "497145", "title": "", "text": "$I = -\\int_0^{\\infty} \\dfrac{\\sin^2(x)}{x^2}dx$"}
{"_id": "1231094", "title": "", "text": "$\\sum_{k=0}^{n-1}\\cos\\left(\\frac{2\\pi k}{n}+\\phi\\right)=0$"}
{"_id": "6796678", "title": "", "text": "$\\delta(\\gamma)=\\sup_{s,t\\in S^1}\\frac{d_\\gamma(\\gamma(s),\\gamma(t))}{|\\gamma(s)-\\gamma(t)|}$"}
{"_id": "7852612", "title": "", "text": "$i_j (x) = (x,j)$"}
{"_id": "3545563", "title": "", "text": "$\\displaystyle f(x)=\\frac{e^x-1}{e^x+1}$"}
{"_id": "6764106", "title": "", "text": "$ (\\gamma ^\\mu )^*=\\gamma _\\mu=\\begin{cases}(-\\gamma ^0,\\vec{\\gamma}) & \\text{for signature }(-,+,\\cdots ,+) \\\\ (\\gamma ^0,-\\vec{\\gamma}) & \\text{for signature }(+,-,\\cdots ,-)\\end{cases}, $"}
{"_id": "7824399", "title": "", "text": "$2\\sum_{n\\geq1}\\frac{\\left(-1\\right)^{n}}{4n^{2}-1}=\\sum_{n\\in\\mathbb{Z}}\\frac{\\left(-1\\right)^{n}}{4n^{2}-1}+1.$"}
{"_id": "439133", "title": "", "text": "$\\int_{0}^{\\infty} {f(x)} dx = C$"}
{"_id": "8749214", "title": "", "text": "$R=\\lambda-1$"}
{"_id": "4423061", "title": "", "text": "$ \\left[       \\begin{array}{cccc|c}         1&1&4&2\\\\         0&3&12&6 \\\\         -1&2&8&4 \\\\         0&1&4&2       \\end{array}     \\right]$"}
{"_id": "607672", "title": "", "text": "$\\frac{1}{2} + \\frac{2}{2^2} +  \\ldots +\\frac{n-1}{2^{n-1}} + \\frac{n}{2^{n}} <2-\\frac{n+1}{2^{n-1}}<2$"}
{"_id": "519740", "title": "", "text": "$ f(x) = \\frac{e^x}{1+ \\lceil x\\rceil } $"}
{"_id": "4381971", "title": "", "text": "$A_1 ... A_c$"}
{"_id": "3936211", "title": "", "text": "$p_1 \\mid q_2 \\cdot (q_3 \\ldots q_n)$"}
{"_id": "2458538", "title": "", "text": "$f_3=$"}
{"_id": "8003238", "title": "", "text": "$T_x(X)\\times T_x(X)$"}
{"_id": "90043", "title": "", "text": "$|f(z)|\\leq 1$"}
{"_id": "1108815", "title": "", "text": "$\\text{span}(v_1,v_2,\\ldots,v_n,\\ldots)=\\text{span}(e_1,e_2,e_3,\\ldots,e_n,\\ldots)$"}
{"_id": "2748637", "title": "", "text": "$ f(a_1) = g(a_1) - g(-a_1) $"}
{"_id": "2689663", "title": "", "text": "$P\\cfrac{(1+r/12)^{n}-(1+r/12)^{k}}{(1+r/12)^{12}-1}$"}
{"_id": "2041697", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-\\pi n^2 z}=\\frac{1}{\\sqrt{z}}\\sum_{n=-\\infty}^{\\infty}e^{-\\pi n^2/z}.$"}
{"_id": "3451089", "title": "", "text": "$A = A_1 \\times \\ldots \\times A_{n+1}$"}
{"_id": "276450", "title": "", "text": "$\\lim_{n \\rightarrow\\infty}\\inf f_n(k)$"}
{"_id": "7670059", "title": "", "text": "$i_1 = A, i_2 = B, i_3 = C, i_4=i_A = D, i_5 = E, i_T=i_6 = F, i_7=G, i_8 = H, i_9=I.$"}
{"_id": "7683031", "title": "", "text": "$x^n\\delta^{(n)}(x)=(-1)^n\\, n! \\, \\delta(x)$"}
{"_id": "5959839", "title": "", "text": "$G = \\left\\{\\begin{bmatrix} \\alpha & \\beta \\\\ . & \\gamma \\end{bmatrix} : \\alpha,\\beta,\\gamma \\in K, \\alpha\\gamma = 1 \\right\\} \\leq B = \\left\\{\\begin{bmatrix} \\alpha & \\beta \\\\ . & \\gamma \\end{bmatrix} : \\alpha,\\beta,\\gamma \\in K, \\alpha\\gamma \\neq 0\\right\\}$"}
{"_id": "1505123", "title": "", "text": "$ [1,-1,-1,1,1,-1,-1,1,1,-1, \\ldots]$"}
{"_id": "1547314", "title": "", "text": "$f(x)=\\frac{2x}{(x-1)^2}$"}
{"_id": "8265487", "title": "", "text": "$b^x=a^{x\\log_a(b)}$"}
{"_id": "3778660", "title": "", "text": "$ab = r p$"}
{"_id": "1822117", "title": "", "text": "$M,M',M''$"}
{"_id": "3680575", "title": "", "text": "$[t_{i-1}^n, t_{i}^n]$"}
{"_id": "91455", "title": "", "text": "$\\left[ \\begin{array} {rrr|r}  1 &  0 &  2 &  3 \\\\ -1 & -1 &  1 & -6 \\\\  0 &  1 & -2 &  1 \\\\ \\end{array}\\right]$"}
{"_id": "3053410", "title": "", "text": "$xRy \\wedge yRx \\rightarrow x = y$"}
{"_id": "6600930", "title": "", "text": "$y(x)=(x-a)/(b-a)$"}
{"_id": "7604936", "title": "", "text": "$f(x) = x^3-x+2$"}
{"_id": "6338762", "title": "", "text": "$ \\gcd\\{a,b\\}=\\gcd\\{|a|,b\\}=\\gcd\\{a,|b|\\}=\\gcd\\{|a|,|b|\\} . $"}
{"_id": "9333385", "title": "", "text": "$\\iff Cov(X,Y)=0$"}
{"_id": "1725516", "title": "", "text": "$\\vartheta (z;\\tau)$"}
{"_id": "3347267", "title": "", "text": "$\\frac{a+b+c}{b-a}$"}
{"_id": "781604", "title": "", "text": "$A = \\{ \\langle x , y \\rangle \\in \\mathbb{R} \\times \\mathbb{R} : y > 0 \\text{ or } x = 0 \\},$"}
{"_id": "5360834", "title": "", "text": "$(x,y,z)=(\\log 2)\\cdot(1,1,1)$"}
{"_id": "794138", "title": "", "text": "$ \\begin{pmatrix} \\overline{a} & b \\\\ -\\overline{b} & a \\end{pmatrix} $"}
{"_id": "635147", "title": "", "text": "$\\|A\\|_2 = [\\operatorname{Tr}(A^TA)]^{1/2}$"}
{"_id": "4419538", "title": "", "text": "$\\sum_{k=1}^n\\frac{1}{2^k}$"}
{"_id": "7680044", "title": "", "text": "$\\sum_{m=-\\infty}^{\\infty}e^{-\\pi m^{2}/x}=\\lim_{N\\rightarrow\\infty}\\sum_{n=-N}^{N}\\int_{-\\infty}^{\\infty}e^{-\\left(\\pi t^{2}/x\\right)-2\\pi int}dt=\\lim_{N\\rightarrow\\infty}\\sum_{n=-N}^{N}\\sqrt{x}e^{\\pi n^{2}x}=  $"}
{"_id": "2755509", "title": "", "text": "$a_n=\\frac{1}{3^{\\frac{n(n+1)}{2}}}$"}
{"_id": "716109", "title": "", "text": "$λ(m) = [40, 10] = 40$"}
{"_id": "224419", "title": "", "text": "$\\Delta = |Â| = -m^2$"}
{"_id": "7161371", "title": "", "text": "$r^2\\sin(\\varphi)\\cos(\\varphi)\\frac{r^2(\\sin(\\varphi)^2-\\cos(\\varphi)^2)}{r^2(\\sin(\\varphi)^2+\\cos(\\varphi)^2)}=r^2\\sin(\\varphi)\\cos(\\varphi)\\frac{r^2(\\sin(\\varphi)^2-\\cos(\\varphi)^2)}{r^2\\cdot 1}\\\\=r^2\\sin(\\varphi)\\cos(\\varphi)\\cdot (\\sin(\\varphi)^2-\\cos(\\varphi)^2)$"}
{"_id": "2576092", "title": "", "text": "$\\sum_{m=0}^\\infty \\sum_{k=1}^1 \\frac{1}{(k+m^2)\\prod_{n=0}^{k-2}(n+m^2)}=\\frac{1}{2} (\\pi  \\coth (\\pi )-1)$"}
{"_id": "6681648", "title": "", "text": "$\\langle_{\\mathbb{R}}=\\{\\langle x,y \\rangle \\in \\mathbb{R}^2: \\text{ x is greater that y}\\}$"}
{"_id": "7509038", "title": "", "text": "$a:\\left\\{  \\begin{array}{c} x=2\\lambda+3\\mu  \\\\ y=1\\lambda-2\\mu\\\\ z=-1\\lambda-\\mu\\\\ t=3\\lambda+\\mu \\end{array} \\right. $"}
{"_id": "9327038", "title": "", "text": "$D=\\{(x,y)\\in\\Bbb R^2: a\\le x\\le b, l(x)\\le y\\le u(x)\\},$"}
{"_id": "3827394", "title": "", "text": "$ \\begin{cases} x+2y=z\\\\ x+2y=2\\\\ x+z=4 \\end{cases}\\implies\\begin{cases} z=2\\\\ x=2\\\\ y=0 \\end{cases}$"}
{"_id": "801008", "title": "", "text": "$P_1P_2P_4$"}
{"_id": "1802431", "title": "", "text": "$A\\subset  U\\subset X$"}
{"_id": "1870949", "title": "", "text": "$ (1+2+3+4+5+6+7+.....)=-\\frac{1}{12} $"}
{"_id": "5523648", "title": "", "text": "$\\mathbf{x} = (x,y).$"}
{"_id": "6465529", "title": "", "text": "$f'(x)=\\frac{r(1+x)-r(1+rx)}{(1+x)^{r+1}}=\\frac{(r-r^2)x}{(1+x)^{r+1}}$"}
{"_id": "2224454", "title": "", "text": "$d(\\alpha)=\\alpha^+$"}
{"_id": "2838318", "title": "", "text": "$x \\in y \\vee y \\in x$"}
{"_id": "7777699", "title": "", "text": "$\\|x_n - x_{n-1} \\| < \\frac{1}{3^n}$"}
{"_id": "8856812", "title": "", "text": "$x\\ge 1, x \\in \\mathbb N$"}
{"_id": "7779805", "title": "", "text": "$\\det (\\mathrm A) = \\left(1 + \\frac{n b}{a-b}\\right) (a-b)^n = (a + (n-1) b) \\, (a-b)^{n-1}$"}
{"_id": "5894268", "title": "", "text": "$x^3 - x - x + 1$"}
{"_id": "2899283", "title": "", "text": "$\\left(a+b\\right)^r\\leqslant 2^{r-1}\\left(a^r+b^r\\right)$"}
{"_id": "4060932", "title": "", "text": "$\\mu (A)=\\lim_{n\\to \\infty}\\mu(A_n)+G(n)=\\lim_{n\\to \\infty}\\mu(A_n)=\\sup_{n\\in \\Bbb N}\\mu(A_n).$"}
{"_id": "7866187", "title": "", "text": "$|ab|=-ab$"}
{"_id": "6651880", "title": "", "text": "$f^{n+1}-f^n$"}
{"_id": "3846247", "title": "", "text": "$\\|A'\\|_2^2=\\rho(A'^TA)\\leq\\rho(A^TA)=\\|A\\|_2^2$"}
{"_id": "5527731", "title": "", "text": "$F^{++}$"}
{"_id": "6797928", "title": "", "text": "$ \\lim_{n\\to\\infty} \\frac{1}{\\log n}\\sum_{m=1}^n \\frac{1}{m^2} = 0, $"}
{"_id": "2208743", "title": "", "text": "$*M*M*M*M*M*M*$"}
{"_id": "7977076", "title": "", "text": "$\\Jac \\left( \\mathbb{Q}[x] / (x^8-1) \\right) = \\{0\\}$"}
{"_id": "217404", "title": "", "text": "$n^2-n+2$"}
{"_id": "5057679", "title": "", "text": "$b_n=\\int_{-\\pi}^{\\pi} f(x)\\,\\sin(nx)dx$"}
{"_id": "1866682", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-n^2\\pi x}=\\vartheta _3\\left(0,e^{-\\pi  x}\\right)$"}
{"_id": "5568218", "title": "", "text": "$I=\\lim_{t\\to 0}\\dfrac{\\dfrac{(t+1)^{t+1}-(t+1)}{\\ln{(t+1)}-t}+2}{t}=\\lim_{t\\to  0}\\dfrac{(t+1)^{t+1}-(t+1)+2\\ln{(t+1)}-2t}{t(\\ln{(t+1)}-t)}=\\lim_{t\\to 0}\\dfrac{(t+1)^{t+1}+2\\ln{(t+1)}-3t-1}{-\\dfrac{t^3}{2}}$"}
{"_id": "113459", "title": "", "text": "$ G=\\{ e,x,y,y^2,y^3,y^4,z,z^2,z^3,z^4 \\}$"}
{"_id": "3933583", "title": "", "text": "$\\frac{(\\gamma)_k}{(\\gamma+1)_k} = \\frac{\\gamma}{\\gamma+k}$"}
{"_id": "1210274", "title": "", "text": "$r^+$"}
{"_id": "6879126", "title": "", "text": "$|\\gamma'(t)|=|X(\\gamma(t))| \\le C|\\gamma(t)|.$"}
{"_id": "7408502", "title": "", "text": "$\\dfrac{\\pi}{\\sin{p\\pi}}$"}
{"_id": "1716799", "title": "", "text": "$f_Y(y,\\theta) = 2y/\\theta^2$"}
{"_id": "6753782", "title": "", "text": "$c_0(I, C(X_i)) = C_0(X)$"}
{"_id": "9292566", "title": "", "text": "$f(x+f(y)+xf(y)) = y+f(x)+yf(x)\\forall x,y \\in \\mathbb{R}-\\{-1\\}\\;,$"}
{"_id": "1810896", "title": "", "text": "$ds=\\sqrt{dx^2+dy^2}=dx\\sqrt{1+(\\frac {dy}{dx})^2}$"}
{"_id": "4850542", "title": "", "text": "$P(n) \\land P(m) \\rightarrow P(mn)$"}
{"_id": "5574931", "title": "", "text": "$\\left(\\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\\\ -1 & 1 & 2 & 7 \\\\ 2 & 2 & -5 & -8  \\end{array}\\right) = \\left(\\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\\\ 0 & 1 & -1 & 17  \\\\ 0 & 3 & 0 & -11  \\end{array}\\right) \\begin{array}{l}  \\\\  R_1 + R_2 \\\\  R_3 + R_2 - R_1  \\end{array}= \\left(\\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\\\ 0 & 1 & -1 & 17  \\\\ 0 & 0 & 3 & -62  \\end{array}\\right) \\begin{array}{l}  \\\\  \\\\  R_3 - 3R_2  \\end{array}= \\left(\\begin{array}{ccc|c} 1 & 0 & 0 & -52 \\\\ 0 & 1 & 0 & -7  \\\\ 0 & 0 & 1 & -24  \\end{array}\\right) \\begin{array}{l}  R_1 + R_3 \\\\  R_2 + R_3/3 \\\\  R_3/3  \\end{array}$"}
{"_id": "1733039", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-\\pi n^2x}=x^{-1/2}\\sum_{n=-\\infty}^\\infty e^{-\\pi n^2/x}.$"}
{"_id": "8087878", "title": "", "text": "$4a(x)[(2-x)a(x)-2a(x)^2]=-xa(x)+x(1-x)a'(x)-x^2a''(x)$"}
{"_id": "4081161", "title": "", "text": "$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$"}
{"_id": "6550162", "title": "", "text": "$\\|A\\|_F = \\sqrt{\\operatorname{tr}(A^\\dagger A)}$"}
{"_id": "1746879", "title": "", "text": "$[x,y]=xu=yv$"}
{"_id": "5331193", "title": "", "text": "$\\int_{-\\infty}^{\\infty} e^{-|x+1|} dx$"}
{"_id": "1474643", "title": "", "text": "$\\{\\gamma,-\\gamma\\}=\\{\\alpha,-\\alpha\\}$"}
{"_id": "1418491", "title": "", "text": "$|a-b| +|b| = -a + 2b = -a + 2a = a$"}
{"_id": "6441096", "title": "", "text": "$ A = \\left[\\begin{array}{rrr|r} 1 & 1 & a & -1 \\\\ 3 & (a+1) & (a-1) &-1\\\\ a&2&1&0  \\end{array}\\right] $"}
{"_id": "6104313", "title": "", "text": "$\\int \\frac{1}{(x^2 +a)^2} dx $"}
{"_id": "5605926", "title": "", "text": "$366*(1/(1-e^{-1/365})$"}
{"_id": "6279239", "title": "", "text": "$\\mathbb{Cov}(X,W)=0$"}
{"_id": "5360432", "title": "", "text": "$F=A\\oplus B$"}
{"_id": "5984638", "title": "", "text": "$g(x):=\\int_a^xf'(t)\\,dt-(f(x)-f(a))$"}
{"_id": "8956619", "title": "", "text": "$P(E|A)=1/6$"}
{"_id": "1523197", "title": "", "text": "$\\mathbb P(Y_n=0|\\mathcal{F_n})=1-X_n$"}
{"_id": "3514891", "title": "", "text": "$W_0\\subseteq W_1\\subseteq W_2\\subseteq\\ldots~$"}
{"_id": "6655350", "title": "", "text": "$\\zeta(s)=\\frac{(2\\pi)^{s}}{\\pi}\\sin(\\frac{\\pi s}{2})\\Gamma(1-s)\\zeta(1-s)$"}
{"_id": "2290531", "title": "", "text": "$\\{k,k+d,k+2d,\\ldots\\}$"}
{"_id": "467834", "title": "", "text": "$\n F(x)=\\int_a^xf(t)dt\\Rightarrow F'(x)=f(x)\n $"}
{"_id": "1853909", "title": "", "text": "$\\sum^{\\infty}_{n=1}\\left\\|f_n\\right\\|<\\infty$"}
{"_id": "4627362", "title": "", "text": "$\\lim_n \\frac{x_1+x_2+...x_n}{n}=0$"}
{"_id": "35029", "title": "", "text": "$\\frac1{2^{n-1}}$"}
{"_id": "1600880", "title": "", "text": "$A_1 \\subseteq A_2 \\subseteq A_3 \\subseteq \\cdots$"}
{"_id": "4725949", "title": "", "text": "$ k^2 = \\begin{vmatrix}a,b\\\\-b,a\\\\\\end{vmatrix} $"}
{"_id": "5440223", "title": "", "text": "$E(G) = \\{e_1, e_2, ..., e_m\\}$"}
{"_id": "5947180", "title": "", "text": "$\\langle u^\\top xx^\\top u\\rangle-\\langle u^\\top x\\rangle\\langle x^\\top u\\rangle=u^\\top\\langle xx^\\top\\rangle u-u^\\top\\langle x\\rangle\\langle x^\\top\\rangle u=u^\\top\\left(\\langle xx^\\top\\rangle-\\langle x\\rangle\\langle x^\\top\\rangle\\right)u\\;.$"}
{"_id": "7300319", "title": "", "text": "$B^* = (AA^*)^* = (A^*)^* A^* = AA^* = B$"}
{"_id": "755772", "title": "", "text": "$|f(x) - f(a)| < \\epsilon, |x - a| < \\delta$"}
{"_id": "2387915", "title": "", "text": "$ \\sum_{k=1}^{n-1}\\sin\\frac{\\pi k}{n} = \\cot \\frac{\\pi}{2 n} $"}
{"_id": "8099839", "title": "", "text": "$\\frac{K}{\\pi}=\\frac{1}{2\\pi}\\int_0^{\\pi/2} \\frac{x}{\\sin(x)} \\ dx=\\frac{1}{2}\\int_0^{1/2} \\frac{\\pi x}{\\sin(\\pi x)} \\ dx$"}
{"_id": "3270277", "title": "", "text": "$\\;\\displaystyle\\phi =  \\color{red}{1 + \\frac{1}{1 + \\frac{1}{1 + \\frac{1}{...}}}} = \\color{blue}1 + \\frac{1}{\\color{red}{1 + \\frac{1}{1 + \\frac{1}{...}}}} = \\color{blue}{1}+\\frac{1}{\\color{red}{\\phi}}\\,$"}
{"_id": "3169380", "title": "", "text": "$X=[X, y]$"}
{"_id": "5307234", "title": "", "text": "$f(\\pi) = f(0) = 0$"}
{"_id": "2769945", "title": "", "text": "$\\mathcal{A}(X)$"}
{"_id": "649610", "title": "", "text": "$P(n) \\implies P(n+1).$"}
{"_id": "7109860", "title": "", "text": "$\\frac{1}{\\Gamma(s)} \\int_0^{\\infty} \\frac {x^{s-1}}{e^x -1} \\ dx = \\zeta(s)$"}
{"_id": "2000279", "title": "", "text": "$\\frac 14 (k + 1)((k + 1) + 1)((k + 1) + 2)((k + 1) + 3)  = \\frac 14(k + 1)(k + 2)(k + 3)(k + 4)$"}
{"_id": "1582602", "title": "", "text": "$\\frac{ds}{dx} = \\sqrt{1 + {\\frac{dy}{dx}}^2}$"}
{"_id": "7161336", "title": "", "text": "$f(k) = \\binom{n}{k}$"}
{"_id": "7373916", "title": "", "text": "$ (y - 4) = \\frac{4}{10}(x - 10) \\\\ y = \\frac{2x}{5} \\\\ $"}
{"_id": "5940620", "title": "", "text": "$ \\begin{align} \\int_{0}^{\\infty} \\frac{\\sin^{2n+1} (x)}{x} \\ dx &= \\frac{1}{2} \\int_{-\\infty}^{\\infty} \\frac{\\sin^{2n+1} (x)}{x} \\ dx = \\frac{1}{2} \\int_{-\\infty}^{\\infty} \\frac{\\sin x}{x} \\sin^{2n} (x) \\ dx \\\\ &= \\frac{1}{2} \\int_{0}^{\\pi} \\sin^{2n} (x) \\ dx = \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2n} (x) \\ dx \\\\ &= \\frac{\\pi}{2^{2n+1}} \\binom{2n}{n}. \\tag{1} \\end{align}$"}
{"_id": "3143130", "title": "", "text": "$|f(z^3)| \\leq 1 + |z|$"}
{"_id": "1726225", "title": "", "text": "$ \\ x^4 -6\\, x^2 + 9\\ =\\, -4\\, x^2,\\ $"}
{"_id": "5691724", "title": "", "text": "$d=ax+yb$"}
{"_id": "4468427", "title": "", "text": "$ [T_x,T_y] = 0$"}
{"_id": "5542744", "title": "", "text": "$B=\\{\\{X\\}\\}$"}
{"_id": "8601886", "title": "", "text": "$\\frac 1{2^{n-2}} \\le 1$"}
{"_id": "3057782", "title": "", "text": "$f'(x) = \\dfrac{1}{5}+ e^{25x} (25) - e^3 -\\dfrac{7}{3}\\sqrt[3]{x^4}$"}
{"_id": "9246159", "title": "", "text": "$\\lim_{N \\to \\infty}\\sum\\limits_{n=1}^{N} \\frac{n^2+1}{2^n}$"}
{"_id": "3936208", "title": "", "text": "$p_1 \\mid q_{1} \\cdot (q_{2}\\cdots q_{n}) \\implies p_1 \\mid q_1\\,$"}
{"_id": "8361018", "title": "", "text": "$f(x) = \\dfrac1\\theta e^{\\frac{-x}{\\theta}}$"}
{"_id": "559038", "title": "", "text": "$f(x) = \\sum_{i=1}^k x_ic_i$"}
{"_id": "1637970", "title": "", "text": "$x\\in{\\langle m\\rangle}$"}
{"_id": "5810919", "title": "", "text": "$ X''(x)Y(y) + X(x)Y''(y)=X(x)Y(y) $"}
{"_id": "534916", "title": "", "text": "$x!=\\Gamma(x+1)=\\sqrt{2\\pi x}\\left(\\frac{x}{e}\\right)^x \\left(1+O\\left(\\frac{1}{x}\\right)\\right)$"}
{"_id": "552585", "title": "", "text": "$\\dfrac{\\sqrt{1-x^2}\\cdot(1-x^2-1+2x^2)}{\\sqrt{1-x^2}+\\sqrt{1 - 2x^2}}+\\dfrac{(1-1+2x^2)\\sqrt{x}}{1+\\sqrt{1 - 2x^2}} = 0$"}
{"_id": "3117830", "title": "", "text": "$=\\frac{1}{a}(1-e^{\\frac{-1}{a}})e^{\\frac{-1}{a}}\\Big(0-(\\frac{-a}{2})\\Big)=\\frac{1}{2}(1-e^{\\frac{-1}{a}})e^{\\frac{-1}{a}(u)}$"}
{"_id": "5936106", "title": "", "text": "$2\\pi\\int \\rho\\,ds,$"}
{"_id": "8144530", "title": "", "text": "$\\mu(\\lim\\inf A_n)\\le \\lim\\inf\\mu(A_n)$"}
{"_id": "2513868", "title": "", "text": "$\\lim_{x \\to c} = L$"}
{"_id": "1664908", "title": "", "text": "$\\displaystyle∫^∞_0f(x)\\,dx$"}
{"_id": "5880747", "title": "", "text": "$p_1 p_2=lcm(p_1,p_2) \\mid (aX-1)$"}
{"_id": "1589706", "title": "", "text": "$\\lim_{n \\to \\infty }\\mu(E_n) = \\mu(E)$"}
{"_id": "6051316", "title": "", "text": "$T_x H + T_x Y = T_x X$"}
{"_id": "1120764", "title": "", "text": "$\\begin{vmatrix}x_1&y_1&z_1\\\\x_2&y_2&z_2\\\\x_3&y_3&z_3\\end{vmatrix}$"}
{"_id": "3032612", "title": "", "text": "$b^{\\mathrm{log}_b(x)} = x$"}
{"_id": "9167775", "title": "", "text": "$L \\equiv \\begin{cases} x+z=16\\\\ 2y-3z=-30 \\end{cases}$"}
{"_id": "2367078", "title": "", "text": "$\\mathbb {\\bar{R}^+}$"}
{"_id": "8124923", "title": "", "text": "$ \\int_{0}^{2\\pi}\\frac{\\sin(x)}{x} = \\int_0^{\\pi} \\frac{\\sin(x)}{x}\\; dx + \\int_{\\pi}^{2\\pi} \\frac{\\sin(x)}{x}\\; dx = \\dots $"}
{"_id": "3671700", "title": "", "text": "$X \\supset X_1 \\supset X_2 \\supset \\cdots \\supset X_\\alpha \\supset X_{\\alpha+1} \\supset \\cdots$"}
{"_id": "6972438", "title": "", "text": "$\\{e_1,e_2,e_3,..., e_n\\}$"}
{"_id": "4010087", "title": "", "text": "$x_{n+1}=2018^{1+[2/n(n-1)]}x_1^{-1/n(n-1)}.$"}
{"_id": "4233880", "title": "", "text": "$f_n(x_n)=(\\frac{1}{2})$"}
{"_id": "3766110", "title": "", "text": "$t=1/|x|$"}
{"_id": "336686", "title": "", "text": "$1+2+3+\\dots = {-1\\over 12}$"}
{"_id": "8724373", "title": "", "text": "$ P\\left( \\sup_{t \\geq 0 } M_t > x \\mid \\mathcal{F}_0 \\right) = 1 \\wedge \\frac{M_0}{x}. $"}
{"_id": "7481749", "title": "", "text": "$\\operatorname{H}^{0,1}(X)=\\Omega ^{0,1}(X)/ \\bar{\\partial}( \\Omega^0(X)  ) $"}
{"_id": "6938596", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac{1}{n2^n(3n-1)}=\\frac{1}{\\sqrt[3]{2}}\\int_{0}^{\\frac{1}{\\sqrt[3]{2}}}\\frac{1}{x^2}\\log\\frac{1}{1-x^3}dx$"}
{"_id": "2123833", "title": "", "text": "$P(2) \\implies P(n)$"}
{"_id": "5357565", "title": "", "text": "$((a+b),(a-b))=e$"}
{"_id": "2748814", "title": "", "text": "$\\sum\\limits_{k=1}^n \\cos \\left(\\frac {2 \\pi k}{n} \\right) =0= \\sum \\limits_{k=1}^n \\sin \\left(\\frac {2 \\pi k}{n} \\right)$"}
{"_id": "8546644", "title": "", "text": "$ =\\left(1+\\frac{\\gamma}{x+\\alpha}\\right)^{ x+\\alpha+1/2} (1+\\beta/x)^{\\gamma}\\left(\\frac{x}{e}\\right)^{\\gamma}  \\approx e^\\gamma (x/e)^\\gamma = x^\\gamma.  $"}
{"_id": "4858463", "title": "", "text": "$ A_1 \\subset A_2 \\subset \\cdots \\subset A_i \\subset A_{i+1} \\subset \\cdots $"}
{"_id": "2025322", "title": "", "text": "$\\int_{a}^{a}f(x)dx?$"}
{"_id": "3713000", "title": "", "text": "$f^2(x) = \\frac x9 + 1 + 3 = \\frac x9 + 4$"}
{"_id": "2297836", "title": "", "text": "$S(a)=\\frac12\\int_0^1u^{\\frac{a-1}2}(1-u)^{-1/2}du$"}
{"_id": "51590", "title": "", "text": "$C_n = C_1 - 1 + \\frac {n (n+1)} 2 = 3 + \\frac {n (n+1)} 2$"}
{"_id": "318862", "title": "", "text": "$H(\\alpha^+)$"}
{"_id": "2924399", "title": "", "text": "$F(\\Delta x) = \\frac{f(a + \\Delta x) - f(a - \\Delta x)}{2\\Delta x}$"}
{"_id": "8742250", "title": "", "text": "$(x^r+y^r)/(x+y)^r$"}
{"_id": "7680608", "title": "", "text": "$1/\\lvert x\\rvert$"}
{"_id": "7125861", "title": "", "text": "$\\int_{0}^{\\infty}f(x)dx=0.5$"}
{"_id": "253479", "title": "", "text": "$(a,b)  + (a,b)= (2a,2b)$"}
{"_id": "2494671", "title": "", "text": "$f(a_1) = g(a_1)$"}
{"_id": "4183857", "title": "", "text": "$I_n = [n\\pi, (n+1)\\pi]$"}
{"_id": "1291480", "title": "", "text": "$\\det\\left(  I_{n}+A\\right)  =\\det\\left( \\begin{array} [c]{ccc} 1+a_{1,1} & a_{1,2} & a_{1,3}\\\\ a_{2,1} & 1+a_{2,2} & a_{2,3}\\\\ a_{3,1} & a_{3,2} & 1+a_{3,3} \\end{array} \\right)  $"}
{"_id": "7675493", "title": "", "text": "$\\frac{x_1+x_2+\\dots+x_n}n \\le \\sqrt{\\frac{x_1^2+x_2^2+\\dots+x_n^2}n}.$"}
{"_id": "1866211", "title": "", "text": "$\\frac{n(n+1)}{2} = \\frac{n^2+n}{2}$"}
{"_id": "5286358", "title": "", "text": "$ (\\frac{(-1) X^{n-1}e^{-x}}{(n-1)!}) $"}
{"_id": "6094953", "title": "", "text": "$-2i\\sin(\\pi s) \\Gamma(s)\\zeta(s)= -2i\\sin(\\pi s) \\Gamma(s)\\frac{\\Gamma(1-s)}{\\Gamma(1-s)}\\zeta(s) = \\frac {-2\\pi i}{\\Gamma(1-s)}\\zeta(s) = \\Large\\int_{\\gamma} \\frac{(-z)^{s-1}}{e^z-1}dz$"}
{"_id": "1491626", "title": "", "text": "$ \\int_{0}^{x}f(x-t)g(t)dt = \\int_{0}^{x}f(t)g(x-t)dt. $"}
{"_id": "4433959", "title": "", "text": "$4\\pi\\int_0^{\\pi/2}\\sin(\\varphi)\\cos(\\varphi)\\,d\\varphi$"}
{"_id": "5597552", "title": "", "text": "$K = cov(X) - cov(X,Y) cov(Y)^{-1} cov(Y,X)$"}
{"_id": "7606342", "title": "", "text": "$x_n=p_n(x)$"}
{"_id": "7899286", "title": "", "text": "$\\forall a,b \\in \\Bbb N, f(ab)=f(a)+f(b)$"}
{"_id": "9372212", "title": "", "text": "$\\newcommand\\cov{\\operatorname{Cov}}\\cov[X,E(Y|X)]=\\cov[X,Y]$"}
{"_id": "3868408", "title": "", "text": "$f(x) \\equiv \\frac{e^{x^2}}{10^x}$"}
{"_id": "2204854", "title": "", "text": "$x_0=\\sqrt[3]{\\frac{3\\pi}{2^{n+2}}}$"}
{"_id": "8092359", "title": "", "text": "$= \\frac{(1+i)^{n+1} - (1+i)^n}{(1+i)^n-1}$"}
{"_id": "6510937", "title": "", "text": "$\\lim_{t\\to 0} \\frac{(1+t)\\ln{(1+t)}-t}{t^2(1+t)+t^2-t(1+t)\\ln{(1+t)}}=Taylor=$"}
{"_id": "5840470", "title": "", "text": "$E[|XY|^2] \\leq \\displaystyle \\sup_{\\omega \\in \\Omega}|X(\\omega)|^2 E[|Y|^2]$"}
{"_id": "2717564", "title": "", "text": "$1/|x|=1+t$"}
{"_id": "6140495", "title": "", "text": "$f(f(n))=f(f(n+2)+2)=n$"}
{"_id": "6136968", "title": "", "text": "$x^4 + 1 = 2y^2$"}
{"_id": "9243545", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta>0:|y-x|<\\delta \\implies  |f(y)-f(x)| < \\epsilon$"}
{"_id": "7834392", "title": "", "text": "$ \\begin{cases}   x\\equiv 1\\ [u] \\\\   x\\equiv 1 \\ [v] \\end{cases}$"}
{"_id": "7662066", "title": "", "text": "$[t_x,t_s]$"}
{"_id": "4306929", "title": "", "text": "$f(x) = \\frac{2x}{x^2 - 4}$"}
{"_id": "5805428", "title": "", "text": "$ \\sum_{n=1}^\\infty |a_n| = \\sum_{n=1}^\\infty |b_n| + \\sum_{n=1}^\\infty |c_n| < \\infty $"}
{"_id": "2008566", "title": "", "text": "$X_\\gamma=\\{a+b\\gamma: a, b\\in\\mathbb{Z}\\}$"}
{"_id": "8448611", "title": "", "text": "$\\mathbb{E}[X|\\mathcal{F}_0]:=P_{\\mathcal{H}}(X)$"}
{"_id": "79792", "title": "", "text": "$\\lim _{n \\to \\infty} \\frac{1}{n} \\sum_{r=1} \\frac{1}{r/n}$"}
{"_id": "211324", "title": "", "text": "$(x-a)^{n-1}$"}
{"_id": "556864", "title": "", "text": "$A=\\begin{pmatrix}1&1&1\\\\a&b&c\\\\a^2&b^2&c^2\\end{pmatrix}$"}
{"_id": "8703672", "title": "", "text": "$\\int_0^{\\pi} \\Phi(\\theta) \\delta(\\cos \\theta-\\cos \\theta_0)\\sin \\theta d\\theta=\\int_0^{\\pi} \\Phi(\\theta) \\frac{\\delta(\\theta- \\theta_0)}{|\\sin \\theta|}\\sin \\theta d\\theta$"}
{"_id": "6694801", "title": "", "text": "$d(x,A)\\leqslant d(x,a)$"}
{"_id": "5909515", "title": "", "text": "$T^{1,0}M\\cap T^{0,1}M=(0)$"}
{"_id": "653649", "title": "", "text": "$(E(|X+Y|^r))^{1/r}\\le (E(|X|^r))^{1/r}+(E(|Y|))^{1/r}$"}
{"_id": "3250623", "title": "", "text": "$\\tan(\\alpha) = \\frac{s}{2}.$"}
{"_id": "9241398", "title": "", "text": "$s = (rs)(r^2s)^{-1}(rs)$"}
{"_id": "1204827", "title": "", "text": "$K \\subseteq D \\subseteq \\bar{D} \\subseteq U$"}
{"_id": "1117894", "title": "", "text": "$\\bar {x}_n =\\frac {x_1+x_2+...+x_n} n$"}
{"_id": "7575112", "title": "", "text": "$\\left( \\begin{array}{ll}  k+1 & k-1\\\\  k-1 & k+1 \\end{array}\\right)^5$"}
{"_id": "2674126", "title": "", "text": "$\\lfloor\\frac{\\lfloor \\frac{n}{s} \\rfloor}{r}\\rfloor = \\lfloor \\frac{n}{rs} \\rfloor$"}
{"_id": "5159838", "title": "", "text": "$\\gamma(s+t)=\\gamma(s)\\gamma(t)$"}
{"_id": "1757337", "title": "", "text": "$\\lim \\limits_{x \\to c^+} f(x) = f(c)$"}
{"_id": "6757742", "title": "", "text": "$\\Bbb R[x]/(x^4+1)$"}
{"_id": "5677203", "title": "", "text": "$[x,y] = \\alpha$"}
{"_id": "4938828", "title": "", "text": "$\\begin{align} n^3 \\frac{1}{n}\\sum_{k=1}^n \\sqrt{1+\\frac{k}{n^4}}\\sin\\frac{2\\pi k}{n} &=  n^3 \\frac{1}{n}\\sum_{k=1}^n \\left(\\sqrt{1+\\frac{k}{n^4}}-1\\right)\\sin\\frac{2\\pi k}{n} + n^3\\cdot\\frac{1}{n}\\sum_{k=1}^n\\sin\\frac{2\\pi k}{n} \\\\ &\\approx n^3 \\frac{1}{n}\\sum_{k=1}^n \\frac{k}{2n^4}\\sin\\frac{2\\pi k}{n} + n^3\\cdot\\frac{1}{n}\\sum_{k=1}^n\\sin\\frac{2\\pi k}{n} \\\\ &= \\frac{1}{2}\\cdot\\frac{1}{n}\\sum_{k=1}^n \\frac{k}{n}\\sin\\frac{2\\pi k}{n} + n^3\\cdot\\frac{1}{n}\\sum_{k=1}^n\\sin\\frac{2\\pi k}{n} \\\\ &\\xrightarrow[n\\to\\infty]{} \\frac{1}{2}\\int_0^1 x\\sin(2\\pi x)\\, dx+\\frac{1}{2}\\int_0^1 \\sin(2\\pi x)\\, dx =  \\frac{1}{2}\\int_0^1 x\\sin(2\\pi x)\\, dx \\\\&= \\boxed{-\\frac{1}{4\\pi}} \\end{align}$"}
{"_id": "7897793", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{x_1+x_2+...+x_n}{n}=a$"}
{"_id": "7311603", "title": "", "text": "$(n+1)!+2, (n+1)!+3,\\ldots, (n+1)!+n$"}
{"_id": "1012520", "title": "", "text": "$\\tfrac{1}{2^{2n-1}}$"}
{"_id": "5170419", "title": "", "text": "$|AB|=r$"}
{"_id": "7313251", "title": "", "text": "$Cov(X,B)=0$"}
{"_id": "6443494", "title": "", "text": "$ G=\\{A\\in M_{2\\times2}(\\Bbb{R})\\mid \\det A\\neq0, XA=X\\}. $"}
{"_id": "4191747", "title": "", "text": "$L=P\\oplus S$"}
{"_id": "4270034", "title": "", "text": "$\\ \\sin\\theta = \\frac{x}{2}$"}
{"_id": "6453462", "title": "", "text": "$\\int_0^\\pi{f(x)\\sin xdx} = \\int_0^\\pi{f(x)\\cos xdx} = 1.$"}
{"_id": "5634866", "title": "", "text": "$\\frac{1-r}{(1+r)^2}=\\frac{2-(1+r)}{(1+r)^2}=\\frac{2}{(1+r)^2}-\\frac{1}{(1+r)}$"}
{"_id": "8856768", "title": "", "text": "$I_n(\\pi) = 0$"}
{"_id": "3117828", "title": "", "text": "$f_{U}(u)=\\int_{-\\infty}^{\\infty}f_{X}(u+v)\\cdot f_{Y}(v)\\cdot 1 \\, dv=\\int_{0}^{\\infty}\\frac{1}{a}e^{\\frac{-1}{a}(u+v)}(1-e^{\\frac{-1}{a}})e^{\\frac{-1}{a}(v)}\\, dv$"}
{"_id": "1619456", "title": "", "text": "$\\int_0^\\infty f(x) = \\int_0^\\infty g(x) =\\infty.$"}
{"_id": "95292", "title": "", "text": "$\\{r+1,r+2,\\ldots,2r\\}$"}
{"_id": "5184507", "title": "", "text": "$V=\\{\\langle x,y\\rangle\\in G\\times G:xy\\in U\\}$"}
{"_id": "3347272", "title": "", "text": "$\\frac{a+b+c}{b-a}=3$"}
{"_id": "8025685", "title": "", "text": "$ \\sum_{k=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{k^2 + n^{1/\\gamma}} \\geq \\sum_{k=1}^{\\infty}\\sum_{n=1}^{\\infty} \\frac{1}{k^2 + n^2}. $"}
{"_id": "4202809", "title": "", "text": "$\\lvert f(z)\\rvert \\leqslant m\\lvert g(z)\\rvert.$"}
{"_id": "6997278", "title": "", "text": "$f(\\pi),f(0)=0,f(\\pi)>0$"}
{"_id": "1448835", "title": "", "text": "$\\sum_{i=1}^n \\frac{1}{3^i}\\tag{displayed}$"}
{"_id": "6172717", "title": "", "text": "$\\mathbb{R}[x]/{\\left< x^2+1\\right>}$"}
{"_id": "6413206", "title": "", "text": "$a^{log_ax} = a^b$"}
{"_id": "8501225", "title": "", "text": "$(x,y)\\in D:=\\{(x,y):a\\leq x \\leq b, c\\leq y\\leq d \\}$"}
{"_id": "3163735", "title": "", "text": "$f(z) = \\sqrt r e^{\\frac {i\\theta}2}$"}
{"_id": "6739350", "title": "", "text": "$\\frac{\\sin\\theta}{\\tan\\theta}=\\frac{\\sin\\theta}{\\frac{\\sin\\theta}{\\cos\\theta}}$"}
{"_id": "2489696", "title": "", "text": "$c_n:=\\frac{n}{2(n-1)}$"}
{"_id": "5938424", "title": "", "text": "$\\lim_{n\\to\\infty}\\inf_{k\\geq n}\\mu(A_k)$"}
{"_id": "1106859", "title": "", "text": "$ \\sum_{n \\geq 1} \\frac{1}{n^2},$"}
{"_id": "1378437", "title": "", "text": "$\\sum||f_n-g_n||_2^2<1$"}
{"_id": "1635023", "title": "", "text": "$\\sum\\limits_{r=1}^{n} r(r!) = (n+1)!-1$"}
{"_id": "5962787", "title": "", "text": "$(a+b, a-b)=2$"}
{"_id": "4414408", "title": "", "text": "$(6^k-3^k-3(2^k-1))/6$"}
{"_id": "8576267", "title": "", "text": "$g_1(x,y,z)=x^2+y^2+z^2$"}
{"_id": "1130889", "title": "", "text": "$C_r = C_r(0)$"}
{"_id": "7730315", "title": "", "text": "$\\frac{27}{9}=3$"}
{"_id": "6933636", "title": "", "text": "$s=t+1, s=t+2, s=t+4, s=t+17, s=t+34, s=t+68$"}
{"_id": "8793451", "title": "", "text": "$\\langle a,b\\rangle := \\lim\\limits_{N \\to \\infty} \\frac{1}{N}\\sum\\limits_{i=1}^{N}a_i b_i$"}
{"_id": "6169012", "title": "", "text": "$\\int\\frac{1}{(1+x^2)^k} dx$"}
{"_id": "7875673", "title": "", "text": "$\\left\\lfloor \\frac{1}{b} \\left\\lfloor \\frac{a}{b} \\right\\rfloor \\right\\rfloor \\leq \\left\\lfloor \\frac{a}{b^2} \\right\\rfloor.$"}
{"_id": "8124920", "title": "", "text": "$ \\int_0^{2\\pi} \\frac{\\sin x}{x} dx = \\int_0^{\\pi} \\frac{\\sin x}{x} dx  + \\int_\\pi^{2\\pi} \\frac{\\sin x}{x} dx = \\int_0^{\\pi} \\frac{\\sin x}{x} dx + \\int_0^{\\pi} \\frac{\\sin (v+\\pi)}{v+\\pi} dv $"}
{"_id": "8319404", "title": "", "text": "$\\alpha^+ > \\alpha$"}
{"_id": "1596902", "title": "", "text": "$\\frac{21}x+\\frac{70}y=k$"}
{"_id": "7654761", "title": "", "text": "$ d(x,z) \\leq d(x,y)+d(y,z)?$"}
{"_id": "6909718", "title": "", "text": "$(x R y\\wedge y R x)\\to x = y$"}
{"_id": "7321475", "title": "", "text": "$\\zeta(s)=\\sum_{k=1}^N\\frac{1}{k^s}+\\frac{N^{1-s}}{s-1}-s\\int_{N}^\\infty \\frac{x-\\lfloor x \\rfloor}{x^{s+1}} dx,$"}
{"_id": "683648", "title": "", "text": "$\\mp 7= \\color{blue}{\\pm \\mp} (\\mp 7)=\\begin{cases}+(-(-7)) & =+7\\\\ -(+(+7))  & =-7\\end{cases}=\\pm 7\\ne \\mp 7\\tag{*}$"}
{"_id": "6069373", "title": "", "text": "$\\lambda =1/|x|$"}
{"_id": "4041808", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1).$"}
{"_id": "30577", "title": "", "text": "$\\displaystyle F(x)=\\int_0^x f(t)dt$"}
{"_id": "817243", "title": "", "text": "$I_n = \\dfrac1{2\\pi} \\displaystyle\\int_{-\\pi}^{\\pi} \\dfrac{\\sin(nx)}{\\sin(x)}dx$"}
{"_id": "2514802", "title": "", "text": "$(y+i,y-i)\\ne1$"}
{"_id": "1530167", "title": "", "text": "$P[X_i=0\\mid A]=1-(w/n)$"}
{"_id": "5199743", "title": "", "text": "$A=\\{\\gamma\\in\\Bbb N:\\gamma=0\\text{ or }S\\alpha\\le\\alpha+\\gamma\\}$"}
{"_id": "3774841", "title": "", "text": "$a_n = \\frac{x_1+x_2+\\ldots + x_n}{n}$"}
{"_id": "6072564", "title": "", "text": "$f_X(x; \\theta) = (\\theta +1)x^\\theta$"}
{"_id": "7133881", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty \\int e^{-\\pi{n^2}/ x}  dx $"}
{"_id": "3703255", "title": "", "text": "$A_1A_n$"}
{"_id": "307781", "title": "", "text": "$|f(z)|\\leq k|z|$"}
{"_id": "5587545", "title": "", "text": "$\\displaystyle F(x) = \\int \\limits_a ^x f(t) dt \\rightarrow \\frac{dF(x)}{dx} = f(x)$"}
{"_id": "8692175", "title": "", "text": "$\\pi:\\mathbb{R}^{n+m}\\to \\mathbb{R}^n$"}
{"_id": "5566262", "title": "", "text": "$P(E(X|{\\mathcal F}_0)=1)=0$"}
{"_id": "1053783", "title": "", "text": "$z=\\frac{cx(b+1)}{1-bc}$"}
{"_id": "5679222", "title": "", "text": "$\\sum_{k=1}^n \\cos(2 \\pi j k/n) = n$"}
{"_id": "334773", "title": "", "text": "$(1+r+r^2)-(1-r+r^2)=2r,$"}
{"_id": "8775038", "title": "", "text": "$1\\leq\\sum_{n=1}^{+\\infty}\\|e_n-f_n\\|,$"}
{"_id": "3845830", "title": "", "text": "$\\text{ind}_{\\gamma}(1)=1\\;\\;\\;\\text{and}\\;\\;\\;\\text{ind}_{\\gamma}(2)=2$"}
{"_id": "8784082", "title": "", "text": "$g_i(x) = \\sum_{j=1}^n c_{ij}f_j(x)$"}
{"_id": "1931146", "title": "", "text": "$(1+x)^{2n-1}=(x+1)^n(1+x)^{n-1}$"}
{"_id": "784198", "title": "", "text": "$A = \\int_0^\\pi\\frac{\\cos x}{(x+2)^2}\\,dx$"}
{"_id": "3922550", "title": "", "text": "$|g(z^2)| \\le |g(z)|$"}
{"_id": "9184116", "title": "", "text": "$\\frac{x^{n-1}e^{-x}}{(n-1)!}$"}
{"_id": "6969250", "title": "", "text": "$\\begin{bmatrix}z&w\\\\-\\overline{w}&\\overline{z}\\end{bmatrix}$"}
{"_id": "508494", "title": "", "text": "$||x||^2 = \\sum_{n=1}^{\\infty} |\\langle x, e_n \\rangle |^2 $"}
{"_id": "4819191", "title": "", "text": "$D\\left[X\\right] = E\\left[(X- E\\left[X\\right])^2\\right] = E\\left[X^2- 2\\cdot X \\cdot E\\left[X\\right] + E\\left[X\\right]^2\\right] = E\\left[X^2\\right] - 2 \\left[X \\cdot E\\left[X\\right]\\right] + E\\left[E\\left[X\\right]^2\\right]$"}
{"_id": "1062014", "title": "", "text": "$d_1\\mid d_2$"}
{"_id": "4681242", "title": "", "text": "$ \\det\\begin{pmatrix} A-B & B-A\\\\ B & A \\end{pmatrix}=\\det\\begin{pmatrix} A-B & 0\\\\ B & A+B \\end{pmatrix}.$"}
{"_id": "1371713", "title": "", "text": "$f_{X}(x|\\theta)$"}
{"_id": "3984230", "title": "", "text": "$\\forall \\epsilon_2>0, \\exists \\delta_2>0: |x-L|<\\delta_2\\implies |f(x) - f(L)|<\\epsilon_2$"}
{"_id": "1392232", "title": "", "text": "$ (0^x+4^x)<3^x+4^x<4^x+4^x $"}
{"_id": "6197368", "title": "", "text": "$ \\gamma = \\frac{\\gamma + \\overline{\\gamma}}{2} + \\frac{\\gamma - \\overline{\\gamma}}{2},$"}
{"_id": "5759987", "title": "", "text": "$\\lim_{n \\to \\infty} 2^{-n} \\lim_{\\delta \\to 0} {\\left( \\cos(3^nx) ( \\cos(3^n\\delta) - 1 ) - \\sin(3^nx)\\sin(3^n\\delta) \\right) \\over \\delta}$"}
{"_id": "7867854", "title": "", "text": "$ \\sum_{r = 0}^n\\frac1{r!} = 1 + \\sum_{r = 1}^n\\frac{1}{r!} \\leq 1+\\sum_{r = 1}^n\\frac{1}{2^{r-1}}\\\\  = 1 + \\sum_{r = 0}^{n-1}\\frac{1}{2^r}\\\\ \\leq 1 + \\sum_{r = 0}^\\infty\\frac{1}{2^r}\\\\ = 1 + \\frac{1}{1-\\frac12} = 3 $"}
{"_id": "6208124", "title": "", "text": "$V_{i,x} \\subseteq \\overline{V_{i,x}} \\subseteq U_i$"}
{"_id": "1311071", "title": "", "text": "$ W=\\int_{t_0}^{t_1}\\mathrm d W=\\int_{t_0}^{t_1}F_T(t)v(t)\\mathrm dt=\\int_{0}^{\\pi/\\omega}m4B^2t\\,\\mathrm dt=\\frac{m2B^2\\pi^2}{\\omega^2} $"}
{"_id": "9141805", "title": "", "text": "$(c + d, c - d) = 1$"}
{"_id": "2100005", "title": "", "text": "$|f(z)|\\leq M |P(z)|$"}
{"_id": "8909595", "title": "", "text": "$(M,I_M)$"}
{"_id": "4209441", "title": "", "text": "$(I_j)_{j\\geq1}$"}
{"_id": "5515917", "title": "", "text": "$|\\mathbb{R}^n|=|\\mathbb{R}^{n-1}\\times\\mathbb{R}|=|\\mathbb{R}\\times\\mathbb{R}|=|\\mathbb{R}|.$"}
{"_id": "7598734", "title": "", "text": "$\\begin{align} \\lim_{N\\to \\infty}\\sum_{k=1}^N \\frac{1}{k+N}&=\\lim_{N\\to \\infty}\\sum_{k=1}^{2N}\\frac{(-1)^{k-1}}{k}\\\\\\\\ &=\\sum_{k=1}^\\infty \\frac{(-1)^{k-1}}{k}\\\\\\\\ &=\\log(2) \\end{align}$"}
{"_id": "4752453", "title": "", "text": "$f(a_1) = a_1$"}
{"_id": "3768957", "title": "", "text": "$y:=\\frac{(1+x)^2}{2(1+x^2)}$"}
{"_id": "3421963", "title": "", "text": "$d(x,A)=\\inf\\{d(a,x):x∈A\\}$"}
{"_id": "4673684", "title": "", "text": "$x! > \\sqrt{2\\pi x}(x/e)^x$"}
{"_id": "4334979", "title": "", "text": "$[x,x]=0, [x,y]=x$"}
{"_id": "754185", "title": "", "text": "$e^x-(1+\\frac{x}{n})^n>0$"}
{"_id": "6692404", "title": "", "text": "$\\{X,Y,Z\\}=\\{A,-B,0\\}$"}
{"_id": "8984512", "title": "", "text": "$\\lim_{\\gamma \\rightarrow 1} \\frac{x^{1-\\gamma}-1}{1-\\gamma} = \\frac{\\lim_{\\gamma \\rightarrow 1}\\frac{\\mathrm{d}}{\\mathrm{d\\gamma}}\\left(x^{1-\\gamma}-1\\right)}{\\lim_{\\gamma \\rightarrow 1}\\frac{\\mathrm{d}}{\\mathrm{d\\gamma}}\\left(1-\\gamma\\right)} = \\frac{\\lim_{\\gamma \\rightarrow 1}-\\log(x)x^{1-\\gamma}}{\\lim_{\\gamma \\rightarrow 1}-1} = \\log(x)$"}
{"_id": "8965211", "title": "", "text": "$\\lim_{n\\to \\infty}\\dfrac{\\sum\\limits_{k=2}^n k\\cos\\dfrac{π}{k}}{n^2} $"}
{"_id": "9033911", "title": "", "text": "$\\tan x=\\frac{\\sin x}{\\cos x}\\Rightarrow \\cos x=\\frac{\\sin x}{\\tan x}=\\left(\\frac{2\\sqrt{ab}}{a+b}\\right)/\\left(\\frac{2\\sqrt{ab}}{a-b}\\right)=\\frac{a-b}{a+b}.$"}
{"_id": "4296590", "title": "", "text": "$y_n = \\frac{x_1 + x_2 + x_3 + x_4 + \\ldots +x_n}{n} $"}
{"_id": "7332114", "title": "", "text": "$\\{a, a+1, b, b+2\\}$"}
{"_id": "3521778", "title": "", "text": "$g(n)=\\binom{nr}{s}$"}
{"_id": "6119674", "title": "", "text": "$(C_n)_{n\\in N}\\ with \\ c_n = (-1)^n\\frac{n^3 + 2}{n^2+1}$"}
{"_id": "898121", "title": "", "text": "$(\\neg \\neg A\\vee \\neg \\neg B)$"}
{"_id": "1157517", "title": "", "text": "$z_2=\\sqrt 2e^{\\frac {17i\\pi}{12}}$"}
{"_id": "7148447", "title": "", "text": "$J:=\\{a\\in B| \\forall n \\in \\Bbb N: a(n+1)\\geq a(n)\\} $"}
{"_id": "2391056", "title": "", "text": "$o(ab)=rs$"}
{"_id": "8755003", "title": "", "text": "$\n f(x)=\\sum_{j=i}^n c_j f(e_{i_j}).\n $"}
{"_id": "1467074", "title": "", "text": "$\\tan\\theta=\\frac{x+1}2,$"}
{"_id": "8149851", "title": "", "text": "$P(X_1 = 0 | X_0 = 1) = 1/2$"}
{"_id": "4136477", "title": "", "text": "$\\{x,x+k,x+2k,\\cdots\\}$"}
{"_id": "1822781", "title": "", "text": "$r=z^{\\frac{1}{n}}e^{\\frac{2i\\pi k}{n}}$"}
{"_id": "8911900", "title": "", "text": "$a_{m+n} = a_m + a_n + mn$"}
{"_id": "8085295", "title": "", "text": "$xRy, yRx,$"}
{"_id": "1268849", "title": "", "text": "$1-1/|x|$"}
{"_id": "5324124", "title": "", "text": "$\\lvert f(z)\\rvert = \\lim_{\\epsilon \\searrow 0} \\lvert f_\\epsilon(z)\\rvert \\leqslant 1$"}
{"_id": "2796954", "title": "", "text": "$5Y^2=X^3-X, X\\ne 0, \\pm 1.$"}
{"_id": "5705274", "title": "", "text": "$C_r=\\{|z|=R\\}$"}
{"_id": "7126780", "title": "", "text": "$|ab|=m$"}
{"_id": "4839470", "title": "", "text": "$F_3\\times F_3$"}
{"_id": "1512090", "title": "", "text": "$\\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{c}{m^2+n^2}.$"}
{"_id": "4110149", "title": "", "text": "$\\intop_{0}^{-\\infty}f(x)dx$"}
{"_id": "5673746", "title": "", "text": "$ ds \\ = \\ \\sqrt{ \\frac{dx^2}{dx^2} \\ + \\ \\frac{dy^2}{dx^2}} \\ \\ dx \\ \\ = \\ \\ \\sqrt{ 1 \\ + \\ \\left( \\frac{dy}{dx} \\right)^2 } \\ \\ dx   $"}
{"_id": "6427348", "title": "", "text": "$\\text{D}=\\frac{\\left|\\text{a}^{\\Re\\left(\\text{s}\\right)-1}\\right|}{\\exp\\left(\\text{a}\\text{n}\\right)}=\\frac{\\left|\\text{a}^{\\Re\\left(\\text{s}\\right)}\\cdot\\text{a}^{-1}\\right|}{\\exp\\left(\\text{a}\\text{n}\\right)}=\\frac{1}{\\text{a}}\\cdot\\frac{\\left|\\text{a}^{\\Re\\left(\\text{s}\\right)}\\right|}{\\exp\\left(\\text{a}\\text{n}\\right)}=\\frac{1}{\\text{a}}\\cdot\\frac{\\text{a}^{\\Re\\left(\\text{s}\\right)}}{\\exp\\left(\\text{a}\\text{n}\\right)}=\\frac{\\text{a}^{\\Re\\left(\\text{s}\\right)-1}}{\\exp\\left(\\text{a}\\text{n}\\right)}\\tag3$"}
{"_id": "4434758", "title": "", "text": "$\\,\\gcd\\{a,b\\} = \\gcd\\{a\\} = a.\\,$"}
{"_id": "14346", "title": "", "text": "$X^3-X, X^2-1$"}
{"_id": "5792625", "title": "", "text": "$\\beta:T=\\emptyset\\rightarrow\\emptyset$"}
{"_id": "8852933", "title": "", "text": "$\\mathbb{E}s[\\mathbf{1}_{\\{\\tau=0\\}\\cup\\{s\\geq z\\}}]=\\mathbb{E}s-\\gamma\\mathbb{E}s\\mathbf{1}_{\\left(-\\infty,z\\right)}\\left(s\\right)=\\mu_s-\\gamma\\int_{-\\infty}^{z}xf_s(x)dx$"}
{"_id": "1096518", "title": "", "text": "$\\lim_{x \\to a} f_1(x) = L_1, \\lim_{x \\to a} f_2(x) = L_2, ..., \\lim_{x \\to a} f_n(x) = L_n$"}
{"_id": "469556", "title": "", "text": "$A_1 \\times A_2 \\times \\cdots \\times A_n$"}
{"_id": "7040863", "title": "", "text": "$     \\left(  \\begin{array}{rr}   A  &  pB  \\\\    B   &  A   \\end{array}    \\right)  ,   $"}
{"_id": "4638950", "title": "", "text": "$A_1\\subseteq A_2\\subseteq \\cdots \\subseteq A_n $"}
{"_id": "143448", "title": "", "text": "$\\mathrm{Cov}(X,Y)=0$"}
{"_id": "2481088", "title": "", "text": "$T((a,b)+(x,y))=T((a+x,b+y))=(a+b+x+y,a+x)=(a+b,a)+(x+y,x)=T((a,b))+T((x,y))$"}
{"_id": "573911", "title": "", "text": "$a_n=\\frac{1}{\\sqrt{n(n+1)}}$"}
{"_id": "1253715", "title": "", "text": "$\\;m=14\\cdot x\\;,\\;\\;x\\in\\Bbb N\\;$"}
{"_id": "2550198", "title": "", "text": "$\\left\\lfloor\\frac{\\left\\lfloor\\frac{N}{a}\\right\\rfloor}{b}\\right\\rfloor = \\left\\lfloor\\frac{N}{a*b}\\right\\rfloor$"}
{"_id": "952418", "title": "", "text": "$\\left(1+\\frac xn\\right)^n\\le e^x$"}
{"_id": "5540825", "title": "", "text": "$\\overline{\\frak a^+}$"}
{"_id": "1064696", "title": "", "text": "$Cov(X,Y)=c$"}
{"_id": "4230791", "title": "", "text": "$G(x) := \\int_a^x f(t) \\,dt,$"}
{"_id": "1471718", "title": "", "text": "$f(f(x)^2)=xf(x)$"}
{"_id": "391741", "title": "", "text": "$ \\lim_{N \\rightarrow \\infty} \\sum_{n=1}^N \\frac{n^t}{e^n} = \\sum_{n=1}^{N_0} \\frac{n^t}{e^n} + \\lim_{N \\rightarrow \\infty} \\sum_{n={N_0}}^N \\frac{n^t}{e^n} $"}
{"_id": "8565330", "title": "", "text": "$\\sum_{j=1}^{m}c_j\\chi_{K_j}$"}
{"_id": "1905079", "title": "", "text": "$c^Tx,x \\in P$"}
{"_id": "2074181", "title": "", "text": "$\\int_0^{2\\pi} f(x)dx = \\int_0^{\\pi} f(x)dx -\\int_{0}^{\\pi} f(t)dt=0 $"}
{"_id": "5473664", "title": "", "text": "$P(0)\\implies P(1)\\implies P(2)\\implies P(3)\\implies P(4)$"}
{"_id": "6610461", "title": "", "text": "$ mn = (\\tan\\theta+\\sin\\theta)(\\tan\\theta-\\sin\\theta) = \\tan^2\\theta-\\sin^2\\theta $"}
{"_id": "687068", "title": "", "text": "$(\\gamma \\circ \\rho \\circ \\gamma)^n=\\gamma \\circ \\rho \\circ \\gamma$"}
{"_id": "2170556", "title": "", "text": "$ \\beta_1\\geq\\gamma_1\\geq\\beta_2\\geq\\gamma_2\\geq\\cdots\\geq\\gamma_{n-1}\\geq\\beta_n. $"}
{"_id": "1716154", "title": "", "text": "$\\lim_{k \\to \\infty}\\sum_{n=1}^{2^k-1}\\frac{ \\left\\lfloor\\sqrt{4^k-n^2}\\right\\rfloor\\ } {2^{2k}}$"}
{"_id": "5650991", "title": "", "text": "$f(x)=\\frac{x^4}{9+x^3}$"}
{"_id": "5701862", "title": "", "text": "$p_1^2 p_2 p_3$"}
{"_id": "4850672", "title": "", "text": "$ \\frac{2}{x} + \\frac{3}{y} = 1,\\quad x, y \\in  \\mathbb{Z}^{*}$"}
{"_id": "3454300", "title": "", "text": "$log_a(b) = \\frac{log(b)}{log(a)}$"}
{"_id": "2264050", "title": "", "text": "$B(n)=\\{n!+2, n!+3,\\cdots,n!+n\\}$"}
{"_id": "4268452", "title": "", "text": "$ y''' + y' - y'' -y =  \\\\ y''' -y'' + y' - y = 0 $"}
{"_id": "2112284", "title": "", "text": "$7|a^7-a$"}
{"_id": "6026500", "title": "", "text": "$\\frac{1}{\\eta(s)}=\\frac1{(1-2^{1-s})\\cdot\\zeta(s)}=\\frac1{1-2^{1-s}}\\cdot\\sum_{n=1}^{\\infty}\\frac{\\mu(n)}{n^{s}}=\\sum_{n=1}^{\\infty}\\frac{b(n)}{n^{s}}\\iff b(n)=\\frac{\\mu(n)}{1-2^{1-s}}$"}
{"_id": "2687913", "title": "", "text": "$\\int_0^\\infty g(x) \\, dx$"}
{"_id": "3356986", "title": "", "text": "$\\lim_{n\\to\\infty} \\frac 1n \\sum_{r=1}^n\\frac{\\lfloor 2rx \\rfloor} {n}=x $"}
{"_id": "5048701", "title": "", "text": "$u= (\\gamma (b) - \\gamma (a))/|\\gamma (b) - \\gamma (a)|,$"}
{"_id": "1301734", "title": "", "text": "$ \\frac{6}{x}+\\frac{-4}{y}+\\frac{1}{z}=0$"}
{"_id": "3443612", "title": "", "text": "$\\lim_{x\\to a}f(x)=\\lim_{x\\to a^+}f(x)=\\lim_{x\\to a^-}f(x).$"}
{"_id": "5504309", "title": "", "text": "$f(x) = \\dfrac{e^x}{x^2+1}$"}
{"_id": "8189026", "title": "", "text": "$E((\\xi-E\\xi)^2)=E(\\xi^2-2\\xi E(\\xi)+E(\\xi)^2)=E(\\xi^2)-2E(\\xi)E(\\xi)+E(\\xi)^2=E(\\xi^2)-E(\\xi)^2=D\\xi.$"}
{"_id": "5316120", "title": "", "text": "$y=ax + \\frac{b+c}{2} + b - \\frac{b+c}{2}$"}
{"_id": "3735450", "title": "", "text": "$Cov(u,v)=0$"}
{"_id": "6033064", "title": "", "text": "$\\int\\tan^{-1}{\\left(1-x+x^2\\right)}\\,dx=x\\tan^{-1}{\\left(1-x+x^2\\right)}-\\int x\\cdot\\frac{-1+2x}{\\left(1-x+x^2\\right)^2+1}dx\\\\ =x\\tan^{-1}{\\left(1-x+x^2\\right)}-\\int\\frac{x\\left(2x-1\\right)}{\\left(1-x+x^2\\right)^2+1}dx.$"}
{"_id": "6329323", "title": "", "text": "$\\frac{1}{\\gamma}-\\frac{f(1/\\gamma)}{f'(1/\\gamma)}=\\frac{1}{\\gamma}-\\frac{1}{2\\gamma^3}+O\\left(\\frac{1}{\\gamma^5}\\right).$"}
{"_id": "7792419", "title": "", "text": "$f(n)=2n^2-2n+1$"}
{"_id": "577600", "title": "", "text": "$\\zeta(s) = \\sum_{n = 1}^{+\\infty} \\frac{1}{n^s} = s\\int_1^{+\\infty} \\frac{\\lfloor x\\rfloor}{x^{s+1}}\\,dx\\,,$"}
{"_id": "3429646", "title": "", "text": "$\\gcd(a,b) = \\gcd(a,4)$"}
{"_id": "1203757", "title": "", "text": "$f(x) = \\binom x2$"}
{"_id": "3357578", "title": "", "text": "$\\dfrac{2\\pi(1-\\cos(\\theta))}{4\\pi} = \\dfrac{1-\\cos(\\theta)}{2}$"}
{"_id": "5092134", "title": "", "text": "$ 2[x^2(x+y)(x+z)+y^2(x+y)(y+z)+z^2(x+z)(y+z)]\\ge 3(x+y)(x+z)(y+z). $"}
{"_id": "3830906", "title": "", "text": "$(k+1)-k=((k+1)^{\\frac{1}{3}}-k^{\\frac{1}{3}})((k+1)^{\\frac{2}{3}}+(k+1)^{\\frac{1}{3}}k^{\\frac{1}{3}}+(k)^{\\frac{2}{3}})$"}
{"_id": "4353356", "title": "", "text": "$\\sum_{a=0}^m \\sum_{b=0}^a \\sum_{c=0}^b,\\sum_{a=0}^m \\sum_{b=0}^a   \\sum_{c=b}^a,  \\sum_{a=0}^m \\sum_{b=0}^a \\sum_{c=a}^m,\\\\ \\sum_{a=0}^m \\sum_{b=a}^m \\sum_{c=0}^a \\sum_{a=0}^m \\sum_{b=a}^m \\sum_{c=a}^b, \\sum_{a=0}^m \\sum_{b=a}^m \\sum_{c=b}^m  $"}
{"_id": "7511549", "title": "", "text": "$Corr\\left( x,y\\right) = \\dfrac {cov\\left( xy\\right) }{\\sigma _{x}\\sigma _{y}} $"}
{"_id": "1649339", "title": "", "text": "$R=M\\oplus T$"}
{"_id": "4816289", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\left\\| x_{n+1} - x_n \\right\\| = 0$"}
{"_id": "6168532", "title": "", "text": "$F(n, p)=\\frac{\\binom{n+1}{p+1 }}{\\binom{n}{p }}=\\frac{n+1}{p+1}$"}
{"_id": "2101845", "title": "", "text": "$C=\\frac{N-AB}{A+B}$"}
{"_id": "7217010", "title": "", "text": "$R : a < x < b, \\:\\: c < y<d$"}
{"_id": "5775490", "title": "", "text": "$p(M)\\leq \\lambda(M)$"}
{"_id": "9339975", "title": "", "text": "$\\text{a}+\\left(\\text{a}+1\\right)+\\dots+\\left(\\text{b}-1\\right)+\\text{b}=\\sum_{\\text{q}=\\text{a}}^\\text{b}\\text{q}=\\frac{\\left(\\text{a}+\\text{b}\\right)\\left(1-\\text{a}+\\text{b}\\right)}{2}$"}
{"_id": "1278869", "title": "", "text": "$F=\\{f_1,f_2,f_3,...,f_n\\}$"}
{"_id": "843887", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac1n \\sum_{r=1}^n r^{\\frac1r}$"}
{"_id": "1446215", "title": "", "text": "$P[e^{sX} \\ge e^{sa}] \\le \\dfrac{E(e^{sX})}{e^{sa}}=e^{-sa}\\cdot M_X(s)$"}
{"_id": "5495480", "title": "", "text": "$S=1+2+3+4+\\cdots = -\\frac{1}{12}$"}
{"_id": "7779702", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{-z}$"}
{"_id": "7419363", "title": "", "text": "$cov(Y,X)=0$"}
{"_id": "7546568", "title": "", "text": "$\\therefore \\frac{dy}{dx}=\\frac{2^{x}}{2^{x}+5}$"}
{"_id": "3961845", "title": "", "text": "$xRy→yRx$"}
{"_id": "6617885", "title": "", "text": "$\\{\\gamma\\in[0,\\infty] | r_{\\gamma}\\mathscr{U}\\subseteq\\mathscr{V}\\}$"}
{"_id": "2406136", "title": "", "text": "$\\sin x+1=0\\implies \\sin x=-1 \\implies x=\\frac {3\\pi}{2}+2n\\pi$"}
{"_id": "2759603", "title": "", "text": "$n^{log_n(a)}$"}
{"_id": "9250982", "title": "", "text": "$\\lim_{x\\to0^-}f'(x)=\\lim_{x\\to 0^+}f'(x)$"}
{"_id": "285537", "title": "", "text": "$\\int_{0}^{\\infty}\\frac{\\sin^{2n+1}(x)}{x} \\ dx=\\frac{\\pi \\binom{2n}{n}}{2^{2n+1}}$"}
{"_id": "3720986", "title": "", "text": "$\\boxed{\\sqrt{1!\\sqrt{2!\\sqrt{\\ldots\\sqrt{n!}}}}<e^{1.097...}=2.996...<3}$"}
{"_id": "8313264", "title": "", "text": "$A =\\frac{αa + βb + γc}{α + β + γ}.$"}
{"_id": "8636388", "title": "", "text": "$\\lim\\limits_{n \\to \\infty}\\sum_{i=1}^{n}{i^r \\over n^{r+1}}={1 \\over 1+r}$"}
{"_id": "7878400", "title": "", "text": "$ \\frac{\\sqrt{1-2 \\epsilon  \\log \\left(\\sqrt{2} l\\right)}}{\\sqrt{1-k\\, l^2    \\left(1-2 \\epsilon  \\log \\left(\\sqrt{2} l\\right)\\right)}} $"}
{"_id": "1833606", "title": "", "text": "$f(xf(y)+x)=xy+f(x), \\; \\forall x,y \\in \\mathbb{R}.$"}
{"_id": "7093320", "title": "", "text": "$P(X) = \\frac{1}{6}$"}
{"_id": "8951262", "title": "", "text": "$p_1p_2\\cdots p_r\\mid k$"}
{"_id": "4521313", "title": "", "text": "$X = \\sum^k_{j=1} I_j$"}
{"_id": "3066395", "title": "", "text": "$x^3, x^6, x^9, x^{12}$"}
{"_id": "1596886", "title": "", "text": "$\\frac {21}{x} + \\frac {70}{y} = n$"}
{"_id": "1089885", "title": "", "text": "$p\\in V \\subset  \\bar{V}\\subset U$"}
{"_id": "5401544", "title": "", "text": "$||x|| \\geq \\frac{\\alpha}{||F||}$"}
{"_id": "7881021", "title": "", "text": "$x^2 - 3x -5 \\lt 0, x \\in \\mathbb{N}$"}
{"_id": "2451448", "title": "", "text": "$r, r' \\in R, c_r \\neq c_r' $"}
{"_id": "4262634", "title": "", "text": "$\\{\\langle x,y\\rangle\\in\\Bbb R^2:x,y\\ge 0\\text{ and }x+y\\le 1\\}\\;.$"}
{"_id": "2059260", "title": "", "text": "$b=slog_a(c)$"}
{"_id": "52267", "title": "", "text": "$f(x) = \\frac{e^x}{1+e^x}$"}
{"_id": "7663379", "title": "", "text": "$a(4+k)=1*6^{k+2}*a(2)-1*6^{k+1}*a(1)-(k+1)*6^{k}*a(2)+(k)*6^{k-1}*a(1)+(1+2+3(k-2))*6^{k-2}*a(2)-(1+2+3(k-3))*6^{k-3}*a(1)-...+(-1)^{2^{(l(l+1))mod4}}*((l+2>=k)+2*(l+1>=k)+(3-(lmod2))*(k-l))*a(lmod2+2)$"}
{"_id": "8936783", "title": "", "text": "$I_j := f^{-1}(a_j)$"}
{"_id": "7039729", "title": "", "text": "$ \\zeta(s) = 2^s \\pi^{s-1} sin(\\frac{\\pi}{2}s) \\Gamma(1-s) \\zeta(1-s) $"}
{"_id": "6159195", "title": "", "text": "$d(x,\\mathbb{R}-V)=\\frac{1}{a}$"}
{"_id": "8565329", "title": "", "text": "$\\sum_{j=1}^{m}c_j\\chi_{S_j} = \\sum_{j=1}^{m}c_j\\chi_{K_j}$"}
{"_id": "3205336", "title": "", "text": "$ \\psi(x)=\\frac{1}{x^{\\frac32}}e^{\\frac{-1}{4x^2}}$"}
{"_id": "4692978", "title": "", "text": "$V \\cong \\mathbb{R}[x]/(x^4 - 1)$"}
{"_id": "5446897", "title": "", "text": "$x \\in (M,d)$"}
{"_id": "6804964", "title": "", "text": "$ \\left(\\frac{1}{x^{a-b}}\\right)^{\\frac{1}{a-c}}\\times\\left(\\frac{1}{x^{b-c}}\\right)^{\\frac{1}{b-a}}\\times\\left(\\frac{1}{x^{c-a}}\\right)^{\\frac{1}{c-b}} =x^{-\\left(\\frac{a-b}{a-c}+\\frac{b-c}{b-a}+\\frac{c-a}{c-b}\\right)} $"}
{"_id": "8344939", "title": "", "text": "$ \\sin(\\theta)=\\frac{\\tan(\\theta)}{\\sqrt{1+\\tan^2(\\theta)}}. $"}
{"_id": "7140751", "title": "", "text": "$ \\sin\\theta+\\sin\\theta\\tan^2\\theta=\\sin\\theta(1+\\tan^2\\theta)=\\sin\\theta\\sec^2\\theta=\\sec\\theta\\tan\\theta $"}
{"_id": "270001", "title": "", "text": "$(217,246)=1$"}
{"_id": "6786652", "title": "", "text": "$\\int_0^1 \\frac{\\sin(x)-x}{x^2} dx.$"}
{"_id": "544342", "title": "", "text": "$ E_n(x)=\\frac M {(n+1)!}(x-a)^{n-1} $"}
{"_id": "3020511", "title": "", "text": "$3^{k+1}=3^k+3^k+3^k>3^k+1$"}
{"_id": "5260672", "title": "", "text": "$\\|\\overline{x}_n-x_n\\| < 1/n$"}
{"_id": "7736906", "title": "", "text": "$(xRy \\wedge yRx) \\implies x=y$"}
{"_id": "4933713", "title": "", "text": "$\\left(1+\\frac{x}{n} \\right)^n < e^x $"}
{"_id": "5717468", "title": "", "text": "$\\cdots=\\frac{1}{n}\\sum_{j=0}^{n-1}\\omega^{-j\\mu} \\sum_{l=0}^\\infty a_l(\\omega^j x)^l=\\frac{1}{n}\\sum_{j=0}^{n-1}\\omega^{-j\\mu} f(\\omega^j x).$"}
{"_id": "7128719", "title": "", "text": "$(\\gamma\\gamma' )f(x)=f(x\\cdot(\\gamma\\gamma'))=f((x\\gamma')\\gamma)=\\gamma f(x\\gamma')=\\gamma(\\gamma' f)(x)$"}
{"_id": "2369287", "title": "", "text": "$|AB|=15$"}
{"_id": "2637833", "title": "", "text": "$|ab|=|c|$"}
{"_id": "752865", "title": "", "text": "$\\overline{\\Bbb R}_+$"}
{"_id": "7455066", "title": "", "text": "$\\gamma(a) = \\gamma(\\bar a) = a$"}
{"_id": "7897304", "title": "", "text": "$F(x) = \\int_{a}^{x}f(t)\\,dt, G(x) = \\int_{a}^{x}g(t)\\,dt$"}
{"_id": "2740151", "title": "", "text": "$        \\begin{pmatrix}        2  & 0 &  1  \\\\        1 &-2 & 1   \\\\         1  &-1 &0 \\\\      2 &-1 &1\\\\         \\end{pmatrix} $"}
{"_id": "2396622", "title": "", "text": "$3^x+4^x=5^x.$"}
{"_id": "4128865", "title": "", "text": "$\\,f(n) = 2^n-n^2\\,$"}
{"_id": "6913188", "title": "", "text": "$\\frac{1}{x-3}+\\frac{1}{y}=1$"}
{"_id": "3956407", "title": "", "text": "$ \\forall \\epsilon \\gt 0, \\exists \\delta \\gt 0,\\forall a \\in D, \\forall x \\in D: |x-a| \\lt \\delta \\Longrightarrow |f(x) - f(a)|\\lt \\epsilon  $"}
{"_id": "1334047", "title": "", "text": "$u_n = \\frac{n(n+1)}{2}$"}
{"_id": "6286104", "title": "", "text": "$|\\tau(a)| = r(a)$"}
{"_id": "9293367", "title": "", "text": "$f(\\pi) + f(0) = \\int_0^\\pi  g(v) \\sin(\\pi - v) dv = \\int_0^\\pi  g(v) \\sin v dv \\ge 0$"}
{"_id": "488504", "title": "", "text": "$t_n = \\frac{n(n+1)}{2} $"}
{"_id": "3618293", "title": "", "text": "$  S = \\sqrt{1+\\left(\\dfrac{dy}{dx}\\right)^2 + \\left(\\dfrac{dy}{dz}\\right)^2} $"}
{"_id": "7672818", "title": "", "text": "$J^{\\ast}VE(M, \\omega) \\cong T^{0,1}M $"}
{"_id": "6763094", "title": "", "text": "$s = \\sum_{j=1}^n \\alpha_j \\ \\chi_{A_j}$"}
{"_id": "5932581", "title": "", "text": "$P(A \\cap B)=1/6$"}
{"_id": "2115525", "title": "", "text": "$\\frac{1}{n^2}\\sum_{i=1}^n\\sum_{j=1}^n \\mathbb E [X_i Y_j] = \\frac{1}{n^2}\\sum_{i=1}^n\\sum_{j=1, i\\ne j}^n \\mathbb E [X_i]\\mathbb E[Y_j] +\\frac{1}{n^2}\\sum_{i=1}^n \\mathbb E [X_i Y_i]$"}
{"_id": "2003913", "title": "", "text": "$1+2+3+4+5+...=-\\frac{1}{12}$"}
{"_id": "254113", "title": "", "text": "$b_n=\\frac1\\pi\\int_{-\\pi}^\\pi g(x)\\sin nx\\ dx=\\frac{2}{\\pi}\\int_0^\\pi g(x)\\sin nx\\ dx$"}
{"_id": "5714678", "title": "", "text": "$e^{log(|z|)}=r$"}
{"_id": "8184415", "title": "", "text": "$\\sum\\limits_{n=0}^{\\infty}\\sum\\limits_{r=0}^{n}\\left(\\frac{1}{(n-r)!}a^{n-r}\\right)\\left(\\frac{1}{r!}b^{r}\\right)=\\left(\\sum\\limits_{n=0}^{\\infty}\\frac{1}{n!}a^n\\right)\\left(\\sum\\limits_{n=0}^{\\infty}\\frac{1}{n!}b^n\\right)$"}
{"_id": "84526", "title": "", "text": "$(\\frac12)^n$"}
{"_id": "9214868", "title": "", "text": "$\\int_{0}^{1} (e^{\\frac{-x}{a}}-a(1-e^{-\\frac{1}{a}}))^2 dx$"}
{"_id": "4398345", "title": "", "text": "$ E[s^N]=\\frac{1-\\sqrt{1-4p(1-p)s^2}}{2(1-p)s}, $"}
{"_id": "1997681", "title": "", "text": "$\\left\\{ \\matrix{   \\hat p\\left( \\theta  \\right) = \\cos \\theta \\hat x + \\sin \\theta \\hat y \\hfill \\cr    \\hat \\theta \\left( \\theta  \\right) =  - \\sin \\theta \\hat x + \\cos \\theta \\hat y \\hfill \\cr}  \\right.$"}
{"_id": "3218590", "title": "", "text": "$[x,y]=l$"}
{"_id": "7219867", "title": "", "text": "$\\mathbb{R}^{n+m}-D^m$"}
{"_id": "96339", "title": "", "text": "$\\alpha m + \\beta n = g$"}
{"_id": "4735244", "title": "", "text": "$\\sum_{n=1}^{\\infty} {\\frac{1}{n^2}} = \\sum_{n=1}^{N} {\\frac{1}{n^2}} + \\sum_{n = N + 1}^{\\infty} {\\frac{1}{n^2}}$"}
{"_id": "8026842", "title": "", "text": "$x_1,x_2,x_3 \\in K$"}
{"_id": "194208", "title": "", "text": "$\\sum_{n=1}^\\infty\\frac{1}{\\sqrt n(n+a)}=\\frac{1}{\\sqrt{a}}\\int_0^\\infty \\frac{\\text{erfi}(\\sqrt{ar}) e^{-ar}}{e^{r}-1}  ~dr$"}
{"_id": "7580124", "title": "", "text": "$\\gcd(n!+1,(n+1)!)$"}
{"_id": "9163434", "title": "", "text": "$ \\int \\sum \\chi_{I_j}(F(t)) F'(t) =\\sum \\int \\chi_{I_j}(F(t))F'(t) =\\sum \\lambda(I_j)<1/n, $"}
{"_id": "5929123", "title": "", "text": "$(1+y)^r \\sim 1 + r y$"}
{"_id": "5915215", "title": "", "text": "$\\Bbb Q(a,b,c,d)$"}
{"_id": "2013759", "title": "", "text": "$ \\int \\frac{1}{(1+\\sqrt x)^2}\\,dx $"}
{"_id": "8281068", "title": "", "text": "$\\int_2^n {\\frac{1}{{(x^2  + 1)^n }}\\,{\\rm d}x}  \\le \\frac{{n - 2}}{{5^n }}$"}
{"_id": "9065974", "title": "", "text": "$\\dfrac{x}{6}+\\dfrac{x}{10}=\\dfrac{4}{5}$"}
{"_id": "2588533", "title": "", "text": "$ f(x,0) = f(x,\\pi) = 0 $"}
{"_id": "4738375", "title": "", "text": "$\\frac1x + \\frac1y = \\frac4{15}$"}
{"_id": "6118359", "title": "", "text": "$\\gamma\\subset \\omega(\\gamma)$"}
{"_id": "3127835", "title": "", "text": "$ \\begin{align} \\sum_{n=1}^\\infty\\frac1{n^2+z^2} &=\\frac i{2z}\\sum_{n=1}^\\infty\\left(\\frac1{n+iz}+\\frac1{-n+iz}\\right)\\\\ &=\\frac i{2z}\\left[\\mathrm{PV}\\sum_{n\\in\\mathbb{Z}}\\frac1{n+iz}-\\frac1{iz}\\right]\\\\ &=-\\frac1{2z^2}+\\frac{\\pi i}{2z}\\cot(\\pi iz)\\\\[6pt] &=-\\frac1{2z^2}+\\frac\\pi{2z}\\coth(\\pi z)\\tag{2} \\end{align} $"}
{"_id": "2472191", "title": "", "text": "$Â=(Â^{*})^{t}$"}
{"_id": "3524464", "title": "", "text": "$x^n=a^n+na^{n-1}(x-a)+\\frac{1}{2}n(n-1)(\\xi_n)^{n-2}(x-a)^2 \\ \\ \\ \\ (2)$"}
{"_id": "8262616", "title": "", "text": "$\\left\\{\n     \\begin{array}{ll}\n       \\displaystyle\\frac{\\partial f}{\\partial x}=0  \\\\\n       \\displaystyle\\frac{\\partial f}{\\partial y}=0  \\\\\n       \\displaystyle\\frac{\\partial f}{\\partial z}=0 \n     \\end{array}\n   \\right.$"}
{"_id": "5640787", "title": "", "text": "$\\frac{(W-p)^{1-\\gamma }}{1-\\gamma}= \\frac{1}{2}\\frac{(W-x)^{1-\\gamma}}{1-\\gamma}+\\frac{1}{2}\\frac{(W+x)^{1-\\gamma}}{1-\\gamma}$"}
{"_id": "6076893", "title": "", "text": "$ f(x) = \\sum_{j=1}^n x_j g_j(x) $"}
{"_id": "1876594", "title": "", "text": "${ \\xi  }_{ n }=\\frac { { x }_{ 1 }{ x }_{ 2 }+...+{ x }_{ n } }{ n } $"}
{"_id": "5890854", "title": "", "text": "$\\frac {n}{n+1}=e^x.$"}
{"_id": "2273234", "title": "", "text": "$x^n - x^{n+1}$"}
{"_id": "8513307", "title": "", "text": "$\\int_0^\\infty[ f(x)+g(x)]\\, dx=\\int_0^\\infty f(x)\\, dx +\\int_0^\\infty g(x)\\, dx $"}
{"_id": "1046283", "title": "", "text": "$\\frac{(1+x)^{n+1}-(1+x)^m}{x}$"}
{"_id": "3956144", "title": "", "text": "$a=\\{\\{a\\},a\\}$"}
{"_id": "4110148", "title": "", "text": "$\\begin{align*} \\intop_{0}^{-\\infty}f(x)dx & =\\intop_{0}^{\\infty}f(-x)dx\\\\ \\end{align*}$"}
{"_id": "6066486", "title": "", "text": "$|a\\times b | = \\sqrt{3} $"}
{"_id": "5023890", "title": "", "text": "$\\ldots\\subseteq A_5\\subseteq A_3\\subseteq A_1\\subseteq A_2\\subseteq A_4\\subseteq A_6\\subseteq\\ldots\\;,$"}
{"_id": "2695463", "title": "", "text": "$ \\mbox{angle }x = \\arccos \\frac{|\\langle u,v\\rangle|}{\\Vert u \\Vert \\,\\Vert v \\Vert}. $"}
{"_id": "3936961", "title": "", "text": "$a_1 a_2a_3...a_n \\mid k$"}
{"_id": "143978", "title": "", "text": "$f(n) = n^2 - n$"}
{"_id": "4139063", "title": "", "text": "$S\\times A_1 \\times A_2 \\times ...\\times A_n$"}
{"_id": "3278411", "title": "", "text": "$E=\\{\\{a\\},\\{a,b\\}\\} $"}
{"_id": "8213258", "title": "", "text": "$\\{I_\\gamma\\}_{\\gamma\\in\\Gamma}\\subseteq\\mathcal{K}$"}
{"_id": "3689356", "title": "", "text": "$f(x+f(y))=f(x)+y$"}
{"_id": "7656351", "title": "", "text": "$\\int_0^\\infty\\frac{1}{(1+x^2)^s}\\,dx$"}
{"_id": "1238585", "title": "", "text": "$\\sin\\theta = \\frac{\\tan\\theta}{\\sqrt{1+\\tan^2\\theta}}$"}
{"_id": "4414410", "title": "", "text": "$6^k-3^k-3(2^k-1)$"}
{"_id": "843853", "title": "", "text": "$ \\int_{0}^{2\\pi} \\dfrac{1}{2\\pi} \\cdot \\dfrac{1-r^2}{1-2r \\cos(\\varphi) + r^2 } d\\varphi$"}
{"_id": "4641188", "title": "", "text": "$\\frac{\\sin^2\\theta}{\\theta^2} = \\frac{\\sin\\theta}{\\theta}\\cdot\\frac{\\sin\\theta}{\\theta}$"}
{"_id": "794137", "title": "", "text": "$ \\begin{pmatrix} a & -b \\\\ \\overline{b} & \\overline{a} \\end{pmatrix} $"}
{"_id": "9012620", "title": "", "text": "$P(M \\text{and} H) = P(M) P(H) $"}
{"_id": "3693435", "title": "", "text": "$\\forall \\epsilon >0\\,, \\exists \\delta > 0, |x| < \\delta \\implies |f(x) -f(0)| < \\epsilon$"}
{"_id": "6793746", "title": "", "text": "$U, V \\subseteq \\overline{A}$"}
{"_id": "1928181", "title": "", "text": "$(a-b,a+b) =1$"}
{"_id": "4148948", "title": "", "text": "$\\mu = \\frac{x_1 + x_2 + ... + x_n}{n}$"}
{"_id": "3945121", "title": "", "text": "$\\frac 2x + \\frac 1y = \\frac 17,$"}
{"_id": "3663890", "title": "", "text": "$\\tilde \\gamma = \\gamma \\oplus [\\gamma(1), \\gamma_n(1)] \\oplus \\gamma_n^- \\oplus [\\gamma_n(0), \\gamma(0)]$"}
{"_id": "6045382", "title": "", "text": "$f_n(x)\\xrightarrow[n\\to\\infty]{}f(x):=\\begin{cases}0&,\\;\\;0\\le x<1\\\\{}\\\\1&,\\;\\;x=1\\end{cases}$"}
{"_id": "709537", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = \\frac{x+y}{xy}$"}
{"_id": "4095091", "title": "", "text": "$P^n\\setminus P^{n-1}$"}
{"_id": "2451451", "title": "", "text": "$I_R(c_r) = r$"}
{"_id": "8011044", "title": "", "text": "$W^{1,p}(\\Omega)\\subset C^{0,1-n/p}$"}
{"_id": "1072556", "title": "", "text": "$\\sum_{n\\geq1}\\sum_{k\\geq1}\\frac{1}{k^{a}n^{b}\\left(n^{2}+k^{2}\\right)}=\\sum_{n\\geq1}\\sum_{k\\geq1}\\frac{1}{k^{b}n^{a}\\left(n^{2}+k^{2}\\right)}  $"}
{"_id": "631273", "title": "", "text": "$(X, T_x)$"}
{"_id": "3950380", "title": "", "text": "$D=\\{(x,y)\\in\\mathbb{R}^2|1\\le x\\le3, 0\\le y\\le \\ln(x)\\}$"}
{"_id": "3268658", "title": "", "text": "$\\frac{a^2(a+2b)-b^2(2a+b)+(a+b)^2(a-b)}{(2a+b)(a+2b)(a-b)}=\\cdots$"}
{"_id": "4684107", "title": "", "text": "$C_2\\subseteq V'\\subseteq\\overline{V'}\\subseteq U$"}
{"_id": "7593509", "title": "", "text": "$x=c_0 y +c_1 z \\; , \\; c_0 \\neq 1$"}
{"_id": "6174218", "title": "", "text": "$\\displaystyle\\sum_{x=1}^\\infty\\sum_{y=1}^\\infty\\frac{(-1)^{x+y}}{x^2+y^2}$"}
{"_id": "7927028", "title": "", "text": "$|G|=pqk$"}
{"_id": "629964", "title": "", "text": "$\\gamma:[a, b] \\to \\Bbb C, \\; \\gamma(a) = \\gamma(b) = z_0, \\tag 5$"}
{"_id": "3484542", "title": "", "text": "$x + y = f^{-1}(f(x) + f(y))$"}
{"_id": "3020512", "title": "", "text": "$3^k +1 < 3^k +3^k +3^k = 3*3^k = 3^{k+1}$"}
{"_id": "2726991", "title": "", "text": "$(\\mathbb{R}^n,\\mathcal{O}_{\\mathbb{R}^n}) $"}
{"_id": "2400150", "title": "", "text": "$y = (ax + b)/(cx + d)$"}
{"_id": "6102220", "title": "", "text": "$T_1=\\prod_{k=1}^{n-1} \\cos\\frac{k\\pi}{2n}$"}
{"_id": "2605714", "title": "", "text": "$C_R(0,R)$"}
{"_id": "6447419", "title": "", "text": "$\\lim_{N\\to\\infty}2\\sum_{k=1}^N\\frac{\\mu(k)}{k!},$"}
{"_id": "534917", "title": "", "text": "$x! \\sim C \\sqrt{x}\\left(\\dfrac{x}e\\right)^x + \\mathcal{O}(1/x)$"}
{"_id": "6884378", "title": "", "text": "$u = [a_1, a_2, a_3, ...., a_n]$"}
{"_id": "274016", "title": "", "text": "$|zw|=rs=1$"}
{"_id": "855237", "title": "", "text": "$e^{log(X)} = X$"}
{"_id": "7158984", "title": "", "text": "$P(\\liminf{X_n\\leq x})\\leq \\liminf P({X_n\\leq x})\\leq \\limsup P({X_n\\leq x})\\leq P(\\limsup{X_n≤x})$"}
{"_id": "9154571", "title": "", "text": "$\\{\\langle x,y\\rangle\\in\\Bbb R^2:a\\le x+y<b\\}\\;;$"}
{"_id": "6804134", "title": "", "text": "$y^A_\\gamma = \\begin{cases} x_\\gamma, &\\text{if }\\gamma \\in A \\\\ 0, &\\text{otherwise.} \\end{cases}$"}
{"_id": "1637959", "title": "", "text": "$(\\frac{1}{2})^{n/2}$"}
{"_id": "5617466", "title": "", "text": "$y=\\frac{a+b x}{c+d x}$"}
{"_id": "1569126", "title": "", "text": "$ \\lim_{n\\rightarrow\\infty}\\left\\lfloor\\sum_{k=1}^{n}9\\cdot 10^{-k}\\right\\rfloor=0, $"}
{"_id": "7535694", "title": "", "text": "$\\sum_{-\\infty}^{\\infty}1^{-n}$"}
{"_id": "8583968", "title": "", "text": "$cov(X,Y)=-0.7$"}
{"_id": "4378696", "title": "", "text": "$I_n=\\int_0^{\\pi}\\frac{\\sin nx}{\\sin x}dx$"}
{"_id": "8419654", "title": "", "text": "$ G(x)=P[X\\geq x]. $"}
{"_id": "6504029", "title": "", "text": "$B = \\{ (x,y) \\in \\mathbb{R}^2 : y \\leq x \\leq y^2,  0 \\leq y \\leq 1 \\},  $"}
{"_id": "8242023", "title": "", "text": "$P(E23)=1/2$"}
{"_id": "3095475", "title": "", "text": "$\\frac{d^2y}{dx^2}=-\\frac{\\frac{d^2x}{dy^2}}{(\\frac{dx}{dy})^3}$"}
{"_id": "9266968", "title": "", "text": "$\\int_{0}^{1} \\frac{\\sin^{-1}(x)}{x} dx = \\frac{\\pi}{2}\\ln2$"}
{"_id": "987755", "title": "", "text": "$ a u + b v = d $"}
{"_id": "3273511", "title": "", "text": "$\\int_0^1p_{n\\geq M}(t^\\lambda)f(t)dt=0=\\int_0^1f(t)^2dt=0$"}
{"_id": "1498425", "title": "", "text": "$p \\not | q_1 q_2$"}
{"_id": "5549470", "title": "", "text": "$Cov(x,u)=0$"}
{"_id": "1379415", "title": "", "text": "$p_1 \\mid p_2^nb$"}
{"_id": "5258252", "title": "", "text": "$(a,a) = \\{\\{a\\}, \\{a, a\\}\\} = \\{\\{a\\}\\}$"}
{"_id": "4607509", "title": "", "text": "$\\Rightarrow 0<|f(z)|<2|f(z_0)|$"}
{"_id": "672250", "title": "", "text": "$\\lim_{k \\to \\alpha^+} \\phi(k)$"}
{"_id": "3487691", "title": "", "text": "$\\liminf_{n \\to \\infty} \\mu(B_1(x_n)) \\ge \\int \\liminf_{n \\to \\infty} \\chi_{x_n}(y)\\ d\\mu(y) \\ge \\mu(B_1(x))$"}
{"_id": "8054534", "title": "", "text": "$M''=M/M'$"}
{"_id": "5495326", "title": "", "text": "$\\color{red}{\\int \\frac{1}{(x+1)^2}dx}$"}
{"_id": "1814496", "title": "", "text": "$S3=1+2+3+4+5+6+7+8...=-\\frac{1}{12}$"}
{"_id": "3704747", "title": "", "text": "$x^5-1=(x-1)\\left(x-e^{\\frac{2\\pi i}5}\\right)\\left(x-e^{\\frac{-2\\pi i}5}\\right)\\left(x-e^{\\frac{4\\pi i}5}\\right)\\left(x-e^{\\frac{-4\\pi i}5}\\right)$"}
{"_id": "4044116", "title": "", "text": "$\\overline{v}(\\lnot \\alpha) = \\top$"}
{"_id": "1151638", "title": "", "text": "$f_X(x) = \\frac{1}{50}e^{\\frac{-x}{50}}$"}
{"_id": "6815905", "title": "", "text": "$\\phi(G) = (a+b,a-b)$"}
{"_id": "9171343", "title": "", "text": "$S = \\{ a_1, a_2, a_3, \\dots, a_n\\}$"}
{"_id": "6954422", "title": "", "text": "$f(x) = \\left\\{ {\\matrix{    0 \\hfill & {x = a} \\hfill  \\cr     {{x^2}} \\hfill & {x \\ne a,b} \\hfill  \\cr     0 \\hfill & {x = b} \\hfill  \\cr   } } \\right.$"}
{"_id": "4730203", "title": "", "text": "$a(1-a)^{n-1} = s$"}
{"_id": "2805762", "title": "", "text": "$f(n):=n^2-n+41.$"}
{"_id": "1418492", "title": "", "text": "$|a-b| + |b|=a-2b = a-2a = -a$"}
{"_id": "3064524", "title": "", "text": "$\\frac{a}{b}=\\frac{a+b}{a}$"}
{"_id": "4918219", "title": "", "text": "$\\varphi = \\sum_{j=1}^{p}c_{j}\\chi_{E_{j}}$"}
{"_id": "8053336", "title": "", "text": "$ \\left(- \\frac{\\ell}{2}\\right)\\frac{2}{\\sqrt{-\\gamma}} \\frac {\\delta S_{ct}}{\\delta \\gamma^{tt}}  ~=~\\frac{2}{\\sqrt{-\\gamma}} \\frac {\\partial \\sqrt{-\\gamma}}{\\partial \\gamma^{tt}} ~=~\\frac{1}{\\gamma} \\frac {\\partial \\gamma}{\\partial \\gamma^{tt}} ~=~ -\\frac{1}{\\gamma^{-1}} \\frac {\\partial \\gamma^{-1}}{\\partial \\gamma^{tt}}$"}
{"_id": "136071", "title": "", "text": "$\\varphi \\colon K[T_1^{\\pm 1}, \\dots, T_r^{\\pm 1}] \\to \\mathcal O_X(U), T_i \\mapsto \\varphi_i.$"}
{"_id": "6808478", "title": "", "text": "$ a \\leq x_1 \\leq b, \\qquad c \\leq y_1 \\leq d$"}
{"_id": "1878153", "title": "", "text": "$d(y,a)\\leq d(y,x)+d(x,a)$"}
{"_id": "5349320", "title": "", "text": "$ \\int_{-\\pi}^{\\pi}\\frac{\\sin(nx)}{x}\\cos(mx)\\,dx = 2\\pi $"}
{"_id": "891170", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = c$"}
{"_id": "2666931", "title": "", "text": "$\\{t\\gamma\\} = t\\gamma - \\lfloor t\\gamma\\rfloor$"}
{"_id": "6908322", "title": "", "text": "$ a\\alpha -\\beta  +a\\gamma =0  \\Rightarrow  a\\alpha -2\\gamma  +a\\gamma =0  \\Rightarrow  a( \\alpha+ \\gamma )=2\\gamma$"}
{"_id": "7612728", "title": "", "text": "$\\gamma(t+s) = \\gamma(t)\\gamma(s)$"}
{"_id": "6914604", "title": "", "text": "$F(x) = \\int_a^x f(t) dt + F(a),$"}
{"_id": "3026457", "title": "", "text": "$\\mathbb{R}^n \\to \\mathcal{L}(\\mathbb{R}^n,\\mathbb{R})$"}
{"_id": "4190122", "title": "", "text": "$ a+ar+ar^2+\\cdots+ar^n=a\\frac{r^{n+1}-1}{r-1}=a\\frac{1-r^{n+1}}{1-r} $"}
{"_id": "355273", "title": "", "text": "$(M,M,M)$"}
{"_id": "805075", "title": "", "text": "$|A×B|= mn$"}
{"_id": "5784107", "title": "", "text": "$ F(A)=Pr(a\\le A)=\\frac{\\ln A-\\gamma}{\\ln x-\\gamma}. $"}
{"_id": "5540818", "title": "", "text": "$\\frak p\\oplus t$"}
{"_id": "5616834", "title": "", "text": "$\\frac{p(\\cos(x) - p)}{1 - 2p\\cos(x)+p^2} = \\frac{1}{2} \\cdot \\left( \\frac{1 - p^2}{1 - 2p\\cos(x) + p^2} - 1 \\right)$"}
{"_id": "4543235", "title": "", "text": "$\\sqrt{2}e^{\\frac{i\\pi}{n}(2k+1)}$"}
{"_id": "6744530", "title": "", "text": "$f_{x}(x)= \\large \\frac{1}{5} e^{\\frac{-x}{5}}$"}
{"_id": "8263913", "title": "", "text": "$ \\lnot (\\exists c_0, c_1) (\\forall x) (x = c_0 \\lor x = c_1) $"}
{"_id": "7449183", "title": "", "text": "$\\xi = 1-2p^{2}\\sin^{2} \\omega -ip\\sin w$"}
{"_id": "7822353", "title": "", "text": "$\\;\\pm\\sin\\varphi = \\sqrt{1 - \\cos^2\\varphi}$"}
{"_id": "3231523", "title": "", "text": "$f_n(x)={\\lceil x\\rceil\\over n^2}$"}
{"_id": "2759253", "title": "", "text": "$P(A_i)=1/6$"}
{"_id": "4581275", "title": "", "text": "$ (x - a,y - b) (x + a,y + b) = (x + a - a,y + b - b) = (x,y), $"}
{"_id": "8366295", "title": "", "text": "$561|a^{561}-a$"}
{"_id": "5903259", "title": "", "text": "$\\displaystyle Y_n=\\frac{X_1+X_2+...+X_n}{y_n}$"}
{"_id": "5348805", "title": "", "text": "$ \\theta(t)=\\sum_{n=-\\infty}^{\\infty}e^{-\\pi n^2 t}=1+2\\sum_{n=1}^{\\infty} e^{-\\pi n^2 t} $"}
{"_id": "4400412", "title": "", "text": "$\\color{#C00000}{ax+by=1}$"}
{"_id": "5254447", "title": "", "text": "$ \\frac{a}{1-a^2}\\frac{a^{-2}-1}{a^{-1}}= \\frac{a}{1-a^2}\\frac{\\dfrac{1}{a^2}-1}{\\dfrac{1}{a}}= \\frac{a}{1-a^2}\\frac{1-a^2}{a^2}\\,a=1 $"}
{"_id": "1276606", "title": "", "text": "$\\log_a(A)>log_a(B)$"}
{"_id": "4738933", "title": "", "text": "$\\gcd(a,b)=\\gcd(a,\\gcd(a,b))=\\gcd(b,\\gcd(a,b))$"}
{"_id": "3728369", "title": "", "text": "$f(a+b)=f(a)+f(b) + 2\\sqrt{f(a)\\cdot f(b)}$"}
{"_id": "2747407", "title": "", "text": "$\\tau=\\vartheta$"}
{"_id": "5989861", "title": "", "text": "$a\\le x\\le b,\\qquad c\\le y\\le d,\\qquad x\\ge y$"}
{"_id": "812236", "title": "", "text": "$ \\lim_{x\\to b^-}\\dfrac{f(x)-f(b)}{x-b}= \\lim_{x\\to b^-}f'(c_x)=\\lim_{x \\to b^-}f'(x) $"}
{"_id": "6004426", "title": "", "text": "$p(n,m)=p(n,m-1)+p(n-m,m)\\text{ if }m,n\\ne 0\\;.$"}
{"_id": "6806396", "title": "", "text": "$ds=\\sqrt {1+(\\frac {dx}{dy})^2}dy$"}
{"_id": "8207587", "title": "", "text": "$\\begin{cases}a_n=b_{n-1}^2+4a_{n-1}\\\\b_n=4b_{n-1}\\\\c_n=4\\end{cases}$"}
{"_id": "7242170", "title": "", "text": "$P_1\\mid P_2\\mid\\cdots\\mid P_r.$"}
{"_id": "1077450", "title": "", "text": "$E(X+Y)^p \\le 2^p (E(X^p)+E(Y^p))$"}
{"_id": "5381610", "title": "", "text": "$\\tag 2 |g(z_1)| \\lt |g(z_0)|$"}
{"_id": "662702", "title": "", "text": "$(a, a + d, a + 2d)$"}
{"_id": "5263836", "title": "", "text": "$f(n) \\leq {n \\choose 2}$"}
{"_id": "9178263", "title": "", "text": "$ \\gamma\\xi\\gamma^{-1}(\\gamma(a_k))= \\gamma\\xi(a_k)=\\gamma(a_1) $"}
{"_id": "6808480", "title": "", "text": "$ a \\leq x_3 \\leq b, \\qquad c \\leq y_3 \\leq d$"}
{"_id": "4826985", "title": "", "text": "$\\frac{f(a+\\Delta x)-f(a)}{(a+\\Delta x) - a}=\\frac{m(a+\\Delta x)-ma}{\\Delta x}=\\frac{m\\Delta x}{\\Delta x}=m$"}
{"_id": "6713634", "title": "", "text": "$\\int_0^{2\\pi} \\int_0^\\pi \\frac{d}{dR}\\int_0^R f(r,\\theta,\\phi)\\;r^2 \\sin(\\phi)\\,dr\\,d\\phi\\,d\\theta= \\int_0^{2\\pi} \\int_0^\\pi f(R,\\theta,\\phi)R^2 \\sin(\\phi)\\,d\\phi\\,d\\theta= \\iint_{S(R)}f(R,\\theta,\\phi)R^2\\;dS \\;\\;\\;\\text{ where }\\;\\;\\; dS=\\sin(\\phi)\\,d\\phi\\,d\\theta,\\;\\; S(R) \\text{ is the sphere of radius R.} $"}
{"_id": "6293544", "title": "", "text": "$ab = ma + nb$"}
{"_id": "642374", "title": "", "text": "$\\begin{cases}x=t,\\\\y=\\pm{\\sqrt{1-t^2}}, \\\\z=1-t^2,\\end{cases}$"}
{"_id": "2176425", "title": "", "text": "$R = \\lbrace (x,y): a < x < b, c < y < d\\rbrace$"}
{"_id": "4129951", "title": "", "text": "$ \\{v + \\alpha(v) \\mid v \\in T^{0,1} M\\} \\subset T_{\\mathbb{C}} M $"}
{"_id": "337067", "title": "", "text": "$(X-1)(X-\\lambda)^{N-1}$"}
{"_id": "3727273", "title": "", "text": "$B= \\{b_1, b_2, b_3,.....\\}$"}
{"_id": "6335605", "title": "", "text": "$\\forall \\in \\Bbb N^+$"}
{"_id": "2447993", "title": "", "text": "$7\\mid n^7-n$"}
{"_id": "2826646", "title": "", "text": "$f=\\sum_{i=1}^nc_i\\chi_{A_i}$"}
{"_id": "2789668", "title": "", "text": "$\\{\\emptyset, a, bc, abc\\}$"}
{"_id": "4565963", "title": "", "text": "$[a]+[b]=[a+b]=[b+a]=[b]+[a]$"}
{"_id": "1288380", "title": "", "text": "$\\mathbb{Z}_2[x]/(x^4+x+1)$"}
{"_id": "673818", "title": "", "text": "$x^\\frac{1}{(\\log_a x)} = a^{log_a(x^\\frac{1}{(\\log_a x)})}$"}
{"_id": "5090244", "title": "", "text": "$x^{m} \\in <x^{d}>$"}
{"_id": "8697449", "title": "", "text": "$P(k): 0^3+1^3+⋯+k^3=(k(k+1)/2)^2$"}
{"_id": "1117991", "title": "", "text": "$(1+x/n)^{n} \\leq e^{x}$"}
{"_id": "177334", "title": "", "text": "$f'(h)= \\frac{d}{dh}\\langle \\gamma(s+h) - \\gamma(s) , \\gamma(s+h) - \\gamma(s) \\rangle = \\frac{d}{dh} [\\langle \\gamma(s+h), \\gamma(s+h) \\rangle + \\langle \\gamma(s), \\gamma(s) \\rangle -2 \\langle \\gamma(s+h), \\gamma(s) \\rangle ] = 2[\\langle \\frac{d}{dh}\\gamma(s+h), \\gamma(s+h) \\rangle - \\langle \\frac{d}{dh}\\gamma(s+h), \\gamma(s) \\rangle] = 2[\\langle \\frac{d}{dh}\\gamma(s+h), \\gamma(s+h) - \\gamma(s) \\rangle ]$"}
{"_id": "1665649", "title": "", "text": "$\\displaystyle\\implies \\sum_{0\\le k\\le r-1}\\cos\\frac{2k\\pi}r=0$"}
{"_id": "1360933", "title": "", "text": "$\\displaystyle \\int_0^\\infty f(x)dx=+\\infty$"}
{"_id": "9112890", "title": "", "text": "$C_n := \\int_{-\\pi}^\\pi \\lvert D_n(x)\\rvert\\,dx = \\frac{2}{\\pi}\\int_0^{\\pi/2} \\frac{\\lvert\\sin ((2n+1)y)\\rvert}{\\sin y}\\,dy.$"}
{"_id": "143363", "title": "", "text": "$2^{n\\gamma}\\le t^{-\\gamma} \\le 2^{(n+1)\\gamma}$"}
{"_id": "49901", "title": "", "text": "$x^2 + x = y^2 + y = z^2 + z = x + y  + z$"}
{"_id": "7918819", "title": "", "text": "$x^3 - x^2, x - x^2$"}
{"_id": "6773250", "title": "", "text": "$P(A\\mid B)\\leq \\frac{a+b-1}{b}$"}
{"_id": "3716888", "title": "", "text": "$\\mathscr{E} =\\{ \\{1\\}\\}$"}
{"_id": "1096784", "title": "", "text": "$A = \\left(\\begin{array}{crc}  1 &  2 &  -1 & -1\\\\ 1 & 0 & 1 &  1\\\\  2 &  -4 &  6 & 2\\\\  \\end{array}\\right)$"}
{"_id": "6430394", "title": "", "text": "$log_ab=c$"}
{"_id": "4021516", "title": "", "text": "$k|P = A \\left(\\frac{(1+i)^n-1}{i(1+i)^n}\\right) \\left(\\frac{1}{(1+i)^k} \\right)$"}
{"_id": "8976377", "title": "", "text": "$ \\cos k\\frac{2\\pi}{n}\\cos (k+1)\\frac{2\\pi}{n}+\\sin k\\frac{2\\pi}{n}\\sin (k+1)\\frac{2\\pi}{n} $"}
{"_id": "4122313", "title": "", "text": "$\\mid \\frac{\\mid y-p \\mid}{\\mid y-p \\mid} \\cdot \\frac{1}{\\mid x-p \\mid} - \\frac{1}{\\mid y-p \\mid} \\cdot \\frac{\\mid x-p \\mid}{\\mid x-p \\mid} \\mid < \\epsilon$"}
{"_id": "4205394", "title": "", "text": "$|f(z_n)|\\to |f(z)|$"}
{"_id": "580440", "title": "", "text": "$\\left | AB \\right |=c$"}
{"_id": "4386397", "title": "", "text": "$f_2(n) = n^2$"}
{"_id": "7981209", "title": "", "text": "$ \\frac{\\lvert \\langle x, y \\rangle \\rvert}{\\lVert y \\rVert} =\\lVert x \\rVert ~\\overset{1/\\lVert y \\rVert}{\\implies}~\\frac{\\lvert \\langle x, y \\rangle \\rvert}{\\lVert y \\rVert^2} =\\frac{\\lVert x \\rVert}{\\lVert y \\rVert} =\\lvert a \\rvert.$"}
{"_id": "7610778", "title": "", "text": "$a,a',b,b',c,c',d,d'\\in\\mathbb R$"}
{"_id": "6668476", "title": "", "text": "$\\mathscr{L}(\\zeta,\\,\\mathscr{L}(\\gamma,\\,x))(t)=\\mathscr{L}(\\zeta\\,\\gamma,\\,x)(t);\\quad\\zeta,\\,\\gamma \\in \\mathrm{SL}(2,\\,\\mathbb{R})$"}
{"_id": "3467304", "title": "", "text": "$|\\mathbb P\\{X_n\\leq x\\}-\\mathbb P\\{X\\leq x\\}|\\leq \\mathbb P\\{X_n\\leq x,X>x\\}+\\mathbb P\\{X_n>x,X\\leq x\\}\\ \\ ?$"}
{"_id": "8023540", "title": "", "text": "$Q_n(x)=\\frac{(1-x)^{n+1}-(1+x)^{n+1}+2x(n+1)(1+x)^{n+1}}{2x^2},$"}
{"_id": "886049", "title": "", "text": "$\\,r,s\\mid k\\,\\Rightarrow\\,rs\\mid k\\,$"}
{"_id": "4031983", "title": "", "text": "$J_1=\\int_{-\\pi/2}^{\\pi/2} \\frac{\\gamma (1+\\sin^2 x)  \\cos^4 x~dx}{\\left(\\cos^4 x+(\\gamma~ cos^2 x-\\sin x)^2 \\right)^{7/4}}$"}
{"_id": "3416465", "title": "", "text": "$x(x-\\lambda)^{n-1} = (x-\\lambda)^n + \\lambda (x-\\lambda)^{n-1}$"}
{"_id": "7964219", "title": "", "text": "$\\tan \\theta = \\frac{x}{1},$"}
{"_id": "782812", "title": "", "text": "$\\int_0^1f(t)\\ dt=0$"}
{"_id": "5777693", "title": "", "text": "$f_{3}(n) = n + 1$"}
{"_id": "2616313", "title": "", "text": "$\\,L=\\lim_{x\\to a^+}{f'(x)}$"}
{"_id": "3110394", "title": "", "text": "$x^2,x^3,x^4\\in [0,1]$"}
{"_id": "6446400", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\mu_n(A)=\\mu(A)$"}
{"_id": "3585421", "title": "", "text": "$x^2+x, x\\in F$"}
{"_id": "5197818", "title": "", "text": "$ \\langle x - \\gamma y, x - \\gamma y\\rangle = \\langle x, y\\rangle - \\gamma\\langle y,x\\rangle - \\bar\\gamma\\langle x, y\\rangle + |\\gamma|^2\\langle y, y\\rangle $"}
{"_id": "6866017", "title": "", "text": "$\\lim_{x\\to a}f_1(x)=\\lim_{x\\to a}f_2(x)=...=\\lim_{x\\to a}f_n(x)=N$"}
{"_id": "8577783", "title": "", "text": "$A=\\frac {\\Delta f}{\\Delta x} .$"}
{"_id": "881329", "title": "", "text": "$\\int_0^\\infty f_n(x)dx$"}
{"_id": "9320134", "title": "", "text": "$\\left \\| A \\right \\|=\\sqrt{\\lambda_{\\text{max}}(A^*A)}$"}
{"_id": "2835179", "title": "", "text": "$a|d_1d_2$"}
{"_id": "173036", "title": "", "text": "$ \\begin{bmatrix} 0 & -1\\\\  1 & 0  \\end{bmatrix}$"}
{"_id": "3690547", "title": "", "text": "$E(\\hat{\\theta^2}) = E\\left(\\frac{1}{n^2}\\sum_{i=1}^n X_i^2\\right) =E(X^2) = \\int_0^\\infty f_\\theta dx = ... =  \\frac{2\\theta^2}{n^2} $"}
{"_id": "3628774", "title": "", "text": "$f_X(x|\\theta) = \\frac{1}{\\theta}\\ \\ \\ \\text{for $x\\in [0,\\theta]$}$"}
{"_id": "2593621", "title": "", "text": "$[t_i',t_i'']$"}
{"_id": "1284376", "title": "", "text": "$(x,y) = \\{\\{x\\},\\{x,y\\}\\}\\\\ (y,x) = \\{\\{y\\},\\{x,y\\}\\} $"}
{"_id": "6883275", "title": "", "text": "$ \\theta = \\aleph_{\\theta} = \\aleph_{\\aleph_{\\aleph_{\\aleph_{\\ddots}}}} = \\Theta = \\kappa = \\aleph_\\kappa = \\beth_\\kappa = \\beth_{\\beth_{\\beth_{\\beth_{\\ddots}}}} $"}
{"_id": "826750", "title": "", "text": "$ \\tan \\theta_1 = \\frac{l}{2L} = \\frac{\\tan \\theta}{2} $"}
{"_id": "7972930", "title": "", "text": "$x=\\langle D|M^{\\perp}\\rangle $"}
{"_id": "2132915", "title": "", "text": "$q^{n+1} - q^n$"}
{"_id": "5595030", "title": "", "text": "$n \\geq N \\implies\\ |\\frac{x_1 + x_2 + ... + x_n}{n} - L|< \\epsilon \\quad (♫)$"}
{"_id": "4684580", "title": "", "text": "$[X,Y] = XY - YX \\in A$"}
{"_id": "3524466", "title": "", "text": "$ \\dfrac{f(x)-f(a)}{x-a} = \\dfrac{1}{x-a} \\sum_{n=1}^{+\\infty} c_n[na^{n-1}(x-a)+\\frac{1}{2}n(n-1)(\\xi_n)^{n-2}(x-a)^2] $"}
{"_id": "1333294", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}\\mu(A_n)$"}
{"_id": "4790632", "title": "", "text": "$V=\\{\\langle x, y\\rangle \\in A\\times A \\mid x\\le y\\}.$"}
{"_id": "7630966", "title": "", "text": "$L(s,\\chi)=\\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}=\\int_1^\\infty \\frac{S(x)}{x^{s+1}}\\; dx \\tag 2$"}
{"_id": "743937", "title": "", "text": "$\\textbf{m}= \\gamma^2 \\textbf{p}+\\gamma\\ |\\textbf{p}|\\ \\textbf{P} \\nabla_n \\gamma$"}
{"_id": "444335", "title": "", "text": "$(y,x,*,y),(y,*,x,y),(x,y,y,*),(*,y,y,x),(x,y,*,y)$"}
{"_id": "898078", "title": "", "text": "$=\\begin{vmatrix} x &x^2  &1 \\\\  y &y^2  &1 \\\\  z &z^2  &1  \\end{vmatrix}+\\begin{vmatrix} x &x^2  &x^3 \\\\  y &y^2  &y^3 \\\\  z &z^2  &z^3  \\end{vmatrix}$"}
{"_id": "1358719", "title": "", "text": "$     (a, a+b \\bmod N, a+2b \\bmod N, a+3b \\bmod N, \\dots, a + (N-1)b \\bmod N) $"}
{"_id": "5341484", "title": "", "text": "$1 = \\sin^2 \\varphi + \\cos^2 \\varphi$"}
{"_id": "5974110", "title": "", "text": "$\\zeta=1/|X|$"}
{"_id": "8472836", "title": "", "text": "$\\rho(A^TA)=\\lambda\\leq\\|A^TA\\|_\\infty$"}
{"_id": "4432820", "title": "", "text": "$ \\tilde F (x) := \\int_{\\tilde a}^x f(t)\\,dt $"}
{"_id": "7067912", "title": "", "text": "$d(x,A) \\leq d(x,a) \\leq d(x,y)+d(y,a).$"}
{"_id": "3889784", "title": "", "text": "$A=\\begin{pmatrix}1 &3 & 9& 27\\\\3 & 3 & 9& 27\\\\ 9 & 9 & 9& 27 \\\\ 27 & 27 &27& 27\\end{pmatrix}$"}
{"_id": "9223560", "title": "", "text": "$\\mathbb R^m - \\mathbb R^n$"}
{"_id": "3888797", "title": "", "text": "$ E_1\\subset  E_2\\subset\\cdots \\subset E_i\\subset E_{i+1}\\subset\\cdots $"}
{"_id": "4460753", "title": "", "text": "$\\frac{d}{dx} ( (x-a)^{n+1} p^{(n)}(x) ) = (x-a)^n q^{(n+1)}$"}
{"_id": "1400740", "title": "", "text": "$f(a+h,b+k) = f((a,k)+(h,b)) = f((a,k)) + f((h,b))$"}
{"_id": "1543637", "title": "", "text": "$\\left(\\frac12\\right)^{n-2}$"}
{"_id": "8451809", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1&1&k&1\\\\0&k-1&1-k&0\\\\k&1&1&1\\end{array}\\right]$"}
{"_id": "753875", "title": "", "text": "$t = p \\oplus q$"}
{"_id": "2492234", "title": "", "text": "$d(A,x) := \\inf\\{|x-y|; y \\in A\\}$"}
{"_id": "3539826", "title": "", "text": "$k=\\left\\lfloor \\frac{n}{p}\\right\\rfloor+\\left\\lfloor \\frac{n}{p^2}\\right\\rfloor+\\left\\lfloor \\frac{n}{p^3}\\right\\rfloor+\\ldots$"}
{"_id": "4404901", "title": "", "text": "$f(a) > k > f(b)$"}
{"_id": "6180254", "title": "", "text": "$\\frac{2\\tan\\left(\\frac{x}{2}\\right)}{\\left(\\tan\\left(\\frac{x}{2}\\right)-1\\right)^2} + \\frac{\\tan^2\\left(\\frac{x}{2}\\right)+1}{2\\tan\\left(\\frac{x}{2}\\right)}$"}
{"_id": "1267141", "title": "", "text": "$ f(x,y)=\\frac{1}{2\\pi\\sqrt{1-p^2}} \\mathrm e^{-\\frac{(y-px)^2}{2(1-p^2)}-\\frac{x^2}{2}}=\\frac{1}{\\sqrt{2\\pi}\\sqrt{1-p^2}}\\exp\\left(-\\frac{(y-px)^2}{2(1-p^2)}\\right)\\times \\frac{1}{\\sqrt{2\\pi}}\\exp\\left(-\\frac{x^2}{2}\\right) $"}
{"_id": "7180966", "title": "", "text": "$f_X(x;\\theta) = \\frac{1}{2\\theta}$"}
{"_id": "5607891", "title": "", "text": "$p_1p_2\\mid \\chi(1)$"}
{"_id": "5975963", "title": "", "text": "$P(r)\\implies P(r+1)$"}
{"_id": "4493147", "title": "", "text": "$y = \\frac {a} {b} + b x$"}
{"_id": "4965256", "title": "", "text": "$\\frac{7}{x}+\\frac{3}{y}=\\frac{1}{4}$"}
{"_id": "4515923", "title": "", "text": "$|f(z)| \\leq k|z|^2$"}
{"_id": "4885564", "title": "", "text": "$\\begin{bmatrix} 2&-m&1 \\\\ 1&-1&2 \\\\ -1&3&1 \\end{bmatrix} $"}
{"_id": "1431166", "title": "", "text": "$[(f_1f_2)f_3]'=(f_1f_2)'f_3+(f_1f_2)f_3'$"}
{"_id": "3568435", "title": "", "text": "$a_n = \\frac{n(n+1)}{2} * (\\frac{(n+2)}{3})$"}
{"_id": "4217678", "title": "", "text": "$S=\\sum_{k=1}^{n-1}\\sin\\frac{2\\pi k}{n}$"}
{"_id": "5541519", "title": "", "text": "$ ax+by=dn$"}
{"_id": "1780426", "title": "", "text": "$\\lim\\limits_{t\\to-\\infty}x = 0^-$"}
{"_id": "6245442", "title": "", "text": "$(x-z)^2=(x-y+y-z)^2=(x-y)^2+ 2(x-y)(y-z) +(y-z)^2$"}
{"_id": "427219", "title": "", "text": "$ |E(\\gamma_k \\mid \\gamma_1,\\dots,\\gamma_{k-1})|\\le 2^{-k+1} $"}
{"_id": "2715873", "title": "", "text": "$E = (e_1,e_2....e_m)$"}
{"_id": "6231448", "title": "", "text": "$|f(z)|\\le |f(z^2)|$"}
{"_id": "4333982", "title": "", "text": "$(w,x,y,z) = (1,1,1,2)$"}
{"_id": "1707869", "title": "", "text": "$f^{-1}(f(x))=x \\to (f^{-1}(f(x)))' = f'(x)(f^{-1}(f(x)))^{(1)}=1 \\to (f^{-1}(f(x)))^{(1)} = \\frac{1}{f'(x)}$"}
{"_id": "1157516", "title": "", "text": "$z_1=\\sqrt 2e^{\\frac {5i\\pi}{12}}$"}
{"_id": "392914", "title": "", "text": "$F=\\{\\{1\\}\\}$"}
{"_id": "5962782", "title": "", "text": "$(a + b, a − b) = 1$"}
{"_id": "8558667", "title": "", "text": "$20/a + 12/b = 1$"}
{"_id": "4263480", "title": "", "text": "$f(\\#1, \\#2, \\dots)\\&$"}
{"_id": "6445040", "title": "", "text": "$\\lim_{n \\to \\infty}\\inf a_n = \\lim_{n \\to \\infty}\\inf b_n$"}
{"_id": "3308403", "title": "", "text": "$\\lim_{x\\to 0}\\frac{\\sqrt{1-x^2}}{2}=\\frac{1}{2}$"}
{"_id": "2142023", "title": "", "text": "$\\{ e_1,e_2,..,e_{\\vert H\\vert \\vert G \\vert}  \\}$"}
{"_id": "8420530", "title": "", "text": "$N(z),N(w)$"}
{"_id": "4569256", "title": "", "text": "$f(n) = 3*f(n-2) + 2*f(n-3)$"}
{"_id": "6913896", "title": "", "text": "$P\\oplus Q=F\\cong F^\\ast =P^\\ast \\oplus Q^\\ast$"}
{"_id": "5053089", "title": "", "text": "$ \\begin{align} \\prod_{k=1}^{n-1}\\cos\\left(\\frac{2k\\pi}n\\right) &=\\frac{\\cos(n\\pi/2)-(-1)^n}{2^{n-1}}\\\\[3pt] &=\\frac{\\cos(n\\pi/2)-\\cos(n\\pi)}{2^{n-1}}\\\\[6pt] &=\\frac{\\sin(n\\pi/4)\\sin(3n\\pi/4)}{2^{n-2}} \\end{align} $"}
{"_id": "7693799", "title": "", "text": "$\\frac{4}{x}+\\frac{6}{2}=x$"}
{"_id": "2557818", "title": "", "text": "$(\\det M)^2=\\det \\begin{pmatrix} A & -B \\\\ B & A \\end{pmatrix}.$"}
{"_id": "3857729", "title": "", "text": "$\\frac{g}{||f||+||g||}$"}
{"_id": "9331698", "title": "", "text": "$G=A_1 \\times ...\\times A_n$"}
{"_id": "8249196", "title": "", "text": "$\\forall \\epsilon > 0,  \\exists N > 0 \\text{ such that } \\forall x, y \\in I,  \\frac{|f(x)-f(y)|}{|x-y|} > N \\Rightarrow |f(x)-f(y)| < \\epsilon$"}
{"_id": "1207944", "title": "", "text": "$I=\\int\\frac{1}{(x^2-1)^2}\\,dx$"}
{"_id": "3235863", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = \\frac{4}{3} \\tag{2}$"}
{"_id": "3348263", "title": "", "text": "$\\forall y (xRy \\lor yRx)$"}
{"_id": "5048968", "title": "", "text": "$x!\\sim {\\sqrt {2\\pi x}}\\left({\\frac {x}{e}}\\right)^{x}.$"}
{"_id": "8779992", "title": "", "text": "$r_1x+by=d$"}
{"_id": "8395334", "title": "", "text": "$n_a=n_b=n_c=h_a=h_b=h_c=m_a=m_b=m_c$"}
{"_id": "9059149", "title": "", "text": "$f\\left ( f(x)^{2}y \\right )=x^{3}f(xy).$"}
{"_id": "6908824", "title": "", "text": "$f_{Y}(y:θ) = \\frac{2y}{θ^{2}}$"}
{"_id": "3903235", "title": "", "text": "$\\mathrm{Cov}(\\epsilon,x)=0$"}
{"_id": "584026", "title": "", "text": "$ x_n = \\frac{n(n-1)}{2} $"}
{"_id": "7324642", "title": "", "text": "$S(x) = \\frac{x^2}{x^4+1}$"}
{"_id": "2194173", "title": "", "text": "$\\forall n(n\\geq m\\implies P(n))$"}
{"_id": "2411483", "title": "", "text": "$(1+ \\frac {x}{n})^n < e^x$"}
{"_id": "8757991", "title": "", "text": "$\\sum_{n=1}^\\infty n - \\sum_{n=1}^\\infty n = \\sum_{n=1}^\\infty (n-n)$"}
{"_id": "1781375", "title": "", "text": "$\\frac{x^9+27}{3}$"}
{"_id": "7454517", "title": "", "text": "$\\int_0 ^\\infty \\frac{|sin(x)|}{(x^2+1)^n}$"}
{"_id": "79791", "title": "", "text": "$ \\lim _{n \\to \\infty} \\sum_{r=1}^n\\frac1r $"}
{"_id": "7500865", "title": "", "text": "$\\int_0^1 x(t)f(t)\\,\\mathrm dt = 0$"}
{"_id": "2431308", "title": "", "text": "$\\Vert x_{n+1}-x_n\\Vert$"}
{"_id": "7284742", "title": "", "text": "$R(x-a):= -r(x-a)(x-a)$"}
{"_id": "5816153", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\|x_n-y_n\\|^2<1$"}
{"_id": "6197327", "title": "", "text": "$\\{1,x, y, xy, x^2, y^2, x^2y, xy^2, \\cdots\\},$"}
{"_id": "6559387", "title": "", "text": "$\\lim_{n \\to \\infty} \\dfrac{\\sqrt 1 + \\sqrt 2 + \\dots + \\sqrt{n}}{n\\sqrt{n}} =\\lim_{n\\to \\infty} \\dfrac1{n}\\sum^{n}_{k = 1} \\sqrt{\\dfrac k n} $"}
{"_id": "6126426", "title": "", "text": "$\\left(\\begin{array}{cc|c} 1 & 0 & 35 \\\\ 0 & 1 & 99 \\end{array}\\right) \\longrightarrow \\left(\\begin{array}{cc|c} 17 & -6 & 1 \\\\ -14 & 5 & 5 \\end{array}\\right)$"}
{"_id": "7655524", "title": "", "text": "$\\langle x^3-Q(x), x \\rangle =0$"}
{"_id": "5548075", "title": "", "text": "$f_2(n)=n^3$"}
{"_id": "8365828", "title": "", "text": "$y=\\displaystyle{\\frac{ax}{bx+c}}$"}
{"_id": "2998096", "title": "", "text": "$f(x)=\\frac{61}{50}~e^{\\frac{3}{20}x}-\\frac{11}{50}e^{-6x}-3$"}
{"_id": "5305151", "title": "", "text": "$\\left(1+\\frac xn\\right)^n=e^x\\left(1+O\\left(\\frac{x^2}n\\right)\\right)$"}
{"_id": "7017788", "title": "", "text": "$\\frac {S}{(1+r)^T}$"}
{"_id": "8630133", "title": "", "text": "$|2^A|\\leq |B^A| \\leq \\mathcal{P}(A\\times B) \\leq \\mathcal{P}(A^2) = \\mathcal{P}(A) = |2^A|.$"}
{"_id": "1857139", "title": "", "text": "$\\det(A-I\\gamma)=0 \\Rightarrow \\begin{cases}\\gamma=4 \\\\ \\gamma=2 \\end{cases}$"}
{"_id": "8144528", "title": "", "text": "$\\lim\\sup \\mu(A_n)\\le \\mu(\\lim\\sup A_n)$"}
{"_id": "2100295", "title": "", "text": "$\\displaystyle \\zeta(s)=\\dfrac{1}{\\Gamma(s)}\\int_0^{+\\infty} \\dfrac{x^{s-1}}{e^x-1}dx$"}
{"_id": "2465975", "title": "", "text": "$S_N=\\frac{n(n+1)}{2}$"}
{"_id": "1242567", "title": "", "text": "$\\gcd(a) = \\gcd(a, \\gcd())$"}
{"_id": "1173138", "title": "", "text": "$Cov(X,X)=0$"}
{"_id": "6087808", "title": "", "text": "$D = \\{(x, y):1 \\le x^2 + y^2 \\le 4, x \\le y \\le \\sqrt3x, x \\ge 0 \\}$"}
{"_id": "2234727", "title": "", "text": "$x \\in V \\subseteq \\overline{V} \\subseteq O$"}
{"_id": "430148", "title": "", "text": "$f(n) = 2^{n+1} - n - 2.$"}
{"_id": "3583971", "title": "", "text": "$A_1 \\times A_2 \\times \\ldots \\times A_n \\times \\ldots$"}
{"_id": "7322988", "title": "", "text": "$\\frac{1}{\\gamma(a\\gamma-1)}\\int \\frac{(t+a\\gamma)-(t+1)dt}{(t+1)(t+a\\gamma)}$"}
{"_id": "7144257", "title": "", "text": "$\\Big\\lfloor{x}/({y}\\cdot{z})\\Big\\rfloor=\\Big\\lfloor{\\lfloor{x}/{y}\\rfloor}/{z}\\Big\\rfloor$"}
{"_id": "3800578", "title": "", "text": "$ (x_{1} x_{2} \\cdots x_{n})^{1/n} \\leq \\frac{x_{1} + x_{2} + \\cdots + x_{n}}{n}. $"}
{"_id": "7079900", "title": "", "text": "$P(E_n)=1/2$"}
{"_id": "4698006", "title": "", "text": "$ \\bbox[lightyellow] {   \\sum\\nolimits_{\\;x} {x^{\\,\\overline {\\,n\\,} } }  = \\sum\\nolimits_{\\;x} {\\left( {x + n - 1} \\right)^{\\,\\underline {\\,n\\,} } }  = \\left\\{ {\\matrix{    {{1 \\over {n + 1}}\\;\\left( {x - 1} \\right)^{\\,\\overline {\\,n + 1\\,} }  + c} & { - 1 \\ne n}  \\cr     {\\psi (x - 1) + c} & { - 1 = n}  \\cr   } } \\right.  } \\tag{2.b}$"}
{"_id": "6119296", "title": "", "text": "$A=\\{\\{a\\},\\{b\\}\\}$"}
{"_id": "2443945", "title": "", "text": "$|f_{n}(x)| \\leq k(x-a)^{n-1} \\leq k(b-a)^{n-1}$"}
{"_id": "6290130", "title": "", "text": "$rs - r - s$"}
{"_id": "4713652", "title": "", "text": "$A=\\{\\{q\\}:q\\in \\mathbb Q\\}.$"}
{"_id": "4930432", "title": "", "text": "$\\begin{cases} x_1=x_3\\\\x_2=-x_3\\\\x_3\\ \\text{is free}\\end{cases}$"}
{"_id": "6229926", "title": "", "text": "$ P(X>x) = \\frac{(1-p)^x p}{1-(1-p)} = (1-p)^x, $"}
{"_id": "7032188", "title": "", "text": "$ \\zeta(s) = \\frac{1}{\\Gamma(s)}  \\int_0^1 \\frac{x^{s-1}}{e^{x}-1} dx + \\frac{1}{\\Gamma(s)}  \\int_1^\\infty \\frac{x^{s-1}}{e^{x}-1} dx $"}
{"_id": "35486", "title": "", "text": "$X = \\pmatrix{-1\\\\3}$"}
{"_id": "4389105", "title": "", "text": "$α+β+γ+δ+2(\\sqrt{αβ}+\\sqrt{αγ}+\\sqrt{αδ}+\\sqrt{βγ}+\\sqrt{βδ}+\\sqrt{γδ})=0$"}
{"_id": "496287", "title": "", "text": "$P_1 P_2= P_1P_2=0$"}
{"_id": "7931781", "title": "", "text": "$\\mathbb R[X]/(X^{4}+1)$"}
{"_id": "6200491", "title": "", "text": "$\\tilde \\gamma (s) = \\int \\limits _a ^s \\dot {\\tilde \\gamma} (t) \\ \\Bbb d t + \\tilde \\gamma (a) = \\int \\limits _a ^s M_0 \\dot \\gamma (t) \\ \\Bbb d t + \\tilde \\gamma (a) = M_0 \\big( \\gamma (s) - \\gamma (a) \\big) + \\tilde \\gamma (a) = \\\\ M_0 \\gamma (s) + \\big( \\tilde \\gamma (a) - M_0 \\gamma (a) \\big) .$"}
{"_id": "2553622", "title": "", "text": "$(\\tfrac12)^{n-3}$"}
{"_id": "2471707", "title": "", "text": "$|Df(x)|=1/|x|$"}
{"_id": "5470979", "title": "", "text": "$(k + 1) - 3 ≥ 14$"}
{"_id": "8715164", "title": "", "text": "$\\text{FV}=\\text{PV}(1+r)^{n+1}=\\frac {C(1+r)}r\\left[(1+r)^n-1\\right]$"}
{"_id": "8068947", "title": "", "text": "$T({{1}_{[a,b]}})(s)=\\left\\{ \\begin{matrix}    0\\,\\,\\,if\\,\\,s<a  \\\\    \\frac{1}{s}(s-a)\\,\\,\\,if\\,\\,\\,a\\le s<b  \\\\    \\frac{1}{s}(b-a)\\,\\,\\,if\\,\\,\\,s\\ge b  \\\\ \\end{matrix} \\right. $"}
{"_id": "5909968", "title": "", "text": "$u_0=P(X_T=0|X_0=0)=1$"}
{"_id": "7354054", "title": "", "text": "$ P(U_{K_t}<x\\sqrt{K_t})\\leq P\\left(K_{t}\\notin ((1-\\epsilon) a(t),(1+\\epsilon) a(t))\\right)+P\\left(U_{a_t}<x\\sqrt{(1+\\epsilon)a(t)}+m\\cdot \\sqrt{\\epsilon a(t))}\\right)+ P\\left(\\sup_{s\\in ((1-\\epsilon)a(t),(1+\\epsilon)a(t))}|U_{s}-U_{a(t)}|>m\\cdot \\sqrt{\\epsilon a(t))}\\right) $"}
{"_id": "7548804", "title": "", "text": "$f(x)=\\int_a^x g(t)\\ \\mathbb{d}t$"}
{"_id": "1587172", "title": "", "text": "$d(x,A)=\\inf\\{d(x,a):a\\in A\\}\\le \\inf\\{d(x,y):y\\in A\\}+\\inf\\{d(y,a):a\\in A\\}$"}
{"_id": "7732943", "title": "", "text": "$E(P^2|T)-(E(P|T))^{2}$"}
{"_id": "4901819", "title": "", "text": "$4n + 2, 4n + 1, \\dots, 2n + 2$"}
{"_id": "8921343", "title": "", "text": "$ \\sum_{n=0}^\\infty\\frac{(2\\sqrt a)^n}{n!}\\sqrt \\pi= \\sqrt\\pi \\;\\exp(2\\sqrt a)$"}
{"_id": "6436230", "title": "", "text": "${\\displaystyle \\|A\\|_{2}={\\sqrt {\\lambda _{\\max }(A^{^{T}}A)}}}$"}
{"_id": "3535805", "title": "", "text": "$\\beta<\\alpha^{+}$"}
{"_id": "7281629", "title": "", "text": "$100(1+r)^2+110(1+r)-(1+r)-232=0$"}
{"_id": "3961963", "title": "", "text": "$\\int(2\\pi y dx)$"}
{"_id": "252004", "title": "", "text": "$\\int_0^{\\infty} g(x)dx$"}
{"_id": "3686148", "title": "", "text": "$1+2+3+4+5+\\dots=-1/12$"}
{"_id": "2123879", "title": "", "text": "$U=\\langle u_1,u_2,u_3,....,u_n \\rangle$"}
{"_id": "2380162", "title": "", "text": "$f(p(t)) = p(t) + p'(t) + p''(t), \\tag{1}$"}
{"_id": "2544799", "title": "", "text": "$ \\left\\{ \\matrix{   x = 2u + 2n - 1 \\hfill \\cr    y = 1 - u - 2n \\hfill \\cr    z = n \\hfill \\cr}  \\right.\\quad  \\Rightarrow \\quad \\left\\{ \\matrix{   0 \\le x + y = u \\hfill \\cr    u \\le 1 - 2n \\hfill \\cr    0 \\le n \\hfill \\cr}  \\right.\\quad  \\Rightarrow \\quad \\left\\{ \\matrix{   u_{\\,\\min }  = 1 \\hfill \\cr    n = 0 \\hfill \\cr    x = 1,\\;y = 0,z = 0 \\hfill \\cr}  \\right. $"}
{"_id": "5644758", "title": "", "text": "$\\sum_{n \\geq 1} \\frac{1}{n(n+1)} = \\underbrace{\\sum_{n \\geq 1} \\frac{1}{n^2 + n} \\leq \\sum_{n \\geq 1} \\frac{1}{n^2}}_{\\text{Every element is smaller}} = \\frac{\\pi^2}{6}$"}
{"_id": "8612556", "title": "", "text": "$\\eta(s)=\\frac1{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{x^{s-1}}{e^x+1}dx$"}
{"_id": "8338170", "title": "", "text": "$y=a(x-A)(x-B)$"}
{"_id": "3078773", "title": "", "text": "$S = \\{e_1, e_2, \\cdots, e_n\\}$"}
{"_id": "5247084", "title": "", "text": "$\\varphi_n(x) = \\sum_{j=1}^n \\frac{1}{j} \\chi_{A_j}(x)$"}
{"_id": "2197274", "title": "", "text": "$ar=b$"}
{"_id": "9054464", "title": "", "text": "$\\begin{cases} x=2-t\\\\y=1+t\\\\z=t\\end{cases}$"}
{"_id": "6470328", "title": "", "text": "$d(x_0,M)=\\bigg|\\frac {1}{||g||}g(x_0)\\bigg|$"}
{"_id": "6609278", "title": "", "text": "$|f(a)−f(b)|≤K|a−b$"}
{"_id": "4157564", "title": "", "text": "$(\\frac{1}{2})^{n-1}$"}
{"_id": "6108119", "title": "", "text": "$e_i + 1 \\le 2^{e_i}$"}
{"_id": "5012857", "title": "", "text": "$F(x)\\equiv \\int_a^x f(t)\\,dt$"}
{"_id": "2215420", "title": "", "text": "$\\frac{1}{4^{n-2}}$"}
{"_id": "8123836", "title": "", "text": "$\\frac2a+\\frac3b=1$"}
{"_id": "148300", "title": "", "text": "$F=P\\oplus Q$"}
{"_id": "1089483", "title": "", "text": "$x!\\approx \\sqrt{2\\pi x}\\left(\\frac xe \\right)^x\\implies \\log(x!)\\approx (x+\\frac 12)\\log(x)-x+\\log(\\sqrt{2\\pi})$"}
{"_id": "7138838", "title": "", "text": "$ S = \\{ E_1, E_2, E_3, E_4\\}$"}
{"_id": "5713113", "title": "", "text": "$\\bigcup\\mathcal A=X.$"}
{"_id": "3089704", "title": "", "text": "$\\tan \\theta = \\frac{x}{1}$"}
{"_id": "7852524", "title": "", "text": "$K \\subset V \\subset \\overline V \\subset W$"}
{"_id": "8211369", "title": "", "text": "$ z^{1/n} = r^{1/n}e^{\\frac{i \\theta}{n}}$"}
{"_id": "8501235", "title": "", "text": "$\n \\left[\\matrix{1&0&2&0\\cr 0&1&-4&3\\cr 0&0&-3&5\\cr 0&0&0&7\\cr }\\right].\n $"}
{"_id": "3005726", "title": "", "text": "$\\zeta(s)=\\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{t^{s-1} dt}{e^t-1}$"}
{"_id": "6692062", "title": "", "text": "$a(1,1,1,-1) + b(1,-1,0,1) = c(1,1,-1,1) + d(1,3,4,-5)$"}
{"_id": "3551591", "title": "", "text": "$ b^{\\log_b(x)} = x. $"}
{"_id": "5254219", "title": "", "text": "$ \\tan\\theta={x\\over 2-y} $"}
{"_id": "1739300", "title": "", "text": "$d=(a+b,a-b)$"}
{"_id": "1607409", "title": "", "text": "$2\\mid n^2-n$"}
{"_id": "1695948", "title": "", "text": "$\\displaystyle x= b^{\\log_b(x)}$"}
{"_id": "8605604", "title": "", "text": "$z_1^6 = 2^{3/2}e^{\\frac{i\\pi}{2}}$"}
{"_id": "2311630", "title": "", "text": "$\\det A_{a,b,n} = \\det \\begin{pmatrix}a & b\\dotsc  b \\\\ 0 & A_{a-b,0,n-1}\\end{pmatrix}$"}
{"_id": "3011842", "title": "", "text": "$\\begin{cases} x=6 \\\\ y=2 \\\\ z=8 \\end{cases}$"}
{"_id": "3675701", "title": "", "text": "$ \\varphi(\\delta)=\\frac{f(x+\\delta)-f(x)}{\\delta}-a $"}
{"_id": "5811878", "title": "", "text": "$I(a) = \\int_{-\\infty}^{\\infty} \\dfrac{\\cos(ax)}{\\pi(1+x^2)}dx$"}
{"_id": "5471799", "title": "", "text": "$f(x) = \\sin(1/|x|)$"}
{"_id": "927833", "title": "", "text": "$\\zeta(2) = \\frac{1}{\\Gamma(2)}\\int_0^\\infty \\frac{x}{e^x-1} dx$"}
{"_id": "3252990", "title": "", "text": "$-x^2 y+x^2 z+x y^2-x z^2-y^2 z+y z^2=(x-z)(-x y+x z+y^2-y z)= (x - z)(y - x) (y - z)$"}
{"_id": "1814905", "title": "", "text": "$                 f'(0)=0,\\;\\; f(\\pi)=0 $"}
{"_id": "4067198", "title": "", "text": "$\\left(\\int_0^\\infty f(x) \\, dx\\right)^s = C^s $"}
{"_id": "1603302", "title": "", "text": "$\\inf d(x,a)$"}
{"_id": "3833399", "title": "", "text": "$\\mathbb R^{n-1} \\hookrightarrow \\mathbb R^n$"}
{"_id": "2169240", "title": "", "text": "$f(n) = (\\frac{n^2-n}{(3300^2-3300)}*0.63)\\quad,\\quad n\\lt3300$"}
{"_id": "3225501", "title": "", "text": "$\\sum_n \\sum_m \\mu(\\{m\\})\\nu(\\{n-m\\})<\\infty$"}
{"_id": "4025485", "title": "", "text": "$x^n = (1+ \\delta)^n > 1+ \\delta n.$"}
{"_id": "5506553", "title": "", "text": "$cRa\\wedge aRc$"}
{"_id": "3680", "title": "", "text": "$\\|x_n - x_m \\| < \\epsilon$"}
{"_id": "1255405", "title": "", "text": "$\\mathsf P(E)=\\tfrac 1 4$"}
{"_id": "2691050", "title": "", "text": "$\\mathbb R^n\\times \\mathbb R = \\mathbb R^{n+1},$"}
{"_id": "8674129", "title": "", "text": "$ \\|A^{\\dagger}\\|_2^2=\\rho[A^\\dagger(A^\\dagger)^T]=\\rho[(A^TA)^{-1}(A^TA)(A^TA)^{-1}]=\\rho[(A^TA)^{-1}]=\\|(A^TA)^{-1}\\|_2. $"}
{"_id": "1624566", "title": "", "text": "$S= \\int 2 \\pi x ds$"}
{"_id": "7360895", "title": "", "text": "$\\int \\frac {1}{(x^2 + 1)(x^2-1)+2} dx$"}
{"_id": "2425598", "title": "", "text": "$\\int_{0}^{\\infty} f_n(x)dx \\to \\int_{0}^{\\infty} f(x)dx$"}
{"_id": "8258213", "title": "", "text": "$ f(n-1)-f(n-2) = (n^2 + n-2) - (n^2-n-2)= +2n  $"}
{"_id": "8532904", "title": "", "text": "$P = \\{ \\{a\\} , \\{b, c\\} \\}$"}
{"_id": "2133234", "title": "", "text": "$P[X_i \\in A | \\mathcal F] = P[ X_j \\in A | \\mathcal F]$"}
{"_id": "3691001", "title": "", "text": "$S = \\{s_1, s_2, s_3 .... s_n\\}$"}
{"_id": "7782007", "title": "", "text": "$ X = \\{\\, (2\\cos{\\varphi}, \\sqrt{1+4\\cos^2{\\varphi}}, 2\\sin{\\varphi})\\, |\\, \\varphi \\in [0, 2\\pi) \\, \\} \\cup \\{\\, (2\\cos{\\varphi}, -\\sqrt{1+4\\cos^2{\\varphi}}, 2\\sin{\\varphi})\\, |\\, \\varphi \\in [0, 2\\pi) \\, \\}. $"}
{"_id": "6854649", "title": "", "text": "$Payments=P \\frac {R(1+R)^N}{(1+R)^N-1}$"}
{"_id": "3712960", "title": "", "text": "$f(x)=\\frac{x}{3} + 3$"}
{"_id": "4680520", "title": "", "text": "$Log_{a} (x^{n}) = nlog_{a} x$"}
{"_id": "3145694", "title": "", "text": "$(40-x_1)+(40-x_2)+(40-x_3)+(40-x_4)=160-(x_1+x_2+x_3+x_4)$"}
{"_id": "6310202", "title": "", "text": "$K \\subseteq U \\subseteq \\overline{U} \\subseteq O$"}
{"_id": "4737470", "title": "", "text": "$d(X,Y)\\leqslant d(X,Z) +d(Z,Y)$"}
{"_id": "3767421", "title": "", "text": "$\\wp(\\Bbb N)\\setminus\\{\\}=\\wp(\\Bbb N)\\setminus\\varnothing=\\wp(\\Bbb N)$"}
{"_id": "7806951", "title": "", "text": "$\\sum_{c=0}^{n-1}\\cos(2\\pi \\frac{c}{n})=0$"}
{"_id": "3795987", "title": "", "text": "$\\tau_{a,ab}\\equiv\\{\\emptyset,\\{a\\},\\{a,b\\},X\\}$"}
{"_id": "2283078", "title": "", "text": "$a_n = nc_n(x-a)^{n-1}$"}
{"_id": "5462517", "title": "", "text": "$ \\begin{align} \\det\\begin{bmatrix} x_1&y_1&z_1\\\\ x_2&y_2&z_2\\\\ x_3&y_3&z_3 \\end{bmatrix} &= \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix} \\times \\begin{bmatrix} y_1\\\\ y_2\\\\ y_3 \\end{bmatrix} \\cdot \\begin{bmatrix} z_1\\\\ z_2\\\\ z_3 \\end{bmatrix}\\\\ &= \\begin{bmatrix} (x\\times y)_1\\\\ (x\\times y)_2\\\\ (x\\times y)_3 \\end{bmatrix} \\cdot \\begin{bmatrix} z_1\\\\ z_2\\\\ z_3 \\end{bmatrix}\\tag{1} \\end{align} $"}
{"_id": "5456830", "title": "", "text": "$\\large x! \\sim \\sqrt{2\\pi x}(\\frac{x}{e})^x$"}
{"_id": "5961597", "title": "", "text": "$z = \\{ \\{ u \\} , \\{ v \\} \\}$"}
{"_id": "850871", "title": "", "text": "$\\forall \\epsilon, \\exists \\delta > 0 s.t. |x-y| < \\delta, x,y \\in D \\implies |f(x) - f(y)| < \\epsilon$"}
{"_id": "4597715", "title": "", "text": "$[x,y] = ax$"}
{"_id": "6906534", "title": "", "text": "$\\lim_{\\text n \\rightarrow \\infty}\\sum_{k=1}^\\infty \\frac{\\text 1_k}{n}=1$"}
{"_id": "6801471", "title": "", "text": "$\\int_0^\\infty dx \\ f(x) = \\int_0^\\infty dx \\ g(x) \\Rightarrow \\int_0^\\infty dx \\ f(x)h(x) = \\int_0^\\infty dx \\ g(x)h(x)$"}
{"_id": "9166303", "title": "", "text": "${1 \\over F(n)} \\le {1 \\over 2^{n-1}}$"}
{"_id": "7978018", "title": "", "text": "$ y = \\frac{a+x}{b+cx} $"}
{"_id": "6667230", "title": "", "text": "$\\alpha\\subseteq\\gamma\\subseteq\\beta$"}
{"_id": "779020", "title": "", "text": "$f(a)+f(b)=f(a+b)$"}
{"_id": "5444758", "title": "", "text": "$z=be^{\\frac{i\\pi(1 + 4k)}{2n}}$"}
{"_id": "1247769", "title": "", "text": "$\\gamma':\\gamma'\\geq\\beta,\\gamma'\\geq\\gamma$"}
{"_id": "947080", "title": "", "text": "$f(n)=2n^2-5n+1$"}
{"_id": "1046280", "title": "", "text": "$=\\frac{(1+x)^{m+n}-(1+x)^{m}}{x}$"}
{"_id": "4724607", "title": "", "text": "$\\|A\\| = \\sqrt{\\operatorname{tr}(A^TA)}$"}
{"_id": "1917946", "title": "", "text": "$ax+by=c.$"}
{"_id": "493737", "title": "", "text": "$f(x^2+f(y))=(f(x))^2+y$"}
{"_id": "1495233", "title": "", "text": "$a(x)=x^2+x+2$"}
{"_id": "8343262", "title": "", "text": "$\\frac{e^x(e^{(1/n)} - 1)}{\\frac{1}{n}}$"}
{"_id": "4689505", "title": "", "text": "$\\lim_{n \\to \\infty}\\left(\\frac{\\sqrt[n]a}{n+1}+\\frac{\\sqrt[n]{a^2}}{n+\\frac12}+\\frac{\\sqrt[n]{a^3}}{n+\\frac13}+\\cdots+\\frac{\\sqrt[n]{a^n}}{n+\\frac1n}\\right)$"}
{"_id": "8709081", "title": "", "text": "$g(x)=\\int_a^x F(t) dt$"}
{"_id": "7551737", "title": "", "text": "${\\int\\limits_0^{\\pi /2} {dx} }$"}
{"_id": "266335", "title": "", "text": "$\\overline{a_1\\ldots a_n}$"}
{"_id": "6490383", "title": "", "text": "$a_1 = f(a_1)$"}
{"_id": "423637", "title": "", "text": "$B(x,y)=0$"}
{"_id": "4895896", "title": "", "text": "$\\int _{-1}^1\\:\\frac{\\sqrt{1-x^2}}{x^2+1}$"}
{"_id": "5685962", "title": "", "text": "$R: a\\le x\\le b,  c\\le y\\le d,$"}
{"_id": "4847474", "title": "", "text": "$f:(\\mathbb{R}^{n}, \\mathcal{R}^{n})\\to (\\mathbb{R},\\mathcal{R})$"}
{"_id": "4327098", "title": "", "text": "$C_R(x) + r$"}
{"_id": "4700734", "title": "", "text": "$E[T|C<c^*]$"}
{"_id": "6622165", "title": "", "text": "$ \\left[\\begin{array}{ccc|c} 1 & 9 & -5 & 2 \\\\ 2 & -9 & -1 & -23 \\\\ 3 & 1 & 1 & 24 \\end{array}\\right] $"}
{"_id": "2826739", "title": "", "text": "$\\lim_{n \\to \\infty} \\sum\\limits_{k = 1}^n \\frac 1 {2^k}$"}
{"_id": "5024218", "title": "", "text": "$\\sum_{n\\geq 1}\\frac{1}{n^2}<2$"}
{"_id": "7469544", "title": "", "text": "$P(\\{s\\}) = 1/6$"}
{"_id": "8250529", "title": "", "text": "$A_1\\subseteq A_2\\subseteq A_3\\subseteq ...\\subset X$"}
{"_id": "4506769", "title": "", "text": "$f(x)=\\frac{1}{2} e^{\\frac{-x}{2}}$"}
{"_id": "2004160", "title": "", "text": "$\\mathop{GCD}(m,n)=1$"}
{"_id": "1261464", "title": "", "text": "$P(E_{1})=1/6$"}
{"_id": "3344924", "title": "", "text": "$V_0 \\subseteq V_1 \\subseteq V_2 \\subseteq V_3 \\subseteq \\cdots$"}
{"_id": "5964135", "title": "", "text": "$\\sum_{n=1}^\\infty e^{i n x} = \\frac{1}{2} \\sum_{n=-\\infty}^\\infty e^{i n x} + \\frac{1}{2} \\sum_{n=-\\infty}^\\infty \\mathop{\\rm sgn}(n) e^{i n x} - \\frac{1}{2}.$"}
{"_id": "8775043", "title": "", "text": "$(\\diamondsuit)<\\sum_{n=1}^{+\\infty}\\|f_n-e_n\\|<1\\Rightarrow 1<1. $"}
{"_id": "9169621", "title": "", "text": "$U\\subseteq C\\subseteq V$"}
{"_id": "3496507", "title": "", "text": "$\\int_1^2\\sqrt{1+(\\frac{dy}{dx})^2}$"}
{"_id": "4802881", "title": "", "text": "$ \\lim_{x\\to\\infty}\\big(2x-x\\big)=\\infty-\\infty=0 $"}
{"_id": "1196604", "title": "", "text": "$2Cov(X,Y)=0$"}
{"_id": "354717", "title": "", "text": "$\\mathbb{Z}[i]\\cong \\mathbb{Z}[x]/(x^2+1)$"}
{"_id": "7454516", "title": "", "text": "$\\int_0 ^\\infty \\frac{sin(x)}{(x^2+1)^n} < \\infty $"}
{"_id": "7119546", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}= \\frac{1}{2007}$"}
{"_id": "2139236", "title": "", "text": "$\\{a\\} \\times \\{a\\} = \\{\\{\\{a\\}\\}\\}$"}
{"_id": "881119", "title": "", "text": "$\\int_{0}^{\\frac{1}{2}}\\frac{\\sin^{-1}(x)}{\\sqrt{1-x^2}} dx$"}
{"_id": "5264495", "title": "", "text": "$ \\lim_{n\\to \\infty} \\sum_{r=1}^{n}\\frac{1}{n+r}$"}
{"_id": "7607851", "title": "", "text": "$as=r+b$"}
{"_id": "707460", "title": "", "text": "$v^+$"}
{"_id": "505491", "title": "", "text": "$\\sum_{n\\geq 1}\\frac{1}{n^2}-\\sum_{n=1}^{1000}\\frac{1}{n^2}=\\sum_{n\\geq 1001}\\frac{1}{n^2}\\leq\\frac{1}{1000+\\frac{1}{2}}\\tag{3} $"}
{"_id": "5044717", "title": "", "text": "$x^2, \\, x\\in \\mathbb{R}$"}
{"_id": "7532451", "title": "", "text": "$\\sqrt{1+\\frac{d^2y}{dx^2}}=x+\\frac{dy}{dx}$"}
{"_id": "7078014", "title": "", "text": "$M = \\begin{pmatrix} A & B \\\\ \\mathbf{0} & D \\end{pmatrix}$"}
{"_id": "9161170", "title": "", "text": "$h'(\\sum x_n)=\\sum y_n$"}
{"_id": "9180336", "title": "", "text": "$\\ A^B$"}
{"_id": "950373", "title": "", "text": "$X^{a/b}=\\{ x^\\gamma \\mid a\\in \\gamma, b\\not\\in\\gamma,\\gamma\\in\\mathcal R \\}$"}
{"_id": "9001066", "title": "", "text": "$ x_1=0+x_0=h\\\\ x_2=0+x_1=h\\\\ x_3=0+x_2=h\\\\ x_4=0+x_3=h\\\\ x_5=1+x_4=h+1\\\\ x_6=6+x_5=h+7 $"}
{"_id": "3405845", "title": "", "text": "$ \\alpha(\\tau)=-\\tfrac{1}{2}-i\\tfrac{(2+2p-p^2) (1-2 p -2p^2) (1+4p+p^2)}{6 \\sqrt{3} ~p(p+2) (2 p+1)(1-p^2)}=\\tfrac1{4\\sqrt{27}}\\big(\\lambda^3-\\sqrt{27}\\,\\lambda^{-3}\\big)^2 $"}
{"_id": "5546096", "title": "", "text": "$\\tan^2\\theta-\\sin^2\\theta=(\\tan\\theta-\\sin\\theta)(\\tan\\theta+\\sin\\theta)\\gt 0.$"}
{"_id": "7202183", "title": "", "text": "$ \\sum_{n\\leq x}\\frac{1}{n}\\sum_{d\\mid n}d=\\sum_{d\\leq x}\\frac{1}{d}\\sum_{\\substack{n\\leq x\\\\ d\\mid n}}1=\\sum_{d\\leq x}\\frac{1}{d}\\left\\lfloor\\frac{x}{d}\\right\\rfloor=O(\\log x)+x\\sum_{d\\leq x}\\frac{1}{d^2}$"}
{"_id": "317364", "title": "", "text": "$\\begin{pmatrix}a & b\\\\ -b & a \\end{pmatrix}$"}
{"_id": "5240642", "title": "", "text": "$\\partial^2 = \\partial \\bar\\partial + \\bar\\partial \\partial = \\bar\\partial^2 = 0.$"}
{"_id": "3547061", "title": "", "text": "$||A||_F = \\sqrt {Tr(A^TA)}$"}
{"_id": "6910348", "title": "", "text": "$H_1(X;\\mathbb{Z}/2\\mathbb{Z}) = (\\mathbb{Z}/2\\mathbb{Z})^3$"}
{"_id": "8609857", "title": "", "text": "$r(u,v)=\\begin{cases}  x=u\\\\  y=v\\\\  z=2-u-v \\end{cases}$"}
{"_id": "2723675", "title": "", "text": "$\\mathbb{R}[x]/(x^2+3)$"}
{"_id": "1389508", "title": "", "text": "$E_2^{0,1} = E_\\infty^{0,1} = F^0H^1$"}
{"_id": "1750730", "title": "", "text": "$\\gamma^{n}=\\gamma\\ast\\cdots\\ast\\gamma$"}
{"_id": "1249532", "title": "", "text": "$d(x,A)-d(x,y)\\le d(y,A)$"}
{"_id": "5934602", "title": "", "text": "$ \\lim\\limits_{n\\rightarrow\\infty} \\left\\lfloor \\frac{-1}{n}\\right\\rfloor $"}
{"_id": "5410240", "title": "", "text": "$  \\sum_{k=0}^n|\\cos k|>\\frac{2}{\\pi}n.$"}
{"_id": "2768606", "title": "", "text": "$f_n(x) = \\dfrac{nx+x^2}{n^2}$"}
{"_id": "6288005", "title": "", "text": "$\\det(B) = \\det(A-C)^{N-1}*\\det(A+(N-1)C)$"}
{"_id": "3276322", "title": "", "text": "$\\begin{align} a x + b y = c\\\\  d x + e y = f \\end{align}$"}
{"_id": "3969609", "title": "", "text": "$p(m+N)=p'(m)$"}
{"_id": "6880685", "title": "", "text": "$ax - by=d$"}
{"_id": "8826140", "title": "", "text": "$\\hat\\gamma(t):=\\begin{cases}\\gamma(t)&0\\le t\\le 1\\\\ 2\\gamma(1)-\\gamma(t)&1\\le t\\le 2\\\\ 2\\gamma(0)-\\gamma(t)&-1\\le t\\le 0\\end{cases} $"}
{"_id": "9372301", "title": "", "text": "$15^x+26^x=30^x$"}
{"_id": "7650043", "title": "", "text": "$f(x)=\\int_a^x f(t)dt$"}
{"_id": "4366476", "title": "", "text": "$I=\\pi\\int_0^{\\pi/2}\\frac{\\sin x dx}{\\sin x +1}=\\frac{\\pi^2}{2}-\\pi\\int_0^{\\pi/2}\\frac{dx}{1+\\sin x}.$"}
{"_id": "4149336", "title": "", "text": "$\\vert d(x,z)-d(y,t)\\vert\\leq d(x,y)+d(z,t)$"}
{"_id": "7806438", "title": "", "text": "$ {x}^2 \\left(1-2 \\sqrt{2}\\, {x} \\right) \\sin ^2(\\psi )-{y}^2    \\left(1-2 \\sqrt{2} \\,{y} \\right)=0 $"}
{"_id": "1017014", "title": "", "text": "$   xf(x) = f(y) f(f(y)) = f(f(y)^2) = f \\left( f(y)^2 + f(y) f \\left( y - f(y) \\right) \\right) $"}
{"_id": "7003803", "title": "", "text": "$T(n) = \\sqrt1 + \\sqrt2 + \\sqrt{3} + \\cdots + \\sqrt n.$"}
{"_id": "6758524", "title": "", "text": "$y=\\alpha+\\beta=\\alpha+\\beta+\\gamma-\\gamma=-\\frac ba-\\gamma\\implies \\gamma=-\\left(y+\\frac ba\\right)$"}
{"_id": "8002070", "title": "", "text": "$\\,(a\\!-\\!b,ab)=1\\,$"}
{"_id": "4277329", "title": "", "text": "$a^\\gamma b^\\gamma =(ab)^\\gamma$"}
{"_id": "1508836", "title": "", "text": "$\\mathcal{N}_{a} = \\{\\{a\\}\\}$"}
{"_id": "4867271", "title": "", "text": "$\\int^1_0 \\frac{\\sin(x)}{x(1+x)^n} dx$"}
{"_id": "801113", "title": "", "text": "$e=e(P;p)$"}
{"_id": "6996001", "title": "", "text": "$= Cov(2X, X) + Cov(2X, -3Y ) + Cov(Y , X) + Cov(Y ,  -3Y )$"}
{"_id": "6739566", "title": "", "text": "$(A_1 \\times A_2 \\times ... \\times A_{n-1}) \\times A_n$"}
{"_id": "1999152", "title": "", "text": "$\\lim_{x\\to c} \\frac{f(x) - f(c)} {x-c} =f'(c) =L=\\lim_{x\\to c} f'(x)\\tag{1} $"}
{"_id": "4615372", "title": "", "text": "$\\log_b (c^{\\log_b a})=\\log_b a\\log_b c$"}
{"_id": "5396194", "title": "", "text": "$\\# R_X = \\max (\\# X, \\aleph_0)$"}
{"_id": "4583707", "title": "", "text": "$\\frac{f}{||f||}$"}
{"_id": "4130121", "title": "", "text": "$a(n) x^n + a(n-1) x^{n-1} + \\dots + a(1) x + a(0)$"}
{"_id": "5205468", "title": "", "text": "$a<y<b, c<x<d$"}
{"_id": "3805195", "title": "", "text": "$\\delta_v(A):=\\max_i \\ \\frac{|\\langle a_i,v\\rangle|^2}{||v||_2^2}.$"}
{"_id": "3587776", "title": "", "text": "$GF(2^8)[x]/(x^4+1)$"}
{"_id": "8481989", "title": "", "text": "$\\|x_n-y_n\\|<2^{-n}$"}
{"_id": "5963175", "title": "", "text": "$(1-r)(1+r+r^2+\\cdots +r^{n-1})=1-r^n.$"}
{"_id": "4534103", "title": "", "text": "$ f(n + 2) = 3 f(n + 1) + 2 f(n) $"}
{"_id": "3719992", "title": "", "text": "$|f(x)|\\leq M/|x|^{2+\\epsilon}$"}
{"_id": "6068381", "title": "", "text": "$ \\begin{vmatrix} 0 & x & y & z \\\\ x & 0 & z^2 xy & y^2 xz \\\\ y & z^2 xy & 0 & x^2 yz\\\\ z & y^2 xz & x^2 yz & 0 \\\\ \\end{vmatrix}$"}
{"_id": "154290", "title": "", "text": "$Z^+$"}
{"_id": "8742244", "title": "", "text": "$(x+y)^r < x^r + y^r$"}
{"_id": "7147169", "title": "", "text": "$(1+r)^{n-1} + (1+r)^{n-2} + \\dots + (1+r) + 1$"}
{"_id": "7503139", "title": "", "text": "$e^N|\\sum_{n=N+1}^{\\infty}x^n| = e^N|x|^{N+1}\\frac{1}{1-x}.$"}
{"_id": "7210621", "title": "", "text": "$\\beta'(s)=\\frac{\\langle\\gamma'(s),\\gamma(s)-\\gamma(0)\\rangle}{\\beta(s)}=\\frac{\\langle\\gamma'(s),\\gamma(s)-\\gamma(0)\\rangle}{||\\gamma(s)-\\gamma(0)||}$"}
{"_id": "7968713", "title": "", "text": "$\\displaystyle\\lim_{x\\to c}f(x)=\\displaystyle\\lim_{x\\to c}h(x)\\implies \\displaystyle\\lim_{x\\to c}f(x)=\\displaystyle\\lim_{x\\to c}g(x)=\\displaystyle\\lim_{x\\to c}h(x).$"}
{"_id": "8092547", "title": "", "text": "$\\zeta (3) = \\frac{1}{\\Gamma(3)}\\int_{0}^{\\infty} \\frac{x^{3-1}}{e^x-1} dx \\Rightarrow \\int_{0}^{\\infty} \\frac{x^{2}}{e^x-1} dx = 2\\zeta(3) $"}
{"_id": "1180082", "title": "", "text": "$k! \\sim (\\frac{k}{e})^k \\sqrt{2 \\pi k}$"}
{"_id": "8160550", "title": "", "text": "$f(a+_6b)=f(a)+_3f(b).$"}
{"_id": "6430398", "title": "", "text": "$\\therefore log_ab^n=cn=n*log_ab$"}
{"_id": "6741288", "title": "", "text": "$\\left|\\int_a^b f(x)g(x)d\\alpha\\right|\\leq \\int\\left( \\frac {1}{p}|f(x)|^p +\\frac{1}{q}|g(x)|^q\\right)d\\alpha$"}
{"_id": "2613895", "title": "", "text": "$\\forall \\epsilon >0, \\exists \\delta > 0 : |x-a|<\\delta \\implies |f(x) - f(a)|<\\epsilon$"}
{"_id": "9156838", "title": "", "text": "$\\left(\\frac{1}{3}\\right)^{1/3} < \\left(\\frac{1}{2}\\right)^{1/2}$"}
{"_id": "2201", "title": "", "text": "$|abc| = 1$"}
{"_id": "6737583", "title": "", "text": "$\\mathbb{E}(T_s\\,|\\, X_0=s)={1/\\phi_s},$"}
{"_id": "8259961", "title": "", "text": "$\\int\\frac{e^{-x}}{x}dx = \\displaystyle\\frac{1}{x}\\cdot-e^{-x}-\\int-e^{-x}\\cdot\\displaystyle\\frac{-1}{x^{2}} dx = -\\displaystyle\\frac{e^{-x}}{x}-\\int \\displaystyle\\frac{e^{-x}}{x^{2}} dx$"}
{"_id": "5772347", "title": "", "text": "$ z=\\frac{ab+c}{a+b+c},  $"}
{"_id": "5240349", "title": "", "text": "$\\lim_{x\\to a^+}f'(c)=\\lim_{c\\to a^+}f'(c)=f'(a)$"}
{"_id": "5517297", "title": "", "text": "$\\begin{align}s= \\int_0^2 \\sqrt{1+(\\frac{dy}{dx})^2}dt\\end{align}$"}
{"_id": "5550875", "title": "", "text": "$P = \\lim_{n\\rightarrow \\infty}\\sum_{r=0}^{n-1}\\frac{r}{n^2+r^2}$"}
{"_id": "4731769", "title": "", "text": "$\\mathsf{Cov}(X,Y)=0$"}
{"_id": "3264893", "title": "", "text": "$\\psi(x) = \\sum \\limits_{i=1}^n a_i \\chi_{A_i} \\left( x \\right)$"}
{"_id": "3398860", "title": "", "text": "$L = \\mathbb{Q}(a,b,c,d) = \\mathbb{Q}(a)$"}
{"_id": "1836365", "title": "", "text": "$D(M) = P(M)$"}
{"_id": "501734", "title": "", "text": "$\\int \\frac{1}{1+x^2} dx$"}
{"_id": "4675348", "title": "", "text": "$\\overline{\\Bbb R}^2$"}
{"_id": "4600106", "title": "", "text": "$\\displaystyle\\lim_{n\\to\\infty}n\\mu(A_n)=0$"}
{"_id": "1029560", "title": "", "text": "$(m,m,m)$"}
{"_id": "2747009", "title": "", "text": "$x^3,x^2,x^1,x^0$"}
{"_id": "7804885", "title": "", "text": "$I_n(\\pi)$"}
{"_id": "8825607", "title": "", "text": "$[X_M^c,X_M^c]\\subset X_M^c$"}
{"_id": "9211583", "title": "", "text": "$ A y = -(x-a)(x-b) $"}
{"_id": "1422084", "title": "", "text": "$n!,n!+2,n!+3,n!+4....n!+n=n!,2(n!/2+1),3(n!/3+1)...$"}
{"_id": "6739839", "title": "", "text": "$P(M) \\implies P(\\lambda x M)$"}
{"_id": "5136342", "title": "", "text": "$||A||_2=\\sqrt{r(A^TA)}\\leq\\sqrt{||A^TA||_\\infty}\\leq\\sqrt{||A^T||_\\infty||A||_\\infty}=\\sqrt{||A||_1||A||_\\infty}$"}
{"_id": "368236", "title": "", "text": "$X_1\\times...\\times X_n$"}
{"_id": "7423907", "title": "", "text": "$\\sum_{n=1}^\\infty n=1+2+3+4+\\dots=-\\frac 1{12}$"}
{"_id": "5631909", "title": "", "text": "$\\sqrt{(-1)^2}=((-1)^2)^{1/2}$"}
{"_id": "3165908", "title": "", "text": "$f_Y(y) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} e^{-ay} (1-e^{-y})^{b-1}.$"}
{"_id": "234393", "title": "", "text": "$P(i) = 1/6$"}
{"_id": "4490014", "title": "", "text": "$\\large f_X(x)=\\frac{2}{x^3}$"}
{"_id": "7204216", "title": "", "text": "$\\sqrt {1+\\frac {dr}{dx}^2}$"}
{"_id": "6429211", "title": "", "text": "$ x \\sim y \\iff x R_0 y \\wedge y R_0 x. $"}
{"_id": "534901", "title": "", "text": "$x! \\approx \\sqrt{2 \\pi x} \\left(\\frac{x}{e}\\right)^x,$"}
{"_id": "1273823", "title": "", "text": "$Cov(U, V) = 0$"}
{"_id": "7016101", "title": "", "text": "$\\omega ^ {\\omega ^ \\alpha} = \\alpha$"}
{"_id": "9007709", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\left(\\frac ne\\right)^n e^{\\frac{\\theta(n)}{12n}}$"}
{"_id": "1826411", "title": "", "text": "$\\left(\\begin{smallmatrix}a &-b\\\\b&a\\end{smallmatrix}\\right)$"}
{"_id": "6231450", "title": "", "text": "$|f(z)|\\le |f(z^{2^n})|$"}
{"_id": "7938674", "title": "", "text": "$\\forall n,m,p\\in\\Nt:n<m,\\wedge,m<p\\implies n<p$"}
{"_id": "7562618", "title": "", "text": "$\\sum_{\\gamma<\\alpha} a_\\gamma = \\sup_{\\beta<\\alpha} \\sum_{\\gamma<\\beta} a_\\gamma \\le \\sup_{\\beta<\\alpha} \\prod_{\\gamma<\\beta} a_\\gamma \\le \\prod_{\\gamma<\\alpha} a_\\gamma.$"}
{"_id": "6325726", "title": "", "text": "$f(a) < k < f(b)  \\text{ or }  f(a) > k > f(b).$"}
{"_id": "2496291", "title": "", "text": "$\\begin{align} \\sum_{k=1}^n\\cos\\left(\\frac{2\\pi k}{n}\\right)&=\\frac{1}{2\\sin\\left(\\frac{2\\pi}{n}\\right)}\\left(\\sin\\left(\\frac{2\\pi(n+1)}{n}\\right)+\\sin\\left(\\frac{2\\pi(n)}{n}\\right)\\right)\\\\\\\\ &-\\frac{1}{2\\sin\\left(\\frac{2\\pi}{n}\\right)}\\left(\\sin\\left(\\frac{2\\pi(2-1)}{n}\\right)+\\sin\\left(\\frac{2\\pi(1-1)}{n}\\right)\\right)\\\\\\\\ &=\\frac{1}{2\\sin\\left(\\frac{2\\pi}{n}\\right)}\\left(\\sin\\left(\\frac{2\\pi(n+1)}{n}\\right)-\\sin\\left(\\frac{2\\pi(2-1)}{n}\\right)\\right)\\\\\\\\ &=\\frac{1}{2\\sin\\left(\\frac{2\\pi}{n}\\right)}\\left(\\sin\\left(\\frac{2\\pi}{n}\\right)-\\sin\\left(\\frac{2\\pi}{n}\\right)\\right)\\\\\\\\ &=0 \\end{align}$"}
{"_id": "3757019", "title": "", "text": "$Cov(X,Z) = 0$"}
{"_id": "3342385", "title": "", "text": "$P(m) \\implies P(s(m))$"}
{"_id": "7445516", "title": "", "text": "$(\\mathbb{R}[X]/(X))/(2) = \\mathbb{R}[X]/(2,X)$"}
{"_id": "2852466", "title": "", "text": "$Det \\begin{pmatrix} A & B\\\\B& A\\end{pmatrix}$"}
{"_id": "3674881", "title": "", "text": "$\\sigma(x)=\\frac{e^x}{e^x+1}$"}
{"_id": "2244741", "title": "", "text": "$d(B,A)\\leqslant d(B,C)+d(C,A),$"}
{"_id": "326550", "title": "", "text": "$\\frac{478}{221}=2+\\frac{36}{221}=2+\\dfrac1{\\frac{221}{36}}=2+\\frac1{6+\\frac5{36}}=2+\\dfrac1{6+\\dfrac1{\\frac{36}{5}}}=2+\\dfrac1{6+\\dfrac1{7+\\frac15}}$"}
{"_id": "7181649", "title": "", "text": "$y=A^{-1}(b-Bx)$"}
{"_id": "8379633", "title": "", "text": "$\\mathfrak{p}\\oplus \\mathfrak{t}$"}
{"_id": "8207214", "title": "", "text": "$\\{a, a+d, a+2d, \\dots, a+kd\\}$"}
{"_id": "5599493", "title": "", "text": "$f(n)=\\left(\\frac{n}{2}-\\frac{1}{4}\\right)+\\left(\\frac{1}{4}\\right)(-1)^n$"}
{"_id": "7283250", "title": "", "text": "$P\\big((m-1)+1\\big)=P(m)$"}
{"_id": "296914", "title": "", "text": "$n!\\approx\\sqrt{2\\pi n}\\left(\\frac{n}e\\right)^n\\;,$"}
{"_id": "1353104", "title": "", "text": "$Q=(1-\\tan^2(x)) \\left(1-\\tan^2 \\left(\\frac{x}{2}\\right)\\right)\\cdots \\left(1-\\tan^2\\left(\\frac{x}{2^n}\\right)\\right)$"}
{"_id": "5345412", "title": "", "text": "$f(f(x)+x)=2f(x)+x$"}
{"_id": "539273", "title": "", "text": "$ (a,b) \\mapsto (a + b,a - b) $"}
{"_id": "3789508", "title": "", "text": "$\\lim_{n\\to\\infty}f(a_n)=0$"}
{"_id": "1320461", "title": "", "text": "$F(\\theta, \\gamma) = \\{a + b\\theta + c\\gamma + d\\theta\\gamma : a,b,c,d \\in F\\}$"}
{"_id": "1250102", "title": "", "text": "$f[x0] = x2 + zf$"}
{"_id": "6988678", "title": "", "text": "$\\Sigma ||e_n - f_n|| <1 $"}
{"_id": "4546957", "title": "", "text": "$\\displaystyle\\int_1^\\infty\\dfrac{\\sin^2(x)}{x}$"}
{"_id": "5188232", "title": "", "text": "$\\lim (f(x) \\cdot f(y)) = \\lim f(x) \\cdot \\lim f(y)$"}
{"_id": "2026727", "title": "", "text": "$y=\\frac{6-2x}{4-x}$"}
{"_id": "4765687", "title": "", "text": "$f(1)=1, f(a+b)=f(a)+f(b)$"}
{"_id": "6706536", "title": "", "text": "$ 2A=\\int_{0}^{\\pi }\\frac{(\\pi)sin(x)}{1+sin^2(x)}dx=\\pi\\int_{0}^{\\pi }\\frac{sin(x)}{1+sin^2(x)}dx$"}
{"_id": "8363254", "title": "", "text": "$\\begin{align*} 1+2+3+4+\\cdots= \\, -\\frac{1}{12} \\\\ -1-2-3-\\cdots= \\; \\: \\, \\, \\frac{1}{12} \\\\ -1-2-3-\\cdots= \\; \\: \\, \\, \\frac{1}{12} \\\\ 1+2+\\cdots =  -\\frac{1}{12} \\end{align*}$"}
{"_id": "8329170", "title": "", "text": "$\\ \\overbrace{8n\\!+\\!6,\\,6n\\!+\\!3,\\,2n\\!+\\!3,\\,{-}\\color{#c00}6}^{\\Large  a_{k-1} -\\, j\\ a_k\\ =\\ a_{k+1}}\\,$"}
{"_id": "2066833", "title": "", "text": "$f(x)=3^x+5^x-8^x$"}
{"_id": "5299990", "title": "", "text": "$\\alpha \\geq x_{2} \\geq \\beta \\geq x_{1} \\geq \\gamma$"}
{"_id": "466919", "title": "", "text": "$  k = \\left( \\begin{array}{c} k \\\\ 1 \\end{array} \\right),  $"}
{"_id": "5595029", "title": "", "text": "$\\{\\dfrac{x_1 + x_2 + ... + x_n}{n}\\}$"}
{"_id": "2101374", "title": "", "text": "$ [F(a,b):K] = [F(a,b): F(a)][F(a): K] = [F(a,b): F(a)] \\deg{(f)} $"}
{"_id": "2590795", "title": "", "text": "$\\int_0^\\infty F_Z(x) \\, dx$"}
{"_id": "2584092", "title": "", "text": "$z = rs,\\gcd(r,s) = 1$"}
{"_id": "1569660", "title": "", "text": "$\\frac{27}3=9$"}
{"_id": "1913619", "title": "", "text": "$f(x)=\\sum_{j=1}^K\\chi_{I_j}$"}
{"_id": "550948", "title": "", "text": "$\\{\\langle{x,y}\\rangle: 0\\leq x\\wedge 0\\leq y\\wedge x+y\\leq 1\\}$"}
{"_id": "6212853", "title": "", "text": "$ f(x|y) = \\frac{1}{2\\pi\\sqrt{1-y^2}}\\cdot\\mathbb{1}_{[-\\sqrt{1-y^2},\\sqrt{1-y^2}]}, $"}
{"_id": "4077580", "title": "", "text": "$\\sup_{\\|[x]_B\\|_{\\mathbb R^m}=1}\\|[Tx]_{B'}\\|_{\\mathbb R^n}=\\sup_{\\|[x]_B\\|_{\\mathbb R^m}=1}\\|[T]_{B'B}[x]_B\\|_{\\mathbb R^n}.$"}
{"_id": "6320461", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac{1}n\\sum_{i=1}^n\\frac{1}{i}=0,$"}
{"_id": "40117", "title": "", "text": "$\\gamma(a),\\gamma(b)$"}
{"_id": "7676046", "title": "", "text": "$\\begin{cases} a = 1 \\\\ b = 0 \\\\ c = -3 \\\\ d = -4 \\\\ \\end{cases}$"}
{"_id": "605144", "title": "", "text": "$\\neg A \\vee \\neg \\neg A$"}
{"_id": "1975017", "title": "", "text": "$N = \\mathbb R^{n+1}$"}
{"_id": "913947", "title": "", "text": "$ = \\frac{1}{2 \\pi (1 - 2s)} \\int_0^{2 \\pi} \\int_0^{\\infty} e^{-u} \\> du \\> d\\theta$"}
{"_id": "5112588", "title": "", "text": "$\\begin{pmatrix} 1 & -1 & 0 \\\\ 1 & 1 & -1\\\\ 1 & 2 & -1    \\end{pmatrix} $"}
{"_id": "5161503", "title": "", "text": "$(e^{-100\\lambda})^7 (1-e^{-100\\lambda})^3$"}
{"_id": "9048212", "title": "", "text": "$z-2=\\sqrt{2}e^{\\frac{i3\\pi}{4}}$"}
{"_id": "7677026", "title": "", "text": "$R(n) = n^2-n+2$"}
{"_id": "4621426", "title": "", "text": "$\\int_{0}^{\\frac{\\pi}{2}} \\frac{dx}{1 + \\sin(x)}dx = \\int_{0}^{\\frac{\\pi}{2}} \\frac{dx}{1 + \\cos(x)}dx$"}
{"_id": "6923510", "title": "", "text": "$H_n = A_1^C \\cap A_2^C \\cap ... \\cap A_{n-1}^C \\cap A_n$"}
{"_id": "6014588", "title": "", "text": "$f(z) - f(z,2) = f(z) - z$"}
{"_id": "8179512", "title": "", "text": "$cov(X,Y)=-1$"}
{"_id": "1257710", "title": "", "text": "$F[t]/P ((P))$"}
{"_id": "531163", "title": "", "text": "$det\\left((a+b-e)I -buu^\\top\\right)= \\left(a-(n-1)b-e\\right)(a+b-e)^{n-1}.$"}
{"_id": "9001719", "title": "", "text": "$\\Delta f = f'(a)\\Delta x$"}
{"_id": "3183855", "title": "", "text": "$\\overline{p^{-1}(V)}\\subseteq p^{-1}(\\overline{V})\\subseteq U$"}
{"_id": "505715", "title": "", "text": "$\\lim_{x \\to \\infty} f(x)=\\lim_{x \\to \\infty^+} f(x)=\\lim_{x \\to \\infty^-} f(x)$"}
{"_id": "7852970", "title": "", "text": "$K\\subset V\\subset U$"}
{"_id": "3609", "title": "", "text": "$=$"}
{"_id": "8784579", "title": "", "text": "$C_R=\\{|z|=R\\}\\;$"}
{"_id": "2559803", "title": "", "text": "$\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}=1+x^2$"}
{"_id": "6270318", "title": "", "text": "$a_n=\\frac1{2^{n-2}}$"}
{"_id": "7734345", "title": "", "text": "$\\sum_{k=0}^{n-1} (k-n)\\cos\\frac{2k\\pi}{n}=\\frac{n}{2}$"}
{"_id": "3015341", "title": "", "text": "$f_3(n - 9k)$"}
{"_id": "7985093", "title": "", "text": "$|t|^{-n}$"}
{"_id": "528124", "title": "", "text": "$Cov(Y,X)=0$"}
{"_id": "324576", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=1$"}
{"_id": "6736", "title": "", "text": "$\\pmatrix{a_1&a_2&a_3\\\\ b_1&b_2&b_3\\\\ c_1&b_2&c_3}$"}
{"_id": "8722472", "title": "", "text": "$3^k + 3^{k+1} > k^3 + 3^{k+1}$"}
{"_id": "1538821", "title": "", "text": "$f: \\mathbb{R}^{n+1} \\rightarrow \\mathbb{R}^n$"}
{"_id": "1303637", "title": "", "text": "$k^3 *1/3 + k^2(3/2) + k(7/6) + 1/3 + 1/2 + 1/6 = $"}
{"_id": "4163459", "title": "", "text": "$\\sum_{n=1}^\\infty \\sum_{k=n}^\\infty$"}
{"_id": "444820", "title": "", "text": "$\\mu(\\overline{\\mathbb{R}}) = \\overline{\\mathbb{R}}$"}
{"_id": "3024489", "title": "", "text": "$P(E)=\\frac{1}{6}$"}
{"_id": "2952070", "title": "", "text": "$\\mathbb R^n \\otimes \\mathbb R^n$"}
{"_id": "7694050", "title": "", "text": "$lim_{x\\to c^{-}} ~ f(x)=l$"}
{"_id": "2153336", "title": "", "text": "$s < 1/|x|$"}
{"_id": "7337504", "title": "", "text": "$\\frac{x}{x-4} = \\frac{x-4+4}{x-4} = 1 + \\frac{4}{x-4}$"}
{"_id": "3018722", "title": "", "text": "$\\quad \\underbrace{+ \\quad \\underbrace{+ \\quad a \\quad \\underbrace{ * \\quad b \\quad c}_{\\color{red}{b \\,\\cdot\\, c}}}_{\\color{red}{a \\,+\\,} b \\,\\cdot\\, c} \\quad d}_{(a\\,+\\,b \\,\\cdot\\, c) \\color{red}{\\,+\\, d}}$"}
{"_id": "4452822", "title": "", "text": "$A'=\\bigcup_{x\\in A} B(x,r(x))$"}
{"_id": "2317799", "title": "", "text": "$\\int_{0}^{\\infty} \\frac{\\sin(x)}{x}$"}
{"_id": "5200021", "title": "", "text": "$z = \\frac{2ab-c^2}{a^2+b^2}.$"}
{"_id": "1947901", "title": "", "text": "$\\rm(x\\!-\\!y,x\\!+\\!y)=1$"}
{"_id": "1117309", "title": "", "text": "$P(A) = 1/6$"}
{"_id": "6076488", "title": "", "text": "$1+2+3+4+\\cdots = +\\infty$"}
{"_id": "7280447", "title": "", "text": "$\\eta(s)=\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}}{1+e^x}\\,dx$"}
{"_id": "7327643", "title": "", "text": "$\\gamma=\\sqrt{ai}=\\sqrt{a}\\ \\text{cis} \\frac{\\pi}{4}=\\frac{\\sqrt{2a}}{2}+\\frac{\\sqrt{2a}}{2}i$"}
{"_id": "8666729", "title": "", "text": "$\\displaystyle\\sum\\limits_{n\\geq1}\\frac{a_n}{n^\\beta}=\\sum\\limits_{n\\geq1}\\frac{a_n}{n^\\alpha}\\times \\frac{1}{n^{\\beta-\\alpha}}$"}
{"_id": "2597876", "title": "", "text": "$\\begin{cases} a_0=2\\\\ a_1=1\\\\a_{n+2}=a_{n+1}-2a_n\\end{cases}$"}
{"_id": "6350761", "title": "", "text": "$E \\lvert X-Y\\rvert^r = 0 $"}
{"_id": "6146441", "title": "", "text": "$ \\begin{cases} xy=a\\\\ x+y=a\\\\ 0\\leq x\\leq 1\\\\ 0\\leq y\\leq 1\\\\ \\end{cases} $"}
{"_id": "3964681", "title": "", "text": "$\\mathscr B_{[0,1]}\\otimes \\mathscr B_{[0,1]}.$"}
{"_id": "707034", "title": "", "text": "$1/2^{n+2}$"}
{"_id": "3970974", "title": "", "text": "$\\sum\\limits_{i=0}^n r^i = \\frac{1-r^{n+1}}{1-r}. $"}
{"_id": "7557808", "title": "", "text": "$r = \\frac{(100a+10b+c)}{(a+b+c)}$"}
{"_id": "2162365", "title": "", "text": "$\\{(x,y)\\in \\mathbb{R^2}: 0\\le x\\le 1, 0\\le y \\le 1\\}$"}
{"_id": "1516285", "title": "", "text": "$\\sum\\limits_{n=1}^\\infty\\frac{(n!)^n}{n^{4n}}$"}
{"_id": "2176433", "title": "", "text": "$\\Bbb R^{n}× \\Bbb R^{n}$"}
{"_id": "1607688", "title": "", "text": "$\\sin x + \\sin y = 2 \\sin \\frac {x+y} 2 \\cos \\frac {x-y} 2$"}
{"_id": "6467777", "title": "", "text": "$p_1 \\cdot p_2 \\cdots p_k \\mid a$"}
{"_id": "5927972", "title": "", "text": "$d(x,a)\\leq d(x,y)+d(y,a)\\iff$"}
{"_id": "2795291", "title": "", "text": "$\\begin{align} \\left(\\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\\\ 5 & 2 & 6 & \\alpha \\\\ 2 & -1 & \\alpha & -6 \\end{array}\\right) &\\leadsto \\left(\\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\\\ 0 & 7 & -9 & \\alpha + 25 \\\\ 2 & -1 & \\alpha & -6 \\end{array}\\right) \\\\ &\\leadsto \\left(\\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\\\ 0 & 7 & -9 & \\alpha+25 \\\\ 0 & 1 & \\alpha-6 & 4 \\end{array}\\right) \\\\ &\\leadsto \\left(\\begin{array}{rrr|r} 1 & 0 & \\alpha + 3 & -1 \\\\ 0 & 7 & -9 & \\alpha+25 \\\\ 0 & 1 & \\alpha-6 & 4 \\end{array}\\right) \\\\ \\end{align}$"}
{"_id": "9296760", "title": "", "text": "$\\phi^n(z)=(\\phi(z))^n$"}
{"_id": "2542283", "title": "", "text": "$\\forall M>0, \\exists \\delta>0: |x-a|<\\delta \\implies |f(x)|>M$"}
{"_id": "3413416", "title": "", "text": "$A-B=(\\mathbb{R}-\\mathbb{Q})-[(\\mathbb{R}-\\mathbb{Q})-C]=C$"}
{"_id": "5329580", "title": "", "text": "$ \\mathcal{F}:= \\{(x,y) \\in \\mathbb{R^2}:x \\in A,0<y \\leq 1 \\}$"}
{"_id": "2322957", "title": "", "text": "$|a(t)| = r\\omega^2$"}
{"_id": "2536833", "title": "", "text": "$\\displaystyle \\sum_{k=0}^{n-1} \\cos\\left( \\dfrac{2\\pi}{n} \\cdot k\\right) = 0$"}
{"_id": "3021955", "title": "", "text": "$\\frac{e^{\\frac{\\gamma}{1+\\gamma}}}{1+\\gamma} = \\frac{e}{\\gamma} + o\\!\\left(\\frac{1}{\\gamma}\\right)$"}
{"_id": "9205993", "title": "", "text": "$k=\\frac{a+b}{1-a}$"}
{"_id": "7568498", "title": "", "text": "$\\{a, a + \\frac{b-a}{n}, a + 2\\frac{b-a}{n}, \\dots, b\\} =: \\{x_0, \\dots, x_{n}\\}$"}
{"_id": "1520408", "title": "", "text": "$\\inf \\sum_{j=1}^{\\infty}|R_j|=\\inf\\left(\\sum_{rational\\ rect.}|R_j|+\\sum_{irrational\\ rect.}|R_j^{*}|\\right) = \\inf\\left(\\sum_{rational\\ rect.}|R_j|-\\sum_{j=1^\\ }^\\infty\\frac{\\epsilon}{2^j}\\right)$"}
{"_id": "8758608", "title": "", "text": "${\\rm gcd}(a+bi,a-bi) = 1$"}
{"_id": "4887573", "title": "", "text": "$S=2n\\sum_{k=1}^m |y_k|^2\\cos{2k\\pi\\over n}\\ ,$"}
{"_id": "6885803", "title": "", "text": "$A_1 \\subset A_2 \\subset, \\ldots \\subset A_{n-1} \\subset A_n \\subset A_{n+1}$"}
{"_id": "6341085", "title": "", "text": "$\\{a\\}\\cup\\{\\{a\\}\\}$"}
{"_id": "8982100", "title": "", "text": "$\\begin{cases}n_p |s\\\\n_p  \\equiv 1 \\mod p\\end{cases}$"}
{"_id": "1636412", "title": "", "text": "$aRb \\land bRa$"}
{"_id": "5067203", "title": "", "text": "$(x\\in \\{\\varnothing\\}\\,\\Leftrightarrow\\, x=\\varnothing\\,\\not\\Rightarrow\\, x\\in\\varnothing)$"}
{"_id": "8907402", "title": "", "text": "$f_x(x;a) = \\frac{2x}{a^2} \\ \\ \\ \\ , \\ \\ \\ \\ 0<x<a$"}
{"_id": "3379443", "title": "", "text": "$E[P(X_t\\leq x|\\mathcal{F}_s);\\{X_k\\leq y\\}]=E[P(X_t\\leq x|X_s);\\{X_k\\leq y\\}]$"}
{"_id": "6940431", "title": "", "text": "${1\\over x}+{1\\over y}+{1\\over z}=-{-40\\over{40\\over3}}=3$"}
{"_id": "4330659", "title": "", "text": "$(a+b,b)=1$"}
{"_id": "4897637", "title": "", "text": "$\\frac{d}{dz}y=\\frac{1}{\\gamma z}\\frac{\\gamma-a_1}{\\gamma}x^{-\\frac{\\gamma-a_1}{2\\gamma}-1}e^{-\\frac{a_2}{\\gamma^2}}g(\\xi)+x^{-\\frac{\\gamma-a_1}{2\\gamma}}e^{-\\frac{a_2}{\\gamma^2}}g'(\\xi)\\sqrt{\\frac{b_1+\\gamma^2}{\\gamma^2}-\\frac{b_3}{\\gamma^2}}$"}
{"_id": "9200082", "title": "", "text": "$\\det \\gamma = \\det \\beta^T = \\det \\beta$"}
{"_id": "5791059", "title": "", "text": "$\\frac{(1-p)p}{1-(1-p)^3}.$"}
{"_id": "5165097", "title": "", "text": "$\\tan \\theta = \\frac x4$"}
{"_id": "9363263", "title": "", "text": "$j_f(c)=0 \\Leftrightarrow \\lim_{x \\to c-}f(x)=\\lim_{x \\to c+}f(x)=l$"}
{"_id": "1202910", "title": "", "text": "$n!+2,n!+3,...,n!$"}
{"_id": "1482345", "title": "", "text": "$\\int_0^{\\infty} \\frac{\\cos{(x)}}{1+x} \\,\\mathrm dx=\\int_0^{\\infty} \\frac{\\sin{(x)}}{(1+x)^2} \\,\\mathrm dx$"}
{"_id": "1394522", "title": "", "text": "$n! = \\sqrt{2 \\pi n} (n/e)^n$"}
{"_id": "4423059", "title": "", "text": "$ \\left[       \\begin{array}{cccc|c}         1&1&4&2\\\\         0&3&12&6 \\\\         -1&2&8&4 \\\\         1&1&4&2       \\end{array}     \\right]$"}
{"_id": "4417214", "title": "", "text": "$A \\subseteq B \\subseteq \\overline{A} \\subseteq X$"}
{"_id": "4520347", "title": "", "text": "$F(x) = \\int_a^x f(t)dt-(x-a)\\lambda$"}
{"_id": "1341043", "title": "", "text": "$\\|e_m - e_n\\| \\ge 1$"}
{"_id": "6610280", "title": "", "text": "$0+0^+=(0+0)^+=0^+=1$"}
{"_id": "5571236", "title": "", "text": "$\\mathbb{Z}^+ \\cup$"}
{"_id": "3628775", "title": "", "text": "$F_X(x|\\theta) = \\frac{x}{\\theta}\\ \\ \\ \\text{for $x\\in [0, \\theta$]}$"}
{"_id": "1395046", "title": "", "text": "$\\Bbb R^{n+1}-\\{O\\}$"}
{"_id": "1908126", "title": "", "text": "$\\det A=(a+(n-1)b)(a-b)^{n-1}$"}
{"_id": "2576091", "title": "", "text": "$\\zeta(s)=\\sum_{m=0}^\\infty \\sum_{k=1}^\\infty \\frac{1}{(k+m^s)\\prod_{n=0}^{k-2}(n+m^s)}$"}
{"_id": "8771124", "title": "", "text": "$|f'(z)| \\lt 2|f'(a)|$"}
{"_id": "1934955", "title": "", "text": "$f(x)=\\frac{a^x-1}{a^x +1}$"}
{"_id": "5292882", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=0$"}
{"_id": "808977", "title": "", "text": "$ \\lim_{n \\to \\infty}\\sin \\frac{n \\theta}{2}\\sin \\frac{(n+1) \\theta}{2}$"}
{"_id": "7886380", "title": "", "text": "$x\\cdot y = a^{log_ax} \\cdot a^{log_ay} = a^{log_ax\\;+\\;log_ay}$"}
{"_id": "2284456", "title": "", "text": "$9^x+6^{x} > 2^{(x+1)}$"}
{"_id": "6927730", "title": "", "text": "$\\int^{\\pi}_{0}f(x)dx=\\int^{\\pi}_{0}f(\\pi+0-x)dx$"}
{"_id": "2527949", "title": "", "text": "$P(\\bar E) = 1/2$"}
{"_id": "8849232", "title": "", "text": "$\\int_0^\\pi \\int_0^\\pi \\int_0^26\\rho^3sin(\\phi)cos(\\phi)drd\\phi d\\theta = 0 $"}
{"_id": "4226804", "title": "", "text": "$|AB|=mk$"}
{"_id": "7859088", "title": "", "text": "$ \\sum_{k=1}^\\infty\\log\\biggl(1+\\frac1{2^k}\\biggr)<\\sum_{k=1}^\\infty\\frac1{2^k}=1. $"}
{"_id": "2363826", "title": "", "text": "$x^3 - x \\in \\mathbb{Z}$"}
{"_id": "8657619", "title": "", "text": "$ \\mathrm{length}(\\gamma_{|[s_1, s_2]}) = \\int\\limits_{s_1}^{s_2} \\sqrt{\\langle \\gamma^\\prime (s), \\gamma^\\prime (s)\\rangle} = s_2 - s_1. $"}
{"_id": "5956342", "title": "", "text": "$\\frac{1-r^2}{1-2r\\cos t+r^2}$"}
{"_id": "2231352", "title": "", "text": "$\\gcd(a, a+ 2b) = \\gcd(a, (a + 2b) -a) = \\gcd(a,2b)$"}
{"_id": "7582413", "title": "", "text": "$ ||T_u \\times T_v|| = \\sqrt{ \\left( \\frac{\\partial g}{\\partial u} \\right)^2+\\left( \\frac{\\partial g}{\\partial v} \\right)^2+1} $"}
{"_id": "2438325", "title": "", "text": "$[t_{j}, t_{j+1}] \\subset J_{j}$"}
{"_id": "2141017", "title": "", "text": "$\\sum\\limits_{i=1}^{\\log_2 n} [(\\frac{1}{2^{i-1}})^2  \\times    \\frac{1}{\\frac{n}{2^{i-1}}-1}]$"}
{"_id": "5068123", "title": "", "text": "$ (-x)+x+y+x+y+(-y)=(-x)+x+x+y+y+(-y) $"}
{"_id": "1909713", "title": "", "text": "$f(0) = f(\\pi) = (0, 0)$"}
{"_id": "515128", "title": "", "text": "$\\sup _{\\gamma \\in A}|\\gamma (x)-\\gamma (y)|\\leq \\sup _{\\gamma \\in F}|\\gamma (x)-\\gamma (y)|.$"}
{"_id": "8939767", "title": "", "text": "$S_n=\\{n!+2,n!+3,\\cdots,n!+n\\}$"}
{"_id": "4021153", "title": "", "text": "$\\gamma(\\mathbf{v})\\gamma(\\mathbf{u})=\\gamma[\\gamma(\\mathbf{v})(\\mathbf{u})]$"}
{"_id": "9259126", "title": "", "text": "$x-m\\in M^{\\perp}$"}
{"_id": "1144792", "title": "", "text": "$V = W\\oplus Z$"}
{"_id": "3700403", "title": "", "text": "$\\text{Ls}_{n \\rightarrow \\infty} A_n = \\{x \\in X | \\lim_{n \\rightarrow \\infty}\\inf d(x,A_n) = 0\\};$"}
{"_id": "1770157", "title": "", "text": "$\\gamma_{A, B} \\gamma_{B, A} = \\text{id}$"}
{"_id": "5962799", "title": "", "text": "$(a+b, a-b)=4$"}
{"_id": "8214667", "title": "", "text": "$\\ \\gcd(a,0) = \\gcd(a) =$"}
{"_id": "996695", "title": "", "text": "$|e^x-(1+{x \\over n})^n| \\le \\epsilon $"}
{"_id": "6908324", "title": "", "text": "$a( \\alpha+ \\gamma )=2\\gamma \\Rightarrow a(-3\\gamma+ \\gamma )=2\\gamma \\Rightarrow  -2\\gamma  a =2\\gamma$"}
{"_id": "935607", "title": "", "text": "$\\left( \\begin {matrix} 1 & -1 \\\\ 1 & 4 \\end {matrix} \\right)$"}
{"_id": "5624225", "title": "", "text": "$\\displaystyle n\\log\\left(1+\\frac{1}{n}\\right) - \\log\\left(1+\\frac{1}{n+1}\\right) < \\log\\left(1+\\frac{1}{n+1}\\right)$"}
{"_id": "5294031", "title": "", "text": "$\\frac{p}{2\\pi}\\int_{-\\infty}^{+\\infty}\\frac{\\sin xt}{t\\cdot \\sin\\frac12pt}\\sin([\\frac xp]+\\frac12) pt \\, \\mathrm dt=\\cdots$"}
{"_id": "3919726", "title": "", "text": "$y'(x) + a(x)y(x) = a^{-A(x)}d(a^{A(x)}y(x)).$"}
{"_id": "5034485", "title": "", "text": "$\\psi (x)=\\int_a^x f(t) dt$"}
{"_id": "2170555", "title": "", "text": "$\\gamma_1\\geq\\gamma_2\\geq\\cdots\\geq\\gamma_{n-1}$"}
{"_id": "1119193", "title": "", "text": "$\\int_{-\\infty}^{\\infty} f(t)u(t)dt = \\int_{0}^{\\infty} f(t)dt$"}
{"_id": "4385488", "title": "", "text": "$ \\mathbf{P}\\left(\\left. X_n =  x \\right| X_t, t \\in {^\\circ T}\\right) = \\mathbf{P}\\left( X_n =  x \\right), $"}
{"_id": "7751768", "title": "", "text": "$u_x = v_y, v_x = -u_y,u_x=-v_y,v_x=u_y \\implies u_x=v_x=u_y=v_y=f_x=f_y=f'(z)=0$"}
{"_id": "5735933", "title": "", "text": "$\\frac{d}{dt}g\\circ\\gamma=(\\nabla g\\circ\\gamma)\\cdot\\gamma\\,'=0.$"}
{"_id": "739317", "title": "", "text": "$\\forall\\epsilon>0\\exists\\delta>0:|x-a|<\\delta\\implies|f(x)-f(a)|<\\epsilon$"}
{"_id": "9267760", "title": "", "text": "$\\forall m<n \\ P(m) \\implies P(n)$"}
{"_id": "2785289", "title": "", "text": "$1+2+(-3+4)+\\dots+(-1999+2000)=1+2+1+\\dots+1$"}
{"_id": "2155294", "title": "", "text": "$n!+2,n!+3,\\cdots ,n!+n$"}
{"_id": "3121646", "title": "", "text": "$\\frac{n+1}{2}\\left(n+1-\\frac{n+1}{2}\\right)=\\left(\\frac{n+1}{2}\\right)^2$"}
{"_id": "1951592", "title": "", "text": "$\\sum_{n=1}^\\infty \\infty = \\infty$"}
{"_id": "159151", "title": "", "text": "$|\\hat{f}(\\xi)|\\leq C/|\\xi|^{1+\\alpha}$"}
{"_id": "6051628", "title": "", "text": "$(z^2)'' = 2 = f''(f(z))f'(z) + f'(f(z))f''(z)$"}
{"_id": "1899453", "title": "", "text": "$C = \\{\\{a\\}, E\\}$"}
{"_id": "8745873", "title": "", "text": "$\\tan a+\\cot a=\\frac{\\sin a}{\\cos a}+\\frac{\\cos a}{\\sin a}$"}
{"_id": "7824713", "title": "", "text": "$\\zeta(s)=\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\frac{x^{s-1}}{e^x-1}dx.$"}
{"_id": "7655525", "title": "", "text": "$\\langle x^3-Q(x), x^2 \\rangle =0$"}
{"_id": "6292770", "title": "", "text": "$ \\implies s_n = \\frac{n(n+1)}{2}. $"}
{"_id": "8214627", "title": "", "text": "$\\begin{align}\\gamma\\leq \\psi(t)=\\Big\\|\\frac{x}{\\| x\\|_0}\\Big\\|\\implies \\gamma \\| x\\|_0\\leq \\| x\\|\\leq \\beta\\| x \\|_0, \\;\\;\\text{for some} \\;\\;\\gamma,\\beta>0.\\end{align}$"}
{"_id": "5828108", "title": "", "text": "$p\\cdot t$"}
{"_id": "6774774", "title": "", "text": "$S=\\displaystyle \\sum_{n=1}^{\\infty} \\frac{1}{n^2} =\\sum_{n=1}^{5}\\frac{1}{n^2}+\\sum_{n=6}^{\\infty} \\frac{1}{n^2} \\leq \\sum_{n=1}^{5}\\frac{1}{n^2}+\\sum_{n=6}^{\\infty} \\frac{1}{n(n-1)}$"}
{"_id": "4288123", "title": "", "text": "$m(x)=  x^2 + x + 2$"}
{"_id": "2040364", "title": "", "text": "$\\lim_{n \\to \\infty} t_n$"}
{"_id": "2069565", "title": "", "text": "$I(y) = 1/|y|^{n-1}$"}
{"_id": "6360697", "title": "", "text": "$f(f(y) + xf(x)) = y + (f(x))^2$"}
{"_id": "3660109", "title": "", "text": "$\\forall x_1, x_2,..., x_n\\in \\mathbb{N}$"}
{"_id": "6840665", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}i(n)=1$"}
{"_id": "3426156", "title": "", "text": "$|f(x)-f(a)|<\\delta|x-a|<\\delta\\epsilon$"}
{"_id": "3889996", "title": "", "text": "$\\gamma_+ \\cup\\gamma_R \\cup\\gamma_- \\cup\\gamma_{\\epsilon}$"}
{"_id": "4929926", "title": "", "text": "$\\Lambda f=\\int_a^xf(t)dt$"}
{"_id": "167091", "title": "", "text": "$A_{1} \\subseteq A_2 \\subseteq \\dots$"}
{"_id": "3144130", "title": "", "text": "$a,a-b,a-2b,\\cdots$"}
{"_id": "6620931", "title": "", "text": "$\\frac{1}{\\pi^s}\\Gamma(s)\\zeta(2s)$"}
{"_id": "3157829", "title": "", "text": "$[t_1,t_2] \\subset T'$"}
{"_id": "2110704", "title": "", "text": "$M = \\begin{bmatrix} 0&0&0&1\\\\[0.3em] 0&0&0.5&0.5\\\\ 0 & (0.5)^2 & 2(0.5)^2 & (0.5)^2 \\\\[0.3em] (0.5)^3 &3(0.5)^3 &3(0.5)^3 &(0.5)^3\\\\      \\end{bmatrix}$"}
{"_id": "7985091", "title": "", "text": "$x(t)=|t|^{n}$"}
{"_id": "8773282", "title": "", "text": "$\\langle M,{\\in^M}\\rangle$"}
{"_id": "3287909", "title": "", "text": "$b=a^{log_a b} $"}
{"_id": "7177065", "title": "", "text": "$\\lim_{n\\to\\infty}( \\inf A_n) = \\lim_{n\\to\\infty} n = \\lim_{n\\to\\infty} x_n = \\liminf_{n\\to\\infty} x_n = +\\infty$"}
{"_id": "1629857", "title": "", "text": "$\\gamma \\wedge \\gamma = \\gamma \\wedge \\gamma$"}
{"_id": "4341323", "title": "", "text": "$\\frac{1}{n2^{n-1}}$"}
{"_id": "2976386", "title": "", "text": "$ \\leq x(x-1)^{n-1}$"}
{"_id": "8939774", "title": "", "text": "$n\\#+2,n\\#+3,\\ldots, n\\#+n$"}
{"_id": "7210617", "title": "", "text": "$\\frac{d}{ds}f(g(s)) =f'(g(s))g'(s)=\\frac{\\gamma(s)-\\gamma(0)}{\\Vert \\gamma(s)-\\gamma(0)\\Vert}\\gamma'(s) $"}
{"_id": "6346389", "title": "", "text": "$[t_{N},t_{0}]$"}
{"_id": "9121007", "title": "", "text": "$x_1 + x_4 + x_5 = 40,$"}
{"_id": "3671980", "title": "", "text": "$f(n) = {n \\choose4} + {n \\choose 2} + 1$"}
{"_id": "3493694", "title": "", "text": "$as + mt =1$"}
{"_id": "6858809", "title": "", "text": "$E[x_0] = \\mathbf{0}, x_0 \\perp v_t, x_0 \\perp w_t$"}
{"_id": "525774", "title": "", "text": "$\\int_{-\\pi}^{\\pi}f(x)\\sin(2nx)dx= \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}g(t)\\sin(2n(t+\\frac{\\pi}{2}))dt=0$"}
{"_id": "4267868", "title": "", "text": "$A(x) = \\int_a^x f(t) dt$"}
{"_id": "4497379", "title": "", "text": "$dx+by=d$"}
{"_id": "7177155", "title": "", "text": "$ mid = {P_1 + P_2 \\over 2} $"}
{"_id": "6108593", "title": "", "text": "$\\begin{equation}   \\begin{cases}     A = 1\\\\     B = -2\\\\ C = 1   \\end{cases} \\end{equation}$"}
{"_id": "6925579", "title": "", "text": "$ \\Bbb P[X_t=B_t]=1,\\qquad\\forall t\\ge 0. $"}
{"_id": "7073148", "title": "", "text": "$\\int ds=\\int \\sqrt{dx^2+dy^2}.$"}
{"_id": "3432243", "title": "", "text": "$E(A) = 1/6$"}
{"_id": "3976617", "title": "", "text": "$\\lim_{n\\to\\infty}\\mu\\left(B_{n}\\right)=\\mu\\left(\\liminf A_{n}\\right)$"}
{"_id": "138353", "title": "", "text": "$f(x)=\\sum_{i=1}^n a_i \\chi_{A_i}(x)$"}
{"_id": "2162204", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\frac{\\sqrt[n]{3n-1}}{\\sqrt2} = \\frac1{\\sqrt2}.$"}
{"_id": "3733343", "title": "", "text": "$(x, 1)^n = (A, -B)$"}
{"_id": "2848291", "title": "", "text": "$ 1 = \\lim_{N \\to \\infty} \\sum_{n=1}^N \\frac 1 {2^n}.$"}
{"_id": "6892718", "title": "", "text": "$\\lim_{n \\to \\infty}\\dfrac {x_{1}+x_{2}+\\ldots +x_{n}} {n}=x$"}
{"_id": "1756477", "title": "", "text": "$\\sum_{n\\leq x}\\frac{\\phi(n)}{n^{2}}=\\sum_{n\\leq x}\\frac{1}{n}\\sum_{d|n}\\frac{\\mu(d)}{d}.$"}
{"_id": "1144514", "title": "", "text": "$\\sum\\limits_{r=1}^{n+1} f(r)= \\sum\\limits_{r=1}^{n}f(r) + f(n+1)=$"}
{"_id": "2264414", "title": "", "text": "$\\sum_{n=1}^{\\infty}{1/n}=\\infty$"}
{"_id": "562011", "title": "", "text": "$\\gamma=\\frac{a+b}{a-b}$"}
{"_id": "3919725", "title": "", "text": "$a^{-A(x)}d(a^{A(x)}y(x)) = b(x)$"}
{"_id": "4898510", "title": "", "text": "$(1 + \\frac{\\gamma}{a})^n \\cdot (\\gamma +a) \\ge n * \\gamma + \\gamma + a$"}
{"_id": "7027484", "title": "", "text": "$d(a,H)=\\frac{|f(a)|}{||f||}$"}
{"_id": "5765", "title": "", "text": "$\\operatorname{dist}(x,M)=\\frac {|f(x)|}{\\| f \\|}$"}
{"_id": "4899724", "title": "", "text": "$\\gamma J\\subset \\gamma A\\subset R$"}
{"_id": "2298039", "title": "", "text": "$\\begin{cases} x^{,} = y - 2 x \\\\ y^{,} = 2 z - x - 2 y \\end{cases}$"}
{"_id": "8149850", "title": "", "text": "$P(X_1 = 2 | X_0 = 1) = 1/2$"}
{"_id": "7828715", "title": "", "text": "$a^{(\\log_ab)^2}=a^{\\log_a b\\cdot \\log_a b}=(a^{\\log_a b})^{\\log_a b}=b^{\\log_a b}=b^{\\frac{1}{\\log_b a}}.$"}
{"_id": "8879309", "title": "", "text": "$K \\subset \\bigcup_{\\gamma \\in \\Gamma} U_\\gamma$"}
{"_id": "4684373", "title": "", "text": "$ \\forall \\epsilon > 0, \\exists \\delta > 0: |x - y| < \\delta \\implies |f(x) - f(y)| < \\epsilon$"}
{"_id": "654427", "title": "", "text": "$F(a + b) = F(a) + F(b).$"}
{"_id": "2027775", "title": "", "text": "$ \\lim_{x\\to a}f(x)g(x)=\\lim_{x\\to a}f(x) $"}
{"_id": "2302962", "title": "", "text": "$\\#_1 A$"}
{"_id": "8475609", "title": "", "text": "$cov(X,Y) = 1$"}
{"_id": "1559737", "title": "", "text": "$\\min \\{ -(b_1 \\gamma_1+b_2 \\gamma_2) | \\gamma_1 \\ge 0, \\gamma_2 \\ge 0, v_1\\gamma_1+v_1 \\gamma_2 \\le -a, -v_2\\gamma_1+-v_2 \\gamma_2 \\le -1 \\}$"}
{"_id": "728059", "title": "", "text": "$\\int_{-\\pi}{\\pi}$"}
{"_id": "6787756", "title": "", "text": "$\\sum_{n=1}^{N}z^n=e^{\\phi i}+e^{2\\phi i}+...+e^{N\\phi i}$"}
{"_id": "785280", "title": "", "text": "$d(x, z) ≤ d(x, y) + d(y, z).$"}
{"_id": "4091320", "title": "", "text": "$\\|A\\|_2=\\sqrt{\\rho(A^HA)}$"}
{"_id": "8233016", "title": "", "text": "$[\\alpha,\\alpha+\\omega]$"}
{"_id": "1953690", "title": "", "text": "$d(x,y)=|f(x)−f(y)|$"}
{"_id": "1882581", "title": "", "text": "$\\alpha^{-}$"}
{"_id": "6488427", "title": "", "text": "$f(0)=1=\\lim_{x\\to 0^+}f(x)=\\lim_{x\\to 0^-} f(x).........................(1) $"}
{"_id": "5445507", "title": "", "text": "$ \\zeta(s) = \\frac{1}{\\Gamma(s)}\\int\\limits_0^{\\infty} \\frac{x^{s-1}}{e^x - 1} \\text{dx} $"}
{"_id": "2741213", "title": "", "text": "$\\lim_{n\\rightarrow \\infty} \\frac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n}}{n^{3/2}}$"}
{"_id": "3666982", "title": "", "text": "$\\lim_{n\\to\\infty}\\sum_{m=1}^{\\infty}\\frac{1}{m^2 + n^2}$"}
{"_id": "6747312", "title": "", "text": "$(1+i)^{n+1}-\\left (1+\\frac{A}{P} \\right ) (1+i)^{n} + \\frac{A}{P}=0$"}
{"_id": "7095191", "title": "", "text": "$\\begin{array}{c|c|c}\\#1&\\#2&\\#3&\\mbox{Result at }\\#1&\\mbox{result otherwise}\\\\\\hline c&g&g&win&lose\\\\\\hline g&c&g&lose&win\\\\\\hline g&g&c&lose&win\\end{array}$"}
{"_id": "7890132", "title": "", "text": "$|x-a|n(|a|+1)^{n-1}  < \\delta$"}
{"_id": "6470026", "title": "", "text": "$\\frac{1}{48} \\cdot (e^{it}+e^{-it})^3$"}
{"_id": "3034997", "title": "", "text": "$\\mathop {\\lim }\\limits_{n \\to \\infty } \\left( \\frac{\\sqrt[2]{2}+\\sqrt[4]{4}+...+\\sqrt[2n]{2n}}{1+\\sqrt[3]{3}+...\\sqrt[2n-1]{2n-1}} \\right) ^n$"}
{"_id": "538444", "title": "", "text": "$ 1+2+3+\\cdots = -\\frac{1}{12}\\qquad\\mathbf{FALSE} $"}
{"_id": "5976189", "title": "", "text": "$\\frac a x + \\frac by=\\frac 1 z$"}
{"_id": "3360171", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\mathbb{P}(A_{n+1})$"}
{"_id": "7712890", "title": "", "text": "$f(xf(y)-f(x))=2f(x)+xy$"}
{"_id": "3969051", "title": "", "text": "$P\\left({m \\over 2}-1\\right) \\implies P(m - 2) \\implies P(m+2)$"}
{"_id": "725787", "title": "", "text": "$\\frac {6p^3}{(1-p)(1-2p)}$"}
{"_id": "5247595", "title": "", "text": "$ \\int_a^b f(t) \\;dt =\\int_a^b F'(t) \\;dt =F(b)-F(a), $"}
{"_id": "6371298", "title": "", "text": "$E = \\{e_1,..,e_n,... \\}$"}
{"_id": "4407030", "title": "", "text": "$\\frac3x - \\frac4y = 1$"}
{"_id": "1081764", "title": "", "text": "$\\| A \\|=\\sqrt{\\lambda_{\\max}(A^{H}A)}$"}
{"_id": "8978567", "title": "", "text": "$ x= \\begin{bmatrix}   T^{n+1}_1      \\\\   T^{n+1}_2     \\\\   T^{n+1}_3  \\\\   T^{n+1}_4   \\end{bmatrix} $"}
{"_id": "5303295", "title": "", "text": "$X=\\frac{a+b}{b+c}$"}
{"_id": "5344097", "title": "", "text": "$A(x)=\\int_{a}^{x} f(t)dt,$"}
{"_id": "276948", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{k=1}^n(\\frac{1}{k}-\\frac{1}{2^k})$"}
{"_id": "3226124", "title": "", "text": "$A=\\left[\\matrix{0&1&1&1\\cr 1&0&0&0\\cr 1&0&0&-1\\cr 1&0&-1&0\\cr}\\right]\\ .$"}
{"_id": "9358040", "title": "", "text": "$f\\left( a+b\\right) =f\\left( a\\right) \\oplus f\\left(b\\right)$"}
{"_id": "5849064", "title": "", "text": "$\\lim_\\limits{x\\to a^+} f'(x) \\ne  \\lim_\\limits{x\\to a^-} f'(x)$"}
{"_id": "4975198", "title": "", "text": "$z = 2\\ e^{\\frac{i\\pi}{4}}$"}
{"_id": "4557135", "title": "", "text": "$Y=\\{\\{a\\}\\}$"}
{"_id": "6944662", "title": "", "text": "$\\lim_{x\\to c} f(x) = L \\Longleftrightarrow \\lim_{x\\to c^-} f(x) = L = \\lim_{x\\to c^+} f(x).$"}
{"_id": "536379", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=2$"}
{"_id": "7328906", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\sum_{k=1}^{n^2} \\frac{n^{r-1}}{n^r+k^r}$"}
{"_id": "8677814", "title": "", "text": "$f(\\theta_{t}(p))=f(p)+t$"}
{"_id": "6944373", "title": "", "text": "$\\frac{1}{2^{(n-1)/2}}$"}
{"_id": "1541248", "title": "", "text": "$(n+1)!+2,(n+1)!+3,\\cdots ,(n+1)!+n+1$"}
{"_id": "7523525", "title": "", "text": "$\\limsup P[X_n\\leq x]\\leq P[\\limsup X\\leq x]$"}
{"_id": "7601360", "title": "", "text": "$ v_{n+1} = v_1 (1+r)^n + T \\sum_{k=1}^n (1+r)^k = T \\frac{1+r}{r}\\left((1+r)^{n+1}-1\\right) ~~. $"}
{"_id": "2315617", "title": "", "text": "$\\|A\\|_2 = \\sqrt{m} \\|A\\|_{\\infty}$"}
{"_id": "5743005", "title": "", "text": "$\\det(\\gamma,\\gamma',\\gamma'')=\\det(\\gamma,T,\\kappa N)=\\kappa\\det(\\gamma,T,N)$"}
{"_id": "4631329", "title": "", "text": "$\\,\\Bbb Z[x]/(x^2+1)\\cong \\Bbb Z[i],\\,$"}
{"_id": "7561723", "title": "", "text": "$d(x,f(x)) \\leq d(x,y) + d(y,f(y)) + d(f(y),f(x))$"}
{"_id": "7453144", "title": "", "text": "$\\int_0^{2\\pi} f(t)g(t)\\,dt=0$"}
{"_id": "5577627", "title": "", "text": "$D_n = \\frac{n*(n+1)}{2}$"}
{"_id": "1238885", "title": "", "text": "$c = \\frac{P_0  (1+r)^{N-1}r}{(1+r)^N-1} $"}
{"_id": "2050350", "title": "", "text": "$\\chi(n)=\\left(\\frac{n}{p}\\right)$"}
{"_id": "181409", "title": "", "text": "$\\begin{equation} f_{X_n}(x)=\\frac{2x}{R^2}, \\ \\ \\ H\\leq x\\leq \\sqrt{H^2+R^2}, \\end{equation}$"}
{"_id": "7638049", "title": "", "text": "$\\sum\\limits_{k=1}^n \\frac{1}{3^k}$"}
{"_id": "1089726", "title": "", "text": "$n!= \\sqrt{2 \\pi n} (\\frac{n}{e})^n (1+ O(1/n)) $"}
{"_id": "8059285", "title": "", "text": "$ \\sqrt{21} = 5 - \\frac{4}{10-\\frac{4}{10-\\frac{4}{10-\\frac{4}{10-\\frac{4}{10-\\ddots}}}}} $"}
{"_id": "4253364", "title": "", "text": "$\\lim_{x\\to c^-} f(x) \\neq \\lim_{x\\to c^+} f(x)$"}
{"_id": "2775535", "title": "", "text": "$|f(z)| \\leq C|g(z)|$"}
{"_id": "4833306", "title": "", "text": "$ds = \\sqrt {\\frac {dx}{dt}^2+ \\frac {dy}{dt}^2} dt$"}
{"_id": "7209591", "title": "", "text": "$ (\\frac{1}{2^{k/2}})_{j \\in \\mathbb{N}} $"}
{"_id": "1370755", "title": "", "text": "$\\lfloor \\frac{\\lfloor\\frac{a}{b}\\rfloor}{c} \\rfloor = q_2 = \\lfloor \\frac{a}{b \\times c} \\rfloor$"}
{"_id": "8618995", "title": "", "text": "$ds = dx\\,\\sqrt{1 + (\\frac{dy}{dx})^2}$"}
{"_id": "4191688", "title": "", "text": "$d(x, z) \\leqslant d(x, y) + d(y, z),\\ \\forall x, y, z \\in X.$"}
{"_id": "6184375", "title": "", "text": "$0, \\frac{1}{\\pm N}, \\frac{2}{\\pm N}\\cdots\\frac{N - 2}{\\pm N}, \\frac{N - 1}{\\pm N}, \\pm1, \\frac{\\pm N}{N-1}, \\frac{\\pm N}{N-2}\\cdots\\frac{\\pm N}{2}, \\frac{\\pm N}{1}, \\infty$"}
{"_id": "458458", "title": "", "text": "$f(x)=f(x)f(1)-f(x+1)+1=2f(x)-f(x+1)+1,$"}
{"_id": "7782411", "title": "", "text": "$ {\\left(1 + \\frac{x}{n}\\right)}^n \\leq e^x\\mbox{.} \\tag{2} $"}
{"_id": "5326135", "title": "", "text": "$n+0, n+1,\\dots ,n+n$"}
{"_id": "117719", "title": "", "text": "$1+2+3+\\cdots = -\\frac{1}{12}$"}
{"_id": "7373721", "title": "", "text": "$\\vartheta(\\gamma \\tau) = (c\\tau+d)^{12}\\vartheta(\\tau)$"}
{"_id": "10395", "title": "", "text": "$: $"}
{"_id": "2612911", "title": "", "text": "$\\det A = \\prod \\lambda_k = \\left[\\lambda+(n-1)\\mu\\right]\\left[\\lambda-\\mu\\right]^{n-1}$"}
{"_id": "8282494", "title": "", "text": "$[t_{n0},t_{n1})$"}
{"_id": "469553", "title": "", "text": "$A_1\\times\\cdots\\times A_n$"}
{"_id": "3747585", "title": "", "text": "$\\gcd(A)=A$"}
{"_id": "6988937", "title": "", "text": "$\\zeta (s) = (2^s)(\\pi^s-1) \\sin(\\frac {\\pi s} 2) \\Gamma(1-s) \\zeta(1-s) .$"}
{"_id": "8313240", "title": "", "text": "$C_R'=C_R$"}
{"_id": "1464887", "title": "", "text": "$n\\mid p_1p_2$"}
{"_id": "6755774", "title": "", "text": "$P(X_n \\leq X, |X_n - X|< \\epsilon) \\leq P(X \\leq X + \\epsilon)$"}
{"_id": "5351160", "title": "", "text": "$|f(x)|\\leq |f^2(x)|$"}
{"_id": "3562535", "title": "", "text": "$\\begin{cases} b_1+b_2+\\dots+b_{k+1} = n\\\\ 1\\leq b_1\\\\ 1\\leq b_2\\\\ \\vdots\\\\ 1\\leq b_k\\\\0\\leq b_{k+1}\\end{cases}$"}
{"_id": "6084019", "title": "", "text": "$\\frac{(1+r)^n - 1}{r}.$"}
{"_id": "3010603", "title": "", "text": "$2^{x}+4^{x}+9^{x}=3^{x}+2 \\cdot 6^{x}$"}
{"_id": "5342074", "title": "", "text": "$\\left(\\begin{vmatrix}x_2&x_3\\\\y_2&y_3\\end{vmatrix},\\begin{vmatrix}x_1&x_3\\\\y_1&y_3\\end{vmatrix},\\begin{vmatrix}x_1&x_2\\\\y_1&y_2\\end{vmatrix}\\right)$"}
{"_id": "8617928", "title": "", "text": "$ X(\\pi/2)T(t) = \\frac{\\pi}{2}t \\implies  X'(0) = 2\\frac{X(\\frac{\\pi}{2})}{\\pi} $"}
{"_id": "3089909", "title": "", "text": "$\\tag{1}\\sum_{n=1}^{N} \\frac{1}{n} \\leq \\sum_{n=1}^{N} \\frac{1}{n^2}$"}
{"_id": "3303659", "title": "", "text": "$x + \\frac{1}{x + \\frac{1}{x + \\frac{1}{x + \\frac{1}{x}}}} = x + \\frac{1}{x + \\frac{1}{x + \\frac{1}{\\frac{x^2 + 1}{x}}}} = x + \\frac{1}{x + \\frac{1}{x + \\frac{x}{x^2 + 1}}} = x + \\frac{1}{x + \\frac{1}{\\frac{x^3 + 2x}{x^2 + 1}}}$"}
{"_id": "7444410", "title": "", "text": "$   \\theta_g (f)   := \\frac{|\\langle f', g \\rangle|}{\\|g\\|_{L^2}}    = \\frac{|\\langle f, g' \\rangle|}{\\|g\\|_{L^2}} \\, . $"}
{"_id": "8562041", "title": "", "text": "$\\forall \\epsilon \\gt 0 \\ ,\\exists \\delta \\gt 0 \\  ( |x-a|\\lt \\delta \\Rightarrow |f(x)-L| \\lt \\epsilon)$"}
{"_id": "6793245", "title": "", "text": "$ds=\\left(1+\\left(\\frac{dy}{dx}\\right)^2\\right)^{1/2}\\,dx.$"}
{"_id": "2692530", "title": "", "text": "$|f(x_1)| \\leq \\frac{1}{2} |f(x_0)|.$"}
{"_id": "3187065", "title": "", "text": "$\\sum_{n=1}^\\infty\\sum_{k=1}^\\infty\\frac{\\mu(n)\\mu(k)}{n^k k^n}$"}
{"_id": "4182118", "title": "", "text": "$\\frac{1}{2^n-1}+2=\\frac{1}{2^n-1}+\\frac{2^{n+1}-2}{2^n-1}= \\frac{2^{n+1}-1}{2^n-1}$"}
{"_id": "345848", "title": "", "text": "$\\frac 1x+\\frac 1y=\\frac 1n$"}
{"_id": "8328133", "title": "", "text": "$\\alpha = 1/|x|$"}
{"_id": "3293298", "title": "", "text": "$\\phi = \\sum_{i=1}^Nc_i\\chi_{E_i}$"}
{"_id": "3821864", "title": "", "text": "$\\frac{\\pi p}{2(p^2-1)\\sqrt{p^2-1}}$"}
{"_id": "5121837", "title": "", "text": "$K \\subseteq V\\subseteq \\bar{V}\\subseteq U $"}
{"_id": "9124713", "title": "", "text": "$f^{'}(a)<k<f^{'}(b)$"}
{"_id": "1593982", "title": "", "text": "$ \\sum_{n=0}^{\\infty}\\frac{B_n(n+k)!}{k!n!}=\\frac{1}{k!}\\int_{0}^{\\infty}\\frac{x^{k+1}e^{-x}}{e^x-1}dx=(k+1)(\\zeta(k+2)-1) $"}
{"_id": "2935278", "title": "", "text": "$ \\mathcal{B}(X)=\\left\\{\\mu \\in \\Omega^{0,1}(X,K_X^{-1}) : |\\mu|<1\\right\\} $"}
{"_id": "2645470", "title": "", "text": "$\\lim_{x\\to c^-}f(x)=\\lim_{x\\to c^+}f(x)$"}
{"_id": "7679884", "title": "", "text": "$ \\zeta(3)=\\sum_{n\\geq 1}\\frac{1}{n^3}=1+\\sum_{n\\geq 1}\\frac{1}{(n+1)^3}\\color{red}{\\leq}  1+\\sum_{n\\geq 1}\\frac{1}{n(n+1)(n+2)}=1+\\frac{1}{4}=\\color{red}{\\frac{5}{4}}.\\tag{1}$"}
{"_id": "3024490", "title": "", "text": "$P(F)=\\frac{1}{6}$"}
{"_id": "9203277", "title": "", "text": "$\\Phi(x) = e^{-\\gamma x^2} \\left[1, \\sqrt{\\frac{2\\gamma}{1!}}x,\\sqrt{\\frac{(2\\gamma)^2}{2!}}x^2, \\sqrt{\\frac{(2\\gamma)^3}{3!}}x^3, \\ldots\\right]$"}
{"_id": "2448931", "title": "", "text": "$\\nabla_{\\dot{\\alpha}} \\dot{\\alpha}=\\phi''(s)\\cdot \\dot \\gamma(\\phi(s)) + \\big(\\phi'(s)\\big)^2 \\cdot \\nabla_{\\dot{\\gamma}} \\dot{\\gamma} (\\phi(s))= \\big(\\phi'(s)\\big)^2 \\cdot \\Big( \\big(\\nabla_{\\dot{\\gamma}} \\dot{\\gamma} - \\frac{g(\\nabla_{\\dot{\\gamma}}\\dot{\\gamma}, \\dot{\\gamma} )}{g(\\dot{\\gamma}, \\dot{\\gamma})}\\dot{\\gamma} \\big) \\circ \\phi \\Big) =0$"}
{"_id": "4333127", "title": "", "text": "$\\lfloor\\lfloor a/b \\rfloor /c \\rfloor = \\lfloor a/bc \\rfloor$"}
{"_id": "4856487", "title": "", "text": "$ \\frac 1 {2^m} \\frac {1 - e^{2\\pi i \\delta 2^m}} {1 - e^{2\\pi i \\delta }} \\geqslant  \\frac 1 {2^m} \\frac {4 \\delta 2^m} {2 \\pi \\delta} = \\frac 2 {\\pi}. $"}
{"_id": "5994211", "title": "", "text": "$|f(z)^2 |= |f(z) |^2=|f(z)|$"}
{"_id": "1107618", "title": "", "text": "$F_{1/2}(x)=\\int\\frac{dx}{\\left(1+x^{1/2}\\right)^{1/2}}=\\frac43\\left(x^{1/2}-2\\right) \\left(1+x^{1/2}\\right)^{1/2}$"}
{"_id": "8170580", "title": "", "text": "$\\gamma(0) = id \\quad \\text{and} \\quad \\gamma(1) = \\text{Legendre transform}$"}
{"_id": "7670115", "title": "", "text": "$\\lim_{n\\to \\infty} \\dfrac{\\sqrt[n]{e}+\\sqrt[n]{e^2}+\\sqrt[n]{e^3}+...+\\sqrt[n]{e^n}}{n}$"}
{"_id": "4080139", "title": "", "text": "$\\begin{cases}x=t,\\\\y=t,\\\\z=t\\end{cases}$"}
{"_id": "7335339", "title": "", "text": "$F(x) = \\int _a^x f'(t) \\; dt$"}
{"_id": "23526", "title": "", "text": "$T_X(x)$"}
{"_id": "1431520", "title": "", "text": "$d(y,A)\\leq d(x,y)+d(x,A)$"}
{"_id": "7127176", "title": "", "text": "$f(x;\\theta) = \\frac1\\theta e^{\\frac{-x^2}{(2\\theta)}}$"}
{"_id": "364971", "title": "", "text": "$\\gcd(a+b, a-b) = \\gcd(a+b, 2b)$"}
{"_id": "777780", "title": "", "text": "$\\frac{a+b}{a-b}=\\frac{c+d}{c-d}$"}
{"_id": "1225559", "title": "", "text": "$ \\begin{cases} t\\equiv r\\pmod{m}\\\\ t\\equiv s\\pmod{n} \\end{cases} $"}
{"_id": "3488076", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}=\\left(1+\\sqrt{2}+\\sqrt[3]{3}+...+\\sqrt[n]{n}\\right) \\ln\\frac{2n+1}{n}$"}
{"_id": "9200081", "title": "", "text": "$(\\det \\alpha)(\\det \\gamma) = (\\det \\alpha)(\\det \\beta)$"}
{"_id": "6564268", "title": "", "text": "$x-c, c \\in \\mathbb{R}$"}
{"_id": "8392741", "title": "", "text": "$E=\\{\\{a,b\\}\\}$"}
{"_id": "8980369", "title": "", "text": "$f(x) = \\frac{e^x}{2^{x+1}}$"}
{"_id": "1283142", "title": "", "text": "$\\int_0^{\\infty}{f(s)ds}$"}
{"_id": "8743197", "title": "", "text": "$x^{-n}e^x > \\frac{x^{n+1}}{(n+1)!}x^{-n}$"}
{"_id": "8370535", "title": "", "text": "$P(k)=k^2+k+2$"}
{"_id": "1916940", "title": "", "text": "$\\lim_{N\\rightarrow \\infty}\\sum_{n=1}^N\\frac{1}{N+n}=\\log 2$"}
{"_id": "4536052", "title": "", "text": "$\\gamma_3(G)/\\gamma_5(G) \\le Z(\\gamma_2(G)/\\gamma_5(G)$"}
{"_id": "3568236", "title": "", "text": "$=\\left\\lfloor \\{a\\}\\lfloor b\\rfloor+\\{b\\}\\left(\\lfloor a\\rfloor+\\{a\\}\\right)\\right\\rfloor$"}
{"_id": "8411363", "title": "", "text": "$d(\\gamma,\\gamma') = \\max_{0 \\leq t \\leq 1} \\rho(\\gamma(t), \\gamma'(t))$"}
{"_id": "5632171", "title": "", "text": "$I=\\int_0^\\infty \\frac{1}{(x^2+p)^{n+1}}dx$"}
{"_id": "3483088", "title": "", "text": "$\\frac{dy}{dx}=\\sqrt{(1-x)^2+1}$"}
{"_id": "2326209", "title": "", "text": "$f_n(x) = \\frac{e^x}{1 + n + x^2}$"}
{"_id": "3260590", "title": "", "text": "$\\frac{1}{2^{n+2}}$"}
{"_id": "5199772", "title": "", "text": "$z=s^{\\frac 1 n}e^{\\frac{i(\\phi+2k\\pi)}n} $"}
{"_id": "4584306", "title": "", "text": "$\\begin{equation} \\int_0^{2\\pi} f(x) dx = \\int_0^{2\\pi} f(x) \\sin xdx= \\int_0^{2\\pi} f(x) \\cos xdx=...= \\int_0^{2\\pi} f(x) \\sin nx dx=\\int_0^{2\\pi} f(x) \\cos nx dx=0  \\end{equation}$"}
{"_id": "7672829", "title": "", "text": "$T^{0,1}_mM$"}
{"_id": "2381730", "title": "", "text": "$z_k = r^{1\\over n} e^{{i \\varphi + i 2 \\pi k \\over n}}$"}
{"_id": "1953294", "title": "", "text": "$\\overline{N^{n-1}_\\epsilon(y)}\\subseteq V\\subseteq \\overline{V}\\subseteq U$"}
{"_id": "5742363", "title": "", "text": "$\\int_{-\\infty}^{\\infty}\\sin^2(x)/x^2=\\pi$"}
{"_id": "8113987", "title": "", "text": "$\\displaystyle (1+x)^{n+1}-(1+x)^r = \\binom{n+1}{r+1}$"}
{"_id": "5908007", "title": "", "text": "$I = \\int{\\frac{1}{(x^2+1)(x+x^2\\arctan(x))}}dx$"}
{"_id": "4166573", "title": "", "text": "$\\tan(\\tan^{-1}(n+1)-\\tan^{-1}(n-1))=\\tan\\left(\\tan^{-1}\\left(\\frac{2}{n^2}\\right)\\right)$"}
{"_id": "6048390", "title": "", "text": "$|f(z^{2})|\\leq e^{|z|}$"}
{"_id": "2522313", "title": "", "text": "$x \\delta^{(n)}(x) = - n \\delta^{(n-1)}(x)$"}
{"_id": "5567083", "title": "", "text": "$\\frac{\\frac{\\sin \\theta}{\\theta}}{\\frac{4\\theta}{\\theta} +\\frac{\\sin \\theta}{\\theta\\cos\\theta}} = \\frac{\\frac{\\sin \\theta}{\\theta}}{4 +\\frac{\\sin \\theta}{\\theta\\cos\\theta}}$"}
{"_id": "3651748", "title": "", "text": "$ \\alpha + \\beta +\\gamma+\\delta=-\\frac{b}{a}$"}
{"_id": "6803343", "title": "", "text": "$1 \\le (x+y,x-y)$"}
{"_id": "7356881", "title": "", "text": "$1*1!+2*2!+...+n*n! + (n+1)(n+1)! = (n+2)! - 1 \\\\(n+1)! -1 + (n+1)(n+1)! = (n+2)! -1   \\\\(n+1)! + (n+1)(n+1)! = (n+2)! \\\\(n+1)! + (n+1)(n+1)! = (n+1)!(n+2) \\\\n+2=n+2$"}
{"_id": "2810504", "title": "", "text": "$\\Phi(n)=S(n)$"}
{"_id": "5646615", "title": "", "text": "$\\langle Df, Dg\\rangle_{C^\\prime}=\\int_0^1 (Df)^\\prime(t) (Dg)^\\prime(t) dt = \\int_0^1 f(t) g(t) dt = \\langle f,g\\rangle_{L^2[0,1]}$"}
{"_id": "8565899", "title": "", "text": "$2 \\pi R sin(\\varphi)$"}
{"_id": "9333234", "title": "", "text": "$\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}\\underline{}$"}
{"_id": "6290333", "title": "", "text": "$P(m)=P(k)$"}
{"_id": "1709095", "title": "", "text": "$\\le x \\le b, c \\le y\\le d$"}
{"_id": "1486819", "title": "", "text": "$4x-x^4 \\leq 3, x \\in \\Bbb R$"}
{"_id": "5988112", "title": "", "text": "$\\sum_i \\mu(A_i)\\geq \\mu(A_{i^*})=1=\\mu\\Big(\\bigcup_i{A_i}\\Big).$"}
{"_id": "7874357", "title": "", "text": "$\\mu/|x|$"}
{"_id": "5436537", "title": "", "text": "$f(\\theta^{(p)}(t)) = f(p)+t$"}
{"_id": "1069366", "title": "", "text": "$[x, y] = \\emptyset$"}
{"_id": "7761644", "title": "", "text": "$\\partial\\circ\\overline{\\partial}$"}
{"_id": "6197974", "title": "", "text": "$6 \\cdot 4^x - 13 \\cdot 6^x + 6\\cdot 9^x = 0 /:4^x$"}
{"_id": "3847770", "title": "", "text": "$d = gcd(a-b,a+b)$"}
{"_id": "9165487", "title": "", "text": "$\\lfloor\\frac{x-1}{y}\\rfloor=\\lfloor\\frac{x}{y}\\rfloor$"}
{"_id": "3675669", "title": "", "text": "$h=ax+by$"}
{"_id": "8974569", "title": "", "text": "$\\frac{1}{24}n(n-1)(n-2)(n-3)(n+1)!.$"}
{"_id": "1313763", "title": "", "text": "$1/|x|<ε$"}
{"_id": "6980335", "title": "", "text": "$G_2:=\\{x\\in H_2: y\\in H_1, x=2y\\}.$"}
{"_id": "6205563", "title": "", "text": "$d(x,y) = \\arccos(\\langle x, y \\rangle)$"}
{"_id": "4099319", "title": "", "text": "$f(x)=\\frac12+e^{x-\\frac12}-e^{\\frac12-x}$"}
{"_id": "3334604", "title": "", "text": "$1/|x|^s$"}
{"_id": "4071844", "title": "", "text": "$w^+$"}
{"_id": "1139418", "title": "", "text": "$ \\Vert XYZ \\Vert = \\sup_{\\Vert x \\Vert = 1} \\Vert XYZ x\\Vert = \\sup_{\\Vert x \\Vert = 1} \\Vert YZ x\\Vert = \\sup_{z = Zx, \\Vert z \\Vert = 1} \\Vert Y z\\Vert = \\sup_{\\Vert x \\Vert = 1} \\Vert Y x\\Vert = \\Vert Y \\Vert.$"}
{"_id": "2358623", "title": "", "text": "$d(x,\\ker(f))=\\frac{|f(x)|}{\\|f\\|}$"}
{"_id": "523409", "title": "", "text": "$\\xi\\in\\kappa_\\gamma\\subseteq \\lambda_\\gamma$"}
{"_id": "10660", "title": "", "text": "$\\int_0^1f(x)dx$"}
{"_id": "453636", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=15 \\tag{II}$"}
{"_id": "5882049", "title": "", "text": "$\\left(\\frac{1}{2}\\right)^{2n-k}$"}
{"_id": "2638441", "title": "", "text": "$\\displaystyle\\int_0^\\pi f(t) dt=0$"}
{"_id": "5172100", "title": "", "text": "$\\sqrt{A(ABCD)} =\\sqrt{A(ABE)}+ \\sqrt{A(CDE)}$"}
{"_id": "7834534", "title": "", "text": "$B(u,v)=\\begin{pmatrix}  u^1&u^2  &u^3  \\end{pmatrix}\\begin{pmatrix} {v^1+v^2+v^3}\\\\ {v^1+2v^2} \\\\{v^1+3v^3} \\end{pmatrix}$"}
{"_id": "8653264", "title": "", "text": "$0\\oplus T$"}
{"_id": "2130676", "title": "", "text": "$\\det\\begin{pmatrix}A & C' \\\\ 0 & B'\\\\ \\end{pmatrix} = \\det(A)\\det(B')$"}
{"_id": "7746703", "title": "", "text": "$\\lfloor a\\rfloor+\\lfloor b\\rfloor+\\lfloor c\\rfloor$"}
{"_id": "1873975", "title": "", "text": "$\\sum\\limits_{n=1}^\\infty \\left\\lfloor \\frac{N}{p^n}\\right\\rfloor$"}
{"_id": "4091688", "title": "", "text": "$A_1\\times A_2\\times \\ldots \\times A_{n-1}\\times A_{n}=X\\times A_{n}$"}
{"_id": "6694797", "title": "", "text": "$d(x,a)\\leqslant d(x,y)+d(y,a)$"}
{"_id": "3135049", "title": "", "text": "$F(x)=\\int_a^b f(t)\\,dt$"}
{"_id": "2265034", "title": "", "text": "$[256] = [220 + 36] = [220] + [36] = [0] + [36] = [36] = [33] + [3] = [3]$"}
{"_id": "732205", "title": "", "text": "$x/|x|$"}
{"_id": "1576911", "title": "", "text": "$\\mathbb R^2,\\emptyset\\in \\tau$"}
{"_id": "6231819", "title": "", "text": "$g^1\\gamma - \\gamma = \\partial(H\\gamma) + H(\\partial\\gamma).$"}
{"_id": "7897844", "title": "", "text": "$a \\lfloor \\frac{b}{c} \\rfloor \\leq \\lfloor \\frac{ab}{c} \\rfloor$"}
{"_id": "3113193", "title": "", "text": "$p(E|S) = p(E \\& S)/p(S) = p(E).p(T) / (1 - (1 -p(E)).p(T) + p(E).(1 - p(T))$"}
{"_id": "6138283", "title": "", "text": "$\\mathrm{Li}_{s}(z)=\\frac{z}{\\Gamma\\left(s\\right)}\\int_{0}^{\\infty} \\frac{x^{s-1}}{e^{x}-z}dx \\ge \\frac{z}{\\Gamma\\left(s\\right)}\\int_{0}^{\\infty} \\frac{x^{s-1}}{e^{x}}dx = z $"}
{"_id": "2368498", "title": "", "text": "$\\gamma' \\subset \\gamma$"}
{"_id": "79341", "title": "", "text": "$x/2 = \\sec \\theta$"}
{"_id": "3715446", "title": "", "text": "$d_L(h)=|\\langle h,e\\rangle|$"}
{"_id": "4197", "title": "", "text": "$<$"}
{"_id": "8747257", "title": "", "text": "$w(x) = x^2 + x + 2$"}
{"_id": "8036477", "title": "", "text": "$\\sum_{k=0}^{m-1} \\sin \\frac{2 \\pi k}{n}$"}
{"_id": "1557020", "title": "", "text": "$\\frac{d}{dx} \\sum_{n=0}^\\infty e^{-nx} = -\\sum_{n=0}^\\infty n e^{-nx}$"}
{"_id": "65345", "title": "", "text": "$x = (ax+b)/(cx+d),$"}
{"_id": "1243448", "title": "", "text": "$\\binom {n}{2}=n(n-1)/2 =f(n).$"}
{"_id": "6115418", "title": "", "text": "$\\tan\\left((n+1)\\frac{x}{2}\\right) = (n+1)\\tan\\left(\\frac{x}{2}\\right).$"}
{"_id": "2467305", "title": "", "text": "$\\frac{a^{-3}}{(1-a^{-2})}=\\frac{\\frac{1}{a^3}}{(1-\\frac{1}{a^2})}=\\frac{a^2}{a^3(a^2-1)}=\\frac{1}{a(a^2-1)}$"}
{"_id": "7687287", "title": "", "text": "$p'(t) = f(t, p(t))$"}
{"_id": "6007197", "title": "", "text": "$\\{a, \\gamma\\}$"}
{"_id": "6015918", "title": "", "text": "$p'(t)=f(t)\\ p(t)$"}
{"_id": "5971114", "title": "", "text": "$F = \\{e_0, e_1, e_2, ..., e_n\\}$"}
{"_id": "90771", "title": "", "text": "$\\cos\\;\\theta=\\frac{x}{r}$"}
{"_id": "9047748", "title": "", "text": "$f_n(e^{-2/n}) = \\frac{4e^{-2}}{n^2}$"}
{"_id": "4414370", "title": "", "text": "$\\left\\{\\begin{array}{l}x=1+3t\\\\ y=-2+t\\\\z=t  \\end{array} \\right., t\\in\\mathbb{R}.$"}
{"_id": "3964803", "title": "", "text": "$t^n-t^{n-1}$"}
{"_id": "6672017", "title": "", "text": "$g = \\sum_{j=1}^n c_j 1_{A_j}$"}
{"_id": "835904", "title": "", "text": "$|f(x)g(x)-f(y)g(y)| \\leq |g(x)||f(x)+f(y)|+|f(y)||g(x)+g(y)| $"}
{"_id": "3484721", "title": "", "text": "$ n≥3L \\tag{$*$} $"}
{"_id": "1506932", "title": "", "text": "$R = \\{(x,y) \\in \\mathbb{R}^2\\colon\\, 0 \\leq x \\leq a ,\\, -b \\leq y \\leq b \\}$"}
{"_id": "2334589", "title": "", "text": "$\\left\\lfloor\\frac{a+b}{n}\\right\\rfloor=\\left\\lfloor\\frac{a}{n}\\right\\rfloor + \\left\\lfloor\\frac{b}{n}\\right\\rfloor?$"}
{"_id": "3723159", "title": "", "text": "$\\sum_{\\gamma\\le T}\\log |\\gamma| > \\sum_{ f(T) < \\gamma \\le T}\\log |\\gamma| > [ N(T)-N(f(T)) ] \\log f(T).$"}
{"_id": "742635", "title": "", "text": "$(\\mathbb{Z}[x]/(x^2+1))_{(5)}=\\mathbb{Z}_{(5)}[x]/(x^2+1)=\\mathbb{Z}_{(5)}[i]$"}
{"_id": "2942451", "title": "", "text": "$ \\frac{23}{12}(1,2,-1,0) + \\frac{48}{12}(1,0,-1,1)  $"}
{"_id": "1461009", "title": "", "text": "$x! < \\sqrt{w\\pi{x}}(\\frac{x}{e})^x$"}
{"_id": "1421345", "title": "", "text": "$f(n)^{f(n)} = (n^2)^{n^2} = n^{2n^2}$"}
{"_id": "1853024", "title": "", "text": "$I_n=\\int \\frac 1 {(x^2-1) ^n} \\Bbb d x$"}
{"_id": "9208864", "title": "", "text": "$\\sum_{n=1}^H \\frac{1}{n^2}<\\sum_{n=1}^\\infty \\frac{1}{n^2}$"}
{"_id": "5974518", "title": "", "text": "$y_n = \\frac{{x_1}+{x_2}+...+{x_n}}{n}$"}
{"_id": "7611259", "title": "", "text": "$f(n-1)=n^2-n+1$"}
{"_id": "4485394", "title": "", "text": "$S=[a_1,a_2,…,a_{n-1},a_n]$"}
{"_id": "9018759", "title": "", "text": "$f(x) = f(a+b) = f(a) + f(b)$"}
{"_id": "8251845", "title": "", "text": "$\\sin ^{2}\\varphi +\\cos ^{2}\\varphi =1\\iff\\dfrac{\\sin ^{2}\\varphi }{\\cos ^{2}\\varphi }+1=\\dfrac{1}{\\cos ^{2}\\varphi }$"}
{"_id": "3113187", "title": "", "text": "$p(E|S) = p(E \\& S)/p(S)$"}
{"_id": "3274975", "title": "", "text": "$x^n\\left(1+\\dfrac xn\\right)^n$"}
{"_id": "5379876", "title": "", "text": "$z=2+\\lambda$"}
{"_id": "7041218", "title": "", "text": "$e^x\\ge\\left(1+x/2\\right)^2$"}
{"_id": "4585869", "title": "", "text": "$P(E)=P(E\\cap S)+P(E\\cap S^c)$"}
{"_id": "4816436", "title": "", "text": "$f_Y(y)=\\int_{y}^{1} 8xy \\, dx$"}
{"_id": "3807509", "title": "", "text": "$\\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ s.t. } |x - a| < \\delta \\implies |f(x) - f(a)| < \\epsilon.$"}
{"_id": "1824346", "title": "", "text": "$ [a+(n-1)d] r^{n-1} $"}
{"_id": "389998", "title": "", "text": "$\\alpha^+\\in C$"}
{"_id": "6377519", "title": "", "text": "$E(X) = \\int_{-2}^2 x f_x(x)\\, dx = 0$"}
{"_id": "3620522", "title": "", "text": "$\\rm\\ \\mathbb N_+\\: $"}
{"_id": "6082562", "title": "", "text": "$\\Large\\frac{9^k}{9^k+3}$"}
{"_id": "1276541", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac1{n^2} =\\sum_{n=1}^\\infty \\frac1{(2n)^2} + \\sum_{n=1}^\\infty \\frac1{(2n-1)^2}=\\frac14 \\sum_{n=1}^\\infty \\frac1{n^2} + \\sum_{n=1}^\\infty \\frac1{(2n-1)^2}$"}
{"_id": "548750", "title": "", "text": "$f: \\mathbb R \\rightarrow \\mathbb R, f(x+y)=f(x)+f(y), \\forall x,y \\in \\mathbb R$"}
{"_id": "3678545", "title": "", "text": "$\\sum_{k=1}^{n-1}\\lfloor \\frac{mk}{n} \\rfloor$"}
{"_id": "8215904", "title": "", "text": "$\\int_{0}^{\\pi}cos2nxdx=0=\\int_{0}^{\\pi}sin2nxdx,n\\in\\mathbb{N}$"}
{"_id": "1890161", "title": "", "text": "$ \\int{1\\over(x^2+1)^n}\\mathrm{d}x$"}
{"_id": "5153769", "title": "", "text": "$||z|| = \\frac{1}{||f||}$"}
{"_id": "6294584", "title": "", "text": "$d_n={1\\over 2^{n-2}}\\ .$"}
{"_id": "4297693", "title": "", "text": "$\\lim_{x \\to c} f(x)g(x) = \\lim_{x \\to c} f(x) \\lim_{x \\to c} g(x)$"}
{"_id": "8623530", "title": "", "text": "$S \\subseteq \\cup_{\\gamma \\in F}\\ U_\\gamma$"}
{"_id": "4959551", "title": "", "text": "$\\sum_{k=0}^{M-1}\\cos\\frac{2\\pi kj}{M}=0$"}
{"_id": "1777291", "title": "", "text": "$f(Z^2) = f(Z)^2$"}
{"_id": "6905308", "title": "", "text": "$Pxy \\land Pyx$"}
{"_id": "7563861", "title": "", "text": "$R_M(x):=\\frac{\\langle Mx,x\\rangle}{\\lVert x\\rVert^2}$"}
{"_id": "663911", "title": "", "text": "$p_{\\theta}(x)= \\frac{2x}{\\theta^{2}}1_{0 \\leq x \\leq \\theta}.$"}
{"_id": "4191139", "title": "", "text": "$r+s-rs$"}
{"_id": "8421557", "title": "", "text": "$K\\subset V\\subset \\bar{V}\\subset U.$"}
{"_id": "7464236", "title": "", "text": "$I_n=\\int \\tan ^{n}(x)\\,dx=\\frac{\\tan ^{n+1}(x)}{n+1} \\, _2F_1\\left(1,\\frac{n+1}{2};\\frac{n+3}{2};-\\tan   ^2(x)\\right)$"}
{"_id": "7350603", "title": "", "text": "$p_1p_2|a-b$"}
{"_id": "4583075", "title": "", "text": "$T^*M \\otimes \\mathbb{C} = {(T^{1,0})}^*M \\oplus {(T^{0,1})}^*M$"}
{"_id": "1176412", "title": "", "text": "$\\mathbb{P}[X_n \\geq b_n \\text{ infinitely often}] = 1$"}
{"_id": "5671865", "title": "", "text": "$f(x)=\\int_0^\\infty\\frac{e^{-t}}t\\sin (xt)dt=\\int_0^\\infty h(x,t)dt$"}
{"_id": "7147172", "title": "", "text": "$[(1+r)^{n-1} + (1+r)^{n-2} + \\dots + (1+r) + 1][1-(1+r)]=1-(1+r)^{n}$"}
{"_id": "136828", "title": "", "text": "$f_Y(y)=\\frac{4y}{\\sqrt{2\\pi}}e^\\frac{-y^4}{2}=2y{\\sqrt{\\frac2\\pi}}e^\\frac{-y^4}{2}.$"}
{"_id": "8499090", "title": "", "text": "$d(x_0,a) \\leq  d(x_0,y)+d(y,a)$"}
{"_id": "9200377", "title": "", "text": "$[x,y]= s^{2^{n-1}}$"}
{"_id": "9062372", "title": "", "text": "$(p_1,p_2) \\to p_1 + p_2 - p_1p_2$"}
{"_id": "5474206", "title": "", "text": "$\\sum_{n\\geqslant N} \\|f_n-e_n\\|^2 < 1$"}
{"_id": "8429542", "title": "", "text": "$\\{f((0,0)), f((0,\\pi)), f((\\pi,0)), f((\\pi,\\pi))\\}$"}
{"_id": "2147187", "title": "", "text": "$\\int_{0}^{2\\pi} f(t) \\sin ^2 (t-\\theta) dt = g(\\theta)$"}
{"_id": "5854924", "title": "", "text": "$\\int_0^\\infty \\frac{\\sin^{2n-1}x}{x}dx $"}
{"_id": "133875", "title": "", "text": "$\\mathcal{E} = \\left(e_1 = \\frac{x - y}{\\|x - y\\|}, e_2, e_3, \\ldots, e_N\\right).$"}
{"_id": "777981", "title": "", "text": "$c_0 \\subset \\overline{c_{00}}$"}
{"_id": "6211680", "title": "", "text": "$\\begin{cases}x-1=0\\\\ \\dfrac{y-3}1=\\dfrac{t-4}{-1}\\\\z-2=0\\end{cases}\\quad\\text{whence}\\quad\\begin{cases}x=1\\\\z=2\\\\y+t=7\\end{cases}$"}
{"_id": "69030", "title": "", "text": "$\\gamma(st)=r_2\\gamma(s)=\\gamma(t)\\gamma(s)$"}
{"_id": "2748818", "title": "", "text": "$ \\sum\\limits_{n=1}^N \\cos(2\\pi n/N)= 0 $"}
{"_id": "2843862", "title": "", "text": "$ \\forall \\alpha\\in(-\\omega,\\omega), |\\tan(\\alpha)-\\alpha|<\\epsilon |\\alpha| $"}
{"_id": "98793", "title": "", "text": "$f_{X,Y}(x,y) = \\frac{\\exp\\left(-\\frac{1}{1-\\rho^2}\\left(\\frac{x^2}{2}-\\frac{\\rho x y}{\\sigma}+\\frac{y^2}{2\\sigma^2}\\right)\\right)}{2\\pi\\sqrt{1-\\rho^2}\\sigma}.$"}
{"_id": "2107391", "title": "", "text": "$F(x) = \\int_2^x f(t)dt$"}
{"_id": "4875920", "title": "", "text": "$a: \\begin{cases} y=m_1 x\\\\ z=0\\\\ t=0\\end{cases}$"}
{"_id": "1973684", "title": "", "text": "$(a+b,a-b)$"}
{"_id": "6357205", "title": "", "text": "$\\left\\{ \\left( \\begin{matrix} a & b \\\\ -b & a \\end{matrix} \\right) \\right\\}.$"}
{"_id": "8564889", "title": "", "text": "$\\frac{n^2(n-1)}{2}+1+2+...+n=\\frac{n^2(n-1)}{2}+\\frac{n(n+1)}{2}=\\frac{n}{2}(n^2+1).$"}
{"_id": "5021646", "title": "", "text": "$ 105! + 2, 105!+3, 105!+4, \\dots, 105! + 101 $"}
{"_id": "8633601", "title": "", "text": "$C^{î,*}(D)=C^{î+1,*}(D_0) \\oplus C^{î,*}(D_{1})$"}
{"_id": "6118378", "title": "", "text": "$\\gamma \\subseteq \\omega(\\gamma)$"}
{"_id": "6870123", "title": "", "text": "$g_n=\\frac{a}{b^2}\\big((b+1)^{n+1}-(b+1)-bn\\big)$"}
{"_id": "2757149", "title": "", "text": "$e^{-x} \\ge \\bigg{(}1+\\frac{x}{n}\\bigg{)}^{-n}$"}
{"_id": "8392339", "title": "", "text": "$\\begin{vmatrix} 1 & x & x^2 & x^3 \\\\ 1 & y & y^2 & y^3 \\\\ 1 & z & z^2 & z^3 \\\\ 1 & x+y+z & xy+yz+zx & xyz \\end{vmatrix}$"}
{"_id": "7140740", "title": "", "text": "$\\sin\\theta + \\sin\\theta \\tan^2\\theta$"}
{"_id": "4837320", "title": "", "text": "$\\det(\\gamma)=\\det(\\gamma') = 1$"}
{"_id": "8999695", "title": "", "text": "$\\implies \\frac{1}{x}+\\frac{1}{y}=1$"}
{"_id": "5566425", "title": "", "text": "$\\lim_{n\\to\\infty}|\\sin(n\\pi)|=0$"}
{"_id": "8984334", "title": "", "text": "$ \\langle R(\\dot\\gamma,\\gamma')\\dot\\gamma,\\gamma'\\rangle  =  - \\langle R(\\dot\\gamma,\\gamma')\\gamma',\\dot\\gamma\\rangle . $"}
{"_id": "4899731", "title": "", "text": "$\\gamma\\beta\\in\\gamma J\\subseteq R$"}
{"_id": "7019643", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{m^2+n^2}=+\\infty.$"}
{"_id": "7946869", "title": "", "text": "$\\left(a,b\\right)\\mapsto\\left(a-\\lfloor a\\rfloor,b-\\lfloor b\\rfloor\\right)$"}
{"_id": "7503352", "title": "", "text": "$\\omega^{\\alpha}=\\alpha.$"}
{"_id": "8600347", "title": "", "text": "$2^n+3^n=4^n$"}
{"_id": "9079751", "title": "", "text": "$|f(z)|^2 = |f(z)^2| $"}
{"_id": "3605109", "title": "", "text": "$Q(m)\\implies Q(m+1)$"}
{"_id": "5785", "title": "", "text": "$C=\\lim_{x\\to0}\\frac{360^{\\circ}}{x^{\\circ}}\\times r\\times\\sin(x^{\\circ})=360^{\\circ}r\\lim_{x\\to0}\\frac{\\sin(x^{\\circ})}{x^{\\circ}}=360^{\\circ}r$"}
{"_id": "2746726", "title": "", "text": "$[\\mathfrak{g}_{\\mathbb{C}},\\mathfrak{g}_{\\mathbb{C}}] \\subset \\mathfrak{g}_{\\mathbb{C}}$"}
{"_id": "2298040", "title": "", "text": "$\\begin{cases} x^{,} = y - x \\\\ y^{,} = z - x - y \\end{cases}$"}
{"_id": "474158", "title": "", "text": "$x_n = e_n$"}
{"_id": "2751946", "title": "", "text": "$\\int_0^\\infty f(x) dx = 1,$"}
{"_id": "3032438", "title": "", "text": "$P_1P_2=I$"}
{"_id": "973419", "title": "", "text": "$d(x,A) \\leq d(x,y)+d(y,A)$"}
{"_id": "8251013", "title": "", "text": "$ \\sum_{n=1}^{+\\infty}\\frac1{n^2}=\\frac{\\pi^2}{6}\\;\\;. $"}
{"_id": "6051643", "title": "", "text": "$f(z^2)=f(f(f(z)))=(f(z))^2$"}
{"_id": "9113885", "title": "", "text": "$f(x) = \\sum_{m=1}^\\infty\\alpha_m\\chi_{A_m}(x)$"}
{"_id": "7023407", "title": "", "text": "$n+1,n+2,\\ldots, n+50$"}
{"_id": "671280", "title": "", "text": "$\\int_0^x f(t)dt = \\int_x^1 f(t)dt$"}
{"_id": "5325655", "title": "", "text": "$ \\begin{pmatrix}  11 & -1 & 27 & 0 \\\\ -2 & 1 & -12 & -1\\\\ 1 & 16 & -4 & a\\\\ 9 & 5 & 5 & 0 \\end{pmatrix}$"}
{"_id": "6883114", "title": "", "text": "$(2^m m)m < m^m$"}
{"_id": "4841104", "title": "", "text": "$b=a+c;a,b,c,d\\in \\mathbb{R} $"}
{"_id": "7373722", "title": "", "text": "$\\vartheta(-1/\\tau) = \\tau^{12} \\vartheta(\\tau)$"}
{"_id": "8889060", "title": "", "text": "$L=\\lim_{x \\rightarrow c^{-}} f'(x)=\\lim_{\\epsilon \\rightarrow 0} f'(c_1(\\delta(\\epsilon), f))=\\lim_{\\epsilon \\rightarrow 0} \\frac{f(c) - f(c-\\delta(\\epsilon))}{\\delta}=\\lim_{x \\rightarrow c^{-}} \\frac{f(c) - f(x)}{c-x}$"}
{"_id": "1157438", "title": "", "text": "$\\sum^\\infty_{k=1}{\\sum ^\\infty _{n=1}\\frac{a_n}{n^2+k^2}}=\\sum^\\infty_{n=1}{a_n\\sum ^\\infty _{k=1}\\frac{1}{n^2+k^2}}.$"}
{"_id": "4077035", "title": "", "text": "$p_0p_2$"}
{"_id": "9198289", "title": "", "text": "$X, Y \\in \\Gamma(M, T^{0,1}M)$"}
{"_id": "2264234", "title": "", "text": "$(1 + \\frac {k}{N})^N < e^k$"}
{"_id": "1157437", "title": "", "text": "$\\sum\\limits^\\infty_{k=1}{\\sum\\limits^\\infty _{n=1}\\frac{a_n}{n^2+k^2}}$"}
{"_id": "1202606", "title": "", "text": "$P(T_n7)=1/6$"}
{"_id": "3955537", "title": "", "text": "$x \\in U \\subseteq \\overline {U} \\subseteq C^c$"}
{"_id": "1817304", "title": "", "text": "$\\frac1x+\\frac1y=\\frac1{pq}$"}
{"_id": "2215411", "title": "", "text": "$\\dfrac{2}{m} = \\dfrac{1}{2^{n-1}}$"}
{"_id": "1783513", "title": "", "text": "$f(x+f(y))=f(x)+y^n$"}
{"_id": "8651074", "title": "", "text": "$\\int_{\\gamma} \\omega = \\int_{\\gamma} d\\theta = \\theta (\\gamma(1)) - \\theta(\\gamma(0)). $"}
{"_id": "1413576", "title": "", "text": "$ \\mathbb{E}|X - \\mathbb{E}X|^r \\leq \\mathbb{E} |Y|^r, $"}
{"_id": "3113194", "title": "", "text": "$p(E|S) = p(E \\& S)/p(S) = 1$"}
{"_id": "7953522", "title": "", "text": "$\\forall x,y \\, (xRy \\land yRx \\Rightarrow x=y )$"}
{"_id": "698701", "title": "", "text": "$\\mu^+$"}
{"_id": "6246485", "title": "", "text": "$y^{log_8(x)}=2$"}
{"_id": "7579159", "title": "", "text": "$ \\det A= [(n-1)a +1] (1-a)^{n-1}.$"}
{"_id": "679388", "title": "", "text": "$gcd(a,a) = a$"}
{"_id": "7523524", "title": "", "text": "$\\limsup P[X_n\\leq x]\\leq P[X\\leq x]$"}
{"_id": "1944578", "title": "", "text": "$\\{a,a+b,a+2b,a+3b,\\cdots\\}$"}
{"_id": "2796828", "title": "", "text": "$|x_n| \\leq \\| x \\|_{p}$"}
{"_id": "3463567", "title": "", "text": "$\\Delta_n= \\begin{vmatrix} a +(n-1)b& b & b & \\ldots & b\\\\ 0 & a-b & 0 &  \\ldots & 0 \\\\ 0 & 0 & a-b &  \\ldots & 0 \\\\ \\vdots&&&&\\vdots\\\\ 0 & 0 & 0 &  \\ldots & a -b\\end{vmatrix}=[a+(n-1)b](a-b)^{n-1}.$"}
{"_id": "5848352", "title": "", "text": "$ \\log_a x = log_{a^2}(x+6) $"}
{"_id": "8018337", "title": "", "text": "$\\zeta(s)=\\sum_{n=1}^\\infty \\frac{1}{?^s}$"}
{"_id": "6928718", "title": "", "text": "$E[X_{T \\wedge n} - X_0 | \\mathscr{F_m}] =/ \\le X_{T \\wedge m} - X_0 \\ \\forall m < n$"}
{"_id": "4945941", "title": "", "text": "$\\int_0^{\\infty} \\frac{1}{(1+x^2)^k}\\,dx$"}
{"_id": "3105717", "title": "", "text": "$P(k): 1^3+2^3+3^3+....+k^3=[\\frac {k(k+1)}{2}]^2$"}
{"_id": "2004392", "title": "", "text": "$\\sin x=\\frac{x}{2}$"}
{"_id": "3622192", "title": "", "text": "$X_1 \\subseteq X_2 \\subseteq$"}
{"_id": "5845647", "title": "", "text": "$f(x)=\\sum_{n=-\\infty}^\\infty e^{-(x-n)^2} =\\sum_{n=-\\infty}^\\infty e^{-(x+n)^2} = \\sqrt{\\pi}+2\\sqrt{\\pi}\\sum_{k=1}^\\infty e^{-k^2 \\pi^2}\\cos 2\\pi kx$"}
{"_id": "717429", "title": "", "text": "$gcd(a,b) = gcd(a, b-a) = gcd(a, [b-a]-a) = ...$"}
{"_id": "5026022", "title": "", "text": "$x^\\gamma \\implies (x^\\gamma)^\\gamma*(x^\\gamma)^2=1/x \\implies x^{\\gamma^2+2\\gamma}=x^{-1}\\implies x^{\\gamma^2+2\\gamma+1}=1 \\implies x^{(\\gamma+1)^2}=1 \\implies \\gamma=-1\\implies f(x)=1/x$"}
{"_id": "7835222", "title": "", "text": "$\\frac{\\partial l(\\theta)}{\\partial \\theta} = -\\frac{1}{\\theta} + \\frac{|x|}{\\theta^2}$"}
{"_id": "2977368", "title": "", "text": "$B = \\{ e_1, e_2,\\dots, \\}$"}
{"_id": "219419", "title": "", "text": "$G(x)=\\int_a^x g(t)dt$"}
{"_id": "616164", "title": "", "text": "$S=\\{1, 2, 3, ...., n\\}$"}
{"_id": "3556649", "title": "", "text": "$f(x)=\\sum_{q_n<x}\\frac1{n^2}=\\sum_{n\\in Q_x}\\frac1{n^2}\\;.$"}
{"_id": "5173234", "title": "", "text": "$F(0)=0=F(\\pi)$"}
{"_id": "6857312", "title": "", "text": "$ax+by=d.$"}
{"_id": "1836936", "title": "", "text": "${\\bf M} = \\left[\\begin{array}{cccc} 1&0&0&0\\\\ 1&1&0&0\\\\ 1&1&1&0\\\\ 1&1&1&1 \\end{array}\\right] , {\\bf M}^2 = \\left[\\begin{array}{cccc} 1&0&0&0\\\\ 2&1&0&0\\\\ 3&2&1&0\\\\ 4&3&2&1 \\end{array}\\right], \\\\{\\bf M}^3 =  \\left[\\begin{array}{cccc} 1&0&0&0\\\\ 3&1&0&0\\\\ 6&3&1&0\\\\ 10&6&3&1 \\end{array}\\right] , {\\bf M}^4 = \\left[\\begin{array}{cccc} 1&0&0&0\\\\ 4&1&0&0\\\\ 10&4&1&0\\\\ 20&10&4&1 \\end{array}\\right] $"}
{"_id": "2222754", "title": "", "text": "$f\\left( y,\\pm\\frac{\\sqrt{1-y^2}}{2}\\right)$"}
{"_id": "1967066", "title": "", "text": "$\\frac {10^4-1}{10^4}\\,\\frac {10^4-2}{10^4-1}\\,\\frac 1{10^4-2}=\\frac {1}{10^4}$"}
{"_id": "3300598", "title": "", "text": "$ \\begin{align} \\lim_{n\\to\\infty}\\sum_{k=1}^{\\lfloor n/2\\rfloor}\\frac{n}{n^2-k^2} &=\\lim_{n\\to\\infty}\\sum_{k=1}^{\\lfloor n/2\\rfloor}\\frac1{1-\\frac{k^2}{n^2}}\\frac1n\\\\ &=\\int_0^{1/2}\\frac{\\mathrm{d}x}{1-x^2}\\\\ &=\\frac12\\int_0^{1/2}\\left(\\frac1{1-x}+\\frac1{1+x}\\right)\\mathrm{d}x\\\\ &=\\frac12\\left[\\log\\left(\\frac{1+x}{1-x}\\right)\\right]_0^{1/2}\\\\[6pt] &=\\frac12\\log(3) \\end{align} $"}
{"_id": "1353774", "title": "", "text": "$\\sum_{n=m+1}^\\infty \\|e_n-f_n\\|^2  = c <1$"}
{"_id": "1741994", "title": "", "text": "$g(\\text{A N A}) = f(A)+f(A)+1 = 2f(A)+1 =f(\\text{A N A})+1 \\leq 2f(\\text{A N A})+1$"}
{"_id": "2123830", "title": "", "text": "$P(m) \\implies P(n)$"}
{"_id": "3093339", "title": "", "text": "$f'(x)=\\frac{x^2e^x-(e^x-1)^2}{x^2(e^x-1)^2}$"}
{"_id": "4073898", "title": "", "text": "$\\overline{L}(\\gamma) \\ge \\Vert \\gamma(1) - \\gamma(a) \\Vert + \\Vert \\gamma(a) - \\gamma(0) \\Vert > \\Vert \\gamma(1) - \\gamma(0) \\Vert$"}
{"_id": "1230493", "title": "", "text": "$ 9 x^4 (x^2 - 2x + 1) = 9 x^6 - 18 x^5 + 9 x^4;  $"}
{"_id": "8354774", "title": "", "text": "$f(x)=\\frac{2(\\theta-x)}{\\theta^2}$"}
{"_id": "3549281", "title": "", "text": "$a_0 = \\root 6 \\of 3 e^{\\frac{{i\\pi }} {6}}  $"}
{"_id": "3676042", "title": "", "text": "$T=\\mbox{Th}_\\tau (\\cal M) = \\lbrace\\gamma|\\gamma\\mbox{ is a } \\tau \\mbox{- sentence and }\\cal M\\models\\gamma\\rbrace.$"}
{"_id": "3931050", "title": "", "text": "$(s_{1,0} s_{0,1})$"}
{"_id": "1097042", "title": "", "text": "$\\displaystyle \\int_0^x \\sin(t^2) f(t) \\; dt = \\displaystyle \\int_0^1 k(t,x) f(t) \\; dt$"}
{"_id": "4438620", "title": "", "text": "$\\frac{-\\frac{x-4}{4(x+4)}}{x-4}=-\\frac{x-4}{4(x+4)(x-4)}=-\\frac{1}{4(x+4)}$"}
{"_id": "1491239", "title": "", "text": "$\\mathrm{Cov}(X,Z)=0$"}
{"_id": "3974575", "title": "", "text": "$\\left[ \\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\\\  1 & 2 & 3 & 10 \\\\  1 & 2 & \\lambda & \\mu  \\end{array} \\right]$"}
{"_id": "6162969", "title": "", "text": "$\\sum_{k=0}^N e^{\\frac{2\\pi k}{N+1}}=0\\Rightarrow \\sum_{k=0}^N\\cos{\\frac{2\\pi k}{N+1}}=0\\Rightarrow \\sum_{k=1}^N\\cos{\\frac{2\\pi k}{N+1}}=-1.$"}
{"_id": "7859298", "title": "", "text": "$(a-b)\\mid(a^{rs}-b^{rs})$"}
{"_id": "4646512", "title": "", "text": "$\\lim_{L \\rightarrow \\infty} (\\ln{L})^2\\left[1-\\left(1-e^{-\\frac{1}{2}(\\sqrt{L}-2)}\\right)^L\\right]$"}
{"_id": "4687631", "title": "", "text": "$\\log 3 \\log(1+\\sqrt{x}) =\\log 2 \\log x $"}
{"_id": "7136361", "title": "", "text": "$g(y)=4x^{3/4}(x^{1/4}+1)^2$"}
{"_id": "4567501", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac{1}{(1+nx)^2} < \\sum_{n=1}^\\infty \\frac{1}{(nx)^2} = \\frac{1}{x^2}\\sum_{n=1}^\\infty \\frac{1}{n^2}$"}
{"_id": "1464145", "title": "", "text": "$ \\det A=(l-b)^{n-1}(l+(n-1)b). $"}
{"_id": "6544437", "title": "", "text": "$f_1,f_2 \\in \\mathbb{N}^\\mathbb{N}$"}
{"_id": "1550134", "title": "", "text": "$-x^{n+1}$"}
{"_id": "2539146", "title": "", "text": "$ \\forall \\epsilon >0 , \\exists \\delta>0 , \\forall x , 0 <|x-a| <\\delta \\Rightarrow |f(x)-a| <\\epsilon \\ $"}
{"_id": "4141597", "title": "", "text": "$(n+1-r)\\binom{n-r}{r}=((n+1)(n-r)-(n-1)(n+1-r))\\binom{n+1-r}{r}=((n^2+n-rn-r)-(n^2-n+n-1-rn+r))\\binom{n+1-r}{r}=(n-2r+1)\\binom{n+1-r}{r}$"}
{"_id": "1416545", "title": "", "text": "$\\lim_{x \\to c}f(x)g(x) = \\lim_{x \\to c}f(x) \\times \\lim_{x \\to c }g(x)= L \\times \\infty = \\infty$"}
{"_id": "8122869", "title": "", "text": "$f(x)=\\frac{x^x}{(x-3)^3}$"}
{"_id": "1366896", "title": "", "text": "$d(x,y)\\leq d(x,x_n)+d(x_n,y_n)+d(y_n,y)$"}
{"_id": "2046308", "title": "", "text": "$\\dfrac{x_1+x_2+\\cdots+x_n}{n} \\ge \\sqrt[n]{x_1x_2\\cdots x_n}$"}
{"_id": "790721", "title": "", "text": "$ \\int_{--\\frac{b^2}{a}}^{\\frac{b^2}{a}} \\sqrt{1+ {\\frac{dy}{dx}}^{2}} dx = \\int_{--\\frac{b^2}{a}}^{\\frac{b^2}{a}} \\sqrt{1+\\frac{a^{2}x^{2}}{b^{2}(b^{2}+x^{2})}} \\ dx $"}
{"_id": "2743792", "title": "", "text": "$A_1\\subseteq A_2\\subseteq\\cdots \\subseteq A_k\\subseteq \\{1,2,3,\\cdots ,n\\}$"}
{"_id": "5218087", "title": "", "text": "$[X,Y] = X$"}
{"_id": "553272", "title": "", "text": "$N_1\\times\\dots\\times N_n$"}
{"_id": "6829500", "title": "", "text": "$ \\zeta(s) = \\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{x^{s-1}}{e^x-1}\\,dx. $"}
{"_id": "583673", "title": "", "text": "$\\int_{-\\pi}^{\\pi}$"}
{"_id": "5281131", "title": "", "text": "$J=\\int\\frac{\\sin (\\alpha x)}{1+x^2}\\,dx$"}
{"_id": "8493508", "title": "", "text": "$F(x):=\\int_a^x f(t)\\, dt,$"}
{"_id": "2743797", "title": "", "text": "$ A_1 \\subseteq A_2 \\subseteq \\dots \\subseteq A_{k+1} \\subseteq \\lbrace 1, 2, \\dots, n \\rbrace $"}
{"_id": "7461693", "title": "", "text": "$ \\begin{equation} \\left\\{ \\begin{array}{lr}  -(\\tilde{a}(x)u_{x})_x + \\tilde{c}(x) u = \\tilde{f}, 0<x<L \\\\ u(0) = u(L) = 0 \\end{array} \\right. \\end{equation} $"}
{"_id": "4382386", "title": "", "text": "$f = \\{(a)\\} = \\{\\{a\\}\\}$"}
{"_id": "3097454", "title": "", "text": "$\\lfloor\\alpha\\rfloor+\\lfloor\\beta\\rfloor+\\lfloor\\gamma\\rfloor=9-(\\{\\alpha\\}+\\{\\beta\\}+\\{\\gamma\\})=\\color{red}{7,8}$"}
{"_id": "4576990", "title": "", "text": "$\\,p\\mid d\\mid p_1 p_2\\Rightarrow\\,p\\mid p_1p_2\\,$"}
{"_id": "6954847", "title": "", "text": "$ {\\rm Hess} f (\\dot \\gamma, \\dot \\gamma) := \\dot \\gamma (\\dot \\gamma f)-(\\nabla _{\\dot \\gamma} \\dot \\gamma)f=\\dot \\gamma (\\dot \\gamma f).$"}
{"_id": "609603", "title": "", "text": "$b_n=\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi}f(x)\\sin(nx)dx$"}
{"_id": "5162104", "title": "", "text": "$\\lim_{x\\to a^-} f(x)=\\lim_{x\\to a^+}f(x)=\\lim_{x\\to a}f(x)$"}
{"_id": "7647735", "title": "", "text": "$(a_{\\varepsilon},b_{\\varepsilon}) \\subset B_{\\varepsilon}$"}
{"_id": "2008022", "title": "", "text": "$A\\subset U\\subset K\\subset B\\subset Y?$"}
{"_id": "8069935", "title": "", "text": "$ \\lim_{n\\to \\infty}J(u_n) = \\inf_{u\\ne 0}J(u). $"}
{"_id": "5258248", "title": "", "text": "$ (a,a) = \\{\\{a\\}\\} $"}
{"_id": "837808", "title": "", "text": "$\\sigma^{+}$"}
{"_id": "5065821", "title": "", "text": "$\\sum\\limits_{n\\geq 1}\\frac{1}{n^s}=\\sum\\limits_{n\\geq 1}n\\left(\\frac{1}{n^s}-\\frac{1}{(n+1)^s}\\right)$"}
{"_id": "9042454", "title": "", "text": "$X = \\begin{bmatrix} A &B\\\\B&A \\end{bmatrix}$"}
{"_id": "5787284", "title": "", "text": "$\\sqrt z = \\sqrt r e^{-i \\varphi / 2}$"}
{"_id": "5422421", "title": "", "text": "$1+2+3+\\dots n+\\dots=-\\frac1{12}$"}
{"_id": "216932", "title": "", "text": "$a_n=n(n+1)/2$"}
{"_id": "5997739", "title": "", "text": "$ f(x)^2+f(y)^2=f(x+y)(f(f(y))+f(x)) $"}
{"_id": "4899396", "title": "", "text": "$f(x) = \\frac{9^{x}}{9^{x}+3},$"}
{"_id": "9084461", "title": "", "text": "$\\displaystyle \\lim\\limits_{n\\to \\infty} \\sum\\limits_{m=1}^{n} \\log\\left(1+\\log\\left(1+\\frac{m}{n^2}\\right)\\right) = \\lim\\limits_{n\\to \\infty} \\sum\\limits_{m=1}^{n} \\log\\left(1+\\frac{m}{n^2}\\right) = \\lim\\limits_{n\\to \\infty} \\sum\\limits_{m=1}^{n} \\frac{m}{n^2} = \\frac{1}{2}$"}
{"_id": "755542", "title": "", "text": "$p = \\frac {1}{2\\pi(1 - \\cos\\theta)}$"}
{"_id": "2965690", "title": "", "text": "$\\Lambda^+$"}
{"_id": "6101964", "title": "", "text": "$ y + \\lambda = C \\sqrt{1+(\\frac{dy}{dx})^2} $"}
{"_id": "8295175", "title": "", "text": "$\\lim_{N\\rightarrow\\infty}\\sum_{n=1}^N\\frac{1}{n}=\\infty,$"}
{"_id": "4827845", "title": "", "text": "$\\{e_1,e_2,e_3,\\cdots, e_n\\}$"}
{"_id": "2743799", "title": "", "text": "$ A_1 \\subseteq A_2 \\subseteq \\dots \\subseteq A_k \\subseteq A_{k+1} $"}
{"_id": "355977", "title": "", "text": "$ Lie(g) = \\{ \\dot{\\gamma}(0) | \\gamma:(-\\epsilon, \\epsilon) \\rightarrow g, \\gamma \\in C^1,  \\gamma(0) =  \\mathbb{I} \\} $"}
{"_id": "6931922", "title": "", "text": "$S=(x_1,x_2,x_3,\\cdots , x_n)$"}
{"_id": "6817065", "title": "", "text": "$  f = \\frac{ (1-f)^N \\lbrack 1 - (1-f)^2 \\rbrack }{ 1- (1-f)^{N} } $"}
{"_id": "6500392", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{2}+\\frac{1}{z}$"}
{"_id": "3732276", "title": "", "text": "$2^x+3^x>5^x$"}
{"_id": "5884015", "title": "", "text": "$f(a+b)=f(a)+f(b) , \\forall a \\in A , b\\in B$"}
{"_id": "1101116", "title": "", "text": "$x^3, x^2, x$"}
{"_id": "5284916", "title": "", "text": "$\\frac{k(k-1)((k-2)+3)}{3}=\\frac{k(k-1)(k+1)}{3}$"}
{"_id": "2177377", "title": "", "text": "$v((x-a)^mp(x)+(x-a)^nq(x))=v((x-a)^k[(x-a)^{m-k}p(x)+(x-a)^{n-k}q(x)])\\geq v((x-a)^k)=k,$"}
{"_id": "6601505", "title": "", "text": "$\\mathrm{Cov}(X,U)=0$"}
{"_id": "6962091", "title": "", "text": "$\\sin\\left(\\frac{\\pi}{4}+x\\right)\\left(1-2\\sin(x)\\cos(x)\\right)=\\sin\\left(\\frac{\\pi}{4}+x\\right)\\left(1-2\\sin(x)\\cos(x)\\right)$"}
{"_id": "4670768", "title": "", "text": "$-\\frac{x}{4} = \\frac{-x}{4} = \\frac{x}{-4}$"}
{"_id": "7755265", "title": "", "text": "$\\implies \\gamma'=5\\gamma $"}
{"_id": "8846029", "title": "", "text": "$[(x,u)]=x$"}
{"_id": "4532880", "title": "", "text": "$\\int_a^b 1(x)\\, dx$"}
{"_id": "8662665", "title": "", "text": "$f(1)+f(2)+...f(n) = n^{2}f(n)$"}
{"_id": "5748955", "title": "", "text": "$f(ab)=f(a)+af(b)$"}
{"_id": "2439783", "title": "", "text": "$\\sum_{n=1}^\\infty\\frac 1n\\sum_{m=1}^\\infty\\frac 1m=\\sum_{n,m=1}^\\infty\\frac 1{nm}=\\sum_{n=1}^\\infty\\frac1{n^2}+\\sum_{n<m}\\frac2{nm}.$"}
{"_id": "6429062", "title": "", "text": "$F_0=Q\\oplus P$"}
{"_id": "2946958", "title": "", "text": "$\\mathcal{A} = \\{e_1, e_2, e_3, ..., e_n \\}$"}
{"_id": "7856951", "title": "", "text": "$\\;\\begin{cases}x=t\\\\y=t\\\\z=t\\end{cases}\\enspace(0\\le t\\le 1)$"}
{"_id": "6685655", "title": "", "text": "$F\\oplus T$"}
{"_id": "1103019", "title": "", "text": "$(e_1,...,e_k,e_{k+1},..,e_n)$"}
{"_id": "208599", "title": "", "text": "$\\mathbb R^{n+1}\\to\\mathbb R^{n+1}$"}
{"_id": "5691056", "title": "", "text": "$||A||_{2}= \\sqrt{\\rho(A^{*}A)} <1$"}
{"_id": "3689655", "title": "", "text": "$\\frac {50}{x} + \\frac {126}{y} = 1$"}
{"_id": "574499", "title": "", "text": "$3\\mid n^3 - n$"}
{"_id": "8531520", "title": "", "text": "$ \\int_{0}^{2\\pi}\\frac{\\sin(nx)}{\\sin(x)} \\mathrm dx $"}
{"_id": "6625180", "title": "", "text": "$n^2-n+2.$"}
{"_id": "6625310", "title": "", "text": "${\\lim_{n \\to \\infty}}\\inf s_n = s_{*}$"}
{"_id": "7009636", "title": "", "text": "$p|r_1 r_2$"}
{"_id": "4696801", "title": "", "text": "$\\tan \\theta = \\frac{x}{\\sqrt{3}}$"}
{"_id": "1092751", "title": "", "text": "$f(a + b) = f(a) * f(b)$"}
{"_id": "9112836", "title": "", "text": "$y = \\frac{-1 - 2x}{1 - x}$"}
{"_id": "6482511", "title": "", "text": "$\\frac{e^x(x-n)}{x^{n+1}} = \\frac{e^x}{x^n} - \\frac{n}{x}\\frac{e^x}{x^{n}} > \\frac{e^x}{x^n}\\bigg(\\frac{1}{n+1}\\bigg) > \\frac{e^{n+1}}{n^{n+1}}$"}
{"_id": "187780", "title": "", "text": "$A_1...A_n$"}
{"_id": "8320023", "title": "", "text": "$d(x,z)+d(x,y)+d(y,z)$"}
{"_id": "7931782", "title": "", "text": "$(X^4+1)$"}
{"_id": "7483745", "title": "", "text": "$P(E)=P(E)$"}
{"_id": "6912162", "title": "", "text": "$A=\\begin{pmatrix}   1 & 1 & 1 \\\\   x & y & z \\\\   x^2 & y^2 &z^2\\\\ \\end{pmatrix}$"}
{"_id": "7431589", "title": "", "text": "$\\begin{align*} &\\; \\int_0^1 \\frac{\\log^2(x)\\tanh^{-1}(x)}{1+x^2}dx \\\\ &= -2\\int_0^1 \\frac{\\log(x)\\tan^{-1}(x)\\tanh^{-1}(x)}{x}dx-\\int_0^1 \\frac{\\log^2(x)\\tan^{-1}(x)}{1-x^2}dx \\tag{1} \\end{align*}$"}
{"_id": "7598729", "title": "", "text": "$\\lim_{N\\to \\infty}\\sum_{k=1}^{N}\\frac{1}{k+N}=\\ln(2)$"}
{"_id": "4647934", "title": "", "text": "$\\forall x,y \\in X: xIy \\leftrightarrow xRy \\land yRx$"}
{"_id": "1133411", "title": "", "text": "$\\lim_y\\to2^- f(y)=+\\infty$"}
{"_id": "4001599", "title": "", "text": "$\\sum_{n=1}^\\infty\\left\\|x_n\\right\\|_X\\left\\|y_n\\right\\|_Y<\\pi(u)+\\epsilon\\tag1$"}
{"_id": "251883", "title": "", "text": "$\\frac{1}{|x|}$"}
{"_id": "1701916", "title": "", "text": "$\\sum_{n\\ge 1} \\frac{1}{n^2}<\\infty$"}
{"_id": "1196662", "title": "", "text": "$\\begin{pmatrix}a & -b \\\\ b & a\\end{pmatrix}.$"}
{"_id": "4958369", "title": "", "text": "$E(x) = \\int_{a}^{b}3x^2dx$"}
{"_id": "6324897", "title": "", "text": "$y[n-1] = (n-1)(x[n-1] - 5)^2$"}
{"_id": "2449525", "title": "", "text": "$d(x, A) = \\inf \\{d(x, y) \\,|\\, y \\in A\\}$"}
{"_id": "3824166", "title": "", "text": "$ra + sp = 1$"}
{"_id": "342254", "title": "", "text": "$(\\mathbb{Z}/2^n\\mathbb{Z})^{\\times}$"}
{"_id": "8954232", "title": "", "text": "$\\lfloor{a} \\rfloor - \\lfloor{b}\\rfloor = \\lfloor\\lfloor{a}\\rfloor - \\lfloor{b}\\rfloor\\rfloor \\ge \\lfloor\\lfloor{a}\\rfloor - \\lfloor{b}\\rfloor + (\\{b\\} - \\{a\\})\\rfloor = \\lfloor{a - b}\\rfloor$"}
{"_id": "916373", "title": "", "text": "$I(x) = \\int_a^x f(t) \\, dt$"}
{"_id": "5319121", "title": "", "text": "$\\int_{0}^{1}\\frac{\\sin^{-1}\\sqrt x}{x^2-x+1}dx$"}
{"_id": "4156680", "title": "", "text": "$ I_n =\\int_{-1}^{1}\\frac{\\sqrt{n}f(x)}{1+n x^2}dx $"}
{"_id": "282081", "title": "", "text": "$p(k) \\implies p(k+1)$"}
{"_id": "1243913", "title": "", "text": "$a_1 \\cdot \\dotsc \\cdot a_n$"}
{"_id": "110690", "title": "", "text": "$\\cfrac{15}{x}+ \\cfrac{12}{x}+\\cfrac{6}{x}+\\cfrac{3}{x}=1$"}
{"_id": "3791874", "title": "", "text": "$V(X):=E((X-E(X))^2)=E(X^2-2E(X)X+E(X)^2) = E(X^2)-E(X)^2$"}
{"_id": "6256429", "title": "", "text": "$(f(z))^2=f(z^2)$"}
{"_id": "1484773", "title": "", "text": "$d(x,a)\\geq d(y,a)-d(x,y)$"}
{"_id": "3356981", "title": "", "text": "$f(x) = \\lim_{n\\to \\infty} \\sum_{r=1}^n \\frac{\\lfloor2rx\\rfloor}{n^2}$"}
{"_id": "307322", "title": "", "text": "$cov(\\mathbf{e}, \\mathbf{b}) = 0$"}
{"_id": "3113197", "title": "", "text": "$p(E|S) = p(E \\& S)/p(S) = 0$"}
{"_id": "1400077", "title": "", "text": "$(4,7)=1$"}
{"_id": "4316483", "title": "", "text": "$1=x+y\\leq x^r+y^r$"}
{"_id": "6724338", "title": "", "text": "$A,B,X,Y < C < D$"}
{"_id": "5496580", "title": "", "text": "$g(x)=x^2-x-2$"}
{"_id": "8492541", "title": "", "text": "$A_1\\times\\cdots\\times A_n\\subseteq k^n$"}
{"_id": "7283246", "title": "", "text": "$P(2)\\implies P(3)\\implies P(4)\\implies\\cdots\\implies P(18)\\implies P(19)$"}
{"_id": "306306", "title": "", "text": "$d(x,y) = \\dfrac{|x-y|}{1+|x-y|}$"}
{"_id": "2791943", "title": "", "text": "$Cov(x,f(x))=0$"}
{"_id": "4122528", "title": "", "text": "$P[X \\geq k] = 1 - F_n(k)$"}
{"_id": "5531838", "title": "", "text": "$\\displaystyle\\sum_{n=1}^{\\infty} a_{n}^{-}=\\frac{1}{2}\\sum_{n=1}^{\\infty}\\left(a_n-\\left|a_n\\right|\\right)$"}
{"_id": "1954863", "title": "", "text": "$n^{s_{0}}=\\sqrt{n}e^{it_{0}\\log\\left(n\\right)}  $"}
{"_id": "7141", "title": "", "text": "$\\$"}
{"_id": "5582219", "title": "", "text": "$S(n)+an+b=2[S(n-1)+a(n-1)+b]+n$"}
{"_id": "9240088", "title": "", "text": "$\\sqrt{\\gamma}(0)=\\sqrt{\\gamma(0)}$"}
{"_id": "545821", "title": "", "text": "$f(n) = n^2 + n + 41$"}
{"_id": "2126964", "title": "", "text": "$ T^{1,0}M \\times T^{0,1}M \\subset TM^{\\Cpx} \\times TM^{\\Cpx}, $"}
{"_id": "1698349", "title": "", "text": "$P\\left[X_{1}=1\\wedge X_{2}=0\\right]=P\\left[X_{2}=0\\mid X_{1}=1\\right]P\\left[X_{1}=1\\right]=\\frac{g}{r+g+c}\\frac{r}{r+g}$"}
{"_id": "7211535", "title": "", "text": "$\\Delta (x' + y') \\leq \\epsilon(\\Delta| x| + \\Delta |y| + |x| + |y|)$"}
{"_id": "4209473", "title": "", "text": "$A \\subseteq V = \\overline{V} \\subseteq U$"}
{"_id": "8052207", "title": "", "text": "$2|n^2 + n$"}
{"_id": "421732", "title": "", "text": "$\\zeta(s)=2^s\\pi^{s-1}\\Gamma(1-s)\\zeta(1-s)$"}
{"_id": "2566629", "title": "", "text": "$f(n) = (\\frac{n}{q})=\\prod_{p^k \\| q} (\\frac{n}{p})^k$"}
{"_id": "4452738", "title": "", "text": "$ \\|x_n - x_m \\|<{1\\over 2}$"}
{"_id": "687066", "title": "", "text": "$\\rho \\circ \\gamma \\circ \\rho \\subseteq \\gamma \\circ \\rho \\circ \\gamma$"}
{"_id": "6680556", "title": "", "text": "$(y-x) \\begin{vmatrix} x&1&z\\\\ x^2&y+x&z^2\\\\ x^3&y^2+xy+x^2&z^3\\\\ \\end{vmatrix}$"}
{"_id": "5560256", "title": "", "text": "$T: P(F) → P(F),$"}
{"_id": "6642421", "title": "", "text": "$E[Y\\mid X=x_0] = 3x_0/2$"}
{"_id": "2355207", "title": "", "text": "$X = \\left\\{ (x,y) \\in \\mathbb{R^2} : 0 \\leq x \\leq 1, 0 \\leq y \\leq 1 \\right\\}$"}
{"_id": "6799051", "title": "", "text": "$A_1\\times\\dots\\times A_n.$"}
{"_id": "524559", "title": "", "text": "$ \\begin{align} (X^2+X)\\sum_{0}^{\\infty}\\frac1{n!}\\frac1{X^n}&=\\sum_{0}^{\\infty}\\frac1{n!}\\frac1{X^{n-2}}+\\sum_{0}^{\\infty}\\frac1{n!}\\frac1{X^{n-1}}\\\\\\\\ &=\\sum_{-2}^{\\infty}\\frac1{(n+2)!}\\frac1{X^{n}}+\\sum_{-1}^{\\infty}\\frac1{(n+1)!}\\frac1{X^{n}}\\\\\\\\ &=\\frac1{X^{-2}}+\\sum_{-1}^{\\infty}\\left(\\frac1{(n+2)!}+\\frac1{(n+1)!}\\right)\\frac1{X^{n}}\\\\\\\\ &=\\frac1{X^{-2}}+\\sum_{-1}^{\\infty}\\frac1{(n+1)!}\\left(\\frac1{(n+2)}+1\\right)\\frac1{X^{n}}\\\\\\\\ &=\\frac1{X^{-2}}+\\sum_{-1}^{\\infty}\\frac{n+3}{(n+2)!}\\frac1{X^{n}}\\\\\\\\ &=\\sum_{-2}^{\\infty}\\frac{n+3}{(n+2)!}\\frac1{X^{n}} \\end{align} $"}
{"_id": "4118608", "title": "", "text": "$\\matrix{1\\cr 2&3\\cr 4&5&6\\cr 7&8&9&10\\cr 11&12&13&14&15\\cr   \\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\\cr}$"}
{"_id": "3685987", "title": "", "text": "$\\sum_{k=i}^{i+2n-1}\\sin\\frac {k\\pi}{n}=0$"}
{"_id": "9362714", "title": "", "text": "$\\lim_{x\\to c^+}f'(x) = \\lim_{x\\to c^-}f'(x) = f'(c)$"}
{"_id": "5984971", "title": "", "text": "$T\\gamma = b.(\\gamma - Sd\\gamma - T\\partial\\gamma)                          = b.\\gamma -b.Sd\\gamma - b.T\\partial\\gamma $"}
{"_id": "8937511", "title": "", "text": "$\\lbrace T_x : x \\in [0,1]\\rbrace$"}
{"_id": "612524", "title": "", "text": "$P(C) = \\frac{1}{6}$"}
{"_id": "490957", "title": "", "text": "$[t_N^k,t_N^{k + 1}]$"}
{"_id": "5297684", "title": "", "text": "$\\underline{~a~}~\\underline{~b~}~\\underline{~c~}~\\underline{~d~}~\\underline{~e~}~\\underline{~d~}~\\underline{~c~}~\\underline{~b~}~\\underline{~a~}$"}
{"_id": "6524517", "title": "", "text": "$ \\small{\\sum_{k=1}^N\\! \\left( \\cos \\frac{\\pi}{2k}-\\cos \\frac{\\pi}{2(k+2)} \\right)\\!=\\sum_{k=1}^N \\!\\left(\\! \\cos \\frac{\\pi}{2k}-\\cos \\frac{\\pi}{2(k+1)}\\! \\right)\\!+\\!\\sum_{k=1}^N\\! \\left(\\! \\cos \\frac{\\pi}{2(k+1)}-\\cos \\frac{\\pi}{2(k+2)} \\!\\right)} $"}
{"_id": "8594699", "title": "", "text": "$\\gamma''= (\\gamma'')^\\perp = - \\gamma + h(\\gamma', \\gamma') \\nu$"}
{"_id": "6071850", "title": "", "text": "$g(a,b)=(a+b,a-b)$"}
{"_id": "4669788", "title": "", "text": "$n(\\gamma,a)=-n(-\\gamma,a)=n$"}
{"_id": "4618430", "title": "", "text": "$ \\forall m, m < n, P(m) \\implies P(n) $"}
{"_id": "1595871", "title": "", "text": "$\\inf_{g \\in X} || y -g|| = \\sup_{(f,X)=0} \\frac{(y,f)}{||f||},$"}
{"_id": "5279872", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac{\\sqrt[n]e + \\sqrt[n] {e^2} + \\sqrt[n]{e^3}+...+\\sqrt[n]{e^{2n}}}{n}$"}
{"_id": "8116266", "title": "", "text": "$as^{log_s(a^{-1}b)}=b$"}
{"_id": "8590569", "title": "", "text": "$3^x + 10^x = 4^x + 9^x$"}
{"_id": "8988330", "title": "", "text": "$J=\\int_{0}^{\\pi}\\frac{\\sin x}{1+2^x}dx$"}
{"_id": "4438618", "title": "", "text": "$\\frac{\\frac{x+6}{x+4}-\\frac{10}{8}}{x-4}$"}
{"_id": "8738166", "title": "", "text": "$E(n)=n^2-n+41$"}
{"_id": "1985548", "title": "", "text": "$(\\alpha, \\gamma), (\\beta, \\gamma) \\in \\Delta$"}
{"_id": "5686975", "title": "", "text": "$[\\frac52] - [\\frac57] + [\\frac{-2}7] - [\\frac23] + [\\frac53] - [\\frac{-5}2]$"}
{"_id": "6483404", "title": "", "text": "$Y_n = \\frac{1}{n^2} \\sum_{k=1}^{n} X_k \\stackrel{d}{=} \\frac{1}{n^2} \\sum_{k=1}^{n} \\bigg( \\sum_{i=1}^{k} W_i \\bigg) = \\frac{n-1}{n^2} \\sum_{k=1}^{n} W_k - \\frac{1}{n^2} \\sum_{k=1}^{n} k W_k,$"}
{"_id": "8692450", "title": "", "text": "$\\|x_n - x_{n+1}\\| < \\frac{1}{2^n}$"}
{"_id": "4372296", "title": "", "text": "$\\int_0^\\infty f(x)<\\infty$"}
{"_id": "8852056", "title": "", "text": "$0=(-A)+\\lim_{x\\to p}f(x)=\\lim_{x\\to p}(-A)+f(x)=\\lim_{x\\to p}f_1(x),$"}
{"_id": "7904297", "title": "", "text": "$x=x*y=x*(y*z)=(x*z)*y=x*y=x$"}
{"_id": "328144", "title": "", "text": "$ar + bs$"}
{"_id": "6734651", "title": "", "text": "$\\frac{ax + b}{c-x} = y$"}
{"_id": "1348972", "title": "", "text": "$\\sqrt{\\frac{\\pi }{t}}\\sum\\limits_{n=-\\infty }^{\\infty }{{{e}^{-{{\\left( x/2+n\\pi  \\right)}^{2}}/t}}}=\\sum\\limits_{n=-\\infty }^{\\infty }{{{e}^{-{{n}^{2}}t+inx}}}$"}
{"_id": "4303925", "title": "", "text": "$\\frac{k_{n+1}}{k_n}\\le\\frac{(1+\\varepsilon)^{n+1}}{\\frac{1}{2}(1+\\varepsilon)^n}=2+2\\varepsilon$"}
{"_id": "430661", "title": "", "text": "$f(x)=\\sum_{j=1}^n a_j \\chi_{E_j}$"}
{"_id": "6909921", "title": "", "text": "$ \\implies E((X+Y)^p)\\leq 2^p (E(X^p)+E(Y^p))$"}
{"_id": "1399263", "title": "", "text": "$M = \\left(\\begin{matrix} A & B \\\\ B^T & C \\end{matrix}\\right)$"}
{"_id": "5784215", "title": "", "text": "$\\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{n^2-n}{n^3}\\frac{1}{(m^2+n^2)}>0.$"}
{"_id": "7544045", "title": "", "text": "$\\sum_{m=1}^k \\sum_{n=1}^{\\infty} \\frac{1}{m^2+n^2} \\approx \\frac{\\pi \\ln k}{2} + \\frac{6\\gamma \\pi-\\pi^2}{12}$"}
{"_id": "8260518", "title": "", "text": "$(x_1,y_1)=(x_2,y_2)=(x_1^2+y_1^2,x_2^2+y_2^2)=1$"}
{"_id": "7122458", "title": "", "text": "$f_r(r)=\\dfrac{2r}{R^2},$"}
{"_id": "6020468", "title": "", "text": "$ (4n)!+3, (4n)!+7,......,(4n)!+3+4n$"}
{"_id": "7601234", "title": "", "text": "$\\lim_{x\\to 0^-}f'(x) \\ne \\lim_{x\\to 0^+}f'(x) $"}
{"_id": "3697809", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^s} = \\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{x^{s-1}}{e^x+1}\\,dx $"}
{"_id": "9033711", "title": "", "text": "$1/|x| >1$"}
{"_id": "7610889", "title": "", "text": "$g(x) = 3x^2 - x + 2$"}
{"_id": "3626561", "title": "", "text": "$ \\left\\{ \\matrix{   y < 0 \\hfill \\cr    a < x < b \\hfill \\cr    {{b - z\\left( {1 - y} \\right)} \\over y} < a < x < b < {{a - z\\left( {1 - y} \\right)} \\over y} \\hfill \\cr}  \\right. $"}
{"_id": "6287179", "title": "", "text": "$[x,y]=x\\cdot y-y\\cdot x$"}
{"_id": "8293871", "title": "", "text": "$[x,y] = ax + by$"}
{"_id": "2726993", "title": "", "text": "$F: (\\mathbb{R}^n,\\mathcal{O}_{\\mathbb{R}^n}) \\to (F(\\mathbb{R}^n),\\mathcal{O}_{\\mathbb{R}^{m+n} | \\mathbb{R}^n})$"}
{"_id": "7334009", "title": "", "text": "$L = \\overline{p_{1}p_{2}}$"}
{"_id": "8426216", "title": "", "text": "$||f_3||$"}
{"_id": "8664765", "title": "", "text": "$M_{0,1}\\to M_{1,1}$"}
{"_id": "7952337", "title": "", "text": "$\\frac{16}{a^2} + \\frac{9}{b^2} = 1$"}
{"_id": "6428134", "title": "", "text": "$\\phi = (a_1,a_2,a_3,....,a_n)$"}
{"_id": "3739115", "title": "", "text": "$f(a_1)=f(a_1^\\prime) \\implies a_1=a_1^\\prime$"}
{"_id": "4610643", "title": "", "text": "$\\{\\omega\\in\\Omega| \\lim_{n\\rightarrow\\infty} X_n(\\omega)=X(\\omega)\\}\\in\\mathcal{F}$"}
{"_id": "6686404", "title": "", "text": "$I(s) = \\int_0^\\infty \\frac{\\log^{s-1}(1+x)}{x(x+1)} dx = \\int_0^\\infty \\frac{u^{s-1}du}{e^u-1} = \\zeta(s)\\Gamma(s)$"}
{"_id": "1441318", "title": "", "text": "$|f(a_1)|\\le \\frac{1}{2}|f(a_0)|$"}
{"_id": "4089976", "title": "", "text": "$D(x)M\\in\\langle P\\rangle$"}
{"_id": "5762354", "title": "", "text": "$\\int_0^{+\\infty}\\int_0^{+\\infty}F(x,t)dxdt=\\int_0^{+\\infty}\\int_0^{+\\infty}F(x,t)dtdx.$"}
{"_id": "4177042", "title": "", "text": "$A_1\\subset A_2\\subset\\ldots\\subset A_n\\subset \\ldots\\subset X$"}
{"_id": "9117440", "title": "", "text": "$f(n)=(2n-3) f(n-1) -x^2 f(n-2)$"}
{"_id": "3241617", "title": "", "text": "$A_{1}A_{2}A_{4}$"}
{"_id": "2910427", "title": "", "text": "$F=M\\oplus N$"}
{"_id": "6040927", "title": "", "text": "$h(n)=n^3-n+1$"}
{"_id": "591949", "title": "", "text": "$\\begin{pmatrix} 1&0&0\\\\ 0&r^2&0\\\\ 0&0&r^2\\sin(\\xi)^2 \\end{pmatrix}$"}
{"_id": "6079773", "title": "", "text": "$\\frac{1}{e} - \\left( \\frac{x}{x+1} \\right)^x$"}
{"_id": "7717779", "title": "", "text": "$ \\lim_{n\\to \\infty}\\frac{2n}{n}=\\lim_{n\\to \\infty}\\frac{3n}{n}=\\frac{\\infty}{\\infty}=1 $"}
{"_id": "5880470", "title": "", "text": "$lim_{n\\rightarrow \\infty} \\sum_{i=1}^\\infty|x_i^n|=\\sum_{i=1}^\\infty|x_i|$"}
{"_id": "4603789", "title": "", "text": "$P+N=F$"}
{"_id": "3127667", "title": "", "text": "$\\gamma,\\gamma' < \\kappa$"}
{"_id": "2359937", "title": "", "text": "$\\,p_i\\mid a\\, \\Rightarrow\\ p_1\\cdots p_n\\mid a\\ \\ \\ $"}
{"_id": "7237150", "title": "", "text": "$\\root n \\of re^{i\\theta/n}$"}
{"_id": "6084467", "title": "", "text": "$w = \\frac{(ab - d) }{c - a - b}$"}
{"_id": "9188263", "title": "", "text": "$\\int_{-\\pi}^{\\pi}=\\int_{-\\pi}^{0}+\\int_{0}^{\\pi}$"}
{"_id": "977161", "title": "", "text": "$\\Delta f(x)= f'(x)\\Delta x + \\epsilon (\\Delta x) \\Delta x$"}
{"_id": "4950847", "title": "", "text": "$1+2+3+4+5+\\dotsb =\\frac{-1}{12}$"}
{"_id": "3568436", "title": "", "text": "$a_n = \\frac{n(n+1)(n+2)}{6}$"}
{"_id": "8013736", "title": "", "text": "$[x, y, z]\\succ  [1, 1, 1]$"}
{"_id": "4429994", "title": "", "text": "$COV (X,Y) = 0$"}
{"_id": "7000144", "title": "", "text": "$x =c_1e_1+c_2e_2+..c_ne_n$"}
{"_id": "2843480", "title": "", "text": "$36^x=9^x \\times 4^x=3^{2x}\\times 4^x$"}
{"_id": "3939883", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y} = \\frac{2}{40}$"}
{"_id": "2589476", "title": "", "text": "$z = \\sqrt[n]{r} e^{i(\\theta + 2 \\pi k)/n} \\ \\text{ for } \\ k = 0, 1,..., n-1$"}
{"_id": "5234105", "title": "", "text": "$dC=\\sqrt{\\bigg(\\frac{dx}{d\\theta}\\bigg)^2+\\bigg(\\frac{dy}{d\\theta}\\bigg)^2}\\,d\\theta$"}
{"_id": "6763687", "title": "", "text": "$\\frac{19}7=2+\\frac57=2+\\frac1{\\frac75}=2+\\frac1{1+\\frac25}=2+\\frac1{1+\\frac1{\\frac52}}=2+\\frac1{1+\\frac1{2+\\frac12}}$"}
{"_id": "8771151", "title": "", "text": "$|f'(z)| \\leq 2|f'(p)|$"}
{"_id": "8551754", "title": "", "text": "$z = \\sqrt[4]{8} e^{\\frac{5\\pi}{8}i}$"}
{"_id": "1173752", "title": "", "text": "$(\\gamma(i),\\gamma(j))_{\\gamma(0)} - d(\\gamma(0),x) \\leq (\\gamma(i),\\gamma(j))_x$"}
{"_id": "2465082", "title": "", "text": "$\\tan\\theta=y/2$"}
{"_id": "9059184", "title": "", "text": "$f(f(x))^2=x^3f(x)$"}
{"_id": "2246427", "title": "", "text": "$G=\\mathbb{Z}\\oplus \\mathbb{Z}$"}
{"_id": "4531421", "title": "", "text": "$ \\left|\\begin{matrix} 1 & l & l^2 \\\\ 1 & b & b^2 \\\\ 1 & h & h^2 \\end{matrix}\\right| = 0 $"}
{"_id": "6295133", "title": "", "text": "$|f_n (x)| \\leq 2 |f(x)|$"}
{"_id": "8777423", "title": "", "text": "$c/|x|$"}
{"_id": "410913", "title": "", "text": "$h^+$"}
{"_id": "6070675", "title": "", "text": "$ \\sum_{n = 1}^{\\infty}\\sum_{k = 1}^{\\infty} {1 \\over \\left(k^{2} + n^{2}\\right)^{\\left(1+\\epsilon\\right)/2}} $"}
{"_id": "3443975", "title": "", "text": "$G(x) = (x - a)^{n + 1}$"}
{"_id": "623618", "title": "", "text": "$\\frac{1}{n^2}\\sum_{k=1}^nk H_k  = \\frac{1}{n^2}H_n \\sum_{k=1}^n k + \\frac{1}{n^2}\\sum_{k=1}^{n-1}\\left(\\sum_{j=1}^kj \\right)(H_k - H_{k+1}) \\\\ = \\frac{H_n}{n^2}\\frac{n(n+1)}{2} - \\frac{1}{4}\\left(1 - \\frac{1}{n}\\right)  $"}
{"_id": "2544795", "title": "", "text": "$ \\left\\{ \\matrix{   \\left( {2a - b} \\right)x = \\left( {2a - b} \\right)d + 2\\left( {u - ad} \\right) + 2\\left( {3b - 2c} \\right)n \\hfill \\cr    \\left( {2a - b} \\right)y =  - \\left( {u - ad} \\right) + \\left( {c - 3a} \\right)n \\hfill \\cr    z = n \\hfill \\cr}  \\right. $"}
{"_id": "5037649", "title": "", "text": "$f(n) = n^2 + n + 3$"}
{"_id": "8416770", "title": "", "text": "$\\overline{T} = \\begin{pmatrix} \\overline{A} & -B \\\\ \\overline{B} & A \\end{pmatrix}$"}
{"_id": "3495541", "title": "", "text": "$\\int \\kappa\\,ds \\geq 2\\pi$"}
{"_id": "9188336", "title": "", "text": "$x \\in V_x \\subseteq \\overline{V_x} \\subseteq U$"}
{"_id": "1476819", "title": "", "text": "$|abcd|=1$"}
{"_id": "2842860", "title": "", "text": "$ f(x)\\leq \\lim_{n\\rightarrow\\infty} \\inf f(x_n)$"}
{"_id": "9046075", "title": "", "text": "$\\operatorname {Cov}(X,Y)=0.$"}
{"_id": "3899479", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{x_1+x_2+\\dots+x_n}{n}$"}
{"_id": "4639249", "title": "", "text": "$\\frac{1}{n!}\\le \\frac{1}{2^{n-1}}$"}
{"_id": "211328", "title": "", "text": "$p(x)=(x-a)^{n-1}\\bigl((x-a)q(x)\\bigr)$"}
{"_id": "3796906", "title": "", "text": "$a,b,c,d\\in\\mathbb{R}. a<b,c<d$"}
{"_id": "8295006", "title": "", "text": "$(\\Delta\\Delta a)(n-1) = -1\\;.$"}
{"_id": "7483547", "title": "", "text": "$\\frac{2x}{1-x^2} = \\frac{\\sqrt{1 - x^2}}{x}$"}
{"_id": "6572572", "title": "", "text": "$\\lim_{n\\to\\infty}   \\int_{-\\infty}^{\\infty} \\frac{1}{(1+x^2)^n}\\,dx $"}
{"_id": "948006", "title": "", "text": "$f_1(n)=n^{\\log{n}}$"}
{"_id": "8596635", "title": "", "text": "$\\mathbb{A} = \\{a,a+b,a+2b,...\\}$"}
{"_id": "5549458", "title": "", "text": "$Cov(\\mathbf{x},\\mathbf{u})=\\mathbf{0}$"}
{"_id": "260432", "title": "", "text": "$S \\oplus T$"}
{"_id": "2000031", "title": "", "text": "$f(x)=\\frac{e^x}{1+e^{2x}}$"}
{"_id": "580900", "title": "", "text": "$\\Bbb R[x]/(x^2+1)\\cong \\Bbb C$"}
{"_id": "4431672", "title": "", "text": "$(\\cal X, \\cal A)$"}
{"_id": "4287667", "title": "", "text": "$t\\langle \\ddot{\\gamma}(s) \\times \\gamma(s),\\, \\dot{\\gamma}(s)\\rangle/t^2,\\text{ or }(1/t)\\det(\\dot{\\gamma}(s)\\text{ }\\ddot{\\gamma}(s)\\text{ }\\gamma(s)).$"}
{"_id": "324736", "title": "", "text": "$T_x(X).$"}
{"_id": "1693128", "title": "", "text": "$\\int_{0}^{\\pi} \\frac{\\cos(nx)}{(p+\\cos(x))^2+q^2}\\ \\mathrm dx$"}
{"_id": "7428844", "title": "", "text": "$1+\\sum_n (x_n+y_n) =1+\\frac{b}{1-\\frac{1+\\sqrt{5}}4} + \\frac{c}{1-\\frac{1-\\sqrt{5}}{4}} = 1+b(3+\\sqrt{5}) + c(3-\\sqrt{5})$"}
{"_id": "8604225", "title": "", "text": "$\\vert \\vert A \\vert \\vert_2 = \\sqrt{ \\lambda_{max} (A^T A)} $"}
{"_id": "3344921", "title": "", "text": "$\\cdots \\subseteq V_{-1} \\subseteq V_0 \\subseteq V_1 \\subseteq V_2 \\subseteq \\cdots$"}
{"_id": "4143408", "title": "", "text": "$s \\oplus t$"}
{"_id": "6933722", "title": "", "text": "$Cov(X,Y)=-0.75$"}
{"_id": "1351257", "title": "", "text": "$ =(1-q)^2\\int_1^\\infty\\frac{dx}{x(x-q)^{n+1}}(x-1)^{n-1}\\ . $"}
{"_id": "733095", "title": "", "text": "$x^3, x^5, x^7,\\ldots$"}
{"_id": "3717297", "title": "", "text": "$f(x)=\\displaystyle\\frac{2x}{1+x^2}$"}
{"_id": "8325144", "title": "", "text": "$\\sum\\limits_{N\\geq 1} \\frac{1}{N^2}$"}
{"_id": "5166991", "title": "", "text": "$y=\\dfrac{a+c}{b+d}$"}
{"_id": "3261111", "title": "", "text": "$\\text{int}A\\subseteq A\\subseteq\\overline{A}$"}
{"_id": "1216303", "title": "", "text": "$A = \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix}$"}
{"_id": "5305927", "title": "", "text": "$\\sum_{n=1}^{+\\infty}\\frac{1}{n(2n+1)}=2\\sum_{n=1}^{+\\infty}\\int_{0}^{1}\\left(x^{2n-1}-x^{2n}\\right)\\,dx = 2\\int_{0}^{1}\\frac{x}{1+x}\\,dx = 2-\\log 4.$"}
{"_id": "6854659", "title": "", "text": "$P \\frac {R}{1-(1+R)^{-N}}$"}
{"_id": "9098192", "title": "", "text": "$\\sigma_1 = \\sqrt{\\lambda_{max}(A^TA)}$"}
{"_id": "598604", "title": "", "text": "$f(x)=\\frac{e^x}{e^x+1}$"}
{"_id": "4789633", "title": "", "text": "$A(z)-z = z A(z) + \\frac{1}{2} (A(z)^2 - 2z A(z) + A(z^2)) = \\frac{1}{2} (A(z)^2 + A(z^2)).$"}
{"_id": "2465826", "title": "", "text": "$2n , 2n+1 , ... , 3n$"}
{"_id": "6770313", "title": "", "text": "$\\Vert x_n-x_m\\Vert>1$"}
{"_id": "2742812", "title": "", "text": "$1/|\\omega|$"}
{"_id": "4115147", "title": "", "text": "$f\\big(xf(y)+f(x)\\big) = 2f(x)+xy.$"}
{"_id": "6080982", "title": "", "text": "$G \\cong A_{1} \\times ... \\times A_{n}$"}
{"_id": "172055", "title": "", "text": "$|a+b|=s-t$"}
{"_id": "6625568", "title": "", "text": "$ \\sum_{i=1}^n k\\frac{3i}{n}=1. $"}
{"_id": "7245324", "title": "", "text": "$R(M,v):={\\langle v, Mv\\rangle \\over |v|^2}$"}
{"_id": "8136170", "title": "", "text": "$\\frac{n(n-1)(n-2)(n-3)}{24}$"}
{"_id": "5328054", "title": "", "text": "$\\displaystyle \\tan^{-1}\\left(\\frac{1}{x^2+x(2n-1)+n^2-n+1}\\right) = \\tan^{-1}\\left(\\frac{1}{1+(x+n)\\cdot (x+n-1)}\\right) = \\tan^{-1}\\left(x+n\\right)-\\tan^{-1}(x+n-1)$"}
{"_id": "7929389", "title": "", "text": "$A = 2\\pi\\int\\rho\\, ds,$"}
{"_id": "7410802", "title": "", "text": "$y = \\dfrac{ax^2 + bx + c}{dx + e}$"}
{"_id": "3267133", "title": "", "text": "$X\\sim Exp(\\lambda) \\rightarrow f(x) = \\frac{1}{\\lambda} e^{\\frac{-x}{\\lambda}}$"}
{"_id": "5616289", "title": "", "text": "$A=\\pmatrix{1&2&3\\cr 4&5&6\\cr 2&1&0\\cr 1&0&0\\cr 0&1&0\\cr 0&0&1\\cr }$"}
{"_id": "2630011", "title": "", "text": "$\\gamma=\\gamma+ \\beta\\le \\gamma+\\beta$"}
{"_id": "25848", "title": "", "text": "$\\mathbb{C}=\\mathbb{R}[x]/(x^2+1)$"}
{"_id": "8656705", "title": "", "text": "$z_n=\\dfrac{a_1+a_2+...+a_n}{n}$"}
{"_id": "5941319", "title": "", "text": "$L(\\gamma,\\dot{\\gamma},\\ddot{\\gamma})=\\left|\\ddot{\\gamma}\\right|^{2}+K(\\gamma,\\dot{\\gamma})$"}
{"_id": "6868409", "title": "", "text": "$\\frac{\\delta J}{\\delta \\gamma_i} = \\frac{\\partial F}{\\partial \\gamma_i} - \\frac{d}{dt}\\frac{\\partial F}{\\partial \\gamma_i'} = \\frac{\\partial }{\\partial \\gamma_i}H(\\gamma(t)) - \\frac{d}{dt}\\frac{\\partial }{\\partial \\gamma_i'} \\sum_j\\gamma_j'(t)^2 = \\frac{\\partial }{\\partial \\gamma_i}H(\\gamma(t)) - 2\\gamma_i''(t)$"}
{"_id": "6111614", "title": "", "text": "$\\mathsf{P}(\\{s\\})=\\frac16$"}
{"_id": "6898705", "title": "", "text": "$g(t)=F(p+t)-F(p-t)-tf(p-t)-tf(p+t)$"}
{"_id": "7828069", "title": "", "text": "$\\int_{-\\infty}^{\\infty} \\frac{1}{(x^2 + 1)^2 x^{1/2}}dx$"}
{"_id": "1007345", "title": "", "text": "$\\sqrt r e^{\\frac {i \\theta}{2}}$"}
{"_id": "6291131", "title": "", "text": "$[a,b] / [c,d] = \\mathbb{R}$"}
{"_id": "2441579", "title": "", "text": "$E(X) = 1/6$"}
{"_id": "3111588", "title": "", "text": "$xRyR=yRxR$"}
{"_id": "6058311", "title": "", "text": "$x = y = z;\\ x = y > z;\\ x > y = z;\\ x > y > z;\\ x > z > y;\\ x = z > y;\\ y > x = z;\\ y > x > z;\\ y > z > x;\\ y = z > x;\\ z > x = y;\\ z > x > y;\\  z > y > x.$"}
{"_id": "836809", "title": "", "text": "$e(P_3\\mid P_1)=e(P_3\\mid P_2)\\cdot e(P_2\\mid P_1)$"}
{"_id": "4150730", "title": "", "text": "$\\sum\\limits_{k=1}^n\\|e_k-v_k\\|^2<1$"}
{"_id": "3517351", "title": "", "text": "$x = e^{\\frac{i\\pi}{n}} t$"}
{"_id": "8161743", "title": "", "text": "$\\varphi(p)=\\varphi(2p)=p-1$"}
{"_id": "8947174", "title": "", "text": "$\\mathbb R^n / O(n) = \\mathbb R^n$"}
{"_id": "1556382", "title": "", "text": "$f(x)=(x-4)^9$"}
{"_id": "5937156", "title": "", "text": "$2 \\ln |x| = \\ln |x|^2 = \\ln (x^2)$"}
{"_id": "979214", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} X_n(\\omega)=\\lim_{n\\rightarrow\\infty} E(X_n)=c$"}
{"_id": "6609552", "title": "", "text": "$I_1\\supseteq I_2\\supseteq\\dots\\supseteq I_n\\supseteq I_{n+1}\\supseteq\\dots$"}
{"_id": "7513938", "title": "", "text": "$P(S \\mid E) = \\frac{P(E \\mid S)~P(S)}{P(E)}$"}
{"_id": "6008984", "title": "", "text": "$\\displaystyle I_m = \\int \\frac1{(1+t^2)^m}dt$"}
{"_id": "7197608", "title": "", "text": "$y= \\frac{-Ax + Bx +C}{ B +A} $"}
{"_id": "6447431", "title": "", "text": "$\\begin{cases}x_1+x_2+x_3=k\\\\1\\leq x_1\\\\1\\leq x_2\\\\1\\leq x_3\\end{cases}$"}
{"_id": "5488696", "title": "", "text": "$N(\\gamma(t))=\\gamma''-\\left\\langle \\gamma'',\\,\\frac{\\gamma'}{||\\gamma'||}\\right\\rangle\\frac{\\gamma'}{||\\gamma'||}=\\gamma''-\\frac1{||\\gamma'||^2}\\langle\\gamma'',\\gamma'\\rangle\\gamma'$"}
{"_id": "3247651", "title": "", "text": "$  \\int k_g ds + \\int\\int K dA = 2 \\pi \\tag2 $"}
{"_id": "5256910", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       k&1&1&-2\\\\       1&k&1&1\\\\       1&1&k&1     \\end{array} \\right] $"}
{"_id": "6597871", "title": "", "text": "$k=\\frac{a}{2}$"}
{"_id": "4752597", "title": "", "text": "$\\det \\begin{bmatrix} x_1&y_1&0\\\\x_2&y_2&0\\\\x_3&y_3&1\\end{bmatrix}$"}
{"_id": "802359", "title": "", "text": "$\\forall\\epsilon>0, \\exists\\delta>0 \\text{ s.t. } |x - 0|<\\delta \\implies |f(x) - f(0)|<\\epsilon$"}
{"_id": "5675009", "title": "", "text": "$\\kappa(s) = \\|\\gamma''(s)\\| = |\\langle \\gamma''(s),N(s)\\rangle| = |\\frac{d}{ds} \\langle \\gamma'(s),N(s)\\rangle - \\langle \\gamma'(s),\\frac{d}{ds}N(s)\\rangle| = |\\langle \\gamma'(s),\\frac{d}{ds} N(s)\\rangle|.$"}
{"_id": "183699", "title": "", "text": "$\\delta^+$"}
{"_id": "1581082", "title": "", "text": "$\\sin{2\\varphi}=2\\sin{\\varphi}\\cos{\\varphi}.$"}
{"_id": "7028676", "title": "", "text": "$||A||_2 = \\sqrt{\\rho (A^T A)}=\\sqrt{\\rho (I)}=1$"}
{"_id": "4384821", "title": "", "text": "$\\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^N f(x_n) = \\int_0^1 f$"}
{"_id": "4052926", "title": "", "text": "$ f(\\delta) = \\delta - (n - 1)\\delta^{n-1} + n\\delta^{n-1} - n\\delta + n - 2 = \\delta(1- n) + n - 2 + \\delta^{n-1} = \\delta^{n-1} - 1 + (n-1)(1 - \\delta) = (\\delta - 1)(\\delta^{n-2} + \\dots + \\delta + 1 - n + 1) > 0, $"}
{"_id": "6159984", "title": "", "text": "$ f(x^2+xf(y))= xf(y)$"}
{"_id": "5668424", "title": "", "text": "$K^{\\times}=\\left(\\mathbb{F}_3[X]/(X^3-X^2+1)\\right)^{\\times}$"}
{"_id": "3931861", "title": "", "text": "$\\large\\sqrt z=re^{\\frac{\\theta}{2}i}$"}
{"_id": "7663590", "title": "", "text": "$\\frac{b-2a\\cos\\gamma}{a\\sin\\gamma} = \\frac{b-2a\\cos\\gamma}{a\\sin\\gamma}\\cdot\\frac{b}{b}=\\frac{b^2-2ab\\cos\\gamma}{ab\\sin\\gamma}=\\frac{a^2+b^2-2ab\\cos\\gamma-a^2}{ab\\sin\\gamma}=\\frac{c^2-a^2}{ab\\sin\\gamma}$"}
{"_id": "1974170", "title": "", "text": "$\\sum_{n\\geq1}\\frac{\\sin^{2}\\left(n\\right)}{n^{2}}=\\frac{1}{2}\\left(\\sum_{n\\geq1}\\frac{1}{n^{2}}-\\sum_{n\\geq1}\\frac{\\cos\\left(2n\\right)}{n^{2}}\\right)$"}
{"_id": "6694802", "title": "", "text": "$d(x,A)\\leqslant d(x,y)+d(y,a)$"}
{"_id": "5128386", "title": "", "text": "$\\mathbb{Z}_p=\\{a,a+\\beta, a+2\\beta,\\dots , a+(p-1)\\beta\\}\\subseteq A $"}
{"_id": "4735732", "title": "", "text": "$\\ldots <_M \\{3\\} <_M \\{2\\} <_M \\{1\\} <_M \\{0\\}$"}
{"_id": "9239424", "title": "", "text": "$x, y, z \\in S, d(x, z) \\le d(x, y) + d(y, z)$"}
{"_id": "2700876", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{\\sqrt{1} + \\sqrt{2} +\\ldots + \\sqrt{n}}{n\\sqrt{n}}=\\lim_{n \\to \\infty}\\frac{1}{n}\\sum_{k=1}^n\\sqrt{\\frac{k}{n}}=\\int_0^1\\sqrt{x}\\;dx=\\frac{2}{3}.$"}
{"_id": "5508365", "title": "", "text": "$rx = r e^{\\frac{2\\pi i}{n}}$"}
{"_id": "4921255", "title": "", "text": "$P(X \\geq x \\mid H_{0})$"}
{"_id": "5088815", "title": "", "text": "$F_n \\ge (\\frac{1+\\sqrt{5}}{2})^{n-2}$"}
{"_id": "2832759", "title": "", "text": "$3^x+4^x+5^x = 1+x^2$"}
{"_id": "3446095", "title": "", "text": "$\\zeta(A+B) = \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k=1}^N \\frac{1}{k^{A}\\log\\left(1 + \\frac{k^{B}}{N}\\right)}$"}
{"_id": "913951", "title": "", "text": "$f_X(x)=\\dfrac{1}{\\sqrt{2\\pi}}e^{\\frac{-x^2}{2}}$"}
{"_id": "3158856", "title": "", "text": "$f(t)= \\int_0^1 \\frac{\\sin(xt)}{x}\\:dx$"}
{"_id": "3702887", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}\\sum^{n}_{r=1}\\frac{r}{n^2+n+r}$"}
{"_id": "6062065", "title": "", "text": "$\\displaystyle{ \\sum_{n=1}^{\\infty} \\sum_{m=1}^{\\infty} \\frac{\\sin(\\sin(nm))}{n^2+m^2}}.$"}
{"_id": "7572350", "title": "", "text": "$ \\begin{pmatrix} a & b \\\\ -\\bar{b} & \\bar{a} \\end{pmatrix}. $"}
{"_id": "6082564", "title": "", "text": "$\\displaystyle f(k)=\\frac{9^k}{9^k+3},$"}
{"_id": "8263459", "title": "", "text": "$I_n\\to\\pi$"}
{"_id": "7617652", "title": "", "text": "$\\displaystyle tan\\theta = \\frac{z}{2}$"}
{"_id": "6740484", "title": "", "text": "$\\lim\\; \\inf\\; a_n = \\lim_{n\\to\\infty} \\inf(\\bigcup_{k=n}^\\infty a_k)$"}
{"_id": "2902131", "title": "", "text": "$d(x,A) = \\inf \\{d(x,y):y\\in A \\}$"}
{"_id": "8993935", "title": "", "text": "$a\\,x+b\\,y=u$"}
{"_id": "7373821", "title": "", "text": "$\\tan^2 \\theta = \\dfrac{x}{y}$"}
{"_id": "2039009", "title": "", "text": "$a \\cdot b = |A \\times B|$"}
{"_id": "7737252", "title": "", "text": "$f(n) = n^2 - 2n$"}
{"_id": "4578117", "title": "", "text": "$\\sum\\|e_n-f_n\\| < 1$"}
{"_id": "7470692", "title": "", "text": "$\\det\\begin{pmatrix} A & I\\\\B & O\\end{pmatrix}=\\det B$"}
{"_id": "6787548", "title": "", "text": "$X^n+X^{n-1}$"}
{"_id": "3427505", "title": "", "text": "$ \\begin{align} &\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1} }{n(n+1)} \\, \\eta(n) \\\\ &=  \\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1} }{n(n+1)}\\frac{1}{\\Gamma(n)} \\int_{0}^{\\infty} \\frac{x^{n-1}}{1+e^{x}} \\, dx \\\\ &= \\int_{0}^{\\infty} \\frac{1}{1+e^{x}} \\frac{1}{x^{2}} \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n!} \\, \\frac{x^{n+1}}{n+1} \\, dx \\\\ &= \\int_{0}^{\\infty} \\frac{1}{1+e^{x}} \\frac{1}{x^{2}} \\left(e^{-x}+x-1 \\right) \\, dx \\\\ &= \\lim_{s \\to -1^{+}} \\int_{0}^{\\infty} \\frac{x^{s-1}}{1+e^{x}} \\left(e^{-x}+x-1 \\right) \\, dx \\\\ &= \\lim_{s \\to -1^{+}} \\left[2^{-s} \\,  \\Gamma(s) \\left( \\zeta(s) - \\zeta \\left(s, \\frac{3}{2} \\right)\\right) + \\Gamma(s+1) \\eta(s+1 ) - \\Gamma(s) \\eta(s) \\right] \\\\&= \\lim_{s \\to -1^{+}} \\left[2^{-s} \\, \\Gamma(s) \\left(\\zeta(s) - \\zeta \\left(s, \\frac{1}{2} \\right) +2^{s}\\right) + \\Gamma(s+1) \\eta(s+1 ) - \\Gamma(s) \\eta(s) \\right] \\tag{3} \\\\&= \\lim_{s \\to -1^{+}} \\left[2^{-s} \\, \\Gamma(s) \\left(\\zeta(s) - \\left(2^{s}-1\\right)\\zeta (s)  +2^{s}\\right) + \\Gamma(s+1) \\eta(s+1 ) - \\Gamma(s) \\eta(s) \\right] \\tag{4} \\\\ &= \\lim_{s \\to -1^{+}} \\left[\\left(2^{1-s}-1 \\right) \\Gamma(s) \\zeta(s) + \\Gamma(s) + \\Gamma(s+1) \\eta(s+1 ) - \\Gamma(s) \\eta(s)\\right] \\\\&= \\lim_{s \\to -1^{+}} \\left[ \\Gamma(s) + \\Gamma(s+1) \\eta(s+1) -2\\Gamma(s) \\eta(s) \\right] \\tag{5}. \\end{align}$"}
{"_id": "8728079", "title": "", "text": "$\\mathbb{Q}[x]/(x^4+1) \\to \\mathbb{C}$"}
{"_id": "7093169", "title": "", "text": "$\\int_0^{\\infty} F(x)$"}
{"_id": "9022025", "title": "", "text": "$x=\\dfrac{a-.5c+.25d}{a+c+d}$"}
{"_id": "443578", "title": "", "text": "$\\Re(s)=1/2$"}
{"_id": "4051765", "title": "", "text": "$\\begin{align} \\int \\frac{1}{(x+1)^2} \\, dx & = -\\frac{1}{x+1} + c \\end{align}$"}
{"_id": "2874653", "title": "", "text": "$\\sum_{k=1}^\\color{red}{n} \\frac1{2^k}$"}
{"_id": "5064316", "title": "", "text": "$\\begin{align} d(\\alpha, A)-d(x,A) &= \\inf\\{ d(\\alpha, a):a\\in A\\}-\\inf\\{d(x,a): a \\in A\\}\\\\ &= \\inf\\{ d(\\alpha, a) - d(x,a): a \\in A\\}\\\\ &\\leqslant \\inf\\{d(\\alpha, x): a \\in A\\} = d(x,\\alpha) \\end{align}$"}
{"_id": "5251539", "title": "", "text": "$\\mathcal A x = \\mathcal A$"}
{"_id": "5624164", "title": "", "text": "$\\int_0^1 \\frac{\\tanh^{-1}(x)\\ln x}{x(1-x^2)} \\, dx $"}
{"_id": "3728711", "title": "", "text": "$\\tau = \\{ \\emptyset, A_1, A_2, \\dots \\} $"}
{"_id": "4512666", "title": "", "text": "$I_n = \\int_0^{\\pi/2} \\dfrac{\\sin(2n+1)x}{\\sin(x)}dx$"}
{"_id": "273394", "title": "", "text": "$T_xX$"}
{"_id": "2445881", "title": "", "text": "$\\forall \\epsilon > 0, \\exists z  \\neq 0, |z - 0| < \\epsilon \\land z \\in \\{0\\}$"}
{"_id": "3163043", "title": "", "text": "$m=\\Omega(\\sqrt{n})$"}
{"_id": "2090498", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{1}{n^2}I_n \\ge \\frac{1}{N^2}I_N - \\sum_{n\\ne N}\\frac{1}{n^2}|I_n| \\ge \\frac{N^3}{N^2} -  \\pi^2/6,$"}
{"_id": "7045272", "title": "", "text": "$0=f(\\sigma)+\\sigma f(\\sigma)$"}
{"_id": "6236606", "title": "", "text": "$\\sqrt {(3-x)^2 + 2^2} $"}
{"_id": "5214199", "title": "", "text": "$f(x) = b + \\dfrac{a-b^2}{x+b} = \\dfrac{bx + a}{x+b}$"}
{"_id": "7602862", "title": "", "text": "$\\tan\\theta=\\frac{x}{h}$"}
{"_id": "6205863", "title": "", "text": "$ \\bar{a}r + \\bar{b}s = 1 $"}
{"_id": "4217677", "title": "", "text": "$1 + \\sum_{k=1}^{n-1}{\\cos{\\frac{2\\pi k}{n}}} + \\mathrm{i} \\sum_{k=1}^{n-1}{\\sin{\\frac{2\\pi k}{n}}} = 0$"}
{"_id": "1958100", "title": "", "text": "$\\lim_{n\\rightarrow \\infty} \\frac{1+\\sqrt{2}+\\sqrt[3]{3}+\\cdots+\\sqrt[n]{n}}{n}$"}
{"_id": "717425", "title": "", "text": "$ GCD(a,b) = GCD(a, b-a) = GCD (a, r_b) $"}
{"_id": "4492371", "title": "", "text": "$\\lim_{x \\to \\infty}f(x)g(x) = \\lim_{x \\to \\infty}f(x)\\cdot\\lim_{x \\to \\infty}g(x)$"}
{"_id": "2527845", "title": "", "text": "$d_2 \\mid a$"}
{"_id": "5487463", "title": "", "text": "$ \\det\\begin{pmatrix} A & B \\\\ C & D \\\\ \\end{pmatrix}=\\det(A-B)\\det(A+B), $"}
{"_id": "779010", "title": "", "text": "$f(a+b) = f(a)+f(b)$"}
{"_id": "8941575", "title": "", "text": "$L = \\lim_{n\\to \\infty } \\sup(n) = \\lim_{n\\to \\infty } \\inf(n)$"}
{"_id": "4585864", "title": "", "text": "$P(S\\cup E)=P(S)+P(E)-P(S\\cap E)$"}
{"_id": "2377006", "title": "", "text": "$-y''(y'-y'')^2 + \\dfrac{y'''}{y'-y''}(y''-y''') = y''''$"}
{"_id": "6081405", "title": "", "text": "$[\\Lambda, \\partial] = i\\bar{\\partial}^*$"}
{"_id": "5748585", "title": "", "text": "$c'(a,d)=e(a,a+d,a+2d,...,a+(m−1)d)$"}
{"_id": "6462456", "title": "", "text": "$f: P \\to T, P(T) = P(P)$"}
{"_id": "2139692", "title": "", "text": "$b_n=a_n+1=\\frac{n(n+1)}{2}+1=\\frac{n^2+n+2}{2}.$"}
{"_id": "4564199", "title": "", "text": "$ P( E | S ) = \\dfrac{P(S | E) \\times P(E)}{P(S)} $"}
{"_id": "7235338", "title": "", "text": "$\\displaystyle\\sum_{n=1}^{\\infty}|\\langle f, e_n \\rangle|^2 = ||f||^2,$"}
{"_id": "1531067", "title": "", "text": "$b_n=(\\frac{1}{n}-\\frac{1}{n^2})<a_n <\\frac{1}{n}=c_n$"}
{"_id": "1773018", "title": "", "text": "$(e_i)_{1\\leq i\\leq n}$"}
{"_id": "2346597", "title": "", "text": "$F\\colon\\mathbb R^{n+1}\\to\\mathbb R$"}
{"_id": "183469", "title": "", "text": "$X \\subseteq Y \\subseteq \\overline{X}$"}
{"_id": "5784213", "title": "", "text": "$S= \\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{m^2(m^2+n^2)}.$"}
{"_id": "4068322", "title": "", "text": "$f_r(r)= \\frac{2r}{R^{2}}$"}
{"_id": "5772736", "title": "", "text": "$\\sqrt{(1-p)/p^2}$"}
{"_id": "4135906", "title": "", "text": "$\\frac{1}{n^2(n+1)^2}=\\left(\\frac{1}{n}-\\frac{1}{n+1}\\right)^2=\\frac{1}{n^2}+\\frac{1}{(n+1)^2}-\\left(\\frac{2}{n}-\\frac{2}{n+1}\\right),$"}
{"_id": "5742640", "title": "", "text": "$f(x^{-1}) f(x) = f(x^{-1} x) = f(1) = 1 = f(1) = f(x x^{-1}) = f(x) f(x^{-1})$"}
{"_id": "1828163", "title": "", "text": "$\\frac {(1+r)^t}{(1+s)^t}$"}
{"_id": "6092655", "title": "", "text": "$P(K) \\implies P(k + 1)$"}
{"_id": "5292883", "title": "", "text": "$ \\frac{1}{x} + \\frac{1}{y} = \\frac{1}{2} $"}
{"_id": "6219051", "title": "", "text": "$e_n := \\|X_n\\|$"}
{"_id": "8765944", "title": "", "text": "$ f(x)=f(a)+\\int_a^xf'(t)dt=f(a)+\\int_a^xf(t-a)dt=f(a)+\\int_0^{x-a}f(t)dt, $"}
{"_id": "1791754", "title": "", "text": "$(k+1)^3-(k-1)^3-2^3=6(k-1)(k+1)=6k^2-6$"}
{"_id": "2871895", "title": "", "text": "$ \\begin{bmatrix} x & y\\\\ -y & x \\end{bmatrix} $"}
{"_id": "1711147", "title": "", "text": "$JM(A,B)J=M(A,-B),$"}
{"_id": "2394918", "title": "", "text": "$\\lim _{ x\\rightarrow 9 }{ \\frac { (9)^{ 1/2 }+(9)-6 }{ (9)^{ 3/2 }-27 }  } =\\frac { 3+9-6 }{ 27-27 } =\\frac { 6 }{ 0 } =\\infty $"}
{"_id": "5130717", "title": "", "text": "$\\limsup f_n(x)\\le \\lim_{m\\rightarrow \\infty} f(y_m)=f(x)$"}
{"_id": "4588619", "title": "", "text": "$\\frac{dx}{dt} =  \\sqrt{ {150}^2 - \\frac{dy}{dt}^2 }$"}
{"_id": "9164316", "title": "", "text": "$\\sum_{k=1}^{\\infty} \\dfrac{1-e^{-k}(k+1)}{k^2} = \\sum_{k=1}^{\\infty} \\dfrac1{k^2} - \\sum_{k=1}^{\\infty} \\dfrac{e^{-k}}{k} - \\sum_{k=1}^{\\infty} \\dfrac{e^{-k}}{k^2}=\\zeta(2) + \\log(1-1/e) - \\text{Li}_2(1/e)$"}
{"_id": "7662065", "title": "", "text": "$[t_0,t_x]$"}
{"_id": "5438857", "title": "", "text": "$2^{|A \\setminus \\{1,2\\}|} = 2^{|\\{3,\\ldots,n\\}|} = 2^{n - 2}$"}
{"_id": "187620", "title": "", "text": "$A_1\\subseteq A_2\\subseteq \\cdots$"}
{"_id": "5772461", "title": "", "text": "$1/|x|^a$"}
{"_id": "6559386", "title": "", "text": "$\\lim\\limits_{n \\to \\infty} \\frac{\\sqrt 1 + \\sqrt 2 + \\dots + \\sqrt{n}}{n\\sqrt{n}}$"}
{"_id": "3568237", "title": "", "text": "$=\\left\\lfloor \\{a\\}\\lfloor b\\rfloor+a\\{b\\}\\right\\rfloor$"}
{"_id": "7276246", "title": "", "text": "$\\sqrt{1-\\sin^2\\varphi} = \\cos\\varphi$"}
{"_id": "821761", "title": "", "text": "$ax_1+by_1=d$"}
{"_id": "910660", "title": "", "text": "$\\sum_{n=1}^\\infty \\|x_n-e_n\\|^2<1$"}
{"_id": "5382615", "title": "", "text": "$\\{N!+2,N!+3,\\cdots,N!+N-1,N!+N\\}$"}
{"_id": "5125140", "title": "", "text": "$ \\int_{-\\pi}^\\pi \\frac{\\sin^2(nx/2)}{2\\sin^2(x/2)}\\,\\mathrm{d}x =\\int_{-\\pi/2}^{\\pi/2} \\frac{\\sin^2(n u)}{\\sin^2(u)}\\,\\mathrm{d}u \\tag{2} $"}
{"_id": "9014841", "title": "", "text": "$ A_1 \\times ... \\times A_N, $"}
{"_id": "7412831", "title": "", "text": "$\\sigma:P\\rightarrow F=P\\oplus Q$"}
{"_id": "7786606", "title": "", "text": "$S \\equiv R$"}
{"_id": "9336180", "title": "", "text": "$ \\left\\{ \\matrix{   \\left( {m + 2} \\right)a + 2mb = 1 \\hfill \\cr    \\left( {m + 2} \\right)b + 2c + 2md = 0 \\hfill \\cr    2a + \\left( {m + 1} \\right)b = 0 \\hfill \\cr    c = 1 \\hfill \\cr    2b + c + \\left( {m + 1} \\right)d = 0 \\hfill \\cr}  \\right.\\quad  \\Rightarrow \\quad \\left\\{ \\matrix{   a = \\left( {m + 1} \\right)/\\left( {m^{\\,2}  - m + 2} \\right) \\hfill \\cr    b =  - 2/\\left( {m^{\\,2}  - m + 2} \\right) \\hfill \\cr    c = 1 \\hfill \\cr    d = \\left( {2 - m} \\right)/\\left( {m^{\\,2}  - m + 2} \\right) \\hfill \\cr}  \\right. $"}
{"_id": "1812047", "title": "", "text": "$f(A+B)\\subset f(A)+f(B)$"}
{"_id": "1639254", "title": "", "text": "$aRb \\lor bRa$"}
{"_id": "310032", "title": "", "text": "$\\lvert ab\\rvert = \\lvert a\\rvert\\cdot \\lvert b\\rvert$"}
{"_id": "2141159", "title": "", "text": "$COV(X,Y) = 1$"}
{"_id": "3809679", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 2 & 2 & 2 & -1 \\\\ 0 & 4 & -12 & 5 \\\\ 7 & 5 & 13 & -6 \\end{array}\\right]$"}
{"_id": "909971", "title": "", "text": "$\\bigg\\lfloor\\frac{\\lfloor x \\rfloor }{m}\\bigg\\rfloor =\\bigg\\lfloor\\frac{x }{m}\\bigg\\rfloor $"}
{"_id": "2965030", "title": "", "text": "$P(n), P(m)\\implies P({n\\over m})$"}
{"_id": "3018333", "title": "", "text": "$Tor(A,B)=0$"}
{"_id": "7528651", "title": "", "text": "$S_k = \\{v_1, v_2, ..., v_k\\}$"}
{"_id": "782273", "title": "", "text": "$\\frac1T \\int_0^T\\left(\\sum_{k=-\\infty}^{\\infty}c_ke^{j{\\frac{2\\pi kt}{T}}}\\right)^2dt= \\sum_{k=-\\infty}^{\\infty}|c_k|^2$"}
{"_id": "3225496", "title": "", "text": "$\\mathcal{A}=\\{\\{n\\}:n\\in \\mathbb{Z}\\}$"}
{"_id": "9055038", "title": "", "text": "$G=P\\oplus H$"}
{"_id": "8924741", "title": "", "text": "$ \\bbox[lightyellow] {   N_b (s,r,q + 1) = {\\rm No}{\\rm .}\\,{\\rm of}\\,{\\rm solutions}\\,{\\rm to}\\;\\left\\{ \\matrix{   {\\rm 0} \\le {\\rm integer}\\;x_{\\,j}  \\le r \\hfill \\cr    x_{\\,1}  + x_{\\,2}  + \\; \\cdots \\; + x_{\\,q + 1}  = s \\hfill \\cr}  \\right.  }$"}
{"_id": "8079044", "title": "", "text": "$ax + by = f$"}
{"_id": "3279121", "title": "", "text": "$\\gamma=a\\otimes \\gamma=\\gamma a+\\lambda a+\\lambda \\gamma +\\mu$"}
{"_id": "1159135", "title": "", "text": "$(ab,a+b)=1$"}
{"_id": "9274492", "title": "", "text": "$[([x] + \\{x\\})([y]+\\{y\\})] = [[x][y] + \\{x\\}[y] + [x]\\{y\\} + \\{x\\}\\{y\\}] = [x][y]$"}
{"_id": "5010136", "title": "", "text": "$\\mathrm{d}x\\sqrt{1+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}x}\\right)^2}$"}
{"_id": "561101", "title": "", "text": "$f({}n) = \\frac{n{}}{2}$"}
{"_id": "2721100", "title": "", "text": "$ \\left[\\begin{array}{rrr|r} 1&3&3&6\\\\2&2&1&4\\\\-1&-1&2&3\\end{array}\\right] $"}
{"_id": "7868354", "title": "", "text": "$\\begin{cases}x\\equiv r\\mod a\\\\x\\equiv s\\mod b\\end{cases}$"}
{"_id": "5112360", "title": "", "text": "$\\frac4x + \\frac9y + \\frac{16}z$"}
{"_id": "758490", "title": "", "text": "$|a_n-a_{n-1}|=\\frac{1}{2^{n-2}}$"}
{"_id": "8748495", "title": "", "text": "$\\int_0^\\pi \\frac{\\sin(nx)}{n^2} dx$"}
{"_id": "2432508", "title": "", "text": "$f(x)=\\frac{2x}{\\theta^n}.$"}
{"_id": "9075495", "title": "", "text": "$\\sum_{m=0}^{\\infty}\\sum_{n=1}^{\\infty}\\frac{1}{3^{mn}}$"}
{"_id": "7794828", "title": "", "text": "$\\sqrt{1-(1-\\frac{\\gamma}{\\beta})\\cos(\\theta)^2} = \\sqrt{1-(1-\\frac{\\gamma}{\\beta})(1- \\sin(\\theta)^2)}$"}
{"_id": "4872184", "title": "", "text": "$ f(x+y+f(y))=f(x)+ny$"}
{"_id": "8480319", "title": "", "text": "$w=e^{i/n}z$"}
{"_id": "8257613", "title": "", "text": "$\\sum_{k=1}^5\\cos\\frac{2k\\pi}{11}=-\\frac{1}{2}$"}
{"_id": "6262300", "title": "", "text": "$\\lim \\limits_{x \\to x_0+}f(x)g(x) = \\lim \\limits_{x \\to x_0+}f(x)*\\lim \\limits_{x \\to x_0+}g(x)$"}
{"_id": "5010637", "title": "", "text": "$\\{\\dots,b-p,b,b+p,b+2p,\\dots\\}$"}
{"_id": "2666354", "title": "", "text": "$\\int \\frac{x^2}{\\sqrt{x^3+2}}\\, dx$"}
{"_id": "5256185", "title": "", "text": "$|f(x_0)|\\le x_0^n|f(z_n)|$"}
{"_id": "3766147", "title": "", "text": "$c_n = \\sqrt1 + \\sqrt2 + \\cdots + \\sqrt{n}$"}
{"_id": "8457755", "title": "", "text": "$L = \\lim_{x\\to c^{+}} f'(x)$"}
{"_id": "6681649", "title": "", "text": "$\\langle_{\\mathbb{R}}=\\{\\langle x,y \\rangle \\in \\mathbb{R}^2: \\text{ x is smaller that y}\\}$"}
{"_id": "2988260", "title": "", "text": "$3^x + 9^x = 27^x$"}
{"_id": "6582828", "title": "", "text": "$\\int_\\gamma \\omega = \\int_{\\gamma} df = f(\\gamma(2\\pi)) - f(\\gamma(0)) = 0.$"}
{"_id": "5050506", "title": "", "text": "$T\\left[ \\begin{array}{}    1  \\\\    x  \\\\    x^2  \\\\   \\end{array}  \\right]$"}
{"_id": "6172505", "title": "", "text": "$\\mathbb{E}(\\overline{X}(1-\\overline{X})) = \\mathbb{E}(\\overline{X}-(\\overline{X})^2)= \\mathbb{E}\\overline{X}-\\mathbb{E}(\\overline{X})^2$"}
{"_id": "3609914", "title": "", "text": "$V\\subseteq \\overline{V}\\subseteq W$"}
{"_id": "947189", "title": "", "text": "$ \\zeta(a)=\\frac{1}{\\Gamma(a)}\\int_{0}^{+\\infty}\\frac{x^{a-1}}{e^{x}-1}\\,dx$"}
{"_id": "4756727", "title": "", "text": "$\\gamma^{4}+\\gamma+1=(\\gamma+1)(\\gamma^{3}+d\\gamma^{2}+e\\gamma+\\gamma)$"}
{"_id": "3751708", "title": "", "text": "$\\lim_{x\\to c^-} f'(x) = L_1, \\lim_{x\\to c^+} f'(x) = L_2,$"}
{"_id": "8247534", "title": "", "text": "$P[X \\geq \\lambda] = 1/2$"}
{"_id": "4905925", "title": "", "text": "$x_\\omega =\\lim_{n\\rightarrow \\omega}\\ x_n$"}
{"_id": "8086143", "title": "", "text": "$A(x+y,x) = A(x,x)+ A(y,x)$"}
{"_id": "7583033", "title": "", "text": "$(x,-y),(-x,-y),(-x,y),(y,x),(y,-x),(-y,x)$"}
{"_id": "336133", "title": "", "text": "$M=R\\oplus T$"}
{"_id": "4935303", "title": "", "text": "$(1.f)(x)=f(1^{-1}.x)=f(1.x)=f(x)$"}
{"_id": "3300593", "title": "", "text": "$\\lim \\limits_{n \\to \\infty} \\sum_{k=1}^{\\lfloor n/2 \\rfloor} \\frac {n^2}{n^2-k^2}$"}
{"_id": "550391", "title": "", "text": "$\\lim_{n\\to\\infty}\\sum_{r=1}^{n}\\frac{1}{(r+2)r!}\\:$"}
{"_id": "1430658", "title": "", "text": "$\\int_0^{\\infty} dx\\: f(x)$"}
{"_id": "2034213", "title": "", "text": "$ 3 > 2 \\\\ 3 \\cdot -1 = -3 \\\\ 2 \\cdot -1 = -1 \\\\ -3 < -1 \\\\ 3 \\cdot -1 < 2 \\cdot -1 $"}
{"_id": "548161", "title": "", "text": "$\\#_a(w)$"}
{"_id": "2568730", "title": "", "text": "$\\int \\int dS$"}
{"_id": "2484697", "title": "", "text": "$ \\mathbb R^n \\mathrm x \\mathbb R^n$"}
{"_id": "3690026", "title": "", "text": "$(x,x,y,y)+(x',x',y',z')=(x+x',x+x',y+y',y+z')=(x'',x'',y'',z'').$"}
{"_id": "8540276", "title": "", "text": "$i(f)=\\displaystyle\\binom n2$"}
{"_id": "4466885", "title": "", "text": "$\\frac{\\sin ^{-1}(x)}{x}$"}
{"_id": "1047237", "title": "", "text": "$d(x,z)\\le d(x,y)+d(y,z)\\;.$"}
{"_id": "7910210", "title": "", "text": "$\\int_{0}^{\\infty}f(x)\\ dx=c$"}
{"_id": "5356919", "title": "", "text": "$A\\subseteq U,\\ B\\subseteq V,$"}
{"_id": "3713994", "title": "", "text": "$F(x)=\\int_a^xf(x)dt$"}
{"_id": "258172", "title": "", "text": "$Cov(a,a)=Cov(bt,bt)=Cov(a,X)=Cov(bt,X)=0,$"}
{"_id": "496576", "title": "", "text": "$f(p)=p^2-p+1$"}
{"_id": "7777397", "title": "", "text": "$x+\\frac{1}{y}=10$"}
{"_id": "5356903", "title": "", "text": "$f(x)=2e^{\\frac{-x}{2}} $"}
{"_id": "4583061", "title": "", "text": "$ar + bs =1.$"}
{"_id": "1892417", "title": "", "text": "$ x = \\frac{ \\ell^2 - a b }{ a + b - 2 \\ell }, $"}
{"_id": "889254", "title": "", "text": "$6! + 2, 6!+3, 6!+4, 6!+5$"}
{"_id": "3962424", "title": "", "text": "$x! \\approx (\\frac xe)^x \\sqrt {2 \\pi x}$"}
{"_id": "7486238", "title": "", "text": "$l(x)=\\int_{a}^{x}f(t)dt$"}
{"_id": "3462315", "title": "", "text": "$n!=\\sqrt{2\\pi x} \\left({\\frac{x}{e}}\\right)^x$"}
{"_id": "1178760", "title": "", "text": "$A = \\begin{bmatrix}x&y&z&1\\\\x_1&y_1&z_1&1\\\\x_2&y_2&z_2&1\\\\x_3&y_3&z_3&1\\end{bmatrix}$"}
{"_id": "826784", "title": "", "text": "$\\begin{pmatrix}x&-y\\\\y&x\\end{pmatrix}$"}
{"_id": "2277932", "title": "", "text": "$T^{0,1}_pM$"}
{"_id": "1077267", "title": "", "text": "$a_n=\\frac{n(n-1)}{2}$"}
{"_id": "301877", "title": "", "text": "$\\;\\color{blue}{\\bf a} + \\dfrac{1}{\\color{red}{\\bf b}+1/\\color{green}{\\bf c}} = \\dfrac{37}{16} =  2 + \\dfrac 5{16} = 2 + \\dfrac 1{\\frac{16}{5}} = \\color{blue}{\\bf 2} + \\dfrac 1{\\color{red}{\\bf 3} + 1/\\color{green}{\\bf 5}}$"}
{"_id": "702887", "title": "", "text": "$x = a^{\\log_a (x)}.$"}
{"_id": "183038", "title": "", "text": "$2|n^2-n$"}
{"_id": "1517365", "title": "", "text": "$\\mathbb{P}(\\sup_{t}M_{t}= x \\mid\\mathcal{F}_{0})$"}
{"_id": "4046802", "title": "", "text": "$x^*, x_n \\in \\mathbb{R}$"}
{"_id": "272762", "title": "", "text": "$A_n \\subseteq A_{n+1}$"}
{"_id": "3928151", "title": "", "text": "$1 = \\frac 1x + \\frac 2y - \\frac 3z \\le \\frac 3y - \\frac 3z$"}
{"_id": "824545", "title": "", "text": "$\\int_0^\\infty f(x)\\,\\mathrm{d}x$"}
{"_id": "1853461", "title": "", "text": "$ (x-a)^n  \\overset{\\alpha}{\\mapsto}  (x-a)[n(x-a)^{n-1} - 0] - 2[(x-a)^n - 0] = (n-2)(x-a)^n $"}
{"_id": "6203702", "title": "", "text": "$ \\|B\\|_2 = \\sqrt{\\lambda_\\max(B^t B)} $"}
{"_id": "7721027", "title": "", "text": "$x!=(x/e)^x\\sqrt {2\\pi x}\\;(1+d(x))$"}
{"_id": "5325585", "title": "", "text": "$f(x) = \\frac{2x}{x+2}$"}
{"_id": "5668423", "title": "", "text": "$\\left(\\mathbb{F}_3[X]/(X^3-X^2+1)\\right)^{\\times}$"}
{"_id": "1715980", "title": "", "text": "$T^{0,1}$"}
{"_id": "5077984", "title": "", "text": "$\\tan \\theta = \\dfrac 5 2$"}
{"_id": "8330023", "title": "", "text": "$ A = \\begin{bmatrix} 1 & -1 & -7 \\\\ 1 & 3 & -1 \\\\ 2 & -1 & k \\end{bmatrix} $"}
{"_id": "3253843", "title": "", "text": "$\\mathbb{R}^n \\oplus \\mathbb{R}^n$"}
{"_id": "6244194", "title": "", "text": "$f(x) = \\dfrac{a+x}{b + cx}$"}
{"_id": "3341027", "title": "", "text": "$|g'(a_1)|\\leq |f'(a_1)|.$"}
{"_id": "2629955", "title": "", "text": "$P(x)=(n-1)(1-x)^{n-2}$"}
{"_id": "8096319", "title": "", "text": "$a<(a+b)/2<b\\iff 2a<a+b<2b$"}
{"_id": "4979599", "title": "", "text": "$ ((x,y),z)=(\\{\\{x\\},\\{x,y\\}\\},z)=\\{ \\{\\{\\{x\\},\\{x,y\\}\\}\\},\\{\\{\\{x\\},\\{x,y\\}\\},z\\} \\}$"}
{"_id": "130710", "title": "", "text": "$4. d(x,z) \\leq d(x,y)+d(y,z)$"}
{"_id": "226583", "title": "", "text": "$ \\forall \\epsilon >0, \\exists \\delta >0, \\ s.t. \\ 0<|x-a|<\\delta \\implies |f(x)-L|< \\epsilon$"}
{"_id": "618442", "title": "", "text": "$Cov(X,Y)=0.8$"}
{"_id": "8510767", "title": "", "text": "$x=\\big\\{\\{a\\},\\{a,b\\}\\big\\}$"}
{"_id": "9284020", "title": "", "text": "$I_n=\\int{\\tan^{\\frac 1n} (x)}\\,dx=\\frac{n \\tan ^{\\frac{n+1}{n}}(x)}{n+1}\\,\\,\\, _2F_1\\left(1,\\frac{n+1}{2 n};\\frac{3n+1}{2 n};-\\tan    ^2(x)\\right)$"}
{"_id": "5639533", "title": "", "text": "$\\| \\triangle ABC \\| = rs$"}
{"_id": "4267634", "title": "", "text": "$f(x)=\\frac{1}{4}e^\\frac{-|x|}{2}$"}
{"_id": "8966822", "title": "", "text": "$\\displaystyle I=\\int_0^{2π}(4\\cos(\\theta)+4i\\sin(\\theta)+1)^3\\,d\\theta$"}
{"_id": "416661", "title": "", "text": "$\n \\sum_{i=1}^n\\cos\\left(\\frac{2\\pi}{n}i\\right)=0\\tag{10a}\n $"}
{"_id": "8792335", "title": "", "text": "$(\\gamma_1,-\\overline{\\gamma_1}),\n (\\gamma_2,-\\overline{\\gamma_2}),\\ldots,(\\gamma_k,-\\overline{\\gamma_k})$"}
{"_id": "5306757", "title": "", "text": "$T_X,x$"}
{"_id": "8222863", "title": "", "text": "$\\partial_i\\frac{\\gamma}{\\|\\gamma\\|}=\\frac{1}{\\|\\gamma\\|}(\\partial_i\\gamma)  -\\frac{\\gamma\\cdot (\\partial_i\\gamma)}{\\|\\gamma\\|^3}\\gamma,$"}
{"_id": "3259107", "title": "", "text": "$S_n=n(n+1)/2$"}
{"_id": "6530931", "title": "", "text": "$\\cos \\theta = \\frac{x}{3}$"}
{"_id": "5276468", "title": "", "text": "$g_n(x)=\\sum_{j=1}^na_j\\chi_{V_j}(x)>\\alpha$"}
{"_id": "6627305", "title": "", "text": "$\\|B\\|_2 = \\sqrt{\\rho(B^\\top B)}$"}
{"_id": "2937616", "title": "", "text": "$I_1=\\int_{\\frac{1}{2}}^1 \\frac{\\sinh^{-1}(x)}{x}\\,dx$"}
{"_id": "4122009", "title": "", "text": "$A \\meta\\to B = \\quote(+A+\\quote\\to+B+\\quote)$"}
{"_id": "6231037", "title": "", "text": "$ (3)^{\\frac{1}{3}}(9)^{\\frac{1}{9}}(27)^{\\frac{1}{27}}....=x$"}
{"_id": "9142392", "title": "", "text": "$\\int_0^\\pi\\sin(x)\\cos(x)dx=2\\int_0^\\pi\\sin(2x)dx=0$"}
{"_id": "1454356", "title": "", "text": "$\\begin{split}\\{  \\gamma  \\in\\tau : r(f(\\gamma))\\le  1-\\epsilon\\} &=  \\{  \\gamma  \\in\\tau : d(f(\\gamma)) \\ge  \\phi(1-\\epsilon)\\}  \\\\ &\\subseteq \\{  \\gamma  \\in\\tau : d( \\gamma ) \\ge  \\delta\\} \\\\  &\\subseteq \\{  \\gamma  \\in\\tau : r( \\gamma ) \\le  1-\\tau\\}  \\end{split}$"}
{"_id": "6146499", "title": "", "text": "$A=\\big\\{2x - 1 \\wedge y^2 \\; | \\; x \\in \\mathbb{N}, y \\in \\mathbb{N}, x \\leq 10\\}$"}
{"_id": "7872915", "title": "", "text": "$n^{12}\\equiv1\\mod p$"}
{"_id": "5517960", "title": "", "text": "$\\vert f(z) \\vert = r^{-n} \\vert g(z) \\vert, \\tag{5}$"}
{"_id": "4447880", "title": "", "text": "$|abc|=308$"}
{"_id": "3707470", "title": "", "text": "$x=b,\\, y=t, c\\leq t\\leq d$"}
{"_id": "5329341", "title": "", "text": "$A=\\{\\langle x,y\\rangle\\in S^2:x+y\\in S\\setminus\\Bbb Z\\}$"}
{"_id": "5130718", "title": "", "text": "$\\liminf f_n(x)\\ge \\lim_{m\\rightarrow \\infty} f(x_m)=f(x).$"}
{"_id": "1445559", "title": "", "text": "$f(a+b) = f(a)f(b)$"}
{"_id": "1109301", "title": "", "text": "$\\Gamma(\\frac{s}{2})\\zeta(s) \\pi^{-\\frac{s}{2}}=\\Gamma(\\frac{1-s}{2})\\zeta(1-s)\\pi^{-\\frac{1-s}{2}}=\\Lambda(s)$"}
{"_id": "2108858", "title": "", "text": "$ax+by=c\\ ,$"}
{"_id": "4762373", "title": "", "text": "$|a+b|=2k$"}
{"_id": "6689631", "title": "", "text": "$x_4=\\frac{40}{81}$"}
{"_id": "1658133", "title": "", "text": "$d(x,A)\\leq d(x,a)\\leq d(x,y)+d(y,a)$"}
{"_id": "7860566", "title": "", "text": "$s(x) = \\sum \\limits_{i = 1}^{n} a_{i} \\chi_{A_{i}}(x)$"}
{"_id": "1371510", "title": "", "text": "$ L=f(a+h,b+k)-f(a,b)=f(a+h,b+k)-f(a,b+k)+f(a,b+k)-f(a,b)$"}
{"_id": "2824189", "title": "", "text": "$\\zeta(s)=2^s\\pi^{s-1}\\sin\\frac{\\pi s}2\\Gamma(1-s)\\zeta(1-s)$"}
{"_id": "6606031", "title": "", "text": "$[x,y] = [1766319049, 226153980].$"}
{"_id": "7213227", "title": "", "text": "$Q(n) \\implies P(n)$"}
{"_id": "1387757", "title": "", "text": "$\\sqrt[m]{\\alpha}\\mathcal{O}_L = \\prod_{i=1}^s \\mathfrak{p}_i^{mk_i / m} = \\mathfrak{a}\\mathcal{O}_L$"}
{"_id": "2702058", "title": "", "text": "$z=\\frac{a+b+c}{b-a}$"}
{"_id": "7084284", "title": "", "text": "$A_1 \\subset A_2 \\subset \\ldots \\subset A_m \\subset A_{m+1} \\subset \\ldots$"}
{"_id": "5339178", "title": "", "text": "$\\tan^2\\theta=\\frac ba$"}
{"_id": "3452480", "title": "", "text": "$ \\displaystyle\\lim_{x\\to a^-} f(x) = \\displaystyle\\lim_{x\\to a^+} f(x) = L $"}
{"_id": "5494720", "title": "", "text": "$\\theta(x)=(\\sum_{n=-\\infty}^1 e^{-n^2\\pi x}) + 1 + (\\sum_{n=1}^\\infty e^{-n^2\\pi x}) =\\sum_{n=\\infty}^\\infty e^{-n^2\\pi x}$"}
{"_id": "9243158", "title": "", "text": "$\\frac {1-(1+R)^{T+1}}{R}$"}
{"_id": "2781772", "title": "", "text": "$ z = e^{\\frac{2i\\pi}{5}} $"}
{"_id": "7813845", "title": "", "text": "$||A||_2=\\sigma_1(A)=\\sqrt{\\lambda_{max}(A^TA)}$"}
{"_id": "3017567", "title": "", "text": "$\\mathbb{R}^{n+1} - S^n$"}
{"_id": "7598308", "title": "", "text": "$\\gamma( a \\vee b) = \\gamma(a) \\wedge \\gamma(b)$"}
{"_id": "1604838", "title": "", "text": "$S\\equiv R^n$"}
{"_id": "1262700", "title": "", "text": "$x\\ R\\ y \\Leftrightarrow y\\ R\\ x$"}
{"_id": "3608107", "title": "", "text": "$\\begin{vmatrix} 1 & 1 & 1\\\\  a & b & c\\\\  a^3 & b^3 & c^3  \\end{vmatrix}$"}
{"_id": "8431074", "title": "", "text": "$g'(x)=2f(x)\\int_0^xf(t)dt-f(x)^3=f(x)\\left[2\\int_0^xf(t)dt-f(x)^2\\right].$"}
{"_id": "2630140", "title": "", "text": "$\\gamma \\leq \\gamma = \\gamma+0$"}
{"_id": "2465157", "title": "", "text": "$P[M_n > x \\mid X_0]$"}
{"_id": "3203631", "title": "", "text": "$\\int_0^1\\frac{x^n(x-1)^n}{n!}e^xdx=\\int-\\frac{x^n(x-1)^{n-1}}{(n-1)!}e^x-\\frac{x^{n-1}(x-1)^n}{(n-1)!}e^xdx $"}
{"_id": "6098585", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1 & 3 & -1 & -4\\\\4 & -1 & 2 & 3\\\\2 & -1 & -3 & 1\\end{array}\\right]$"}
{"_id": "4056397", "title": "", "text": "$f(x,y)=e^\\frac{-x^2}{2t}-e^\\frac{-y^2}{2t}$"}
{"_id": "5430358", "title": "", "text": "$4^x = 3\\cdot 6^x$"}
{"_id": "2059973", "title": "", "text": "$\\mathbb A=\\{\\{a\\}:a\\in A\\}$"}
{"_id": "7203403", "title": "", "text": "$[x,y]=(Ax,y)$"}
{"_id": "5562372", "title": "", "text": "$\\sum_{n=1}^{+\\infty}\\frac{(-1)^n}{n^4} = -\\sum_{n=1}^{+\\infty}\\frac{1}{n^4}+2\\sum_{m=1}^{+\\infty}\\frac{1}{(2m)^4} = \\left(-1+\\frac{2}{16}\\right)\\sum_{n=1}^{+\\infty}\\frac{1}{n^4}=-\\frac{7}{8}\\zeta(4) = -\\frac{7\\pi^4}{720}.$"}
{"_id": "1005867", "title": "", "text": "$(1-2^{1-z})\\zeta(z)=\\sum\\limits_{s=1}^\\infty \\frac{(-1)^s}{s^z}=\\frac{1}{\\Gamma(z)}\\int\\limits_{0}^\\infty \\frac{t^{z-1}}{e^t+1}dt$"}
{"_id": "2742667", "title": "", "text": "$-9(x + 9)^{\\frac{2}{3}} +27(x +9)^{\\frac{1}{3}} =9 $"}
{"_id": "1298397", "title": "", "text": "$C=\\{\\{A\\}\\}$"}
{"_id": "490687", "title": "", "text": "$B = \\{b_1, b_2, b_3, \\ldots, b_n\\}$"}
{"_id": "6010365", "title": "", "text": "$2^{log_3 x}+x^{log_3 2}=4$"}
{"_id": "1406720", "title": "", "text": "$(1-p^2)^{n-2}$"}
{"_id": "927806", "title": "", "text": "$\\zeta (s)=\\sum_{n=1}^\\infty \\frac {1}{n^s}=\\frac {1}{\\Gamma (s)} \\int _0^{\\infty}\\frac { x^{s-1}}{e^x-1} dx$"}
{"_id": "5474205", "title": "", "text": "$\\sum_{n=1}^\\infty \\|e_n-f_n\\|<\\infty.$"}
{"_id": "2968949", "title": "", "text": "$\\lim_{n\\to\\infty}\\left(\\sum_{k=1}^{2n}\\frac{1}{2n}\\left\\lfloor\\frac{2n}{k}\\right\\rfloor-\\sum_{k=1}^{n}\\frac{1}{n}\\left\\lfloor\\frac{n}{k}\\right\\rfloor\\right)=L_1,$"}
{"_id": "7025336", "title": "", "text": "$[x,_0\\, y]=x$"}
{"_id": "8089859", "title": "", "text": "$\\mathbb{E}^+$"}
{"_id": "8817075", "title": "", "text": "$d(x,M) = \\delta > 0$"}
{"_id": "675621", "title": "", "text": "$\\begin{align} \\int_{-\\pi/2}^{\\pi/2} \\frac{x^n \\left(\\cos x\\right)^{n-2}}{\\left(\\sin x\\right)^n} \\, {\\rm d}x &\\stackrel{y=2x}{=} 2^{-n+1} \\, i^n \\int_{-\\pi}^{\\pi} y^n \\, e^{-iy} \\, \\frac{\\left(1+e^{-iy}\\right)^{n-2}}{\\left(1-e^{-iy}\\right)^n} \\, {\\rm d}y \\\\ &\\stackrel{z=e^{iy}}{=} -i \\, 2^{-n+1} \\int_\\gamma \\left(\\log z\\right)^n \\, \\frac{(z+1)^{n-2}}{(z-1)^n} \\, {\\rm d}z \\end{align}$"}
{"_id": "6239802", "title": "", "text": "$b_n =\\frac {a_1+a_2+\\dots+a_n}n.$"}
{"_id": "1314642", "title": "", "text": "$y'''+y''=y'+y$"}
{"_id": "2666564", "title": "", "text": "$c:=-\\max_{0\\leq\\theta\\leq a}\\mathrm{Re}(\\gamma(\\theta))\\quad(>0)$"}
{"_id": "8501616", "title": "", "text": "$f(A + B ) = f(A) + f(B),   $"}
{"_id": "6547679", "title": "", "text": "$ (*) \\Leftrightarrow \\tan(a+\\gamma)=\\frac{\\frac{b}{c}-\\cos\\gamma}{\\sin \\gamma}$"}
{"_id": "2080759", "title": "", "text": "$\\sum_{k=1}^{\\infty} \\sum_{n=1}^{\\infty} = \\sum_{n=1}^{\\infty} \\sum_{k=1}^{\\infty}$"}
{"_id": "5999170", "title": "", "text": "$p_2 \\mid p_1^k-1$"}
{"_id": "5962785", "title": "", "text": "$(a+b,a^2+b^2)=1$"}
{"_id": "8931059", "title": "", "text": "$>10^{40}$"}
{"_id": "9047910", "title": "", "text": "$[x,e]=x$"}
{"_id": "3184146", "title": "", "text": "$||A||_2 = \\sqrt{\\lambda_{max}(A^H\\,A)}$"}
{"_id": "8437166", "title": "", "text": "$\\langle f,m\\rangle\\in\\mathscr{D}$"}
{"_id": "2837116", "title": "", "text": "$ \\frac{y^2}{(x-y)(y-z)}+\\frac{z^2}{(y-z)(z-x)}+\\frac{x^2}{(z-x)(x-y)}=-1 $"}
{"_id": "7700515", "title": "", "text": "$\\int_{0}^{2\\pi}\\frac{\\sin x}{x}\\,dx = \\int_{0}^{\\pi}\\frac{\\sin x}{x}\\,dx+\\int_{0}^{\\pi}\\frac{-\\sin(x)}{x+\\pi}\\,dx = \\pi\\int_{0}^{\\pi}\\frac{\\sin x}{x(x+\\pi)}\\,dx.$"}
{"_id": "6976789", "title": "", "text": "$A= [a_1, a_2, a_3, ... , a_n]$"}
{"_id": "5973402", "title": "", "text": "$\\sum_{n=2}^{k} \\frac{1}{n}= \\sum_{n=2}^{\\infty}\\left(\\frac{1}{n}-\\frac{1}{n+k-1}\\right)$"}
{"_id": "1998113", "title": "", "text": "$\\forall\\epsilon > 0, \\exists\\delta>0 : 0<|x-a|<\\delta \\implies |f(x) - L|<\\epsilon$"}
{"_id": "5474767", "title": "", "text": "$\\tan\\theta=\\frac x3$"}
{"_id": "1230520", "title": "", "text": "$x = \\frac{10}{4}$"}
{"_id": "4721864", "title": "", "text": "$\\begin{cases}x\\equiv r\\pmod a\\\\x\\equiv s\\pmod b\\end{cases}\\iff x\\equiv sua+rvb\\pmod{ab}.$"}
{"_id": "5778326", "title": "", "text": "$\\dim(M)+\\dim(H)=\\dim(M+H)=\\dim(M+K)=\\dim(M)+\\dim(K)$"}
{"_id": "5397173", "title": "", "text": "$\\frac{1}{\\sqrt1+\\sqrt3}=\\frac{\\sqrt3-\\sqrt1}{2}$"}
{"_id": "4304030", "title": "", "text": "$g(f(a_1))=a_1$"}
{"_id": "718701", "title": "", "text": "$c_r(E)$"}
{"_id": "7388549", "title": "", "text": "$[u,v] = [\\gamma(a), \\gamma(b)] \\subseteq \\gamma([a,b]) \\subseteq I.$"}
{"_id": "8844485", "title": "", "text": "$\\hat{M} = \\frac{X_{1} + X_{2} + ... + X_{n}}{n}$"}
{"_id": "5326515", "title": "", "text": "$\\int{\\frac{1}{(x^4 -1)^2}}\\, dx$"}
{"_id": "3646072", "title": "", "text": "$\\forall \\epsilon > 0, \\exists \\delta >0 \\text{ such that, } 0<|x-a|<\\delta\\implies |f(x) - L| < \\epsilon$"}
{"_id": "8050254", "title": "", "text": "$ \\begin{bmatrix} z & -\\bar{w} \\\\ w & \\bar{z} \\end{bmatrix} $"}
{"_id": "9356275", "title": "", "text": "$\\;a_1\\cdot\\ldots\\cdot a_n\\;$"}
{"_id": "5805513", "title": "", "text": "$\\displaystyle\\lim_{n\\to\\infty} \\frac{n+1}{1+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n}}$"}
{"_id": "6941539", "title": "", "text": "$(M,M,M,m)$"}
{"_id": "288183", "title": "", "text": "$\\zeta(s) = 2^s \\pi^{s-1} \\sin\\left(\\frac{\\pi s}{2} \\right) \\Gamma (1-s) \\zeta(1-s)$"}
{"_id": "1538488", "title": "", "text": "$\\bigcup\\mathcal{P}(\\mathcal{A}) = \\bigcup_{{\\mathcal X} \\subseteq {\\mathcal A}} {\\mathcal X}.$"}
{"_id": "715036", "title": "", "text": "$(x+1)^p = (x+1)^m (x-1)^{n-q}$"}
{"_id": "9141765", "title": "", "text": "$F = n - k$"}
{"_id": "5172488", "title": "", "text": "$p_1p_2 > x$"}
{"_id": "50639", "title": "", "text": "$\\underset{n\\to \\infty }{\\mathop{\\lim }}\\,\\sum\\limits_{k=1}^{n}{\\frac{1}{{{2}^{k}}}}$"}
{"_id": "1590516", "title": "", "text": "$f(a+b)=f(a)\\oplus f(b)$"}
{"_id": "1541609", "title": "", "text": "$\\gamma_{3}(G)=[\\gamma_{2}(G),G]=[\\gamma_{3}(G),G]=\\gamma_{4}(G)$"}
{"_id": "2910127", "title": "", "text": "$(\\frac{1}{1-x^2})^2$"}
{"_id": "2970412", "title": "", "text": "$1-p=\\int_0^1(1-z^{2/3})^2dz\\stackrel{z^2=t^3}=\\int_0^1(1-t)^2\\frac32t^{1/2}dt=\\frac32\\frac{\\Gamma(3)\\Gamma(3/2)}{\\Gamma(9/2)}$"}
{"_id": "2863107", "title": "", "text": "$\\forall \\varepsilon >0 \\exists K>0: x>K \\Rightarrow \\left | f(x)-l \\right |<\\varepsilon$"}
{"_id": "3442203", "title": "", "text": "$\\sum_{k = -\\infty}^{\\infty}e^{-k^2\\tfrac{\\pi}{n}} = \\sum_{k = -\\infty}^{\\infty}\\sqrt{n}e^{-n\\pi k^2}.$"}
{"_id": "1937505", "title": "", "text": "$f(x) = 1/|x|^\\alpha$"}
{"_id": "2481670", "title": "", "text": "$1-\\cos(x)(\\cos(2x))^2=1-(1-2\\sin^2 (x/2))(1-2\\sin^2x)^2$"}
{"_id": "6078312", "title": "", "text": "$f_x(X) = \\frac{2}{x^3}$"}
{"_id": "961206", "title": "", "text": "$b^+$"}
{"_id": "3610283", "title": "", "text": "$\\begin{pmatrix}x&y\\\\-y&x\\end{pmatrix}$"}
{"_id": "6018702", "title": "", "text": "$x = \\sin \\theta + \\tan \\theta, y = \\sin \\theta - \\tan \\theta$"}
{"_id": "5244842", "title": "", "text": "$d(x,y)=1/p^a$"}
{"_id": "5697392", "title": "", "text": "$[X,Y] = X\\circ Y - Y\\circ X$"}
{"_id": "8740356", "title": "", "text": "$n!\\delta(a)^n=\\delta^n(a^n)$"}
{"_id": "6670", "title": "", "text": "$a_n=(1+\\frac{1}{n})^n$"}
{"_id": "7604654", "title": "", "text": "$\\frac{\\pi/2+k\\pi}{\\varphi(0)}$"}
{"_id": "2924413", "title": "", "text": "$(a+\\Delta x, f(a+\\Delta x))$"}
{"_id": "3382355", "title": "", "text": "$x! \\approx \\sqrt{2\\pi x} \\left ( \\frac{x}{e} \\right )^{x}$"}
{"_id": "5793883", "title": "", "text": "$ Z := \\left\\{ (x,y) \\in \\mathbb{R}^2 \\; : \\; b \\leq x \\leq T, \\; f(x) \\leq y \\leq R \\right\\} $"}
{"_id": "4968863", "title": "", "text": "$e^x - x - 1, x>0$"}
{"_id": "3280422", "title": "", "text": "$p_1^* = \\frac{1/(1-f)}{1 + 2^{(H_2(f)/(1 - f))}}$"}
{"_id": "2698990", "title": "", "text": "$COV[f,g]=0$"}
{"_id": "6886856", "title": "", "text": "$\\mathbb F_3^N$"}
{"_id": "3906239", "title": "", "text": "$2a=a+a<a+b<b+b=2b$"}
{"_id": "8624726", "title": "", "text": "$\\vec r= \\begin{cases}  x=u\\\\  y=a\\cos v\\\\  z=a\\sin v \\end{cases}$"}
{"_id": "6372642", "title": "", "text": "$a_j = B I_j$"}
{"_id": "3183862", "title": "", "text": "$\\overline{V}\\subseteq p(U')$"}
{"_id": "1422773", "title": "", "text": "$P(\\{ \\omega \\}) = 1/6$"}
{"_id": "146757", "title": "", "text": "$\\{e_1,e_2,e_3,\\ldots\\}\\cup\\{E\\}$"}
{"_id": "1409996", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sqrt{n}=\\infty$"}
{"_id": "4745150", "title": "", "text": "$ A=\\frac{Pr(1+r)^n}{(1+r)^n-1}. $"}
{"_id": "8032291", "title": "", "text": "$|ab| = L$"}
{"_id": "8472841", "title": "", "text": "$ \\|A\\|_2^2=\\rho(A^TA)\\leq\\|A^TA\\|_\\infty\\leq n\\|A\\|_\\infty^2. $"}
{"_id": "2472246", "title": "", "text": "$y!\\approx \\sqrt{2πy}(\\frac ye)^y$"}
{"_id": "7448764", "title": "", "text": "$F(s)L[e^{ax}]=\\int_0^\\infty e^{-sx}e^{ax}dx=\\lim\\limits_{R \\to \\infty} \\int_0^R e^{(a-s)x}dx=\\lim\\limits_{R \\to \\infty}[\\frac {e^{(a-s)}x} {a-s}]=\\lim\\limits_{R \\to \\infty}[\\frac {e^{(a-s)^R} -1} {a-s}]=\\frac {1}{s-a}(s>a)$"}
{"_id": "1243954", "title": "", "text": "$ \\sum_{h=0}^Nhr^h=r+2r^2+3r^3+...+Nr^N=r\\frac{1-r^{N+1}}{(1-r)^2}-\\frac{(N+1)r^{N+1}}{1-r}, \\quad |r|<1, \\tag2 $"}
{"_id": "3230395", "title": "", "text": "$d(x, A) := \\inf\\{d(x, y): y ∈ A\\}$"}
{"_id": "5223318", "title": "", "text": "$\\tan\\alpha=\\dfrac{r}{2}$"}
{"_id": "9171951", "title": "", "text": "$\\int_0^\\infty P(X > t)dt = \\int_0^\\infty\\int_t^\\infty f(x)dxdt = \\int_0^\\infty f(x) \\int_0^xdt dx = \\int_0^\\infty f(x)*x dx = E(X).$"}
{"_id": "3878410", "title": "", "text": "$ \\begin{align} \\int_0^1\\left(\\frac{\\sin^{-1}(x)}{x}\\right)\\mathrm{d}x &=\\int_0^{\\pi/2}\\left(\\frac{t}{\\sin(t)}\\right)\\mathrm{d}\\sin(t)\\tag{22}\\\\ &=\\int_0^{\\pi/2}t\\,\\mathrm{d}\\log(\\sin(t))\\tag{23}\\\\ &=-\\int_0^{\\pi/2}\\log(\\sin(t))\\,\\mathrm{d}t\\tag{24}\\\\ &=\\bbox[5px,border:2px solid #C0A000]{\\frac\\pi2\\log(2)}\\tag{25} \\end{align} $"}
{"_id": "705109", "title": "", "text": "$Cov(X,Y)\\neq 0$"}
{"_id": "6759442", "title": "", "text": "$ \\left[ \\begin{array}{@{}ccc|c@{}} 1 & -2 & 0 & 2 \\\\ 0 & 1 & 2 & -2 \\\\ 5 & -6 & 8 & 6 \\\\ \\end{array} \\right] $"}
{"_id": "149152", "title": "", "text": "$a \\notin \\{\\gamma \\}$"}
{"_id": "8969913", "title": "", "text": "$ax + by = d \\implies d \\in H$"}
{"_id": "5997312", "title": "", "text": "$\\lim_{N\\rightarrow\\infty}\\sum_{k=1}^N\\left(\\frac{k-1}{N}\\right)^N$"}
{"_id": "4082163", "title": "", "text": "$p(T|A,F)=P(T|F)$"}
{"_id": "5798248", "title": "", "text": "$C=\\{ A_1,A_2,A_3,...,A_n \\}$"}
{"_id": "1204880", "title": "", "text": "$(\\frac{1}{2})^{n-1}=\\frac{2}{2^{n}}$"}
{"_id": "2249382", "title": "", "text": "$p_2q_2 \\mid n$"}
{"_id": "765159", "title": "", "text": "$\\int_0^1 \\dfrac{\\sin(x)}{x} ~dx \\approx 0.9459962252$"}
{"_id": "8751980", "title": "", "text": "$f(a + b) = f(a) + f(b)\\;\\; \\forall a, b \\in \\mathrm{dom}(f)$"}
{"_id": "9262261", "title": "", "text": "$\\sqrt[n]{a} = |\\sqrt[n]{a}|e^{2\\pi i k/n}$"}
{"_id": "2893634", "title": "", "text": "$\\Psi\\begin{pmatrix}a&b\\\\b&a\\end{pmatrix}=(a+b,a-b).$"}
{"_id": "301201", "title": "", "text": "$\\lim_{N\\to\\infty}\\sum_{k=1}^N \\frac{1}{k+\\alpha N}$"}
{"_id": "1123292", "title": "", "text": "$  \\operatorname {dim}k[x,y,z]/(x^3-1)=\\operatorname {dim}_k( \\frac {k[x]}{(x^3-1)})[y,z]=0+2=2$"}
{"_id": "942221", "title": "", "text": "$ \\sum_{m = 1}^\\infty \\frac{1}{n^2 - m^2}$"}
{"_id": "624834", "title": "", "text": "$f(a,b,k) = f(0,b-a,k)$"}
{"_id": "1758283", "title": "", "text": "$\\sum\\limits_{r=1}^n (-1)^{r+1}\\binom{n}{r} \\frac{1}{r+1}.$"}
{"_id": "4023372", "title": "", "text": "$(14 - (k-1))^3= k^3$"}
{"_id": "6377265", "title": "", "text": "$\\operatorname{cov}(X,Y)<0$"}
{"_id": "4886386", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty$"}
{"_id": "3659838", "title": "", "text": "$\\frac{\\zeta(s)}{\\zeta(1-s)} = \\frac{1}{\\pi} (2\\pi)^s \\sin(\\frac{\\pi s}{2})\\Gamma(1-s) \\tag{1}$"}
{"_id": "1328631", "title": "", "text": "$\\int_0^\\pi + \\int_\\pi^{2\\pi}$"}
{"_id": "4930694", "title": "", "text": "$[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]$"}
{"_id": "6944413", "title": "", "text": "$\\lim_{n\\to\\infty}\\inf \\ f_n$"}
{"_id": "5368616", "title": "", "text": "$\\forall \\epsilon > 0, \\exists\\delta>0, s.t. 0<|x-x_0|<\\delta \\implies |f(x)-f(x_o)|< \\epsilon$"}
{"_id": "6082563", "title": "", "text": "$\\Large\\frac{9^{1-k}}{9^{1-k}+3}$"}
{"_id": "941014", "title": "", "text": "$ f(x, y) = \\frac{1}{2 \\pi \\sqrt{1 - p^2}} \\exp \\left\\{ - \\frac{1}{2(1 - p^2)} (x^2 - 2pxy + y^2) \\right\\} $"}
{"_id": "8853422", "title": "", "text": "$\\frac{n\\%2}{2} - \\frac{(n-1)\\%2}{2} = \\frac{n\\%2-(n-1)\\%2}{2} = \\frac{(n-n+1)\\%2}{2}=\\frac{1}{2} $"}
{"_id": "8802505", "title": "", "text": "$\\frac{(1+\\sqrt{3})^{n+2}-(1-\\sqrt{3})^{n+2}}{4\\sqrt{3}}.$"}
{"_id": "462991", "title": "", "text": "$P(2k)=P(n)$"}
{"_id": "6068342", "title": "", "text": "$\\sum_{n=0}^\\infty a_n = \\int_0^1\\frac{\\sin(\\pi x)}{(1-x)^2}dx=\\infty$"}
{"_id": "2921755", "title": "", "text": "$e_n^x=\\left(1+\\frac{x}{n}\\right)^n.$"}
{"_id": "829509", "title": "", "text": "$\\{ e_1,e_2,e_3,u \\}$"}
{"_id": "4272118", "title": "", "text": "$a^{log_a(b)}=b$"}
{"_id": "143417", "title": "", "text": "$(x^2-1)^n=(x-1)^n(x+1)^n$"}
{"_id": "699838", "title": "", "text": "$f_k(n) = \\binom{n}{k}$"}
{"_id": "1612033", "title": "", "text": "$ \\int_{0}^{\\pi}f(x)\\cos kxdx=\\int_{0}^{\\pi}f(x)\\sin kxdx=0 $"}
{"_id": "7499754", "title": "", "text": "$S_n = \\{x_1, x_2, x_3, \\dots, x_n\\}$"}
{"_id": "93365", "title": "", "text": "$x\\mapsto x^3-x$"}
{"_id": "523855", "title": "", "text": "$ \\frac{n^3+1}{n^2+1}=\\frac{n^3+n-n+1}{n^2+1}= \\frac{n^3+n}{n^2+1}+\\frac{1-n}{n^2+1}= n+\\frac{1-n}{n^2+1} $"}
{"_id": "4967292", "title": "", "text": "$\\int\\limits_{0}^{1} \\frac{\\tan^{-1}(x)}{x} \\mathrm{d}x$"}
{"_id": "7437246", "title": "", "text": "$ \\dfrac{\\sqrt{n}+1}{n\\sqrt{n}+\\sqrt{n}+1}\\ge \\frac{\\sqrt{n}}{3n\\sqrt{n}}=\\frac{1}{3}\\frac{1}{n} $"}
{"_id": "7710979", "title": "", "text": "$ \\left[\\begin{array}{ccc|c} 1 & -1 & 1 & 1 \\\\ 3 & m & 2 & 3 \\\\ m & -3 & 1 & 2 \\end{array}\\right] $"}
{"_id": "3358341", "title": "", "text": "$t_1 \\in [0, t_0],$"}
{"_id": "4135", "title": "", "text": "$(x^n + x^{n+1})$"}
{"_id": "2011550", "title": "", "text": "$ \\lim_{x \\to a^+} f(x) = \\lim_{x \\to a^-} f(x) $"}
{"_id": "7031721", "title": "", "text": "$\\gamma[\\lambda a(f)]:=\\lambda\\gamma[a(f)]\\\\ \\gamma[a(f)+a(g)]:=\\gamma[a(f)]+\\gamma[a(g)]\\\\ \\gamma[a(f)a(g)]:=\\gamma[a(f)]\\gamma[a(g)]\\\\ \\gamma[a(f)^*]:=\\gamma[a(f)]^*$"}
{"_id": "908457", "title": "", "text": "$d(x,z) \\leqslant d(x,y) + d(y,z)$"}
{"_id": "579069", "title": "", "text": "$\\lim_{n\\to\\infty} \\sum_{k=1}^{n}\\frac{1}{\\sinh 2^k}$"}
{"_id": "2645261", "title": "", "text": "$N(A_1\\dots A_n)$"}
{"_id": "7009143", "title": "", "text": "$\\mathbb{E}\\left[\\left| X + l^{r+1}Y^{r+1}\\right|\\right] \\geq \\mathbb{E}\\left[\\left| X \\right|\\right]$"}
{"_id": "5996725", "title": "", "text": "$\\lfloor a / b\\rfloor = c$"}
{"_id": "1305234", "title": "", "text": "$u_{n}=\\sum\\limits_{r=1}^{n} \\frac{r}{2^r}$"}
{"_id": "8484756", "title": "", "text": "$x R^T y \\Leftrightarrow y R x$"}
{"_id": "6130436", "title": "", "text": "$ \\zeta(s)=\\sum_{n\\geq 1}\\frac{1}{n^s} = \\sum_{n\\geq 1}\\int_{0}^{+\\infty}\\frac{u^{s-1}}{\\Gamma(s)}e^{-nu}\\,du = \\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{u^{s-1}}{e^u-1}\\,du$"}
{"_id": "287694", "title": "", "text": "$p_1 p_2 \\equiv p_3 p_4$"}
{"_id": "9193913", "title": "", "text": "$\\left\\lfloor\\frac{\\left\\lfloor\\frac{a}{b}\\right\\rfloor}{c}\\right\\rfloor = \\left\\lfloor\\frac{a}{bc}\\right\\rfloor$"}
{"_id": "7493043", "title": "", "text": "$ \\left\\{ \\begin{array}{ll} \\frac{\\lambda x\\sqrt{3}}{2}=1\\\\  \\frac{\\lambda y\\sqrt{3}}{2}=1\\\\ \\frac{x^2\\sqrt{3}}{4}+\\frac{y^2\\sqrt{3}}{4}-2=0 \\end{array} \\right.$"}
{"_id": "5817172", "title": "", "text": "$f'(x) = 5 f(x)$"}
{"_id": "7654207", "title": "", "text": "$\\frac{\\gamma(t)-\\gamma(s)}{|\\gamma(t)-\\gamma(s)|} =  \\frac{\\gamma(\\tilde t)-\\gamma(s)}{|\\gamma(\\tilde t)-\\gamma(s)|} \\to -\\gamma'(0)$"}
{"_id": "4612492", "title": "", "text": "$(x+y)^r = \\sum^{\\infty}_{k=0}\\binom{r}{k}x^{r-k}y^{k}$"}
{"_id": "3992605", "title": "", "text": "$\\epsilon(a,\\Delta x) = \\frac{\\Delta y}{\\Delta x}-f'(a),$"}
{"_id": "6187926", "title": "", "text": "$a \\hat{x} + b \\hat{y} = d$"}
{"_id": "3814172", "title": "", "text": "$L = \\lim_{x\\to c} f_1(x)$"}
{"_id": "3287912", "title": "", "text": "$a^{k*\\log_a b}=b^k $"}
{"_id": "3005026", "title": "", "text": "$\\gamma(\\phi(xy)) = \\gamma(\\phi(x)\\phi(y)) = \\gamma(x'y') = \\gamma(x')\\gamma(y') = [\\gamma(\\phi(x))][\\gamma(\\phi(y))]$"}
{"_id": "5351245", "title": "", "text": "$x! \\sim \\sqrt{2\\pi x} (x/e)^x$"}
{"_id": "5068586", "title": "", "text": "$\\det(A) = \\det(A_{n, n-1}) + a_{n,n}\\det(A_{n,n})$"}
{"_id": "6856361", "title": "", "text": "$ \\frac{\\pi(1-\\sin\\theta_0\\cos\\varphi_0)}{\\sin^2\\theta_0\\sin^2\\varphi_0+\\cos^2\\theta_0} $"}
{"_id": "6596718", "title": "", "text": "$u(x)=\\log\\log(1/|x|)$"}
{"_id": "3348262", "title": "", "text": "$\\forall x \\forall y (xRy \\lor yRx)$"}
{"_id": "5705171", "title": "", "text": "$\\begin{align}\\int_1^{\\infty}f(x) dx &= \\lim_{N\\to \\infty}\\int_1^{N} x^{-r} dx\\\\\n &= \\lim_{N \\to \\infty} \\left[\\frac{1}{1-r}x^{1-r}\\right]_1^N \\\\\n &= \\lim_{N\\to \\infty} \\frac{1}{1-r}\\left[\\frac{1}{N^{r-1}} - 1\\right] \\\\\n &= \\frac{1}{r-1}.\n \\end{align}\n $"}
{"_id": "280759", "title": "", "text": "$\\sec\\theta = \\dfrac x3\\quad$"}
{"_id": "437055", "title": "", "text": "$f(x)-f(a)=\\int_a^xf'(t)dt$"}
{"_id": "4091186", "title": "", "text": "$\\forall x,y,z: \\neg (x R y \\wedge y R z)$"}
{"_id": "6611058", "title": "", "text": "$\\int_0^1 \\frac{{f}(x)}{x^p} $"}
{"_id": "5151809", "title": "", "text": "$\\frac {(1-p)p^3}{1-(1-p^4)}$"}
{"_id": "928967", "title": "", "text": "$\\|x-z\\|<\\frac{|\\langle f,x\\rangle|}{\\|f\\|}+\\varepsilon,$"}
{"_id": "2153670", "title": "", "text": "$\\lim _{n\\to \\infty }\\frac{\\left[1x\\right]+\\left[2x\\right]+\\left[3x\\right]+...+\\left[nx\\right]}{n^2} = \\frac{x}{2}$"}
{"_id": "3967721", "title": "", "text": "$<e^{\\lambda x}(T \\circledast S), \\phi (x)> \\\\ = <T_{x}, <S_{y}, e^{\\lambda (x+y)} \\phi (x+y)>>  \\\\ =<e^{\\lambda x} T_{x}, <e^{\\lambda y}S_{y},  \\phi (x+y)>> \\\\ =<e^{\\lambda x} T \\circledast e^{\\lambda x} S, \\phi (x)> \\\\$"}
{"_id": "3356462", "title": "", "text": "$(av)^\\gamma = a^\\gamma v^\\gamma$"}
{"_id": "641276", "title": "", "text": "$ a^{log_cb}=b^{log_ca} $"}
{"_id": "5316119", "title": "", "text": "$y-\\frac{b-c}{2} = -(ax+\\frac{b+c}{2})$"}
{"_id": "5658864", "title": "", "text": "$y(x) = a +(b-a)x$"}
{"_id": "4913780", "title": "", "text": "$\\mathbb R[x]/(x^2+n)$"}
{"_id": "4273076", "title": "", "text": "$b=\\{\\{x\\}\\}$"}
{"_id": "5765082", "title": "", "text": "$(ab, a+b)|(a, a+b)\\cdot(b, a+b) = 1\\cdot 1$"}
{"_id": "913287", "title": "", "text": "$\\alpha = \\omega + \\alpha$"}
{"_id": "6583608", "title": "", "text": "$(1-rs)(1+rs+\\ldots+r^{n-1}s^{n-1})=1+rs-rs+\\ldots-(rs)^n=1-(rs)^n=1-0=1$"}
{"_id": "6244414", "title": "", "text": "$s_n = \\frac{n(n+1)}{2}$"}
{"_id": "3495732", "title": "", "text": "$ \\sum_{k=n-m}^{n-1} \\frac{k \\sin \\frac{k \\pi}{n}}{1 + \\cos^2 \\frac{k \\pi}{n}} = \\sum_{k=1}^{m} \\frac{(n-k) \\sin \\frac{k \\pi}{n}}{1 + \\cos^2 \\frac{k \\pi}{n}}. $"}
{"_id": "7283606", "title": "", "text": "$e^{i\\pi/2(-(1-p/2))}$"}
{"_id": "8244724", "title": "", "text": "$|V| = (r+1)N^r - rN^{r-1}$"}
{"_id": "720919", "title": "", "text": "$1+2+3+4+...=-1/12$"}
{"_id": "3405836", "title": "", "text": "$ \\alpha(\\tau)=-\\tfrac{1}{2}-i\\tfrac{(2+2p-p^2) (1-2 p -2p^2) (1+4p+p^2)}{6 \\sqrt{3} ~p(p+2) (2 p+1)(1-p^2)} $"}
{"_id": "6373843", "title": "", "text": "$|KH|=qr$"}
{"_id": "2039756", "title": "", "text": "$\\int_0^\\infty {R(x)dx}$"}
{"_id": "763976", "title": "", "text": "$\\displaystyle \\int_0^\\infty x f(x) \\; dx$"}
{"_id": "3496343", "title": "", "text": "$\\boxed{\\begin{array}{ll} f_1=n-1\\\\ f_2=1 \\\\ f_3=n- 4 \\\\ f_4=0 \\end{array}} $"}
{"_id": "5323792", "title": "", "text": "$\\frac{q}{\\sin\\gamma} = \\frac{2 a c\\cos\\gamma}{a+c+f} \\qquad \\frac{r}{\\cos\\beta} = \\frac{2 a b \\sin\\beta}{a-b+e} \\qquad \\frac{s}{\\cos\\gamma} = \\frac{2 a c \\sin\\gamma}{a-c+f}$"}
{"_id": "5194205", "title": "", "text": "$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$"}
{"_id": "6831644", "title": "", "text": "$I_k=\\{(x,y)\\in\\mathbb{R}^2:a_1< x< b_1, a_2< y< b_2\\}$"}
{"_id": "2133007", "title": "", "text": "$\\mathcal{B}=\\{\\{a\\},\\{b\\}\\}$"}
{"_id": "1312502", "title": "", "text": "$\\lim_ {n\\to \\infty} \\mu^*(E_n) \\leqslant \\mu^*(E) $"}
{"_id": "3975827", "title": "", "text": "$k | ar+bs=1$"}
{"_id": "6238800", "title": "", "text": "$\\overline{\\mathbb{Q}}\\subset\\overline{\\mathbb{R}}=\\mathbb{R}.$"}
{"_id": "8992969", "title": "", "text": "$f(a)≤k≤f(b)$"}
{"_id": "4293526", "title": "", "text": "$f(x,y,z):=(x^2,y^2,z^2,yz,zx,xy)$"}
{"_id": "6964045", "title": "", "text": "$n(n-1)={n\\choose2}$"}
{"_id": "2378127", "title": "", "text": "$x = 1 + \\sqrt{2} \\to (x-1)^2 - 2 = 0 \\to x^2 - 2x - 1 = 0$"}
{"_id": "2100296", "title": "", "text": "$\\displaystyle \\sum_{k=1}^{\\infty} \\frac{\\zeta(2k)}{k16^{k}}=4\\int_0^{+\\infty} \\dfrac{\\left(\\sinh\\left(\\tfrac{x}{8}\\right)\\right)^2}{x(e^x-1)}dx$"}
{"_id": "4759285", "title": "", "text": "$x= \\frac{a+x}{1-ax}$"}
{"_id": "3687067", "title": "", "text": "$\\sum_{n=1}^\\infty(a_n)=\\sum_{n=1}^\\infty(b_n)=\\sum_{n=1}^\\infty(c_n)$"}
{"_id": "224738", "title": "", "text": "$\\{\\alpha^\\gamma \\mid \\gamma \\in A \\} \\subseteq \\{ \\alpha^\\gamma \\mid \\gamma < \\beta \\}$"}
{"_id": "3235898", "title": "", "text": "$\\frac43=\\frac1x+\\frac1y=\\frac sp$"}
{"_id": "6334994", "title": "", "text": "$f(x) = \\frac{1}{(-1)^x+1}$"}
{"_id": "1081640", "title": "", "text": "$\\int   \\frac{1}{2(x^2+1)^2}dx$"}
{"_id": "9194285", "title": "", "text": "$f(x)={1\\over200}{e^{-x/200}}$"}
{"_id": "7793604", "title": "", "text": "$\\mu(A \\cup B)= \\lim_{n\\to \\infty}\\mu_n(A \\cup B)= \\lim_{n\\to \\infty}(\\mu_n(A) + \\mu_n(B))= \\lim_{n\\to \\infty}\\mu_n(A) + \\lim_{n\\to \\infty} \\mu_n(B)= \\mu(A) +\\mu(B)$"}
{"_id": "8524909", "title": "", "text": "$X_{1,1},$"}
{"_id": "9149220", "title": "", "text": "$\\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} \\frac{t^{s-1}}{e^{t}-1} \\,dt$"}
{"_id": "8064170", "title": "", "text": "$3^a+6^b=9^b$"}
{"_id": "3777243", "title": "", "text": "$ |(D^\\gamma\\psi)(x)| <=R_\\gamma|x|^{N+1-|\\gamma|}. $"}
{"_id": "7426230", "title": "", "text": "$E(x^2) = E[E(X^2|Y)]$"}
{"_id": "3246387", "title": "", "text": "$P[m, n] \\implies P[m, n+1]$"}
{"_id": "7773964", "title": "", "text": "$ \\begin{bmatrix} A & B \\\\ B & A \\end{bmatrix} $"}
{"_id": "4947182", "title": "", "text": "$ \\int_0^\\infty{|f(x+t)|^2dx} = \\int_0^\\infty{|f(x)|^2dx}$"}
{"_id": "2748635", "title": "", "text": "$B=\\{\\{1\\}\\}$"}
{"_id": "94435", "title": "", "text": "$1 - \\sin^2\\varphi = \\cos^2\\varphi$"}
{"_id": "5163004", "title": "", "text": "$f(f(x)) = xf(x)$"}
{"_id": "194264", "title": "", "text": "$\\begin{align*} \\mathbb{P}(X_T =a \\mid X_0 = i) &= \\mathbb{P} \\left( \\bigcup_{n \\in \\mathbb{N}_0} B_n \\mid X_0 = i \\right) \\\\&= \\sum_{n \\in \\mathbb{N}_0} \\mathbb{P}(B_n \\mid X_0 = i) \\\\ &= \\sum_{n \\in \\mathbb{N}_0} \\mathbb{P}(\\{X_n=a\\} \\cap \\{T =n\\} \\mid X_0 = i). \\tag{1} \\end{align*}$"}
{"_id": "7492697", "title": "", "text": "$1+2+3+4+5\\ldots=-\\frac{1}{12}$"}
{"_id": "5505644", "title": "", "text": "$\\mathcal{G}=\\left\\{ \\left\\{ 0\\right\\} \\right\\} $"}
{"_id": "6735593", "title": "", "text": "$f_{\\theta}(x)=\\frac{2x}{\\theta^2}, x>0$"}
{"_id": "5585977", "title": "", "text": "$x=\\sqrt[4]5e^{\\dfrac{i(3+2n\\pi)}4}$"}
{"_id": "6068840", "title": "", "text": "$1/|x|^3$"}
{"_id": "4928581", "title": "", "text": "$\\lim_{n\\to\\infty} \\left(\\frac{1}{\\sqrt{n^2+1}}+\\frac{1}{\\sqrt{n^2+2}}+...+\\frac{1}{\\sqrt{n^2+n}}\\right)$"}
{"_id": "7145648", "title": "", "text": "$\\frac{\\binom{n+2}{2}-3\\left\\lfloor \\frac{n}{2} \\right\\rfloor - 1}{6} + \\left\\lfloor \\frac{n}{2} \\right\\rfloor + 1$"}
{"_id": "538679", "title": "", "text": "$|a \\cap b| = r \\leq k$"}
{"_id": "3608094", "title": "", "text": "$ \\sum_{k=1}^\\infty \\|e_k-e'_k\\|^2 < 1. \\tag{*} $"}
{"_id": "3662828", "title": "", "text": "$f, f_1, f_2, ...:S\\to\\overline{\\mathbb{R}}$"}
{"_id": "4157140", "title": "", "text": "$ ds = \\sqrt{dx^2+dy^2} = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}dx = \\frac{rdx}{y}\\tag{2} $"}
{"_id": "356048", "title": "", "text": "$f(n) = \\binom{n}{d}$"}
{"_id": "5991626", "title": "", "text": "$ \\lim_{x\\to x_0^-} f(x) = \\lim_{x\\to x_0^+} f(x)= \\lim_{x\\to x_0} f(x). $"}
{"_id": "1117896", "title": "", "text": "$(a+b)^{r} \\leq 2^{r} (a^{r} +b^{r})$"}
{"_id": "6204256", "title": "", "text": "$(t_n^0,x_n^0)$"}
{"_id": "5624879", "title": "", "text": "$-(x/L)^\\gamma+(K-L)(-\\gamma)x^\\gamma L^{-\\gamma-1}$"}
{"_id": "127299", "title": "", "text": "$\\begin{cases}x^3-3x=y \\\\ y^3-3y=z \\\\ z^3-3z=x \\end{cases}$"}
{"_id": "2531904", "title": "", "text": "$(\\gamma(b)-\\gamma(a),\\gamma(d)-\\gamma(c))=(\\gamma(b)-\\gamma(a))\\overline{(\\gamma(d)-\\gamma(c))}=0$"}
{"_id": "8103827", "title": "", "text": "$\\in C^n\\setminus C^{n+1}$"}
{"_id": "7621815", "title": "", "text": "$E[(X-E[X])^2] = E[X^2-2 X E[X] + E[X]^2] = E[X^2] -2 E[X] E[X] + E[X]^2 = E[X^2] - E[X]^2.$"}
{"_id": "529629", "title": "", "text": "$\\omega=2\\pi\\left(1-\\frac{\\sqrt{d^2-R^2}}{d}\\right)$"}
{"_id": "6536482", "title": "", "text": "$d(x,u) \\leq d(x,y) + d(y,u)$"}
{"_id": "6061968", "title": "", "text": "$ \\tilde{\\fermi}\\pars{s,x} = {\\expo{x} \\over s - 2} \\quad\\mbox{and}\\quad \\fermi\\pars{t,x} = \\expo{x}\\int_{\\gamma - \\ic\\infty}^{\\gamma + \\ic\\infty} {\\expo{st} \\over s - 2}\\,{\\dd s \\over 2\\pi\\ic}\\quad\\mbox{with}\\quad\\gamma > 2 $"}
{"_id": "1985397", "title": "", "text": "$\\begin{cases} x=t \\\\ y=-2t \\\\ z=t \\end{cases} \\implies \\pmatrix{x \\\\ y \\\\ z} = t\\pmatrix{1 \\\\ -2 \\\\ 1}$"}
{"_id": "1741092", "title": "", "text": "$X = F \\oplus Y$"}
{"_id": "5087551", "title": "", "text": "$m\\in\\langle m\\rangle$"}
{"_id": "4271986", "title": "", "text": "$\\sum_{n = 1}^\\infty \\frac{1}{n^2 + a^2} = \\frac{1}{2}\\sum_{n = -\\infty}^\\infty \\frac{1}{n^2 + a^2} - \\frac{1}{2a^2}$"}
{"_id": "2662492", "title": "", "text": "$ \\left(\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x    }^{x_x^x}}\\right)_{\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x    ^x\\right)_{x_x^x}^{x_x^x}}}^{\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^    {\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}} $"}
{"_id": "5576195", "title": "", "text": "$(a^2+b^2,a+b)=1$"}
{"_id": "2309301", "title": "", "text": "$F(x)= \\int_a^{x^2}f(t)dt$"}
{"_id": "31277", "title": "", "text": "$X^+$"}
{"_id": "3992601", "title": "", "text": "$\\Delta y=\\Delta y(a,\\Delta x)=f(a+\\Delta x)-f(a)$"}
{"_id": "1013500", "title": "", "text": "$\\boxed{\\zeta(-1) = 1 + 2 +3 + 4 + \\dots =  - \\frac{1}{2}}$"}
{"_id": "9119791", "title": "", "text": "$\\|A\\| = \\sqrt{\\rho(A^*A)} = \\sqrt{\\rho(AA^*)} \\leq 1$"}
{"_id": "3839861", "title": "", "text": "$p\\circ T$"}
{"_id": "7056988", "title": "", "text": "$x_n \\in V_n \\subseteq \\overline{V_n} \\subseteq U_n$"}
{"_id": "4668566", "title": "", "text": "$ \\{x_1, x_2,x_3,....,x_n\\} $"}
{"_id": "2626399", "title": "", "text": "$f^2(z)=[f(z)]^2$"}
{"_id": "1345808", "title": "", "text": "$ \\|x\\|\\le \\sum_{i=1}^n |x_i| \\|e_i\\| \\le \\|x\\|_\\infty \\sum_{i=1}^n \\|e_i\\|. $"}
{"_id": "8398328", "title": "", "text": "$ \\liminf_{n\\to\\infty}\\mu_n(X)\\geq \\mu(B(x,r)) $"}
{"_id": "9174993", "title": "", "text": "$z = \\mathbb{R}[x]/(x^2+1)$"}
{"_id": "8402834", "title": "", "text": "$p_1p_2\\mid n$"}
{"_id": "562376", "title": "", "text": "$\\int_0^{2\\pi}=\\int_0^{\\pi}+\\int_{\\pi}^{2\\pi}$"}
{"_id": "2671943", "title": "", "text": "$\\frac{X_1+X_2+...+X_n}{n} $"}
{"_id": "690925", "title": "", "text": "$\\prod_{i=1}^n\\left(\\frac1i+1\\right)=\\prod_{i=1}^n\\left(\\frac{1+i}i\\right)=\\frac{\\displaystyle\\prod_{i=1}^n(1+i)}{\\displaystyle\\prod_{i=1}^ni}=\\frac{\\displaystyle\\prod_{i=2}^{n+1}i}{\\displaystyle\\prod_{i=1}^ni}=n+1$"}
{"_id": "8663043", "title": "", "text": "$\\forall a,b \\in \\mathbb{Z}\\times\\mathbb{Z}, f(a+b) = f(a)+f(b)$"}
{"_id": "1884090", "title": "", "text": "$\\lim_{n\\to\\infty}\\sqrt{1+\\sqrt{2+\\cdots+\\sqrt{n}}}$"}
{"_id": "5643792", "title": "", "text": "$\\text{P}(B) = 1/6$"}
{"_id": "5574822", "title": "", "text": "$\\frac{|\\langle Ax, y\\rangle |}{\\|x\\|\\cdot \\|y\\|}\\le \\frac{\\|Ax\\|}{\\|x\\|}.$"}
{"_id": "1117979", "title": "", "text": "$e^{x} \\geq \\left (1 + \\frac{x}{n} \\right) ^{n}$"}
{"_id": "2451841", "title": "", "text": "$\\int_0^\\infty g(x)\\,\\mathrm dx$"}
{"_id": "687488", "title": "", "text": "$a,\\quad a+d,\\quad a+2d,\\quad\\ldots$"}
{"_id": "3251745", "title": "", "text": "$\\omega \\subset \\alpha$"}
{"_id": "8580236", "title": "", "text": "$ S=2\\pi\\int y\\ ds \\\\ V=\\pi\\int y^2\\ dx  $"}
{"_id": "4536861", "title": "", "text": "$E[X\\mid\\mathcal F_{k_0}]=0$"}
{"_id": "5537449", "title": "", "text": "$\\int_0^1 f'(t) \\phi(t)dt = 0$"}
{"_id": "3282597", "title": "", "text": "$\\left[\\begin{array}{ccc|c}       2&   k&   2&  0\\\\       1&   -1&  1&  1\\\\      0&  1& -1&  k     \\end{array}\\right]$"}
{"_id": "9050768", "title": "", "text": "$PV\\int_{-\\infty}^{\\infty} \\frac{1}{(x^2+1)(x^2+2x+2)}dx$"}
{"_id": "5034562", "title": "", "text": "$\\log_a(b)+\\log_a(c) = \\log_a(bc)$"}
{"_id": "8363935", "title": "", "text": "$a_{k+p}=\\frac{1}{N}\\sum\\limits_{n=-\\infty}^{\\infty}e^{-j\\frac{2 \\pi}{N}(z-p)n}=\\frac{1}{N}\\sum\\limits_{n=-\\infty}^{\\infty}e^{-j\\frac{2 \\pi}{N}zn}e^{j\\frac{2 \\pi}{N}pn}$"}
{"_id": "5048288", "title": "", "text": "$\\frac{4^x}{6^x + 9^x} + \\frac{9^x}{6^x + 4^x} = \\frac{7}{6}$"}
{"_id": "3448644", "title": "", "text": "$ \\left|\\begin{array}{ccc} 1 & 1 & 1 \\\\ x_1 & x_2 & x_3 \\\\ y_1 & y_2 & y_3  \\end{array} \\right| $"}
{"_id": "5473902", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx = \\int_{0}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx + \\int_{-\\pi}^{0}\\frac {\\sin nx}{(1+2^x)\\sin x}dx$"}
{"_id": "355710", "title": "", "text": "$B = \\{v_1, v_2, \\ldots, v_n\\}$"}
{"_id": "113638", "title": "", "text": "$\\{x^2,xy,xz,y^2,yz,z^2\\}$"}
{"_id": "7840688", "title": "", "text": "$Q=\\{(x,y) \\in \\mathbb{R}^2 : 0\\leq x\\leq 1, 0 \\leq y \\leq 1\\}$"}
{"_id": "6890214", "title": "", "text": "$R^*\\equiv S^*$"}
{"_id": "4541616", "title": "", "text": "$f_{n}(n)=\\frac{1}{2}$"}
{"_id": "3782649", "title": "", "text": "$F_3 = F_2 + F_1 = 1 + 1 = \\mathbf{2}\\\\F_4 = F_3 + F_2 = 2 + 1 = \\mathbf{3}\\\\F_5 = F_4 + F_3 = 3 + 2 = \\mathbf{5}\\\\F_6 = F_5 + F_4 = 5 + 3 = \\mathbf{8}\\\\F_7 = F_6 + F_5 = 8 + 5 = \\mathbf{13}\\\\\\vdots$"}
{"_id": "4316762", "title": "", "text": "$\\left(\\frac12\\right)^n\\times2=\\frac1{2^{n-1}}$"}
{"_id": "2665687", "title": "", "text": "$a_n = 1 + \\frac 1{2^{n-2}}$"}
{"_id": "2065760", "title": "", "text": "$xRy \\rightarrow yRx.$"}
{"_id": "390529", "title": "", "text": "$B_1\\times\\cdots\\times B_n$"}
{"_id": "4010980", "title": "", "text": "$F(x) = \\displaystyle \\int_{a}^x f(t)dt=x^2-2x-3$"}
{"_id": "1264564", "title": "", "text": "$F(n)=F(n-2)+F(n-3)+F(n-2)=2*F(n-2)+F(n-3)=2*(F(n-3)+F(n-4))+F(n-3)=3*F(n-3)+2*F(n-4)$"}
{"_id": "2968958", "title": "", "text": "$\\lim_{n\\to\\infty}\\left(\\sum_{k=1}^{2n}\\frac{1}{2n}\\left\\lfloor\\frac{2n}{k}\\right\\rfloor-\\sum_{k=1}^{n}\\frac{1}{n}\\left\\lfloor\\frac{n}{k}\\right\\rfloor\\right)$"}
{"_id": "3435436", "title": "", "text": "$\\|x_m-x_n\\|_1<\\varepsilon$"}
{"_id": "6219250", "title": "", "text": "$g(x) =\\dfrac{c^{x}}{c^{x}+c^{1/2}} $"}
{"_id": "662501", "title": "", "text": "$\\left\\lVert e_n-e_m\\right\\rVert_p<\\epsilon$"}
{"_id": "2779830", "title": "", "text": "$\\lnot F \\lor \\lnot\\lnot F$"}
{"_id": "5458447", "title": "", "text": "$k=n^2-n+2$"}
{"_id": "8954229", "title": "", "text": "$\\left\\lfloor{a}\\right\\rfloor - \\left\\lfloor{b}\\right\\rfloor \\ge \\left\\lfloor{a-b}\\right\\rfloor$"}
{"_id": "8011847", "title": "", "text": "$(c_0+c_1)x+$"}
{"_id": "3258270", "title": "", "text": "$ \\lim_{N\\to\\infty}\\sum_{k=1}^N\\frac{9}{10^k}. $"}
{"_id": "39455", "title": "", "text": "$s(x)=\\sum_{i=1}^n c_i\\chi_{E_i}$"}
{"_id": "559139", "title": "", "text": "$E_3(n) = (\\frac{n}{2})^{(n-1-\\lfloor\\frac{n}{2}\\rfloor)}$"}
{"_id": "7395803", "title": "", "text": "$F(a,b)=K(b)=F(a)(b)$"}
{"_id": "2235225", "title": "", "text": "$f(z) = z \\ ^ 2 + \\lambda$"}
{"_id": "1173067", "title": "", "text": "$\\int \\frac{1}{{(1+x^2)^3}} \\ dx $"}
{"_id": "488814", "title": "", "text": "$C =\\begin{pmatrix} A & -B \\\\ B & A \\end{pmatrix} $"}
{"_id": "5675962", "title": "", "text": "$A_0 \\subseteq A_1 \\subseteq \\ldots \\subseteq A_{\\alpha} \\subseteq \\ldots (\\alpha \\in \\omega_1)$"}
{"_id": "295157", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1)$"}
{"_id": "2409680", "title": "", "text": "$\\{e_1,e_2,e_3,.....,e_n,....\\}$"}
{"_id": "4099034", "title": "", "text": "$g(n)=\\sum_{j=1}^nA_j $"}
{"_id": "1799247", "title": "", "text": "$\\{\\{A\\}\\} \\subseteq B$"}
{"_id": "847140", "title": "", "text": "$=1-P(m=0)-P(m=1)$"}
{"_id": "8338023", "title": "", "text": "$\\int_{-\\pi}^{\\pi}f(t)dt=2\\int_{0}^{\\pi}f(t)dt \\neq 0$"}
{"_id": "4212146", "title": "", "text": "$(x + a) (x - a) = 0$"}
{"_id": "3822422", "title": "", "text": "$\\frac{1}{16}e^\\frac{-t}{4}(t-8) $"}
{"_id": "2618696", "title": "", "text": "$x(t) = \\frac{1}{\\sqrt{1-\\frac{\\epsilon^2}{4}}} e^{\\frac{-\\epsilon t}{2}}\\sin(t\\sqrt{1-\\frac{\\epsilon^2}{4}})$"}
{"_id": "7735612", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty}\\frac{1}{n^2+1}=\\pi\\sum_{n=-\\infty}^{\\infty}e^{-2\\pi|n|}=\\pi coth\\pi$"}
{"_id": "4960538", "title": "", "text": "$ \\det \\begin{pmatrix}A & 0 \\\\ 0 & B\\end{pmatrix}=\\det(A)\\det(B) $"}
{"_id": "6547680", "title": "", "text": "$ \\Leftrightarrow (\\sin\\gamma)(\\tan (a+\\gamma))+\\cos \\gamma=\\frac{b}{c}$"}
{"_id": "415507", "title": "", "text": "$[a+(n-1)b](a-b)^{n-1}$"}
{"_id": "133460", "title": "", "text": "$\\lim_{x\\to a}f(x)g(x)=\\lim_{x\\to a}f(x)\\cdot\\lim_{x\\to a}g(x)$"}
{"_id": "1045043", "title": "", "text": "$|\\mathbb{E}[XY]|\\le\\mathbb{E}[|XY|]\\le(\\mathbb{E}|X|^p)^{\\frac{1}{p}}(\\mathbb{E}|Y|^q)^{\\frac{1}{q}}$"}
{"_id": "5103746", "title": "", "text": "$f(ka+b)=kf(a)+f(b)$"}
{"_id": "883965", "title": "", "text": "$ \\Delta a_{k+1} = r \\Delta a_{k} + \\gamma = r^2 \\Delta a_{k-1} + r \\gamma + \\gamma = r^3 \\Delta a_{k-2} + r^2 \\gamma +  r \\gamma + \\gamma\\\\ =\\ldots = r^{k-1} \\Delta a_2 + \\Gamma $"}
{"_id": "7457633", "title": "", "text": "$(1,1,-1,1,-1,-1,-1,1,-1,1,-1,-1,-1,-1)$"}
{"_id": "2733149", "title": "", "text": "$\\gamma_1 \\geq \\dots \\geq \\gamma_n$"}
{"_id": "3833445", "title": "", "text": "$ \\zeta(s)\\Gamma(s)= \\int_{0}^{\\infty}\\frac{x^{s-1}}{e^x-1}\\,dx \\implies \\frac{d}{ds}(\\zeta(s)\\Gamma(s))= \\int_{0}^{\\infty}\\frac{x^{s-1} \\ln(x)}{e^x-1}\\,dx $"}
{"_id": "3948829", "title": "", "text": "$A_n = \\frac {n(n+1)}2 - 1$"}
{"_id": "3587774", "title": "", "text": "$GF(2^k)[x]/(x^4+1)$"}
{"_id": "8997385", "title": "", "text": "$f(a_1)>a_1$"}
{"_id": "4752697", "title": "", "text": "$\\dfrac{\\sqrt{1-\\rho^2}}{2\\pi}I$"}
{"_id": "6701665", "title": "", "text": "$V= P \\oplus K $"}
{"_id": "3482059", "title": "", "text": "$a\\in V\\subseteq\\cl V\\subseteq U$"}
{"_id": "6402118", "title": "", "text": "$(\\mathbb{Z}/3\\mathbb{Z})[x]/(x^2+x+1)$"}
{"_id": "2639070", "title": "", "text": "$\\|x_m-x_n\\| \\geq 1$"}
{"_id": "5840462", "title": "", "text": "$E|XY|^2 \\leq (E|X|^p)^{1/p} E|Y|^2 \\ $"}
{"_id": "8261324", "title": "", "text": "$n! =\\sqrt{2n\\pi}\\left(\\frac{n}{e}\\right)^n e^r$"}
{"_id": "3974128", "title": "", "text": "$\\int\\int dx^2=\\int \\int 2xdx$"}
{"_id": "3457619", "title": "", "text": "$\\lim_{t\\rightarrow 0} \\frac{||\\gamma(a+t)-\\gamma(a)-\\gamma'(a)t||}{|t|}=\\lim_{t\\rightarrow 0} ||\\frac{\\gamma(a+t)-\\gamma(a)}{t}-\\gamma'(a)||=||\\lim_{t\\rightarrow 0} \\frac{\\gamma(a+t)-\\gamma(a)}{t} -\\gamma'(a)||=0$"}
{"_id": "4561360", "title": "", "text": "$\\begin{bmatrix} \\alpha & \\beta \\\\ -\\beta &\\alpha\\end{bmatrix}$"}
{"_id": "937706", "title": "", "text": "$ y^2-p(x) = (A(x)y+a(x))(B(x)y+b(x))=A(x)B(x)y^2+(A(x)b(x)+B(x)a(x))y+a(x)b(x) $"}
{"_id": "143804", "title": "", "text": "$A\\subseteq B\\subseteq C$"}
{"_id": "4073247", "title": "", "text": "$f(f(x)-x)=f(1-x)=-1=f(1)=f(f(x))$"}
{"_id": "5082724", "title": "", "text": "$A^{\\bot\\bot}\\otimes B^{\\bot\\bot}\\vdash(A\\otimes B)^{\\bot\\bot}$"}
{"_id": "2557455", "title": "", "text": "$(x- a)^{n-1}v,\\ldots (x-a)v, v$"}
{"_id": "273585", "title": "", "text": "$B=\\{e_1,e_2, ...., e_n\\}$"}
{"_id": "86966", "title": "", "text": "$ x!\\sim (x/e)^x\\sqrt{2\\pi x} $"}
{"_id": "4366711", "title": "", "text": "$r(r+1)=\\frac{1}{3}((r+1)^3-r^3-1)$"}
{"_id": "538446", "title": "", "text": "$ 1+2+3+\\cdots \\rightarrow -\\frac{1}{12} $"}
{"_id": "7142768", "title": "", "text": "$ I(A) = \\int_0^2 \\sqrt{1 + \\frac{dy}{dx}^2}dx  = \\int_0^2 \\sqrt{1 + (2x)^2} dx = \\sqrt{17} + \\frac{1}{4} \\sinh^{-1}(4) \\approx 4.64678. $"}
{"_id": "3212883", "title": "", "text": "$\\begin{align}\\int_0^\\infty f(x)\\,\\mathrm dx&=\\int_0^1 f(x)\\,\\mathrm dx+\\lim_{y\\to a^-}\\int_1^y f(x)\\,\\mathrm dx+\\lim_{z\\to a^+}\\int_z^c f(x)\\,\\mathrm dx+\\lim_{w\\to\\infty}\\int_c^wf(x)\\,\\mathrm dx\\\\&=\\int_0^1 f(x)\\,\\mathrm dx+\\infty+\\lim_{z\\to a^+}\\int_z^c f(x)\\,\\mathrm dx+\\lim_{w\\to\\infty}\\int_c^wf(x)\\,\\mathrm dx\\\\&\\equiv\\text{ undefined or diverges to infinity}\\end{align}$"}
{"_id": "5017492", "title": "", "text": "$ \\left[ \\begin{array}{ccc|c} 1 & a & 0 & 1\\\\ 0 & 1 & b & 1\\\\ 0 & -ac & 1 & 1-c \\end{array} \\right]. $"}
{"_id": "2467298", "title": "", "text": "$p\\in \\overline{p_1 p_2}$"}
{"_id": "2409285", "title": "", "text": "$\\displaystyle \\int_0^\\infty \\dfrac{\\sin^3x}xdx=\\dfrac\\pi4$"}
{"_id": "2561824", "title": "", "text": "$(k+1)^3<\\frac{(k+1)^3}{k^3}2^k$"}
{"_id": "6776561", "title": "", "text": "$F=A\\oplus R(T)$"}
{"_id": "9214676", "title": "", "text": "$\\Phi(s,n)=\\phi(s,n)=s\\varphi(n)$"}
{"_id": "5212987", "title": "", "text": "$h(P) = (bg)(P)\\implies b(P)g(P)=0$"}
{"_id": "4226127", "title": "", "text": "$f(n) = \\left(\\frac{\\sqrt 2}{2}(1+i)\\right)^n$"}
{"_id": "765160", "title": "", "text": "$ \\int_0^1 \\dfrac{\\sin(x)}{x} ~dx \\approx 0.9460830704$"}
{"_id": "6569534", "title": "", "text": "$P\\{\\sup_{t\\geq 0} M_t > x \\,| \\,F_0\\} \\geq 1\\wedge \\frac{M_0}{x}.$"}
{"_id": "1177137", "title": "", "text": "$f(k)=k^2-k+1$"}
{"_id": "7433482", "title": "", "text": "$\\forall \\epsilon, \\exists \\delta \\quad 0<|x-a|<\\delta \\quad \\implies \\quad |f(x)-f(a)| < \\epsilon.$"}
{"_id": "5579100", "title": "", "text": "$P(X=k)=\\frac{1}{\\zeta(3)k^3}$"}
{"_id": "8255420", "title": "", "text": "$s\\equiv r\\equiv 2s\\pmod {4}$"}
{"_id": "8883561", "title": "", "text": "$ \\frac{d}{d\\gamma} \\int_{0}^{\\gamma} (\\gamma-s) \\, dF(s) = F(\\gamma) - F(0) = \\int_{0}^{\\gamma} dF(s) $"}
{"_id": "59196", "title": "", "text": "$\\frac {(1 + \\frac 1n)^{n + 1} - (1 + \\frac {1}{n + 1})^{n + 1}}{b - a} < (n + 1)(1 + \\frac 1n)^n$"}
{"_id": "311591", "title": "", "text": "$P(k)\\Rightarrow P(k+1)$"}
{"_id": "4890014", "title": "", "text": "$1+\\frac{n}{1!}+\\frac{n*(n-1)}{2!}+\\frac{n*(n-1)*(n-2)}{3!}+...+\\frac{n*(n-1)...3*2}{(n-1)!}+\\frac{n!}{n!}=2^n$"}
{"_id": "1668920", "title": "", "text": "$\\sum_{i=1}^{n-1} \\cos \\frac{2ik\\pi}{n}\\sin \\frac{2il\\pi}{n}=0$"}
{"_id": "7212254", "title": "", "text": "$xRy\\wedge yRx \\Longrightarrow x=y$"}
{"_id": "6145922", "title": "", "text": "$W=P\\oplus Q$"}
{"_id": "3913596", "title": "", "text": "$P(E)=P(E\\cap E)=P(E)P(E)$"}
{"_id": "7985092", "title": "", "text": "$x(t)=|t|^{-n}$"}
{"_id": "267799", "title": "", "text": "$G(x) = \\int_a^xf(t)\\,dt$"}
{"_id": "1468151", "title": "", "text": "$A=\\{\\langle x,y\\rangle\\in\\Bbb R^2:y=0\\}$"}
{"_id": "7009619", "title": "", "text": "$p \\nmid r_1r_2$"}
{"_id": "2750297", "title": "", "text": "$\\mathcal{C} = (e_1,e_2,e_3,e_4,e_5)$"}
{"_id": "3097441", "title": "", "text": "$\\lfloor \\alpha \\rfloor = \\alpha-\\{\\alpha\\},\\lfloor \\beta \\rfloor = \\beta-\\{\\beta\\},\\lfloor \\gamma \\rfloor = \\gamma-\\{\\gamma\\},0\\leq \\{\\alpha\\},\\{\\beta\\},\\{\\gamma\\}<1$"}
{"_id": "4304201", "title": "", "text": "$A =  \\begin{bmatrix} 1&x&x^{2} \\\\ 1&y&y^{2} \\\\ 1&z&z^{2} \\end{bmatrix}  $"}
{"_id": "5916973", "title": "", "text": "$\\int_{\\Omega}|u_n|^{q_1\\gamma} \\le C\\left(\\int_{\\Omega}|u_n|^{q_1} + |u_n|^{q_1-1}|\\nabla u_n|\\right)^{\\gamma},$"}
{"_id": "4056351", "title": "", "text": "$[p+e,p-e]$"}
{"_id": "1528631", "title": "", "text": "$\\lfloor\\frac{a}{b}\\rfloor - \\lfloor\\frac{a}{b}\\rfloor x + \\lfloor\\frac{c}{d}\\rfloor x + x - x^2$"}
{"_id": "1332492", "title": "", "text": "$f'(c) = \\lim_\\limits{x\\to c^-} p'(x) = \\lim_\\limits{x\\to c^+} q'(x)$"}
{"_id": "9207882", "title": "", "text": "$ | \\int_{\\gamma} f | \\leq \\int_{\\gamma} |f| |dz| $"}
{"_id": "8950363", "title": "", "text": "$\\sigma = \\gamma^{23} = (\\gamma^2)^{11} \\gamma = e^{11} \\gamma = \\gamma .$"}
{"_id": "7914457", "title": "", "text": "$\\rho(A^TA)$"}
{"_id": "3569476", "title": "", "text": "$(x,y,z) \\cdot (1,1,1)=0$"}
{"_id": "1787695", "title": "", "text": "$P(\\{\\omega_4\\})=1/6$"}
{"_id": "7404863", "title": "", "text": "$ \\sum_{n \\in \\mathbb{Z}} e^{-\\pi n^2 t}= \\frac{1}{\\sqrt{t}}\\sum_{n \\in \\mathbb{Z}} e^{-\\pi n^2 / t}$"}
{"_id": "8858288", "title": "", "text": "$F(x)=\\displaystyle\\int\\limits_a^x f(t)\\,\\text{d}t.$"}
{"_id": "8370294", "title": "", "text": "$z=\\sqrt re^{it/2}$"}
{"_id": "7744315", "title": "", "text": "$\\lim_{ n \\to \\infty }f(\\infty)=?$"}
{"_id": "2072672", "title": "", "text": "$f(t)=t^2-t+1$"}
{"_id": "1485309", "title": "", "text": "$f_n(x,t) = \\sum_{j=1}^n f_{j,n}(x)\\mathbf{1}_{T_i}(t)$"}
{"_id": "6246499", "title": "", "text": "$a^{log_b(c)}= p$"}
{"_id": "5604197", "title": "", "text": "$f_X(x)=\\frac{1}{100}e^{-\\frac{x}{100}}$"}
{"_id": "2813618", "title": "", "text": "$\\mathbb{Z}/(9)$"}
{"_id": "816604", "title": "", "text": "$D_\\dot\\gamma \\dot\\gamma = \\nabla_\\dot\\gamma \\dot\\gamma - A(\\dot\\gamma,\\dot\\gamma)\\nu.$"}
{"_id": "7929989", "title": "", "text": "$f(a)=\\lim_{x\\rightarrow a^+}f(x)=\\lim_{x\\rightarrow a^-}f(x)=\\lim_{x\\rightarrow a}f(x)$"}
{"_id": "2962139", "title": "", "text": "$log_a(b)=1.26$"}
{"_id": "5056597", "title": "", "text": "$f'(x)=-\\frac1{x\\log^2x}+\\frac1{(x-1)^2}=\\frac{x\\log^2x-(x-1)^2}{x(x-1)^2\\log^2x}$"}
{"_id": "3097440", "title": "", "text": "$\\lfloor \\alpha \\rfloor+\\lfloor \\beta \\rfloor +\\lfloor \\gamma \\rfloor =$"}
{"_id": "7986583", "title": "", "text": "$f(f(x))=xf(x)+1$"}
{"_id": "3368264", "title": "", "text": "$\\tan(x+y)=a(\\tan(x)+\\tan(y))$"}
{"_id": "3430651", "title": "", "text": "$(x+a)(x-a)^2$"}
{"_id": "9334878", "title": "", "text": "$R_N(x_N) = \\frac{3^N}{3^N-5} > 1$"}
{"_id": "9354428", "title": "", "text": "$d(x,z)\\lt d(x,y)+d(y,z)$"}
{"_id": "5806057", "title": "", "text": "$\\lim_{x\\to a^+}\\frac{f(x)-f(x)}{x-a}=\\lim_{x\\to a^+}f'(x)=f'(a)=\\lim_{c\\to a^+}f'(c)$"}
{"_id": "4912839", "title": "", "text": "$xRy, yRx$"}
{"_id": "1745920", "title": "", "text": "$(a+b, ab)=1$"}
{"_id": "5382631", "title": "", "text": "$f(x+f(x)+2f(y))=f(2x)+f(2y)$"}
{"_id": "7681510", "title": "", "text": "$s \\equiv u \\equiv t \\Rightarrow s \\equiv t \\Rightarrow [s] = [t]$"}
{"_id": "6773", "title": "", "text": "$[x,y]=\\langle x,y\\rangle$"}
{"_id": "3305948", "title": "", "text": "$\\bigl\\lfloor \\frac{a}{bc} \\bigr\\rfloor = \\Bigl\\lfloor \\frac{\\bigl\\lfloor \\frac{a}{b} \\bigr\\rfloor}{c} \\Bigr\\rfloor$"}
{"_id": "6557448", "title": "", "text": "$\\sum|a_n-b_n|<\\delta $"}
{"_id": "6476157", "title": "", "text": "$\\forall x\\in A\\forall y\\in A(xRy\\lor yRx)\\tag{2}$"}
{"_id": "3633944", "title": "", "text": "$ P(m) = \\frac{(1-p)^{m-1}p}{1-(1-p)^{N}} $"}
{"_id": "6626411", "title": "", "text": "$\\frac{13}5=2+\\frac35=2+\\frac1{\\frac53}=2+\\frac1{1+\\frac23}=2+\\frac1{1+\\frac1{\\frac32}}=2+\\frac1{1+\\frac1{1+\\frac12}}$"}
{"_id": "8326451", "title": "", "text": "$T = ([n], E)$"}
{"_id": "4512392", "title": "", "text": "$p_1p_2\\cdots p_m\\mid a$"}
{"_id": "8276813", "title": "", "text": "$f(n) = \\sin\\left(\\frac{n\\pi}{2}\\right)$"}
{"_id": "5150147", "title": "", "text": "$f \\sin(t) - \\int_{0}^{\\pi} f^{\\prime} \\sin(t) dt = 0$"}
{"_id": "5168284", "title": "", "text": "$\\|\\delta x\\|=\\frac{\\|\\Delta x\\|}{\\|x\\|},\\|\\delta A\\|=\\frac{\\|\\Delta A\\|}{\\|A\\|}$"}
{"_id": "2703222", "title": "", "text": "$a^{a+2b}=b^{b+2a}$"}
{"_id": "8883562", "title": "", "text": "$ \\frac{d}{d\\gamma} \\int_{0}^{\\gamma} (\\gamma-s) \\, dF(s) \\quad``\\,=\\text{''}\\quad \\underbrace{(\\gamma - \\gamma)F'(\\gamma)}_{=0} + \\int_{0}^{\\gamma} dF(s).  $"}
{"_id": "1107487", "title": "", "text": "$\\frac{a^2+bc}{b+c}$"}
{"_id": "5409913", "title": "", "text": "$f_v(x)=\\frac{\\langle x, v \\rangle}{\\|v\\|^2}$"}
{"_id": "1939252", "title": "", "text": "$  (y-z)^2 = (y-x + x- z)^2 = (y-x)^2  + (x- z)^2 + 2 (y-x)(x-z)  $"}
{"_id": "5526551", "title": "", "text": "$(A_1\\times ... \\times A_n)$"}
{"_id": "8847902", "title": "", "text": "$\\;\\begin{cases} x=-3+t,\\\\y=4,\\\\z=1. \\end{cases}$"}
{"_id": "7878698", "title": "", "text": "$\\frac{1}{m}\\sum_{n = 1}^\\infty \\mu(A_n) \\ge \\mu(B_m)$"}
{"_id": "8619003", "title": "", "text": "$\\implies \\left|\\frac{ds}{dx}\\right| = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}$"}
{"_id": "244210", "title": "", "text": "$aq_1q_2 |a$"}
{"_id": "4744693", "title": "", "text": "$|d(x,y) -d(u,v)|\\leq d(x,u) +d(y,v)$"}
{"_id": "3725784", "title": "", "text": "$\\forall \\epsilon>0\\quad\\exists\\delta>0\\quad\\forall x,y\\quad|x-y|<\\delta\\Rightarrow|f(x)-f(y)|<\\epsilon$"}
{"_id": "6494089", "title": "", "text": "$\\{1,x,y,xy,x^2,y^2\\}$"}
{"_id": "7015702", "title": "", "text": "$y=\\frac{ab-bx}{a}$"}
{"_id": "7627887", "title": "", "text": "$[x,y]=\\{x\\}$"}
{"_id": "8329904", "title": "", "text": "$ |g(y)-g(x)| \\le  \\int_x^y \\gamma t^{\\gamma-1}dt \\le \\gamma (x-y)^\\gamma \\cdot \\left(\\int_x^y |t|^{(\\gamma-1)\\cdot \\frac{\\gamma}{1-\\gamma}} dt\\right)^{\\frac{1-\\gamma}\\gamma} =  \\gamma (x-y)^\\gamma ( y^{1-\\gamma}-x^{1-\\gamma})^{\\frac{1-\\gamma}\\gamma} \\le \\gamma (x-y)^\\gamma . $"}
{"_id": "765148", "title": "", "text": "$\\displaystyle \\int_0^1 \\frac{\\sin(x)}{x}~ dx$"}
{"_id": "8195995", "title": "", "text": "$\\frac{2^n}{1+2^n}=\\frac{2^n}{2^n}\\frac{1}{\\frac{1}{2^n}+1}=1$"}
{"_id": "9203136", "title": "", "text": "$z^{\\frac{1}{n}} = r^{\\frac{1}{n}}e^{\\frac{\\mathbb{i} \\theta}{n} } $"}
{"_id": "2129767", "title": "", "text": "$ \\frac{1-a^{(1-s)/2}}{(1-a)a^{1/2}} = \\frac{1-a^{1/2}}{(1-a)a^{1/2}} + \\frac{1-a^{-s/2}}{1-a} = \\frac{1}{\\sqrt{a}(1+\\sqrt{a})} + \\frac{1-a^{-s/2}}{1-a}. $"}
{"_id": "1813055", "title": "", "text": "$\\displaystyle \\int\\frac{1}{\\sqrt{x^8+1}}dx$"}
{"_id": "1300123", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta>0,\\, |x-a|\\lt \\delta\\implies |f(x)-f(a)|\\lt \\frac{\\epsilon}{|\\lambda|}$"}
{"_id": "2118241", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{n}$"}
{"_id": "8833728", "title": "", "text": "$x_n\\leq\\frac{1}{2^{n-2}}$"}
{"_id": "7112669", "title": "", "text": "$f_i:\\mathbb{R}^{n+1}\\to \\mathbb{R}^n$"}
{"_id": "8177120", "title": "", "text": "$\\begin{align}\\lim_{n \\to \\infty}\\dfrac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\dots+\\sqrt{n-1}}{n\\sqrt{n}} &= \\lim_{n \\to \\infty} \\frac{\\sqrt{n}}{(n+1)\\sqrt{n+1}-n\\sqrt{n}} \\\\&=\\lim_{n \\to \\infty} \\frac{1}{(n+1)\\sqrt{1+\\frac{1}{n}}-n}\\\\ &=\\frac{1}{1+\\frac{1}{2}}=\\frac{2}{3}\\end{align}$"}
{"_id": "14347", "title": "", "text": "$x^3-x$"}
{"_id": "9075494", "title": "", "text": "$\\sum_{m=1}^{\\infty}\\sum_{n=1}^{\\infty}\\frac{1}{3^{mn}}$"}
{"_id": "2454377", "title": "", "text": "$A=A_1\\times...\\times A_n$"}
{"_id": "7881263", "title": "", "text": "$\\left\\lfloor \\frac{k+1}{p} \\right\\rfloor=\\left\\lfloor \\frac{k}{p} \\right\\rfloor$"}
{"_id": "7851325", "title": "", "text": "$a(n)=n^2+n+2$"}
{"_id": "1472883", "title": "", "text": "$D= \\{(x,y)\\in \\mathbb{R}^2: 0\\le x\\le 1,\\text{ } 0\\le y \\le 1 \\} $"}
{"_id": "6744994", "title": "", "text": "$\\bar{x}=\\frac{x_1+x_2+...+x_n}{n}$"}
{"_id": "8189437", "title": "", "text": "$ \\eqalign{   & {{1 - y^{\\,3/4} } \\over {\\left( {1 - y} \\right)y^{\\,7/8} }} = {{y^{\\, - 7/8}  - y^{\\, - 1/8} } \\over {\\left( {1 - y} \\right)}} =   \\cr    &  = {{1 - y^{\\, - 1/8}  - \\left( {1 - y^{\\, - 7/8} } \\right)} \\over {\\left( {1 - y} \\right)}} \\cr}  $"}
{"_id": "1710044", "title": "", "text": "$x_{1}, x_{2} \\in \\mathbb{N}$"}
{"_id": "2348144", "title": "", "text": "$2^x+3^x=6^x$"}
{"_id": "6526725", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{1}{n^2}=\\sum_{n\\geq 1}\\frac{(n+1)-n}{n^2}\\color{red}{\\neq}\\sum_{n\\geq 1}\\frac{n+1}{n^2}-\\sum_{n\\geq 1}\\frac{n}{n^2}. $"}
{"_id": "7977073", "title": "", "text": "$\\operatorname{\\mathcal{Jac}}\\left( \\mathbb{Q}[x] / (x^8-1) \\right)$"}
{"_id": "7656462", "title": "", "text": "$\\displaystyle S_N = \\frac{N(N-1)}{2}$"}
{"_id": "253463", "title": "", "text": "$\\nu^+$"}
{"_id": "7855632", "title": "", "text": "$\\int_{-\\pi}^{\\pi} f(t) \\sin (nt) dt \\rightarrow 0 , \\int_{-\\pi}^{\\pi} f(t) \\cos (nt) dt \\rightarrow 0$"}
{"_id": "4529239", "title": "", "text": "$\\displaystyle\\int_{-\\infty}^{\\infty}\\displaystyle\\frac{1}{(x^2+1)(x^2+9)}dx$"}
{"_id": "2397446", "title": "", "text": "$|f'(x)| \\leq c |f(x)|$"}
{"_id": "4072375", "title": "", "text": "$P(E) = \\frac{1}{4}$"}
{"_id": "2428517", "title": "", "text": "$\\rho(A) - \\rho(A) = \\infty - \\infty).$"}
{"_id": "5370617", "title": "", "text": "$1-x,\\,x-x^3$"}
{"_id": "4306020", "title": "", "text": "$u(x) = \\sum_{j=1}^k \\frac{1}{|I_j|} \\chi_{I_j}(x)$"}
{"_id": "5363871", "title": "", "text": "$ f(e^{-t}) = \\sum_{n=-\\infty}^{\\infty } e^{-tn^2} = \\sqrt{\\frac{\\pi}{t}}\\sum_{n=-\\infty}^{\\infty } e^{-\\frac{n^2}{t}} =\\sqrt{\\frac{\\pi}{t}}f(e^{-\\frac{1}{t}}) $"}
{"_id": "3099987", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{k=1}^n \\frac{1}{2+k/n}$"}
{"_id": "6668138", "title": "", "text": "$\\sum\\limits_{n=0}^{\\infty}\\frac{1}{n^2+a^2}$"}
{"_id": "8153837", "title": "", "text": "$\\int \\frac{1}{(x^2-1)^{3/2}}dx$"}
{"_id": "5311920", "title": "", "text": "$S\\to T_{0,1}|T_2\\\\T_{0,1}\\to\\epsilon|002|112|012\\\\T_2\\to 00T_{0,1}2|011122|111122$"}
{"_id": "5329582", "title": "", "text": "$P(\\{(x,y) \\in \\mathbb{R^2}:x \\in A,0<y \\leq 1 \\})=\\lambda(A)$"}
{"_id": "6744483", "title": "", "text": "$ \\langle\\dot\\gamma(t),\\dot\\gamma(t)\\rangle \\ddot\\gamma(t) = \\langle\\dot \\gamma(t),\\ddot \\gamma(t)\\rangle\\dot \\gamma(t) $"}
{"_id": "1432338", "title": "", "text": "$P[X_1<\\cdots < X_n] = 1/n!$"}
{"_id": "3977985", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\int^{1}_{k=0}x^{k}\\sum^{n}_{k=0}\\frac{x^{k+1}-(1-x)^{k+1}}{(2x-1)(1-x)^k}dx$"}
{"_id": "7210620", "title": "", "text": "$2\\beta(s)\\beta'(s)=2\\langle\\gamma'(s),\\gamma(s)\\rangle-2\\langle\\gamma'(s),\\gamma(0)\\rangle=2\\langle\\gamma'(s),\\gamma(s)-\\gamma(0)\\rangle$"}
{"_id": "6303241", "title": "", "text": "$I=(x_1,x_2,x_3, ..., x_n)$"}
{"_id": "1661359", "title": "", "text": "$S= \\{s_{1}, s_{2}, s_{3},\\dots,s_{n}\\}$"}
{"_id": "576902", "title": "", "text": "$|AB|=R\\theta$"}
{"_id": "6277483", "title": "", "text": "$\\left\\lfloor\\frac{a+b+c}{d}\\right\\rfloor+\\left\\lfloor\\frac{a+b+d}{c}\\right\\rfloor+\\left\\lfloor\\frac{a+d+c}{b}\\right\\rfloor+\\left\\lfloor\\frac{d+b+c}{a}\\right\\rfloor$"}
{"_id": "5590005", "title": "", "text": "$e_n - f(x_n)/f'(x_n)$"}
{"_id": "1023649", "title": "", "text": "$T^{1,0}M\\oplus T^{0,1}M$"}
{"_id": "4902579", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{100}$"}
{"_id": "2459856", "title": "", "text": "$f(a)+f(b)\\ge f(a+b)$"}
{"_id": "3287485", "title": "", "text": "$\\Bbb F_2[x]/(x^4+x^2)$"}
{"_id": "8224615", "title": "", "text": "$g(x)=(1-x^2)\\frac{\\sin ^{-1}(x)}{x}$"}
{"_id": "6798425", "title": "", "text": "$[rn, r(n+1))$"}
{"_id": "5572183", "title": "", "text": "$rS-S=r^n-1$"}
{"_id": "982716", "title": "", "text": "$xRy, yRz $"}
{"_id": "7969332", "title": "", "text": "$\\left(1 + \\frac{x}{N}\\right)^N \\approx e^x,$"}
{"_id": "6294112", "title": "", "text": "$(-)\\times (\\pm) = (\\mp)\\\\  (\\pm)\\times(\\pm) = (+)\\\\  (\\mp)\\times(\\mp) = (+)\\\\  (\\pm)\\times(\\mp) = (-) $"}
{"_id": "122722", "title": "", "text": "$q = \\bigg\\lfloor{\\frac{\\lfloor{x}\\rfloor}{d}}\\bigg\\rfloor = \\Big\\lfloor{\\frac{x}{d}}\\Big\\rfloor$"}
{"_id": "4408022", "title": "", "text": "$ x'+y=y+x'=(y+x)'=(x+y)'=x+y' $"}
{"_id": "2483322", "title": "", "text": "$R=P\\oplus Q$"}
{"_id": "8032285", "title": "", "text": "$S_N=\\frac{X_1+X_2+ \\cdots +X_N}{N}$"}
{"_id": "4154766", "title": "", "text": "$ dl = \\sqrt{ \\left(\\frac{dx}{dt}\\right)^2+\\left(\\frac{dy}{dt}\\right)^2 } $"}
{"_id": "1477361", "title": "", "text": "$[ ....0, \\lambda, -(\\lambda+\\gamma), \\gamma, 0, ...]$"}
{"_id": "2236754", "title": "", "text": "$[T^{a,x}, (T^{b,y})^*]= [T^{a,x}, T_{b,y}]$"}
{"_id": "5199790", "title": "", "text": "$||f|| = \\int_a^b|f(t)| \\, dt + \\int_a^b|f'(t)| \\, dt$"}
{"_id": "4449274", "title": "", "text": "$D(x,B)\\leqslant d(x,y)+D(y,B)$"}
{"_id": "7362349", "title": "", "text": "$S=\\big\\{\\{a\\}:a\\in A\\big\\}$"}
{"_id": "6902795", "title": "", "text": "$(x_1,x_2,x_3,x_4)=x*(1,0,-1,1)+y*(0,1,2,-1)$"}
{"_id": "8040338", "title": "", "text": "$\\int_0^\\infty \\int_0^\\infty |f(t,x)|dtdx < \\infty$"}
{"_id": "4257264", "title": "", "text": "$ y=\\frac{b+dx}{1-b-dx} $"}
{"_id": "4580644", "title": "", "text": "$ \\overline{D}=D^*=(U^*AU)^*= U^*A^*U=U^*AU=D $"}
{"_id": "6139514", "title": "", "text": "$U_t \\subseteq V\\subseteq \\overline{V}$"}
{"_id": "1663993", "title": "", "text": "$f_X(x)=\\frac{1}{\\sqrt{2\\pi}}e^{\\frac{-x^2}{2}},x \\in \\mathbb{R}$"}
{"_id": "606012", "title": "", "text": "$|AB|=x$"}
{"_id": "2897212", "title": "", "text": "$ar-bs = 1-nz.$"}
{"_id": "4602706", "title": "", "text": "$I=\\int_0^\\infty\\,\\frac{1}{\\left(x^2+1\\right)^3}\\,\\text{d}x$"}
{"_id": "5735382", "title": "", "text": "$\\mathcal{B}_a=\\bigl\\{\\{a\\}\\bigr\\}$"}
{"_id": "452892", "title": "", "text": "$a^{1+sk}\\equiv b^{1+sk}\\equiv b\\cdot b^{sk}\\equiv b\\cdot (b^s)^k\\equiv b \\cdot 1\\equiv b \\bmod 9.$"}
{"_id": "6411658", "title": "", "text": "$T_p \\mathbb{R}^{n+1} = \\mathbb{R}^{n+1}$"}
{"_id": "614755", "title": "", "text": "$I_j=(a_j,a_{j+1})$"}
{"_id": "3228878", "title": "", "text": "$S_n=1+2+...+n+n+(n-1)+...+1$"}
{"_id": "3038682", "title": "", "text": "$f(x)=\\frac{2^{x+1}}{2^x+1}$"}
{"_id": "347801", "title": "", "text": "$\\mathbf{type}(\\gamma\\times\\gamma,\\lhd)\\leq \\gamma$"}
{"_id": "1375652", "title": "", "text": "$f(n) = 2f(\\frac n2)$"}
{"_id": "2820075", "title": "", "text": "$N! + 2, N! + 3, N! + 4, ..., N!+N$"}
{"_id": "318868", "title": "", "text": "$\\alpha^{++}$"}
{"_id": "4981517", "title": "", "text": "$ y=\\frac{b-x}{1-bx} $"}
{"_id": "5168702", "title": "", "text": "$\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}$"}
{"_id": "9036446", "title": "", "text": "$\\frac{-27}{-3} = 9$"}
{"_id": "4569717", "title": "", "text": "$ A_1, A_2, ... , A_k, A_{k+1} $"}
{"_id": "16745", "title": "", "text": "$ S_n = \\frac{n(n-1)}{2}$"}
{"_id": "4959552", "title": "", "text": "$\\sum_{k=1}^{M-1}\\cos\\frac{2\\pi kj}{M}=-1.$"}
{"_id": "7656965", "title": "", "text": "$f_{X,Y}(x,y) = \\frac{1}{100}e^{\\frac{-(x+y)}{10}}$"}
{"_id": "4361491", "title": "", "text": "$\\int_0^{\\infty} dx \\frac{x^{u-1}}{e^x-1} = \\sum_{k=0}^{\\infty} \\int_0^{\\infty} dx\\, x^{u-1} \\, e^{-(k+1) x}  = \\Gamma(u) \\sum_{k=0}^{\\infty} \\frac{1}{(k+1)^u} = \\Gamma(u) \\zeta(u)$"}
{"_id": "6331254", "title": "", "text": "$V \\oplus T$"}
{"_id": "5765081", "title": "", "text": "$(b, a+b) = (b, a) = 1$"}
{"_id": "4279467", "title": "", "text": "$ L'=\\lim_{x \\rightarrow \\infty}f'(x)=\\lim_{x \\rightarrow \\infty}f'(\\xi_x)=\\lim_{x \\rightarrow \\infty}[f(x+1)-f(x)]=0.$"}
{"_id": "8223478", "title": "", "text": "$ \\begin{vmatrix}     1  & 1  & 1 \\\\     x  & y  & z \\\\     yz & xz & xy  \\end{vmatrix}  = 0 $"}
{"_id": "6116412", "title": "", "text": "$\\lim_{x\\to 0} f'(x)=0=\\lim_{x\\to\\delta} f'(x)$"}
{"_id": "2471844", "title": "", "text": "$p_1 p_2 \\ldots p_k + 1$"}
{"_id": "3384109", "title": "", "text": "$\\overline{\\mathbb{R}_+^2}$"}
{"_id": "1435496", "title": "", "text": "$\\int_{0}^{\\pi}\\frac{\\sin^n{x}}{(1+r^2-2r\\cos{x})^{(n+2)/2}}\\,\\mathrm dx=\\frac{1}{1-r^2}\\int_{0}^{\\pi}\\sin^n{x}\\,\\mathrm dx$"}
{"_id": "5190234", "title": "", "text": "$f(x)=1/|x|^{1/2}$"}
{"_id": "9181590", "title": "", "text": "$(x+y)^2(y+z)(x-z)+(y+z)^2(x+z)(y-x)+(x+z)^2(x+y)(z-y) = 2 x y^3+2 x^3 z + 2 y z^3-2 x^2 y z-2 x y^2 z-2 x y z^2.$"}
{"_id": "4605160", "title": "", "text": "$\\mathbb{P}(X_n\\leq x)\\rightarrow \\mathbb{P}(X\\leq x)$"}
{"_id": "2968956", "title": "", "text": "$\\lim_{n\\to\\infty}\\sum_{k=1}^{2n}(-1)^{k-1}\\frac{1}{2n}\\left\\lfloor\\frac{2n}{k}\\right\\rfloor=\\lim_{n\\to\\infty}\\left(\\sum_{k=1}^{2n}\\frac{1}{2n}\\left\\lfloor\\frac{2n}{k}\\right\\rfloor-2\\sum_{k=1}^{n}\\frac{1}{2n}\\left\\lfloor\\frac{2n}{2k}\\right\\rfloor\\right)$"}
{"_id": "8624473", "title": "", "text": "$\\sin a + \\sin^2 a + \\tan a - \\tan a \\sin a  = \\sin a + \\sin^2 a + \\frac{\\sin a} {\\cos a} - \\frac{sin^2 a}  {cos a}$"}
{"_id": "267952", "title": "", "text": "$a = \\frac{187}{200} = \\frac{1}{\\frac{200}{187}} = \\frac{1}{1+\\frac{13}{187}} = \\frac{1}{1+\\frac{1}{\\frac{187}{13}}} = \\frac{1}{1+\\frac{1}{14+\\frac{5}{13}}} = \\frac{1}{1+\\frac{1}{14+\\frac{1}{2+\\frac{3}{5}}}}=\\ldots$"}
{"_id": "3137995", "title": "", "text": "$ T_n = n\\cdot (n+1)/2 $"}
{"_id": "1916208", "title": "", "text": "$\\sum \\limits_{j = 1}^n i_j \\ge k$"}
{"_id": "6610282", "title": "", "text": "$0^++m^+=(0^++m)^+=(0+m^+)^+=0+(m^{+})^{+}$"}
{"_id": "6796665", "title": "", "text": "$\\sum_{k \\le x}\\frac1k= \\frac{[x]}{x}+ \\int_1^x \\frac1{u}du-\\int_1^x \\frac{\\{u\\}}{u^2}du = O(1) + \\ln (x) - O(1/x).$"}
{"_id": "5846447", "title": "", "text": "$f(x) = \\frac{2x}{x^{2}+1}$"}
{"_id": "1474639", "title": "", "text": "$\\{\\gamma,-\\gamma\\}$"}
{"_id": "2630012", "title": "", "text": "$\\beta^+$"}
{"_id": "8724374", "title": "", "text": "$ \\mathbb{E} \\left[ 1_{\\sup_{t \\geq 0 } M_t > x}  \\mid \\mathcal{F}_0 \\right] = 1 \\wedge \\frac{M_0}{x}.$"}
{"_id": "1698193", "title": "", "text": "$|AB| = 2$"}
{"_id": "226121", "title": "", "text": "$T^+$"}
{"_id": "3127322", "title": "", "text": "$v_{p,γ}=v=[(p,γ)]$"}
{"_id": "8873143", "title": "", "text": "$a(n) =\\dfrac{n(n+1)}{2} $"}
{"_id": "1310952", "title": "", "text": "$f_{n}\\left( x\\right) =\\dfrac {x^{n}} {1+x^{n}}$"}
{"_id": "4045644", "title": "", "text": "$ \\begin{cases} x=t\\\\ y=t\\\\ z=t\\\\ \\end{cases} $"}
{"_id": "1841628", "title": "", "text": "$=a+\\sum_{i=1}^n[(b-a)^{-(n-1)} \\binom{n-1}{i-1}(b-x)^{n-1-(i-1)}(x-a)^{i-1}](x-a)=$"}
{"_id": "5293750", "title": "", "text": "$\\frac{\\pi^2}{6} = \\sum_{n=1}^\\infty \\frac{1}{n^2} = \\sum_{n=0}^\\infty\\frac{1}{(2n+1)^2} + \\sum_{n=1}^\\infty\\frac{1}{(2n)^2}$"}
{"_id": "2449363", "title": "", "text": "$g : B \\rightarrow A_1 \\times \\dotsc \\times A_n$"}
{"_id": "2515550", "title": "", "text": "$\\begin{cases} x_1=1+s+2t \\\\ x_2=s \\\\ x_3=t \\end{cases}$"}
{"_id": "5794205", "title": "", "text": "$\\sum_{n=1}^\\infty a_k(n \\bmod 2^k) e^{- \\pi n^2 x} = (2^k x)^{-1/2}  \\sum_{m=1}^\\infty \\frac{h_k(e^{2i \\pi m/2^k})}{2^k} e^{- \\pi m^2 2^k/ x}$"}
{"_id": "3972577", "title": "", "text": "$a_1 = g(f(a_1))$"}
{"_id": "9194504", "title": "", "text": "$\\frac{x!}{\\left( \\frac{x}{e} \\right)^{x}}\\approx \\sqrt{2\\pi\\,x} $"}
{"_id": "5966927", "title": "", "text": "$\\int_0^\\infty f(x) dx = \\int_0^\\infty g(x) dx$"}
{"_id": "6117357", "title": "", "text": "$ u_1 = a - \\frac{1}{u_0} = a - \\frac 1a = \\frac{a^2-1}{a},\\\\ u_2 = a - \\frac{1}{u_1}= a - \\frac{a}{a^2-1}=\\frac{a^3-2a}{a^2-1},\\\\ u_3 = a - \\frac{1}{u_2} = a - \\frac{a^2-1}{a^3-2a}=\\frac{a^4-3a^2+1}{a^3-2a},\\\\ u_4 = a - \\frac{1}{u_3} = a - \\frac{a^3-2a}{a^4-3a^2+1}=\\frac{a^5-4a^3+2a-1}{a^4-3a^2+1}. $"}
{"_id": "4680528", "title": "", "text": "$log_{b^{n}} x = \\frac {log_{c}x}{log_{c} (b^ {n})}$"}
{"_id": "5583849", "title": "", "text": "$\\lim_{x\\to p}f(x)g(x)=\\lim_{x\\to p}f(x)\\cdot\\lim_{x\\to p}g(x)$"}
{"_id": "4085837", "title": "", "text": "$\\int_0^\\infty f(t)g(t) dt \\neq 0$"}
{"_id": "2589481", "title": "", "text": "$z = \\sqrt[n]{r} e^{2 \\pi k i/n} \\ \\text{ for k= 0, 1,..., n-1 }$"}
{"_id": "7288136", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}E(x_{n})\\geq E(x)$"}
{"_id": "7183941", "title": "", "text": "$\\kappa_\\gamma = |h(\\dot \\gamma, \\dot \\gamma)| \\le C,$"}
{"_id": "3333411", "title": "", "text": "$D_{\\pm} \\times \\Bbb R^{n+1}$"}
{"_id": "7189393", "title": "", "text": "$\\frac{1}{2^n+1}+\\frac{1}{2^n+2}+...+\\frac{1}{2^{n+1}}\\geq\\frac{1}{2^{n+1}}+\\frac{1}{2^{n+1}}+...+\\frac{1}{2^{n+1}}=\\frac{2^n}{2^{n+1}}=\\frac{1}{2}.$"}
{"_id": "6758511", "title": "", "text": "$\\alpha+\\beta+\\gamma=-\\frac ba, \\alpha\\beta+\\beta\\gamma+\\gamma\\alpha=\\frac ca,\\alpha\\beta\\gamma=-\\frac da$"}
{"_id": "2345187", "title": "", "text": "$(a^2 - b^2 - 2ab, a^2 - b^2 + 2ab) = 1$"}
{"_id": "4617072", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{4}{5}$"}
{"_id": "1207933", "title": "", "text": "$\\begin{align*} \\int \\frac{1}{(x^2 - 1)^2}\\,\\mathrm{d}x &\\equiv \\int \\frac{1}{\\left((x + 1)\\cdot(x - 1)\\right)^2}\\,\\mathrm{d}x \\\\ &= \\int \\frac{1}{(x + 1)^2\\cdot(x - 1)^2}\\,\\mathrm{d}x\\tag{1} \\end{align*}$"}
{"_id": "4363116", "title": "", "text": "$\\Big | \\frac{S_v}{\\sqrt{v}} \\Big | = \\frac{1}{\\sqrt{v}}\\Big |X_1 +...+ X_v\\Big | \\le \\frac{1}{\\sqrt{v}}\\Big (\\Big |X_1 \\Big | +...+ \\Big | X_v\\Big | \\Big) \\le \\frac{1}{\\sqrt{v}}v = \\sqrt{v}$"}
{"_id": "2004051", "title": "", "text": "$\\left(\\frac{x}{60}e^{\\frac{-47x}{60}}\\right)^{2}$"}
{"_id": "8916992", "title": "", "text": "$  R = R_1 R_2 R_3 = \\left( \\begin{array}{rr} -2 & 1 \\\\ 1 & -1 \\end{array} \\right) $"}
{"_id": "2653541", "title": "", "text": "$z\\dfrac{f^{'}(z)}{f(z)} = z^{2}\\dfrac{f^{'}(z^{2})}{f(z^{2})}$"}
{"_id": "2525190", "title": "", "text": "$\\hat f_n(t)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}\\frac{\\sin x\\sin (nx)\\cos(xt)}{x^2}dx-\\frac{i}{\\sqrt{2\\pi}}\\underbrace{\\int_{-\\infty}^{\\infty}\\frac{\\sin x\\sin (nx)\\sin(xt)}{x^2}dx}_{=0\\text{ by symmetry}}$"}
{"_id": "3951841", "title": "", "text": "$\\forall a \\in D, \\exists \\epsilon \\gt 0, \\exists \\delta \\gt 0, \\forall x \\in D: |x-a| \\lt \\delta \\Longrightarrow |f(x) - f(a)|\\lt \\epsilon  $"}
{"_id": "1905859", "title": "", "text": "$ f (x ) = x^2 - x - 2$"}
{"_id": "2454823", "title": "", "text": "$\\frac{L(b)-L(a)}{b-a} = \\frac{\\sqrt{\\frac{\\gamma Rb}{2\\mu}}-\\sqrt{\\frac{\\gamma Ra}{2\\mu}}}{b-a} =\\sqrt{\\frac{\\gamma R}{2\\mu}}\\frac{\\sqrt{b}-\\sqrt{a}}{b-a}$"}
{"_id": "4660835", "title": "", "text": "$S= \\{v_1,v_2,.......,v_k\\}$"}
{"_id": "3474124", "title": "", "text": "$\\|A\\|_1=\\operatorname{trace}(\\sqrt{A^* A})$"}
{"_id": "422209", "title": "", "text": "${\\bf x} = (x,y)$"}
{"_id": "4817094", "title": "", "text": "$x^2+y^2+z^2= (x-y)(y-z)(z-x)$"}
{"_id": "8557583", "title": "", "text": "$x \\to \\alpha^{+}$"}
{"_id": "1522230", "title": "", "text": "$\\begin{pmatrix} a & b \\\\ b & -a\\end{pmatrix} \\tag 2$"}
{"_id": "9062431", "title": "", "text": "$=\\left[\\begin{array}{ccc|c}1&-2&2&0\\\\2&k&-1&3\\\\0&-1&-1&5\\end{array}\\right]$"}
{"_id": "570971", "title": "", "text": "$\\frac{n(n+1)}{2}+(n+1)=(n+1)\\left[\\frac{n}{2}+1\\right]=(n+1)\\left[\\frac{n}{2}+\\frac{2}{2}\\right]=(n+1)\\frac{n+2}{2}=\\\\=\\frac{(n+1)(n+2)}{2}$"}
{"_id": "1200536", "title": "", "text": "$ ay - xb = d$"}
{"_id": "1679969", "title": "", "text": "${a^{Lo{g_a}x}} = x$"}
{"_id": "3960871", "title": "", "text": "$1974$"}
{"_id": "4637922", "title": "", "text": "$I=\\int_0^1\\int_0^1 \\frac{\\sin^{-1}(xy)}{xy}$"}
{"_id": "9272599", "title": "", "text": "$\\frac{r(1-r^{N-2})}{(1-r)^2}$"}
{"_id": "9228820", "title": "", "text": "$\\gamma \\subseteq \\gamma_0\\forall \\gamma \\in A$"}
{"_id": "5515915", "title": "", "text": "$|\\mathbb{R}^n|=|\\mathbb{R}^{n-1}\\times\\mathbb{R}|$"}
{"_id": "6548681", "title": "", "text": "$ \\frac{3}{\\beta} = \\frac{1}{\\alpha} + \\frac{1}{\\beta} + \\frac{1}{\\gamma} = \\frac{\\alpha\\beta + \\beta\\gamma + \\gamma\\alpha}{\\alpha\\beta\\gamma} $"}
{"_id": "5225503", "title": "", "text": "$f_2(n)=2$"}
{"_id": "474843", "title": "", "text": "$q = (a, a+k, a+2k, ..)$"}
{"_id": "3147331", "title": "", "text": "$x_3, x_4\\in\\mathbb{N}$"}
{"_id": "3198844", "title": "", "text": "$d(\\gamma(y_{i+1}),\\gamma(a)) < d(\\gamma(y_i), \\gamma(a))$"}
{"_id": "6477656", "title": "", "text": "$B_2=(e_1-u,e_2-u,e_3,...,e_n)$"}
{"_id": "9314368", "title": "", "text": "$f'(t)=2(2r \\cos(t)-1)(-2r \\sin(t))+2r \\sin(t)\\cos(t)=$"}
{"_id": "7233018", "title": "", "text": "$d(x,A)\\leqslant|x-a|\\leqslant|x-y|+|y-a|.$"}
{"_id": "3857727", "title": "", "text": "$\\frac{f}{||f||} = \\frac{g}{||g||}.$"}
{"_id": "680518", "title": "", "text": "$f_0(x) = \\frac{2}{x^2}$"}
{"_id": "5289682", "title": "", "text": "$\\left\\lfloor \\dfrac{\\left\\lfloor\\tfrac{x}{m}\\right\\rfloor}{n} \\right\\rfloor = k = \\left\\lfloor\\dfrac{x}{mn}\\right\\rfloor$"}
{"_id": "1960583", "title": "", "text": "$\\sqrt{2\\pi x} \\left(\\frac{x}{e}\\right)^x$"}
{"_id": "7850409", "title": "", "text": "$ M(z) = \\frac{z-\\alpha}{1-\\bar{\\alpha} z} $"}
{"_id": "99537", "title": "", "text": "$\\int_0^\\infty f(x)dx=1$"}
{"_id": "5385505", "title": "", "text": "$|f(x)| = |f(x) - c_0 + c_0| \\geq \\left||c_0| - |f(x) - c_0| \\right|\\geq |c_0| - |f(x) - c_0| > |c_0| - |c_0| = 0$"}
{"_id": "473843", "title": "", "text": "$\\tilde\\gamma = \\arg\\min_\\gamma \\int_0^T L(\\gamma, \\dot\\gamma)\\,dt\\quad  \\textrm{s.t.}\\quad \\gamma(0) = \\gamma_0, \\gamma(T) = \\gamma_T$"}
{"_id": "8581601", "title": "", "text": "$ds=\\sqrt{1+(dy/dx)^2}$"}
{"_id": "3922551", "title": "", "text": "$|f(z_0^2)| \\le |f(z_0)|$"}
{"_id": "6120477", "title": "", "text": "$ a = \\int_{0}^{\\infty} f(x) dx $"}
{"_id": "839244", "title": "", "text": "$f'(x)=1-{e^{-1\\over x}\\over x}-e^{-1\\over x}$"}
{"_id": "6084103", "title": "", "text": "$1 + 2 + 3 + 4 + 5 + ... = \\frac{-1}{12}.$"}
{"_id": "5586881", "title": "", "text": "$\\lim_{x \\to a} f(x)g(x) = \\lim_{x \\to a} f(x) * \\lim_{x \\to a} g(x) = 0 * G = 0, \\forall G \\in \\mathbb R$"}
{"_id": "6293204", "title": "", "text": "$mRm\\Leftrightarrow P(m)=P(m)$"}
{"_id": "1134110", "title": "", "text": "$ax+by=p$"}
{"_id": "6491280", "title": "", "text": "$\\left[ \\begin{array}{ccc|c} 1 & 4 & -1 & 0 \\\\ 0 & -5 & 1 & 1 \\\\ 0 & 0 & 3 & 2 \\\\ 0 & 0 & 2 & -1 \\end{array} \\right]$"}
{"_id": "7431590", "title": "", "text": "$\\begin{align*}\\int_0^1\\frac{\\log(x)\\tan^{-1}(x)\\tanh^{-1}(x)}{x}dx &= \\frac{\\pi^2}{16}G-\\frac{7\\pi\\zeta(3)}{32} \\tag{2}\\\\ \\int_0^1\\frac{\\log^2(x)\\tan^{-1}(x)}{1-x^2}dx &= -\\beta(4)-\\frac{\\pi^2}{24}G+\\frac{7\\pi}{16}\\zeta(3)\\tag{3} \\end{align*}$"}
{"_id": "7893369", "title": "", "text": "$x^3 - 1, x^2 - 1 \\in (x - 1)$"}
{"_id": "7168281", "title": "", "text": "$\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\simeq \\mathbb{R}^{2n}$"}
{"_id": "8799934", "title": "", "text": "$\\left[\\matrix{\\lambda & 0\\cr 0& 1\\cr}\\right],\\ \\left[\\matrix{1 & 0\\cr 0& \\mu\\cr}\\right],\\ \\left[\\matrix{1 & 1\\cr 0& 1\\cr}\\right],\\ \\left[\\matrix{1 & 0\\cr 1& 1\\cr}\\right].$"}
{"_id": "2969131", "title": "", "text": "$\\lfloor c \\rfloor \\geq \\lfloor a \\rfloor + \\lfloor b \\rfloor$"}
{"_id": "1938017", "title": "", "text": "$d(x,A) = \\inf\\{d(x,y) : y \\in A\\}.$"}
{"_id": "1637960", "title": "", "text": "$(\\frac{1}{2})^{(n-1)/2}$"}
{"_id": "5394080", "title": "", "text": "$C=\\{B\\} = \\{\\{A\\}\\}$"}
{"_id": "7764020", "title": "", "text": "$1 < \\frac {x_1 + x_2 +... + x_n}{n} \\leq 2$"}
{"_id": "5434920", "title": "", "text": "$\\dfrac{2r+1}{(r^2+r)^2+1}=\\dfrac{(r+1)^2-r^2}{1+r^2(r+1)^2}$"}
{"_id": "7664626", "title": "", "text": "$=\\frac{(\\sin 2^o\\cos 1^o+\\cos 2^o\\sin 1^o)-(\\cos 2^o\\cos 1^o-\\sin 2^o\\sin 1^o)}{-(\\sin 2^o\\cos 1^o+\\cos 2^o\\sin 1^o)-(\\cos 2^o\\cos 1^o-\\sin 2^o\\sin 1^o)}$"}
{"_id": "9120980", "title": "", "text": "$(1-\\frac{x^2}{n})e^{x} \\leqslant (1+\\frac{x}{n})^n \\leqslant e^{x}. $"}
{"_id": "275377", "title": "", "text": "$ax+by=r$"}
{"_id": "6450169", "title": "", "text": "$ Q =  \\begin{bmatrix} 0 & 1 & 2 & 1 & 2  \\\\ 0 & 1 & -1 & -1 & 1 \\\\ 0 & 2 & -5 & -4 & 1 \\\\ 0 & 4 & 2 & 0 & 6  \\end{bmatrix} $"}
{"_id": "6138709", "title": "", "text": "$\\mathcal{L}(\\gamma) = \\int_\\gamma \\sqrt{g(\\gamma', \\gamma')}$"}
{"_id": "46488", "title": "", "text": "$\\kappa^+$"}
{"_id": "3922542", "title": "", "text": "$|f(z^2)| ≤ |f(z)|$"}
{"_id": "152933", "title": "", "text": "$ \\lim_{n\\rightarrow \\infty} \\mu(A_n)=A, $"}
{"_id": "7019655", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{m^2+n^2}  > 4 \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{(2m)^2+(2n)^2} $"}
{"_id": "1787482", "title": "", "text": "$A_1\\subseteq A_2\\subseteq A_3\\subseteq \\cdots \\subseteq A_n\\subseteq A_{n+1}\\subseteq \\cdots$"}
{"_id": "7354801", "title": "", "text": "$A_n=1+2+..+n=\\frac{n(n+1)}{2}$"}
{"_id": "5766178", "title": "", "text": "$\\left(1+\\frac{x}{n} \\right)^n \\leq e^x \\leq \\left( 1 - \\frac{x}{n} \\right)^{-n} = \\left( 1 + \\frac{x}{n - x} \\right)^n$"}
{"_id": "6188804", "title": "", "text": "$\\int_0^1f(t)P_n(t)dt\\to \\int_0^1f(t)g(t)dt$"}
{"_id": "2445499", "title": "", "text": "$\\partial_z\\vartheta[p,0](z,\\tau)$"}
{"_id": "6332598", "title": "", "text": "$f(T, p) = p$"}
{"_id": "2477851", "title": "", "text": "$ x11=\\left(\\begin{array}{rrr}% d&e\\\\% 0&f\\\\% \\end{array}\\right)% =df, x12=-\\left(\\begin{array}{rrr}% 0&e\\\\% 0&f\\\\% \\end{array}\\right)% =0, x13=\\left(\\begin{array}{rrr}% 0&d\\\\% 0&0\\\\% \\end{array}\\right)% =0 $"}
{"_id": "5662975", "title": "", "text": "$ \\left(  \\left(  R^{=}\\right)  ^{\\sigma}\\right)  ^{+}=\\left(  \\left( R^{\\sigma}\\right)  ^{=}\\right)  ^{+}=\\left(  \\left(  R^{\\sigma}\\right) ^{+}\\right)  ^{=}\\supseteq\\bigcup_{n=-\\infty}^{\\infty}R^{n}. $"}
{"_id": "8994189", "title": "", "text": "$f(x)=\\frac{9x}{9x+3}$"}
{"_id": "6444003", "title": "", "text": "$ \\zeta = \\int_0^\\infty f(x)\\,dx $"}
{"_id": "8734292", "title": "", "text": "$F = K \\oplus P$"}
{"_id": "4385204", "title": "", "text": "$=\\frac{1}{2}-\\frac{n}{n+1}+\\frac{n}{n+2}-\\frac{1}{n+2}=\\frac{1}{2}+\\frac{-n}{(n+1)(n+2)}-\\frac{1}{n+2}=\\frac{1}{2}-\\frac{2n+1}{(n+1)(n+2)}$"}
{"_id": "8381898", "title": "", "text": "$\\frac ad=\\frac{1-\\cos\\gamma\\cos(2\\delta+\\gamma)}{\\sin\\gamma\\cos\\gamma}\\tag{1}$"}
{"_id": "6066802", "title": "", "text": "$\\left(\\frac{1}{n-1}\\right)^2$"}
{"_id": "3982511", "title": "", "text": "$\\displaystyle \\int_1^\\infty f(x) \\quad dx$"}
{"_id": "6521870", "title": "", "text": "$∪!\\mathcal F = \\{x | ∃!A(A ∈ \\mathcal F ∧ x ∈ A)\\}$"}
{"_id": "4820163", "title": "", "text": "$ {}^s({}^ra) = {}^{sr}a $"}
{"_id": "2949955", "title": "", "text": "$f_n(x)=\\sum_{j=-\\infty}^\\infty c_j \\, \\mathbf 1_{A_j}(x)$"}
{"_id": "8429388", "title": "", "text": "$(z-x^2, z+x^2)=d>1$"}
{"_id": "7937371", "title": "", "text": "$ \\lim_{n\\to\\infty} \\frac {1+\\sqrt[n]2 + \\sqrt[n]3 + ... \\sqrt[n]n} {n}  $"}
{"_id": "3374901", "title": "", "text": "$\\frac{a+b}{b+c}$"}
{"_id": "2449387", "title": "", "text": "$g':B\\rightarrow A_1 \\times \\dotsc \\times A_n$"}
{"_id": "3878407", "title": "", "text": "$ \\begin{align} \\int_0^1\\left(\\frac{\\sin^{-1}(x)}{x}\\right)^3\\,\\mathrm{d}x &=\\int_0^{\\pi/2}\\left(\\frac{t}{\\sin(t)}\\right)^3\\,\\mathrm{d}\\sin(t)\\tag{6}\\\\ &=-\\frac12\\int_0^{\\pi/2}t^3\\,\\mathrm{d}\\frac1{\\sin^2(t)}\\tag{7}\\\\ &=-\\frac12\\frac{\\pi^3}8+\\frac32\\int_0^{\\pi/2}\\left(\\frac{t}{\\sin(t)}\\right)^2\\,\\mathrm{d}t\\tag{8}\\\\ &=-\\frac12\\frac{\\pi^3}8-\\frac32\\int_0^{\\pi/2}t^2\\,\\mathrm{d}\\cot(t)\\tag{9}\\\\ &=-\\frac12\\frac{\\pi^3}8+3\\int_0^{\\pi/2}t\\cot(t)\\,\\mathrm{d}t\\tag{10}\\\\ &=-\\frac12\\frac{\\pi^3}8+3\\int_0^{\\pi/2}t\\,\\mathrm{d}\\log(\\sin(t))\\tag{11}\\\\ &=-\\frac12\\frac{\\pi^3}8-3\\int_0^{\\pi/2}\\log(\\sin(t))\\,\\mathrm{d}t\\tag{12}\\\\ &=\\bbox[5px,border:2px solid #C0A000]{-\\frac12\\frac{\\pi^3}8+\\frac{3\\pi}2\\log(2)}\\tag{13} \\end{align} $"}
{"_id": "7119105", "title": "", "text": "$P[X\\geq b] =1$"}
{"_id": "809563", "title": "", "text": "$\\lim_{x \\rightarrow c}{g(x)}=L$"}
{"_id": "3217962", "title": "", "text": "$\\vartheta \\mapsto \\vartheta + \\pi$"}
{"_id": "8369305", "title": "", "text": "$\\begin{cases}x=3\\\\y=0\\\\z=t\\end{cases}$"}
{"_id": "1298276", "title": "", "text": "$\\int_{-\\infty}^{\\infty}|f(t)|^2\\,dt<\\infty$"}
{"_id": "7702138", "title": "", "text": "$f(a+b)\\ge f(a)+f(b)$"}
{"_id": "1635159", "title": "", "text": "$ar + bs= d$"}
{"_id": "415537", "title": "", "text": "$ \\det A=\\chi_B[X:=a] = (a-b)^{n-1}(a+(n-1)b). $"}
{"_id": "4343153", "title": "", "text": "$A_1\\subseteq A_2\\subseteq\\ldots\\subseteq A_n$"}
{"_id": "56275", "title": "", "text": "$\\displaystyle n!\\approx\\sqrt{2\\pi n}\\bigg(\\frac{n}{e}\\bigg)^n$"}
{"_id": "1214547", "title": "", "text": "$ak + bs = d$"}
{"_id": "4874786", "title": "", "text": "$(A,B;X,Y) = (A'',B'';X'',Y'') = (A',B';X',Y')$"}
{"_id": "4832211", "title": "", "text": "$n^{\\frac{n}{9}}\\geq (1+\\frac 1n)^n$"}
{"_id": "257656", "title": "", "text": "$d_2(x,y)=|f(x)-f(y)|$"}
{"_id": "2482534", "title": "", "text": "$n_1n_2|a$"}
{"_id": "24252", "title": "", "text": "$I=\\int\\frac{(1+x^2)dx}{1-2x^2(1-2\\sin^2(\\frac{\\alpha}{2}))+x^4}$"}
{"_id": "2602826", "title": "", "text": "$\\lvert AB \\rvert = s$"}
{"_id": "1050385", "title": "", "text": "$g(x)=x^2+x-2$"}
{"_id": "6085337", "title": "", "text": "$r\\equiv\\begin{cases} x-y=2 \\\\2x-z+1=0\\end{cases}$"}
{"_id": "4681239", "title": "", "text": "$\\begin{pmatrix} A-B & B-A\\\\ B & A \\end{pmatrix}$"}
{"_id": "1162751", "title": "", "text": "$\\gamma=\\gamma_{R} \\wedge \\gamma_{2} \\wedge \\gamma_{\\delta} \\wedge \\gamma_{3},$"}
{"_id": "3215271", "title": "", "text": "$\\det A=(a+(n-1)b)(a-b)^{n-1}.$"}
{"_id": "6252377", "title": "", "text": "$d(x,y)=\\frac1m$"}
{"_id": "8550938", "title": "", "text": "$ \\begin{align} \\sum_{m = 1}^\\infty\\sum_{n = 1}^\\infty \\frac{1}{nm(n+m)^2} & = \\sum_{r = 2}^\\infty \\frac{1}{r^2}\\sum_{s = 1}^{r-1} \\frac{1}{s(r-s)} = 2 \\sum_{r=2}^\\infty \\frac{1}{r^3} \\sum_{s=1}^{r-1}\\frac{1}{s}, \\tag{3} \\end{align} $"}
{"_id": "6347243", "title": "", "text": "$(q,u) \\in \\mathbb{R}^{n+1} \\times \\mathbb{B}^n$"}
{"_id": "171000", "title": "", "text": "$|f(x_1)| \\leq \\frac{1}{2}|f(x_0)|$"}
{"_id": "3965673", "title": "", "text": "$-31=\\frac{-39.2}{x}(1-e^{\\frac{x}{4}})$"}
{"_id": "318872", "title": "", "text": "$0 < i < \\alpha^{+}$"}
{"_id": "559505", "title": "", "text": "$a^{\\log_a(b)} = b$"}
{"_id": "2739014", "title": "", "text": "$\\quad a^{\\large \\log_a(b)} =  b.$"}
{"_id": "1118002", "title": "", "text": "$\\frac{9*(9-3)}{2}=27$"}
{"_id": "5796272", "title": "", "text": "$d(x,y)= \\sup _{f \\in K} \\lvert f(x) - f(y)\\rvert$"}
{"_id": "8780405", "title": "", "text": "$x(x-9)-6(x-9)=0$"}
{"_id": "2549351", "title": "", "text": "$f_1(n)=n^2$"}
{"_id": "5766017", "title": "", "text": "$\\sigma(a)\\,\\rho\\,\\sigma(b)$"}
{"_id": "4850833", "title": "", "text": "$\\mathbf Z/(3)$"}
{"_id": "4266132", "title": "", "text": "$xRy\\,\\,\\text{ and }\\,\\, yRx \\qquad \\Rightarrow \\qquad x = y$"}
{"_id": "4498446", "title": "", "text": "$p(m) \\implies p(m + 1)$"}
{"_id": "6285328", "title": "", "text": "$\\gamma=\\{c_n\\}=\\{\\frac1{2^{n+3}}\\}$"}
{"_id": "1370756", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor\\frac ab\\rfloor}c\\right\\rfloor$"}
{"_id": "9197268", "title": "", "text": "$(x-2n+1)(x+1)^{n-1}(x-1)^n$"}
{"_id": "1542680", "title": "", "text": "$ y = \\frac{1+8x}{3-3x}$"}
{"_id": "7167110", "title": "", "text": "$\\int_{-\\infty}^{\\infty} f(x)10^{x}dx$"}
{"_id": "7778174", "title": "", "text": "$(a,a+n,a+2n)$"}
{"_id": "2867111", "title": "", "text": "$\\begin{bmatrix}1 & 1 & -1 & 1 \\\\ 0 & 1 & a+2 & 1 \\\\ 0 & 0& (3+a) & 1 \\end{bmatrix}$"}
{"_id": "6365272", "title": "", "text": "$R_1-(k-1)R_2\\to R_1\\\\ R_1-R_2\\to R_2\\\\ R_1+R_3\\to R_3\\\\ R_3-R_2\\to R_3\\\\ -\\dfrac{1}{k}R_2\\to R_2\\\\ \\dfrac{1}{k+1}R_3\\to R_3\\\\ R_3-R_2\\to R_2\\\\ (k-1)R_3+R_2\\to R_3\\\\ R_1+R_3\\to R_1\\\\ \\dfrac{1}{k-1}R_3\\to R_3\\\\ R_1-kR_2\\to R_1$"}
{"_id": "536182", "title": "", "text": "$C_n=\\dfrac {(2n)!}{n!(n+1)!}$"}
{"_id": "5031685", "title": "", "text": "$(p^{e_i}), 1 \\le e_i \\le e$"}
{"_id": "2748827", "title": "", "text": "$\\sum_{k=1}^n\\sin(\\frac{2\\pi k}{n})$"}
{"_id": "5446250", "title": "", "text": "$F(x)=\\int_a^xf(t)~dt \\Rightarrow F'(x)=f(x).$"}
{"_id": "9258798", "title": "", "text": "$\\lvert\\lvert A \\rvert\\rvert \\leq \\sqrt{\\rho(A^TA)} $"}
{"_id": "2610689", "title": "", "text": "$F=X+Y$"}
{"_id": "4576002", "title": "", "text": "$Y = \\{0,1,2,3,4\\}$"}
{"_id": "2013523", "title": "", "text": "$ \\begin{cases} x=at\\\\[4px] y=bt\\\\[4px] z=ct \\end{cases} $"}
{"_id": "3168764", "title": "", "text": "$\\exists! g':C_1' \\to C_0'$"}
{"_id": "3847164", "title": "", "text": "$f(x)=9x^3+9x-7$"}
{"_id": "6692766", "title": "", "text": "$f(x;\\theta)=\\frac{2x}{\\theta^2}\\mathbb{I}(\\theta\\geqslant x),$"}
{"_id": "5718127", "title": "", "text": "$A_{1}\\subset ... \\subset A_{i}\\subset A_{i+1}\\subset ...\\subset \\mathbb{R}$"}
{"_id": "1077074", "title": "", "text": "$1+2+3+\\cdots=-1/12$"}
{"_id": "6251113", "title": "", "text": "$\\displaystyle \\bullet\\; a^{\\log_{a}(x)} = x$"}
{"_id": "6905298", "title": "", "text": "$I\\;=\\; \\frac{x\\sqrt{1-x^{2}\\,}}{1-2x^{2}} + C$"}
{"_id": "7050416", "title": "", "text": "$A=\\text{im}(\\gamma)$"}
{"_id": "1934089", "title": "", "text": "$\\left(\\begin{array}{ccc|c} 1& -2& 1& 0 \\\\ 3& 0& -1& 4 \\\\ 1& 1& -1& k \\end{array}\\right).$"}
{"_id": "1030073", "title": "", "text": "$\\begin{bmatrix} 0 & 2 & 3 & 1 \\\\ -2 & 0 & 1 & 4 \\\\ -3 & -1 & 0 & 1 \\\\ -1 & -4 & -1 & 0   \\end{bmatrix}$"}
{"_id": "3218928", "title": "", "text": "$\\frac{1}{\\pi} \\int_{0}^{\\pi} \\sin(x) dx + \\frac{1}{\\pi} \\int_{\\pi}^{2\\pi} \\sin(x) dx$"}
{"_id": "2384417", "title": "", "text": "$s_n(x) = \\sum_{i = 1}^n a_i \\chi_i(x)$"}
{"_id": "3181261", "title": "", "text": "$\\int_0^1 \\frac{\\tanh ^{-1}(y)}y \\,dy=\\sum_{n=0}^\\infty \\frac {1}{(2n+1)^2}=\\frac{\\pi ^2}{8}$"}
{"_id": "1842594", "title": "", "text": "$\\sqrt{1+ \\frac{dy}{dx}^2} \\approx 1 + \\frac{1}{2}\\frac{dy}{dx}^2$"}
{"_id": "6036991", "title": "", "text": "$d(x,y)+d(x,s)\\leqslant d(y,s)$"}
{"_id": "6797194", "title": "", "text": "$f(n)=\\dfrac{n^2-n+4}{2}$"}
{"_id": "6109990", "title": "", "text": "$1/\\vartheta(z;\\tau)$"}
{"_id": "3610600", "title": "", "text": "$b, a+b, 2a+b, …$"}
{"_id": "5026687", "title": "", "text": "$Cov(X,Y)>0$"}
{"_id": "359710", "title": "", "text": "$\\Bbb C\\cong \\Bbb R[x]/(x^2+1)$"}
{"_id": "7121473", "title": "", "text": "$X_1 \\cdot ... \\cdot X_n$"}
{"_id": "376255", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} b_n = \\sqrt2.$"}
{"_id": "5796453", "title": "", "text": "$1/|x|^n$"}
{"_id": "1060424", "title": "", "text": "$\\displaystyle 9^x+3\\leq a(3^x+1)\\Rightarrow a\\geq \\frac{9^x+3}{3^x+1} = \\frac{9^x-1+4}{3^x+1} = 3^x-1+\\frac{4}{3^x+1}$"}
{"_id": "567134", "title": "", "text": "$\\frac{a+b+\\cdots}{\\gamma}=\\frac{a}{\\gamma} + \\frac{b}{\\gamma}+\\cdots.$"}
{"_id": "9042616", "title": "", "text": "$cov(X,Y)=-0.2994\\neq 0$"}
{"_id": "3436227", "title": "", "text": "$\\frac{1}{2^{(n-1)^2}}.$"}
{"_id": "292642", "title": "", "text": "$T_x(X) + T_x(Z) = T_x(Y)$"}
{"_id": "6360702", "title": "", "text": "$f(f(y) + xf(x)) = y + f^2(x)$"}
{"_id": "7388544", "title": "", "text": "$[ \\gamma (a), \\gamma (b)] \\subseteq \\gamma [a,b]$"}
{"_id": "2440582", "title": "", "text": "$Tr(\\gamma^{\\mu}\\gamma^{\\nu}\\gamma^{\\rho}\\gamma^{\\sigma}\\gamma^{\\alpha}\\gamma^{\\beta}\\gamma^{5}) $"}
{"_id": "3309625", "title": "", "text": "$\\{x \\in \\mathbb{R}:a\\leq x\\leq b\\}=\\{y\\in \\mathbb{R}:\\exists s,t\\in [0,1]\\; with\\; s+t=1\\; and\\; y=sa+tb\\}$"}
{"_id": "8283145", "title": "", "text": "$n! = \\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^{n} e^{r_{n}}$"}
{"_id": "4306719", "title": "", "text": "$f(x)=\\frac{4x^2}{x^2+3} $"}
{"_id": "6783283", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}\\sum\\limits_{r=0}^n a_r$"}
{"_id": "7322987", "title": "", "text": "$\\frac{1}{\\gamma}\\int \\frac{dt}{(t+1)(t+a\\gamma)}$"}
{"_id": "3568433", "title": "", "text": "$a_n = \\frac{n(n+1)}{2} * (\\frac{(2n+1 +3)}{6})$"}
{"_id": "8848409", "title": "", "text": "$f(-\\pi)=0=f(\\pi)$"}
{"_id": "3342903", "title": "", "text": "$\\{(x+y)(x+z), (x+y)(y+z), (x+z, y) \\}$"}
{"_id": "228435", "title": "", "text": "$({\\color{red}4},6)=2$"}
{"_id": "5872379", "title": "", "text": "$y'' + y' + y = y''$"}
{"_id": "2270584", "title": "", "text": "$2^x+4^x=8^x$"}
{"_id": "8728375", "title": "", "text": "$P(x)=\\left(x^{\\frac{p-1}{2}}+1\\right)^2$"}
{"_id": "1363397", "title": "", "text": "$[f(f^{-1}(x))]' = f'(f^{-1}(x))*{f^{-1}}'(x) = 1$"}
{"_id": "4547756", "title": "", "text": "$\\lim_{x\\to a}f(x)=\\lim_{x\\to a^+}f(x)=\\lim_{x\\to a^-}f(x)$"}
{"_id": "359991", "title": "", "text": "$\\|A\\|_2 = \\sqrt{\\lambda_{\\max}(A^*A)}.$"}
{"_id": "4520364", "title": "", "text": "$\\int_1^\\infty \\frac{\\lfloor x\\rfloor}{x^{s+1}}\\,dx = \\sum_{n=1}^\\infty \\int_1^\\infty \\frac{\\chi_n(x)}{x^{s+1}}\\,dx = \\sum_{n=1}^\\infty \\int_n^\\infty \\frac{dx}{x^{s+1}} = \\sum_{n=1}^\\infty \\left[-\\frac{1}{sx^s}\\right]_n^\\infty = \\frac{1}{s}\\sum_{n=1}^\\infty \\frac{1}{n^s} = \\frac{\\zeta(s)}{s}$"}
{"_id": "5614900", "title": "", "text": "$\\sum_{n=1}^{\\infty}p^n\\cos(nx)=\\frac{1}{2}\\left(\\frac{1-p^2}{1-2p\\cos(x)+p^2}-1\\right)$"}
{"_id": "4346231", "title": "", "text": "$f_Y(x)=\\frac{1}{\\sqrt{2\\pi}}(e^{\\frac{-(\\frac{x-\\mu}{\\sigma})^2}{2}}-e^{\\frac{-(\\frac{x+\\mu}{\\sigma})^2}{2}})$"}
{"_id": "6438988", "title": "", "text": "$x+z=x+[y+(-x)]=x+[-x+y]=[x+(-x)]+y=0+y=y,$"}
{"_id": "6379482", "title": "", "text": "$\\ln\\left(2\\right) = \\lim_{N \\to \\infty}{\\sum_{n = 1}^{N}{\\frac{1}{N + n}}}$"}
{"_id": "7137304", "title": "", "text": "$\\left(\\frac{1+i}{2}\\right)^{2^{n}} = \\frac{1}{2^{2^{n-1}}},$"}
{"_id": "1265772", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       2&-2&1&10\\\\       3&1&-3&18\\\\       1&-5&5&k\\\\     \\end{array} \\right] $"}
{"_id": "5626786", "title": "", "text": "$\\gamma(ab)\\neq\\gamma(a)\\gamma(b)$"}
{"_id": "4186022", "title": "", "text": "$A = \\sum_{k=0}^{n-1}\\cos\\frac{2k^2\\pi}{n}$"}
{"_id": "5414062", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{2-\\sqrt{2+\\sqrt{2 +\\sqrt{2+ \\cdots n \\text{ times}}}}}{4^{-n}}$"}
{"_id": "2733823", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1) \\simeq \\mathbb{C}$"}
{"_id": "1537854", "title": "", "text": "$\\mathcal{A}x = \\mathcal{b}$"}
{"_id": "3416464", "title": "", "text": "$x(x-a)^{n-1} = (x-a)^n + a(x-a)^{n-1}$"}
{"_id": "4129950", "title": "", "text": "$T^{0,1} M \\to T^{1,0} M$"}
{"_id": "6449386", "title": "", "text": "$(\\mathbb{Z}[x]/(x^2+1))/(p)=\\mathbb{Z}[i]/(p))\\cong \\mathbb{F}_{p^2}$"}
{"_id": "6601226", "title": "", "text": "$\\displaystyle \\frac{2x}{\\theta^2}$"}
{"_id": "3030053", "title": "", "text": "$ D: \\begin{cases} x=-1+6t   \\\\[2ex] y=6-5t\\\\[2ex] z=1-2t \\end{cases} $"}
{"_id": "649603", "title": "", "text": "$P(0) \\implies P(1),$"}
{"_id": "8852934", "title": "", "text": "$\\mathbb E[s\\mid\\{\\tau=0\\}\\cup\\{s\\geq z\\}]=\\frac{\\mu_s-\\gamma\\int_{-\\infty}^{z}xf_s(x)dx}{1-\\gamma F_{s}\\left(z\\right)}=\\frac{(1-\\gamma)\\mu_s+\\gamma\\int_{z}^{\\infty}xf_s(x)dx}{1-\\gamma F_{s}\\left(z\\right)}$"}
{"_id": "3937715", "title": "", "text": "$\\ds{\\sum_{k = 1}^{m}\\omega_{k} = \\sum_{k = 1}^{m}\\exp\\pars{2\\pi k\\ic \\over m} = \\color{red}{0}}$"}
{"_id": "656953", "title": "", "text": "$ [I_U]^{f}_{f'} [L]^{e}_{f} [I_V]^{e'}_{e}[v]_{e'} = [I_U]^{f}_{f'} [L]^{e}_{f} [I_Vv]_{e} = [I_U]^{f}_{f'} [LI_Vv]_{f} = [I_ULI_Vv]_{f'} = [Lv]_{f'} = [L]^{e'}_{f'}[v]_{e'}. $"}
{"_id": "8356914", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}y_n=\\sqrt2-1$"}
{"_id": "4854791", "title": "", "text": "$\\gamma(A)=\\sup\\{\\gamma(K)\\vert K\\subseteq A , K \\space\\text{compact}   \\}$"}
{"_id": "5724811", "title": "", "text": "$ \\begin{pmatrix} a & -b \\newline b & a \\end{pmatrix}.$"}
{"_id": "8919057", "title": "", "text": "$P[X_0=x_0]=1$"}
{"_id": "1700486", "title": "", "text": "$U(a,b,c)=\\{\\langle x,y\\rangle\\in\\Bbb R^2:a<x<b\\text{ and }c<y\\}\\;.$"}
{"_id": "2001549", "title": "", "text": "$\\mu^{*}(A) =       \\begin{cases}        0 &\\quad\\text{if } A = \\emptyset \\\\        1 &\\quad\\text{if } A \\text{ is bounded (and } A \\neq \\emptyset ) \\\\        \\infty &\\quad\\text{if } A \\text{ is unbounded and (} A \\neq \\emptyset )          \\       \\end{cases} $"}
{"_id": "3083631", "title": "", "text": "$\\begin{align} \\int_1^L \\frac{\\sin^2(x)}{x}\\,dx&=\\frac12\\int_1^L \\frac{1-\\cos(2x)}{x}\\,dx\\\\\\\\ &=\\frac12\\log(L)-\\frac12\\int_2^{2L}\\frac{\\cos(x)}{x}\\,dx\\tag 1 \\end{align}$"}
{"_id": "934563", "title": "", "text": "$p_1p_2 = a_3$"}
{"_id": "7925249", "title": "", "text": "$\\gamma_{a,b} + \\gamma_{b,z} = \\gamma_{a,z}$"}
{"_id": "2249722", "title": "", "text": "$(S^{n+1} -S^n )$"}
{"_id": "527477", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\sum_{m=1}^{\\infty} \\frac{\\sin(n-m)}{n^2+m^2}$"}
{"_id": "7748321", "title": "", "text": "$\\lim_{n \\to \\infty} \\inf \\mu_n(U) \\ge \\mu(U)$"}
{"_id": "5118349", "title": "", "text": "$S=\\{r\\in\\Bbb Q:r<2\\}$"}
{"_id": "7701029", "title": "", "text": "$\\alpha=\\frac{1-2p^2}{1-2p}$"}
{"_id": "7792571", "title": "", "text": "$x+h = (x_n)+(-x_n) = (x_n+[-x_n]) = (0) = 0$"}
{"_id": "6270935", "title": "", "text": "$\\begin{vmatrix} x_0 & y_0 &z_0 \\\\  x_1 & y_1 & z_1\\\\  x_2 & y_2 & z_2 \\end{vmatrix}=0.$"}
{"_id": "2742665", "title": "", "text": "$x + 9 -9(x+9)^{\\frac{2}{3}} + 27(x+9)^{\\frac{1}{3}} -27 = x -9$"}
{"_id": "7793808", "title": "", "text": "$s(x)= \\sum_{j=1}^p \\alpha_j 1_{A_j} (x)$"}
{"_id": "5913590", "title": "", "text": "$\\gamma= \\gamma_{1,+} \\cup \\ldots \\cup \\gamma_{n,+}\\cup \\gamma_{\\sigma(1),-} \\cup \\ldots \\gamma_{\\sigma(n),-}$"}
{"_id": "500427", "title": "", "text": "$\\frac 1 {(1-x^2)(1-2x)}=\\frac {a+bx} {1-x^2} +  \\frac c {1-2x}$"}
{"_id": "986213", "title": "", "text": "$ 0 \\le x \\le 1 \\implies 0 \\le x^{n-1} \\le 1 \\implies 0 \\le x^n \\le x.$"}
{"_id": "5676129", "title": "", "text": "$S(n)={n+1\\choose2}C_n$"}
{"_id": "2630058", "title": "", "text": "$\\|x_{n+1}-x_n\\|\\to0$"}
{"_id": "8938509", "title": "", "text": "$|G(p)|^2 \\ge |G(p^2)|$"}
{"_id": "7186896", "title": "", "text": "$\\|Af(x)\\|_1 \\leq \\|f(x^2)\\|_1$"}
{"_id": "864377", "title": "", "text": "$(x,y,z) = (1,1,1)$"}
{"_id": "2914738", "title": "", "text": "$ [A]^\\mathcal{B}_{\\mathcal{B}'} \\cdot [x]_\\mathcal{B} = [A(x)]_{\\mathcal{B}'}$"}
{"_id": "318002", "title": "", "text": "$y = \\frac{ax+b}{cx+d}$"}
{"_id": "5784210", "title": "", "text": "$\\sum_{m=1}^{\\infty}\\sum_{n=1}^{\\infty} \\frac{1}{n(m^2+n^2)}=\\sum_{m=1}^{\\infty}\\sum_{n=1}^{\\infty} \\frac{1}{n^2(m^2+n^2)}=\\frac{\\pi^4}{72}$"}
{"_id": "1514248", "title": "", "text": "$\\gamma_n\\subseteq\\gamma$"}
{"_id": "4666323", "title": "", "text": "$f_X(x) = \\frac{1}{2\\pi}e^\\frac{-(x)^2}{2}$"}
{"_id": "5639641", "title": "", "text": "$P(2) = 1/6$"}
{"_id": "8808441", "title": "", "text": "$=\\sum_{k=1}^n\\sqrt{\\frac{\\color{Red}{k^2(k^2+k+1)}+\\color{Green}{k(k^2+k+1)}+\\color{Blue}{(k^2+k+1)}}{k^2(k+1)^2}} $"}
{"_id": "8453", "title": "", "text": "$\\frac{C}{1+r}+\\frac{C}{(1+r)^2}+...+\\frac{C+FaceValue}{(1+r)^2}$"}
{"_id": "6888154", "title": "", "text": "$\\alpha < \\beta = \\gamma^+ = \\gamma \\cup \\{ \\gamma \\}$"}
{"_id": "3569864", "title": "", "text": "$\\frac{a + b}{b}$"}
{"_id": "4517614", "title": "", "text": "$ f(9) = 3(9) = 27 $"}
{"_id": "1159124", "title": "", "text": "$(a,b)=1\\implies (ab, a+b)=1$"}
{"_id": "7982734", "title": "", "text": "$ d_{t}=\\frac{(1+\\sqrt{2})^{t+1}-(1-\\sqrt{2})^{t+1}}{2\\sqrt{2}} $"}
{"_id": "8545548", "title": "", "text": "$b^{log_b(z)} = log_b(b^z)$"}
{"_id": "1494842", "title": "", "text": "$w=xyz\\,[y,x]\\,[z,y]\\,[[z,y],[y,x]]\\,[z,x]\\,[[z,x],[y,x]]\\,[[z,x],y]\\,[[[z,x],y],[y,x]],$"}
{"_id": "2201885", "title": "", "text": "$\\ \\overbrace{3\\mid\\color{#b0f}{n\\!-\\!1} = k(k\\!+\\!1)/2-1}^{\\large n\\text{ has remainder } 1}$"}
{"_id": "406916", "title": "", "text": "$=\\frac{1-r^2}{2(1-2r\\cos(\\theta-\\phi)+r^2)}$"}
{"_id": "4975200", "title": "", "text": "$w = \\sqrt[3]{2}\\ e^{\\frac{i\\pi}{12}}$"}
{"_id": "4379094", "title": "", "text": "$m+m>m$"}
{"_id": "357070", "title": "", "text": "$ x=t^{\\frac{1}{n}}$"}
{"_id": "9271550", "title": "", "text": "$1+2+3+4+...\\stackrel{?}{=}-\\frac{1}{12}$"}
{"_id": "1831189", "title": "", "text": "$\\mathcal C = (\\{a\\}, \\Omega)$"}
{"_id": "7307528", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{n^3}{e^{n\\pi}-1}-16\\sum_{n=1}^{\\infty}\\frac{n^3}{e^{4n\\pi}-1}=\\frac{1}{16}$"}
{"_id": "3956256", "title": "", "text": "$P(X_n\\leq x)=P(X_n<x)+P(X_n=x)$"}
{"_id": "4554513", "title": "", "text": "$f_n(x)=(1+\\frac{x}{n})^n, x \\in \\mathbb{R}$"}
{"_id": "102669", "title": "", "text": "$\\tan \\theta =\\frac{x \\pm \\sqrt{x^2-4ky-4k^2}}{2k}$"}
{"_id": "9091665", "title": "", "text": "$\\int_0^\\infty \\frac{r^{p-1}}{(1+r)^n}$"}
{"_id": "8862597", "title": "", "text": "$ \\begin{cases} x=3\\\\ y=1+4t\\\\ z=-2+4t \\end{cases} $"}
{"_id": "6169211", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty}\\sum_{m=-\\infty}^{\\infty}{\\left(n^2+m^2\\right)^{-{p}}}$"}
{"_id": "6252376", "title": "", "text": "$d(x,z)=\\frac1n$"}
{"_id": "3199221", "title": "", "text": "$f(f(f(x)))=1-f(f(x))=1-(1-f(x))=f(x);$"}
{"_id": "5251574", "title": "", "text": "$\\forall \\varepsilon>0, \\exists \\delta>0: |z-y|<\\delta\\implies |f(z)-f(y)|<\\varepsilon\\ \\ ?$"}
{"_id": "1607296", "title": "", "text": "$\\underline{x} = (x,y) $"}
{"_id": "5832022", "title": "", "text": "$ax+by=c+d$"}
{"_id": "420796", "title": "", "text": "$\\mathbf \\Sigma^+$"}
{"_id": "1794230", "title": "", "text": "$d(x,A) \\leq d(x,y) + d(y,a)$"}
{"_id": "245635", "title": "", "text": "$\\,f(n) = n^3-n\\,$"}
{"_id": "2639078", "title": "", "text": "$T=\\{t_1, t_2,..,t_{m-n}\\}$"}
{"_id": "4323447", "title": "", "text": "$f(x) = \\frac{(1-p) x}{\\sqrt{1-4 p (1-p) x^2}}$"}
{"_id": "3191539", "title": "", "text": "$ \\begin{cases} x=-4s\\\\ y=s\\\\ z=-5s\\\\ \\end{cases}$"}
{"_id": "1155675", "title": "", "text": "$ \\left[ \\begin{array}{ccc|c}   1&1&-1&-2\\\\   2&-1&3&14\\\\ -1&-2&1&3 \\end{array} \\right] $"}
{"_id": "8778334", "title": "", "text": "$\\mathrm{erf}(x)=\\frac{e^{-x^2}}{\\sqrt{\\pi}}\\sum^{\\infty}_{n=0}\\frac{(2x)^{2n+1}}{(2n+1)!!}$"}
{"_id": "6559868", "title": "", "text": "$det\\begin{pmatrix}b^*&a\\\\ {-a^*}&b\\end{pmatrix}=1$"}
{"_id": "5360893", "title": "", "text": "$\\tan \\theta = \\frac{x}{\\sqrt{81-x^2}}$"}
{"_id": "175734", "title": "", "text": "$A= B\\oplus C$"}
{"_id": "3590444", "title": "", "text": "$a^r \\equiv a^{s + k\\varphi(n)} \\equiv a^s \\cdot (a^{\\varphi(n)})^k \\equiv a^s \\cdot 1^k \\equiv a^s \\pmod n$"}
{"_id": "747819", "title": "", "text": "$\\det \\left( \\begin{smallmatrix} A & -B \\\\ B & A \\end{smallmatrix} \\right) =|\\det(A+iB)|^2$"}
{"_id": "1361580", "title": "", "text": "$\\cos{\\theta} = \\frac{x}{5}$"}
{"_id": "8062872", "title": "", "text": "$ \\forall \\epsilon>0 \\qquad \\exists r_{\\epsilon}>0 \\qquad |y-x|\\leq r_{\\epsilon} \\Rightarrow |h(x)-h(y)|\\leq \\epsilon   $"}
{"_id": "7575622", "title": "", "text": "$S=\\{\\langle x,y\\rangle:0\\le x,y<1\\}$"}
{"_id": "7562615", "title": "", "text": "$\\sum\\limits_{\\gamma<\\alpha} a_\\gamma<\\prod\\limits_{\\gamma<\\alpha} a_\\gamma$"}
{"_id": "5480293", "title": "", "text": "$\\left\\lfloor\\frac{\\left\\lfloor\\frac ac\\right\\rfloor}b\\right\\rfloor=\\left\\lfloor\\frac{\\left\\lfloor\\frac ab\\right\\rfloor}c\\right\\rfloor =\\left\\lfloor\\frac a{bc}\\right\\rfloor.$"}
{"_id": "8259395", "title": "", "text": "$ \\sum\\limits_{n} e^{-n^2 t} = \\frac{1}{{2 \\pi t}} \\sum\\limits_{n} e^{-n^2/4 t}.$"}
{"_id": "2458024", "title": "", "text": "$z(x)=1+b(x−1)+…$"}
{"_id": "864785", "title": "", "text": "$\\gamma,\\tilde\\gamma \\in C^1([a,b], \\Omega)$"}
{"_id": "4025459", "title": "", "text": "$|f(x)|\\leq\\sup|f(x)|$"}
{"_id": "960699", "title": "", "text": "$\\alpha\\in \\Omega\\subset\\Delta$"}
{"_id": "2780679", "title": "", "text": "$I = \\int^{1}_{0}\\frac{\\tan^{-1}(x)}{x}dx = \\tan^{-1}(x)\\ln(x)\\bigg|^{1}_{0}-\\int^{1}_{0}\\frac{\\ln(x)}{1+x^2}dx$"}
{"_id": "8391147", "title": "", "text": "$ \\begin{matrix}    & 1 &  x& x^2 &   & w & x w &   & w^2 &  & \\\\ 1  & 1 &  .& -3   &   & 2 & .   &   & 1   &  & \\\\ y  & . &  .&     &   & . &     & \\\\ y^2& . &   &     &   &   &     & \\\\ \\\\ z  & . & 1 &     &   & -2&     & \\\\ y z& . &   &\\\\ \\\\ z^2& 1 &   &\\\\ \\end{matrix} $"}
{"_id": "5712715", "title": "", "text": "$\\lim\\limits_{n \\to \\infty} \\sum_{k=1}^n \\frac{1}{2^k} = 1$"}
{"_id": "2681635", "title": "", "text": "$\\|x_n - y_n\\| < 1$"}
{"_id": "438212", "title": "", "text": "$P(\\omega_i)=1/6$"}
{"_id": "1337161", "title": "", "text": "$ax + by = e$"}
{"_id": "1591555", "title": "", "text": "$\\left[\\begin{array}{cccc|c}       1 & 1 & 1 & 0 & 75         \\\\[0.55ex]       0 & 0 & -1 & 1 & 5        \\\\[0.55ex]       0 & -1 & 0 & 1 & 0\\\\[0.55ex]       0 & 0 & 1 & 2 & 70 \\end{array}\\right]$"}
{"_id": "455525", "title": "", "text": "$A \\subset V \\subset \\bar V \\subset U$"}
{"_id": "1302066", "title": "", "text": "$\\frac{d}{dx} (-x)^{3/7} = \\frac{-3}{7}(-x)^{-4/7} = \\frac{-3}{7}(-x)^{3/7-1}=\\frac{-3}{7}\\frac{(-x)^{3/7}}{-x}=\\frac{3}{7}\\frac{(-x)^{3/7}}{x}(x<0)$"}
{"_id": "7417545", "title": "", "text": "$\\lim \\limits_{n\\to\\infty}\\frac{\\sum\\limits_{r=1}^{n}r^0}{\\sum \\limits_{r=1}^{n}r} =6$"}
{"_id": "7822212", "title": "", "text": "$\\varphi(2p) = \\varphi(2) \\varphi(p) = \\varphi(p)$"}
{"_id": "369213", "title": "", "text": "$|CK| = r-s$"}
{"_id": "7126784", "title": "", "text": "$|ab|=m\\mid 6$"}
{"_id": "7891343", "title": "", "text": "$C_0 < x < C_1$"}
{"_id": "5038085", "title": "", "text": "$F(x) = \\int_{-\\infty}^{x}f(t)\\text{ d}t$"}
{"_id": "4306265", "title": "", "text": "$\\int_0^\\infty\\frac{\\sin^4x}{x^4}$"}
{"_id": "3006984", "title": "", "text": "$\\frac 6a + \\frac 6b =1$"}
{"_id": "5199768", "title": "", "text": "$z=s^\\frac 1n e^{(i\\frac \\phi n+\\frac{2kpi}n)}$"}
{"_id": "135308", "title": "", "text": "$y''-\\dfrac{(1+x^2)+ax}{b(1+x^2)^3}y=0$"}
{"_id": "1939250", "title": "", "text": "$ (x-y)^2 (x+y) + (y-z)^2 (y+z) + (z-x)^2 (z+x) \\geq  {(y-z)^2} (x+y+z) $"}
{"_id": "2297838", "title": "", "text": "$C(a)=\\frac12\\int_0^1 u^{-1/2}(1-u)^{\\frac{a-1}2}du$"}
{"_id": "3787804", "title": "", "text": "$y = 31 - 10(e^{\\large \\frac{x}{20}} + e^{\\large \\frac{-x}{20}} ).$"}
{"_id": "1235582", "title": "", "text": "$(c_0,c_1)$"}
{"_id": "9225362", "title": "", "text": "$ax + by = d - cz_0$"}
{"_id": "3402465", "title": "", "text": "$1 - \\cos \\varphi = 2\\sin^2 \\frac{\\varphi}{2},$"}
{"_id": "654255", "title": "", "text": "$\\displaystyle \\lim_{x \\rightarrow c}f(x)=l$"}
{"_id": "3568234", "title": "", "text": "$=\\left\\lfloor \\lfloor a\\rfloor\\lfloor b\\rfloor+\\{a\\}\\lfloor b\\rfloor+\\{b\\}\\lfloor a\\rfloor+\\{a\\}\\{b\\}\\right\\rfloor-\\lfloor a\\rfloor\\lfloor b\\rfloor$"}
{"_id": "4704811", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       1&2&1&1\\\\       -1&4&3&2\\\\        2&-2&a&3     \\end{array} \\right] $"}
{"_id": "6314768", "title": "", "text": "$(F(z))^2=f(z^2)$"}
{"_id": "9017203", "title": "", "text": "$\\vartheta(a+\\!\\frac{z+b}\\tau,-i\\tau)$"}
{"_id": "8852926", "title": "", "text": "$f_s(s |\\{\\tau=0\\} \\cup \\{s \\geq z\\})=\\frac{(1-\\gamma)+1_{s\\geq z}\\gamma}{(1-\\gamma)+(1-F_s(z))\\gamma}f_s(s)$"}
{"_id": "7298627", "title": "", "text": "$\\left(\\begin{array}{ccc|c} 1&-2&1&7\\\\ 1&-2&-k&k\\\\ k&-2&k&7 \\end{array}\\right)$"}
{"_id": "320100", "title": "", "text": "$F(x) = \\int_a^x\\!f(t)\\, dt.$"}
{"_id": "5675918", "title": "", "text": "$\\mathbb{N}_{k,n}=\\{k,k+n,k+2n,\\ldots\\},\\ 1\\le k\\le n,\\ \\bigcup_{k=1}^n\\mathbb{N}_{k,n}=\\mathbb{N}.$"}
{"_id": "5626267", "title": "", "text": "$n! \\sim \\sqrt{2\\pi n}\\cdot \\left(\\frac{n}{e}\\right)^n,$"}
{"_id": "6660864", "title": "", "text": "$f(x)=3^x / (3^x+1)$"}
{"_id": "4595514", "title": "", "text": "$ax+by=M.$"}
{"_id": "889258", "title": "", "text": "$m!+2, \\cdots, m!+m$"}
{"_id": "7699613", "title": "", "text": "$|f(z_r)| \\geq |f(z)|$"}
{"_id": "5821908", "title": "", "text": "$G=\\{\\langle x,y\\rangle\\in\\Bbb R^2:y<x\\}$"}
{"_id": "9233009", "title": "", "text": "$f(a_1,g(a_1))=0$"}
{"_id": "1186546", "title": "", "text": "$f(n)=\\frac{3^{2n}}{3^{2n}+3}$"}
{"_id": "6086079", "title": "", "text": "$ I = \\int\\int_D \\frac{1}{(x^2 + y^2)^{n/2}}  dxdy .$"}
{"_id": "223732", "title": "", "text": "$(a+b,a-b)=(x,y)$"}
{"_id": "3534184", "title": "", "text": "$\\mathbb Q(a)=\\mathbb Q(a+c)$"}
{"_id": "2325541", "title": "", "text": "$d(a,x)\\le d(a,y)+d(y,x)$"}
{"_id": "9076936", "title": "", "text": "$g(x) := \\sum_{j=1}^n c_j 1_{A_j}(x)$"}
{"_id": "8726965", "title": "", "text": "$ P\\left( \\sup_{t \\geq 0 } M_t > x \\mid \\mathcal{F}_0 \\right) = 1 \\wedge \\frac{M_0}{x} \\quad\\text{a.s.,} $"}
{"_id": "3210551", "title": "", "text": "$\\mathbb{R}[x]/(x^2+1) \\approxeq \\mathbb{C}$"}
{"_id": "2475372", "title": "", "text": "$ \\lim_{n\\rightarrow\\infty} \\sum_{i=1}^{n} i^{-p}a_i < \\infty$"}
{"_id": "6766065", "title": "", "text": "$2^e p_1p_2\\cdots p_k,$"}
{"_id": "7558447", "title": "", "text": "$ \\mathrm{dist}(x,M)=\\frac{|F(x)|}{\\Vert F\\Vert}=|F(x)| $"}
{"_id": "814542", "title": "", "text": "$1 + 2 + 3 + 4 + \\dots = -\\frac{1}{12}$"}
{"_id": "911525", "title": "", "text": "$T f(p)=F(p-t)$"}
{"_id": "6138721", "title": "", "text": "$\\gamma=(\\gamma_{1},\\gamma_{2},...,\\gamma_{n})$"}
{"_id": "7156291", "title": "", "text": "$Cov(X,Y) = 0.$"}
{"_id": "4517240", "title": "", "text": "$\\lim_{n\\to \\infty} \\frac{1}{n} (\\sqrt{\\frac{1}{n}} + \\sqrt{\\frac{2}{n}} + \\sqrt{\\frac{3}{n}} + ... + \\sqrt{\\frac{n}{n}})$"}
{"_id": "296638", "title": "", "text": "$ \\exists \\varepsilon > 0, \\forall \\delta > 0, \\exists q_\\delta\\in(x-\\delta,x+\\delta)\\text{ s.t. } |f(x)-f(q_\\delta)|=f(q_\\delta)=q_\\delta^2 > \\varepsilon$"}
{"_id": "1930943", "title": "", "text": "$S = 1 + 2 + 3 + 4 + \\ldots $"}
{"_id": "7926165", "title": "", "text": "$13\\mid n^{13}-n$"}
{"_id": "2689434", "title": "", "text": "$F(\\gamma) = \\begin{cases} \\frac{1}{\\ln{\\gamma}} & \\gamma>e\\\\ \\frac{\\gamma}{e} & |\\gamma|< e\\\\ -\\frac{1}{\\ln{-\\gamma}} & \\gamma<-e\\end{cases}$"}
{"_id": "3394110", "title": "", "text": "$F(x) = \\int_{a}^{b} f(t)dt.$"}
{"_id": "6482618", "title": "", "text": "$f (x)\\le d (x,a)\\le d (x,y)+d (y,a) $"}
{"_id": "952245", "title": "", "text": "$\\{a, a+b, a+2b,..., a+ib\\}$"}
{"_id": "7990743", "title": "", "text": "$R \\equiv R \\land S \\equiv R \\lor S \\equiv S$"}
{"_id": "6537758", "title": "", "text": "$ \\lim_{n \\to \\infty} \\mu(A_i \\cap E_n) = \\mu(A_i),$"}
{"_id": "3253173", "title": "", "text": "$(a+b,a-b)=g$"}
{"_id": "54447", "title": "", "text": "$d = ax +by$"}
{"_id": "8455647", "title": "", "text": "$\\min\\left(\\lfloor \\frac{R}{\\gamma}\\rfloor,N\\right)$"}
{"_id": "3303376", "title": "", "text": "$e^x > \\bigg(1+ \\frac xn\\bigg)^n$"}
{"_id": "7437564", "title": "", "text": "$\\int_0^\\infty f(x)\\ dx=0.$"}
{"_id": "4524934", "title": "", "text": "$\\exists c_0\\in (a,x): g'(c_0)=0$"}
{"_id": "7202998", "title": "", "text": "$(a+b,-a+2b,a+b,2a+b)$"}
{"_id": "8360889", "title": "", "text": "$\\lfloor{a^b}\\rfloor = c$"}
{"_id": "2249931", "title": "", "text": "$R=\\{x+iy:a\\leq x\\leq b, c\\leq y\\leq d\\}$"}
{"_id": "3652379", "title": "", "text": "$\\frac{x_1 + x_2 +... + x_n}{n} = a$"}
{"_id": "1361475", "title": "", "text": "$E_n, x_n$"}
{"_id": "2247649", "title": "", "text": "$\\textbf{x}= \\gamma \\textbf{p}_1$"}
{"_id": "8378042", "title": "", "text": "$\\Delta (1/|x|)= \\delta(x)$"}
{"_id": "5537658", "title": "", "text": "$ log_a(x) = b $"}
{"_id": "304108", "title": "", "text": "$1/|x|^\\alpha$"}
{"_id": "2748826", "title": "", "text": "$\\sum_{k=1}^n\\cos(\\frac{2\\pi k}{n})$"}
{"_id": "5327901", "title": "", "text": "$\\int_{0}^{\\pi} f(t) \\sin (nt)\\mathrm dt =0$"}
{"_id": "1052676", "title": "", "text": "$U=span\\{e_1,e_2,e_3,e_4,e_5\\}$"}
{"_id": "62535", "title": "", "text": "$\\int_{1}^{\\infty} f(x)dx$"}
{"_id": "134233", "title": "", "text": "$A = \\{a\\}$"}
{"_id": "7284441", "title": "", "text": "$(x*y)*z =(x+y-[x+y])*z = x+y-[x+y]+z - [x+y-[x+y]+z]$"}
{"_id": "1095370", "title": "", "text": "$p_1\\nmid a, p_2\\mid a$"}
{"_id": "3564783", "title": "", "text": "$ I(a)=-\\log(2\\sin(\\pi/a))-\\gamma+\\log(2 \\pi/a)+\\gamma=\\\\ \\log\\left( \\frac{\\pi/a}{\\sin(\\pi/a)}\\right) $"}
{"_id": "7952612", "title": "", "text": "$p(m-1)<p(m)<p(m+1).$"}
{"_id": "7354939", "title": "", "text": "$f(X) = X^2-X+41$"}
{"_id": "309014", "title": "", "text": "$\\Phi(s \\otimes n) = \\Phi((s, n) + S\\otimes_RN) = s\\varphi(n) = s'\\varphi(n') = \\Phi(s' \\otimes n')$"}
{"_id": "1472138", "title": "", "text": "$\\zeta(s)=\\sum_{n=1}^m\\frac 1{n^s}+\\gamma(1-s)\\sum_{n=1}^m\\frac 1{n^{1-s}} + R(s)$"}
{"_id": "5803414", "title": "", "text": "$\\phi(n):=n^{-s}$"}
{"_id": "790173", "title": "", "text": "$\\{e_i, 1\\le i \\le n\\}$"}
{"_id": "5253271", "title": "", "text": "$0 = L(P_n,f)\\le U(P_n,f) = \\sum_{k=1}^{n^2}M_k\\cdot \\frac{1}{n^2} = \\sum_{k=1}^{n}M_k\\cdot \\frac{1}{n^2} + \\sum_{k=n+1}^{n^2}M_k\\cdot \\frac{1}{n^2}.$"}
{"_id": "6478860", "title": "", "text": "$f_{X,Y}(x,y) = \\frac{2}{\\theta^2}$"}
{"_id": "5487144", "title": "", "text": "$\\begin{align}     (m_1 - m_2)^2 &= 4 \\frac{\\tan^2 \\theta}{\\sin^4 \\theta} - 4 \\frac{(\\tan^2 \\theta + \\cos^2 \\theta)}{\\sin^2 \\theta} \\\\         &= 4 \\frac{\\tan^2 \\theta - \\sin^2 \\theta (\\tan^2 \\theta + \\cos^2 \\theta)}{\\sin^4 \\theta} \\\\         &= 4 \\frac{\\tan^2 \\theta (1 - \\sin^2 \\theta) - \\sin^2 \\theta \\cos^2 \\theta}{\\sin^4 \\theta} \\\\         &= 4 \\frac{(\\tan^2 \\theta - \\sin^2 \\theta) \\cos^2 \\theta}{\\sin^4 \\theta} \\\\         &= 4 \\frac{(\\sec^2 \\theta - 1) \\cos^2 \\theta}{\\sin^2 \\theta} \\\\         &= 4 \\frac{\\tan^2 \\theta \\cos^2 \\theta}{\\sin^2 \\theta} \\\\         &= 4 , \\end{align}$"}
{"_id": "5683870", "title": "", "text": "$\\,(C_r=\\{|z|=r\\})$"}
{"_id": "817337", "title": "", "text": "${n+1\\choose 2}\\frac 1{2^{n-1}}$"}
{"_id": "8799797", "title": "", "text": "$\\frac{1}{2}\\int_{-\\infty}^{\\infty} \\frac{\\cos(ax)}{(1+x^2)^2}\\mathrm dx$"}
{"_id": "4821245", "title": "", "text": "$(A, -B)$"}
{"_id": "1347779", "title": "", "text": "$ ||A||^{2}_2 = \\rho(A^{T}A) $"}
{"_id": "3855656", "title": "", "text": "$f_Z(z) = \\frac{4 \\log z}{z^3}, \\quad z > 1.$"}
{"_id": "1294102", "title": "", "text": "$\\overline{X}=\\overline{Y}=\\mathbb{R}$"}
{"_id": "320941", "title": "", "text": "$P \\equiv R \\equiv S$"}
{"_id": "142232", "title": "", "text": "$ f_x(x) = \\frac {2x + 1}{10^{2}} $"}
{"_id": "7236979", "title": "", "text": "$Log_a(b)$"}
{"_id": "474840", "title": "", "text": "$(a,a+k,a+2k,a+3k, ...)$"}
{"_id": "4506900", "title": "", "text": "$(x-a,x-b)=1$"}
{"_id": "6695243", "title": "", "text": "$\\sum_{m=1}^{\\infty}\\sum_{n=1}^{\\infty}\\frac{1}{10^{mn}}$"}
{"_id": "4097507", "title": "", "text": "$f(x)=\\dfrac{2ax}{k^2}$"}
{"_id": "7082443", "title": "", "text": "$ \\begin{align}    \\int_{-\\infty}^{\\infty}|(\\lambda I-L)^{-1}f|^{2}\\,dx & \\le     \\int_{-\\infty}^{\\infty}\\left(\\int_{-\\infty}^{x}e^{-\\Im\\lambda(x-t)}|f(t)|\\,dt\\right)^{2}\\,dx \\\\     & \\le \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{x}e^{-\\Im\\lambda(x-t)}\\,dt         \\int_{-\\infty}^{x}e^{-\\Im\\lambda(x-t)}|f(t)|^{2}\\,dt\\,dx \\\\     & \\le \\frac{1}{\\Im\\lambda}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{x}e^{-\\Im\\lambda(x-t)}|f(t)|^{2}\\,dt\\,dx \\\\     & = \\frac{1}{\\Im\\lambda}\\int_{-\\infty}^{\\infty}\\int_{t}^{\\infty} e^{-\\Im\\lambda(x-t)}\\,dx\\,|f(t)|^{2}dt \\\\     & = \\frac{1}{(\\Im\\lambda)^{2}}\\int_{-\\infty}^{\\infty}|f(t)|^{2}\\,dx            =\\frac{\\|f\\|^{2}}{(\\Im\\lambda)^{2}}. \\end{align} $"}
{"_id": "3036624", "title": "", "text": "$w = e^{\\frac{2\\pi i} {n}}$"}
{"_id": "1726853", "title": "", "text": "$\\sin\\theta=\\frac{x}{5}$"}
{"_id": "6532897", "title": "", "text": "$  \\left[\\begin{array}{rr|r}7.5 & -5 & 0 \\\\k & 5 & 0 \\\\\\end{array}\\right]$"}
{"_id": "3761001", "title": "", "text": "$||A||_{2}=\\sqrt{\\rho(AA^{t})}=\\sqrt{\\rho(A^{2})}=\\rho$"}
{"_id": "6734655", "title": "", "text": "$\\frac{ax+b}{c-x}=y,$"}
{"_id": "1377317", "title": "", "text": "$\\Bbb R[x]/(x)$"}
{"_id": "6547692", "title": "", "text": "$ \\frac{\\tan(\\gamma)+\\tan(a)}{1-\\tan(a)\\tan(\\gamma)}=\\frac{k - \\cos(\\gamma)}{\\sin(\\gamma) }$"}
{"_id": "1769816", "title": "", "text": "$\\frac{1}{\\pi} \\left( \\arccos(1 - 2p) - \\frac{1}{2} \\sin 2 \\arccos (1 - 2p)\\right), 0 < p < 0.5$"}
{"_id": "1925847", "title": "", "text": "$F = L \\oplus M \\oplus N$"}
{"_id": "5429555", "title": "", "text": "$S=\\{S_1,S_2,S_3,... S_n\\} $"}
{"_id": "4540519", "title": "", "text": "$\\sum_{n=1}^\\infty n =1+2+3+4+\\dots = -\\frac{1}{12}.$"}
{"_id": "4786361", "title": "", "text": "$\\left(\\frac{n(n+1)}2\\right)^2-\\left(\\frac{(n-1)n}2\\right)^2=n^2\\left(\\frac{n+1}2+\\frac{n-1}2\\right)\\left(\\frac{n+1}2-\\frac{n-1}2\\right).$"}
{"_id": "1928182", "title": "", "text": "$(a-b,a+b) =2$"}
{"_id": "8280261", "title": "", "text": "$\\frac{-rB_1}{(1+r)^2}$"}
{"_id": "2913037", "title": "", "text": "$\\|A(A^TA)^{-1}A^T\\|_2=1$"}
{"_id": "6369115", "title": "", "text": "$A_1 \\supseteq A_2 \\supseteq \\cdots \\supseteq A_n \\supseteq A_{n+1} \\supseteq \\cdots$"}
{"_id": "7012679", "title": "", "text": "$(x,y,z)=k(1,1,1)$"}
{"_id": "73542", "title": "", "text": "$f(x)= \\frac{1}{x^2}$"}
{"_id": "1405044", "title": "", "text": "$\\sum_{n = 1}^{\\infty} 1 = \\infty$"}
{"_id": "94431", "title": "", "text": "$\\cos^2\\varphi = 1 - \\sin^2\\varphi$"}
{"_id": "6681647", "title": "", "text": "$(\\forall x \\in A)(\\forall y \\in A) \\rightarrow xRy \\lor yRx \\lor x=y$"}
{"_id": "6998045", "title": "", "text": "$\\sum_{n\\in \\mathbb{N}} \\frac{1}{n^2} = \\sum_{n\\in \\mathbb{N}}^{} \\frac{1}{(2n)^2} + \\sum_{n\\in \\mathbb{N}} \\frac{1}{(2n - 1)^2}$"}
{"_id": "3348265", "title": "", "text": "$\\forall x\\forall y (xRy\\lor yRx)\\to \\forall x(xRx)$"}
{"_id": "5873897", "title": "", "text": "$f(a+b) = fa + fb$"}
{"_id": "3723734", "title": "", "text": "$(P, *, *, *, E, E, E) \\\\ (E, P, *, *, *, E, E) \\\\ (E, *, P, *, *, E, E) \\\\ (E, *, E, P, *, E, *) \\\\ (E, E, *, *, P, *, E) \\\\ (E, *, E, *, *, P, E) \\\\ (E, *, *, E, E, *, P) \\\\ $"}
{"_id": "5808576", "title": "", "text": "$\\mathbf{A}(x,y,z)=(x^2,y^2,z^2).$"}
{"_id": "5282103", "title": "", "text": "$\\eta(s) = \\frac{1}{\\Gamma(s)}\\int_{0}^{\\infty} \\frac{x^{s-1}}{e^{x}+1} \\, dx \\, , \\quad \\text{Re}(s) >0. $"}
{"_id": "3833889", "title": "", "text": "$\\sum_{k\\in\\mathbb Z}\\binom nkk^2x^k=nx\\bigl[(1+x)^{n-1}+(n-1)x(1+x)^{n-2}\\bigr]=F(x)\\,.$"}
{"_id": "2855223", "title": "", "text": "$ P[Z\\geqslant sa]=P[s^Z\\geqslant s^{sa}]\\leqslant s^{-sa}E[s^Z]=\\mathrm e^{-aI(s)}, $"}
{"_id": "543651", "title": "", "text": "$f(a+(f(x)-x))=f(a)+f(x)$"}
{"_id": "364092", "title": "", "text": "$x = \\frac{n(n + 1)}2$"}
{"_id": "5976169", "title": "", "text": "$\\frac{1}{x}+\\frac{4}{y} = \\frac{1}{12}$"}
{"_id": "2383588", "title": "", "text": "$\\{a,a+b,a+c,a+b+c\\}$"}
{"_id": "8694919", "title": "", "text": "$\\inf_{k \\in \\mathbb N} \\mu(A_k)=\\lim_{k \\to \\infty} \\mu(A_k)=0$"}
{"_id": "5607894", "title": "", "text": "$p_1p_2\\mid \\eta(1)$"}
{"_id": "1575555", "title": "", "text": "$\\phi(x)=\\frac{1}{2\\pi}e^{\\frac{-x^2}{2}}$"}
{"_id": "2667298", "title": "", "text": "$\\begin{cases}   x=u\\\\   y=\\cos v\\\\   z=\\sin v \\end{cases}$"}
{"_id": "8948314", "title": "", "text": "$|ab|=\\infty,$"}
{"_id": "2559538", "title": "", "text": "$\\frac{\\Bbb{R[x]}}{\\langle x^2+1\\rangle}$"}
{"_id": "4577", "title": "", "text": "$n_1=n_2=\\ldots =n_l=n_{l+1}+1=n_{l+2}+1=\\ldots =n_k+1$"}
{"_id": "693882", "title": "", "text": "$1 + 2 + 3 + 4.... = -\\frac 1 {12}$"}
{"_id": "1654050", "title": "", "text": "$x^5-x^3,x^2-x^4,x-x^3,1$"}
{"_id": "2924412", "title": "", "text": "$(a-\\Delta x,f(a-\\Delta x))$"}
{"_id": "670620", "title": "", "text": "$x_n=\\frac{n\\cdot(n+1)}2$"}
{"_id": "3447626", "title": "", "text": "$ \\cos^2\\varphi = \\frac{\\sqrt{1 - \\cos^2 l \\cos^2\\theta} - \\sin l}{1 + \\sqrt{1-\\cos^2 l\\cos^2\\theta}},$"}
{"_id": "6552205", "title": "", "text": "$\\frac{k(k+1)(4k-1)}{3}+(2k+1)2(k+1)=\\frac{(k+1)(4k^2+11k+6)}{3}=\\frac{(k+1)(k+2)(4k+3)}{3}$"}
{"_id": "6025160", "title": "", "text": "$n=1 \\ldots 128$"}
{"_id": "4407031", "title": "", "text": "$\\frac7x + \\frac2y = \\frac{11}{12}$"}
{"_id": "7852026", "title": "", "text": "$\\Delta c_0 = c_1 - c_0$"}
{"_id": "4680526", "title": "", "text": "$Log_{a} (x^{n})=bn$"}
{"_id": "3643929", "title": "", "text": "$\\varphi(x)=\\int_0^1f(t)\\cos(xt)dt$"}
{"_id": "7270227", "title": "", "text": "$\\frac{\\sqrt{1-\\rho^2}}{2\\pi(1-\\rho \\sin(2\\theta))}$"}
{"_id": "3781100", "title": "", "text": "$\\vec{p_1p_2}$"}
{"_id": "8324078", "title": "", "text": "$\\mathcal{A}\\mathcal{X}=\\mathcal{B}$"}
{"_id": "1480982", "title": "", "text": "$\\{a,a+C,a+2C,a+3C\\}$"}
{"_id": "2067489", "title": "", "text": "$\\,f(x) = x^2 -x + 1.\\,$"}
{"_id": "6375337", "title": "", "text": "$\\left[\\begin{array}{ccccc|c} 1 & 2 & -2 & 2 & -1 & 0\\\\ 1 & 2 & -1 & 3 & -2 & 0\\\\ 2 & 4 & -7 & 1 & 1 & 0 \\end{array}\\right]$"}
{"_id": "348411", "title": "", "text": "$K=\\mathbb{Q}(\\sqrt{d})$"}
{"_id": "6937322", "title": "", "text": "$\\lim_{n\\to \\infty} n\\sum_{k=1}^n \\frac 1{n^2+k^2}$"}
{"_id": "2943016", "title": "", "text": "$G:=\\left[\\matrix{1&0&-1\\cr 2&1&1\\cr 3&2&1\\cr}\\right]\\ .$"}
{"_id": "8018489", "title": "", "text": "$ f(x)=9^x - 3^x+1$"}
{"_id": "2559343", "title": "", "text": "$x \\, \\mapsto \\, x^{3}-x-1$"}
{"_id": "3088720", "title": "", "text": "$\\lim_{N\\to\\infty}\\sum_{i=1}^N\\frac1i=\\infty.$"}
{"_id": "979748", "title": "", "text": "$f(x) = \\frac{1}{N}e^x - \\frac{1}{N} - e^{x/N}+1$"}
{"_id": "2320744", "title": "", "text": "$\\tilde X=\\cases{z_1^2+z_2^3+z_3^4+z_4^5+z_5^6=1\\\\ z_1z_2z_3z_4z_5=1\\\\ (z_1z_2+z_2z_3+z_3z_4+z_4z_5) \\cdot z_6 =1 }$"}
{"_id": "3175437", "title": "", "text": "$\\int_{0}^5 \\frac{1}{10}e^{-\\frac{1}{10}x} = 1-e^{\\frac{-1}{2}}=.3934$"}
{"_id": "3243708", "title": "", "text": "$\\int_1^\\infty \\frac{\\sin(x)}{x} dx$"}
{"_id": "1426683", "title": "", "text": "$\\delta(x) = \\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty e^{inx}$"}
{"_id": "2242623", "title": "", "text": "$\\exists L >0,\\|\\gamma(x)-\\gamma(y)\\| \\leq L |x-y|.$"}
{"_id": "8639538", "title": "", "text": "$f(y)=y^2-y+1$"}
{"_id": "8313269", "title": "", "text": "$A =\\frac{αa + βb + γc}{α + β + γ}$"}
{"_id": "7214195", "title": "", "text": "$z^{n+1}-z^n-1$"}
{"_id": "3201207", "title": "", "text": "$\\mathcal G=\\{\\{1\\}\\}$"}
{"_id": "76336", "title": "", "text": "$ax+by=c$"}
{"_id": "7724476", "title": "", "text": "$\\sum_{n\\ge1}\\frac{1+(-1)^nni}{n^2}=\\sum_{n\\ge1}\\frac1{n^2}+\\sum_{n\\ge1}\\frac{(-1)^ni}n$"}
{"_id": "1010742", "title": "", "text": "$\\displaystyle P(B)=\\frac{1}{6}$"}
{"_id": "2968646", "title": "", "text": "$|f(a)-f(b)|>k$"}
{"_id": "871544", "title": "", "text": "$f(x) = \\left\\{ {\\matrix{    {{x^\\beta }\\sin \\left( {{1 \\over {{x^\\alpha }}}} \\right)}  \\cr     {0,x = 0}  \\cr   } } \\right.$"}
{"_id": "7354185", "title": "", "text": "$\\alpha^\\omega > \\alpha$"}
{"_id": "2299736", "title": "", "text": "$\\sum_{n=1}^{\\infty} (2n - 1)  - \\sum_{n=1}^\\infty n = \\sum_{n=1}^\\infty n$"}
{"_id": "1794229", "title": "", "text": "$d(x,a) \\leq d(x,y) + d(y,a)$"}
{"_id": "5032448", "title": "", "text": "$6^x-5^x=4^x-3^x$"}
{"_id": "3162029", "title": "", "text": "$\\le\\sum_{n=1}^\\infty\\frac{\\sqrt n}{n^2}=\\sum_{n=1}^\\infty\\frac1{n^{3/2}}=\\zeta(3/2)<\\infty$"}
{"_id": "6420527", "title": "", "text": "$ \\| \\gamma(a)-\\gamma(b)\\| =d(\\gamma(a),\\gamma(b)) \\le L(\\gamma)=\\int_a^b \\| \\gamma '\\|.$"}
{"_id": "6616627", "title": "", "text": "$|f(a)−f(b)|≤K|a−b|$"}
{"_id": "2796460", "title": "", "text": "$F=A\\oplus X$"}
{"_id": "6416208", "title": "", "text": "$\\{a < X \\leq b, c < Y \\leq d\\}$"}
{"_id": "9225936", "title": "", "text": "$\\left[\\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\\\ 0  & 0  & 1  & -2 & 2    & -2  & -4 \\\\ 0 & 0 & -1 & 2  & a^{2}+2a &   2  & 3  \\\\ 0 & 0 &  0  &  0  & -1  & 0  & 1  \\\\ \\end{array}\\right]$"}
{"_id": "5564543", "title": "", "text": "$\\sum\\|f_n-e_n\\|^2< 1$"}
{"_id": "6264310", "title": "", "text": "$Cov(x,y)=0$"}
{"_id": "4337350", "title": "", "text": "$\\|T\\|=\\sup_{\\|x\\|=1}\\|Tx\\|=\\sup_{\\|x\\|=1}\\|x+M\\|=\\sup_{\\|x\\|=1}\\inf_{m\\in M}\\|x-m\\|\\leq\\sup_{\\|x\\|=1}\\|x\\|=1$"}
{"_id": "3275664", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}\\frac{1}{n}\\sum_{k=1}^n \\frac{1}{1+\\frac{k^2}{n^2}}$"}
{"_id": "6432970", "title": "", "text": "$1/|x-y|$"}
{"_id": "8675655", "title": "", "text": "$\\phi(x) = \\sum_{j=1}^N a_j \\chi_{B_j}$"}
{"_id": "456866", "title": "", "text": "$D_1, D_2\\in{\\cal D}\\quad\\Longrightarrow\\quad  D_1\\cap D_2 \\in \\cal D$"}
{"_id": "1892347", "title": "", "text": "$\\sum_{n \\in \\mathbb{N}}\\|e_n-f_n\\|^2 <1.$"}
{"_id": "6310203", "title": "", "text": "$K \\subseteq U \\subseteq \\overline{U} \\text{ compact} \\subseteq O$"}
{"_id": "67982", "title": "", "text": "$z=-2\\lambda$"}
{"_id": "4277708", "title": "", "text": "$F(x)=\\int_a^xf(t)dt-\\int_x^bf(t)dt$"}
{"_id": "571067", "title": "", "text": "$A_1 \\subseteq A_2 \\subseteq A_3 \\subseteq\\ldots$"}
{"_id": "4202129", "title": "", "text": "$\\sum_{k=1}^{\\frac{n-1}{2}}\\cos\\left(\\frac{2\\pi k}{n}\\right)=-\\frac{1}{2}$"}
{"_id": "8036859", "title": "", "text": "$\\det(B)=\\det\\begin{pmatrix}     r & a & a & \\cdots &a \\\\     a & r& a & \\cdots & a \\\\      \\vdots  & \\vdots& \\vdots & \\ddots & \\vdots \\\\      a & a & a & \\cdots & r          \\end{pmatrix}=(r+(n-1)a)(r-a)^{n-1}$"}
{"_id": "5784217", "title": "", "text": "$\\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{1}{n^2} \\left ( \\sum_{m=-\\infty}^{\\infty} \\frac{1}{m^2+n^2} - \\frac{1}{n^2} \\right )$"}
{"_id": "438840", "title": "", "text": "$\\frac{a+c}{b+d}=\\frac{a+xa}{b+xb}=\\frac{a(1+x)}{b(1+x)}=\\frac{a}{b}$"}
{"_id": "686749", "title": "", "text": "$f(n)=\\bigl(\\frac{n-1}{n}\\bigr)^{\\log n}$"}
{"_id": "3528257", "title": "", "text": "$x = \\frac{a^2 + b^2 + ab}{a + b}$"}
{"_id": "5597379", "title": "", "text": "$c_1x + c_0$"}
{"_id": "8494360", "title": "", "text": "$xSy\\iff xRy\\land yRx$"}
{"_id": "5772051", "title": "", "text": "$ \\left(1+\\frac{x}{n}\\right)^n\\ge\\left(1+\\frac{x}{m}\\right)^m\\tag{3} $"}
{"_id": "8833546", "title": "", "text": "$\\det(A) = (a+(n-1)b)(a-b)^{n-1}$"}
{"_id": "7700088", "title": "", "text": "$ a_m = \\frac{2}{2 \\pi} \\int_{-\\pi}^{\\pi} \\sum_{n \\in \\mathbb{N}} 10^{-n} \\frac{\\sin(nx)}{\\sin(x)} \\cos\\bigg(m \\frac{2 \\pi x}{2\\pi}\\bigg) \\mathrm{d}x = \\frac{2}{\\pi} \\sum_{n \\in \\mathbb{N}} \\int_{0}^{\\pi}  10^{-n} \\frac{\\sin(nx)}{\\sin(x)} \\cos(m x) \\mathrm{d}x $"}
{"_id": "8383052", "title": "", "text": "$f(x)= \\binom{x}{2} = \\frac{x^2-x}{2}$"}
{"_id": "389987", "title": "", "text": "$\\alpha^+ = \\alpha\\cup\\{\\alpha\\}$"}
{"_id": "8915123", "title": "", "text": "$\\cos\\theta=\\frac{x}{9}$"}
{"_id": "670564", "title": "", "text": "$2\\times1\\times2\\times1\\times2\\times1 \\cdots=2^{n/2}$"}
{"_id": "5894254", "title": "", "text": "$n^2-1,\\quad n\\in\\mathbb{Z}$"}
{"_id": "503006", "title": "", "text": "$P(m) \\Rightarrow P(m+1)$"}
{"_id": "487125", "title": "", "text": "$\\{\\{a\\}\\}$"}
{"_id": "7112857", "title": "", "text": "$\\lim_{n \\to \\infty} \\sum_{k=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor}{\\binom{n-k}{k}\\frac{1}{2^{n-k}}}.$"}
{"_id": "106036", "title": "", "text": "$\\lVert x_k - x_m\\rVert < 1$"}
{"_id": "4140479", "title": "", "text": "${\\large\\int}_{-1}^1\\frac{dx}{\\sqrt[3]{9+4\\sqrt5\\,x}\\ \\left(1-x^2\\right)^{2/3}}=\\frac{3^{3/2}}{2^{4/3}5^{5/6}\\pi }\\Gamma^3\\left(\\frac13\\right)$"}
{"_id": "183526", "title": "", "text": "$\\begin{matrix} \\\\&\\dfrac1{0!}&+&\\dfrac1{1!}&+&\\dfrac1{2!}&+&\\dfrac1{3!}&+&\\dfrac1{4!}&+&\\cdots \\\\+&&&\\dfrac1{0!}&+&\\dfrac1{1!}&+&\\dfrac1{2!}&+&\\dfrac1{3!}&+&\\cdots \\\\\\hline =&\\dfrac1{0!}&+&\\dfrac{1+1}{1!}&+&\\dfrac{1+2}{2!}&+&\\dfrac{1+3}{3!}&+&\\dfrac{1+4}{4!}&+&\\cdots \\\\=&\\dfrac1{0!}&+&\\dfrac{2}{1!}&+&\\dfrac{3}{2!}&+&\\dfrac{4}{3!}&+&\\dfrac{5}{4!}&+&\\cdots \\\\=&\\dfrac{1^2}{1!}&+&\\dfrac{2^2}{2!}&+&\\dfrac{3^2}{3!}&+&\\dfrac{4^2}{4!}&+&\\dfrac{5^2}{5!}&+&\\cdots \\end{matrix}$"}
{"_id": "6056342", "title": "", "text": "$\\mathbf{C}[\\gamma^2,\\gamma^3]\\subset \\mathbf{C}[\\gamma]$"}
{"_id": "2343539", "title": "", "text": "$\\mathrm{gcd}(a+b,a-b)$"}
{"_id": "482991", "title": "", "text": "$\\lim_{x\\to a}f(x)=L \\iff \\lim_{x\\to a^+}f(x)=L=\\lim_{x\\to a^-}f(x)$"}
{"_id": "5586967", "title": "", "text": "$|ab| = |cd|$"}
{"_id": "6441954", "title": "", "text": "$p\\mid pn_2$"}
{"_id": "8979856", "title": "", "text": "$M=\\pmatrix{A & B\\cr B & C\\cr}.$"}
{"_id": "4540088", "title": "", "text": "$R= \\lbrace (x,y) \\in \\mathbb{R}^2 \\mid a \\leq x \\leq b, \\ c \\leq y \\leq d \\rbrace \\subset \\mathbb{R}^2$"}
{"_id": "3442943", "title": "", "text": "$ \\zeta(s+1)=\\frac1{\\Gamma(s+1)}\\int_0^\\infty\\frac{x^s}{e^x-1}\\:dx, \\qquad s>0, \\tag1 $"}
{"_id": "552397", "title": "", "text": "$\\mathbb Z[i] = \\mathbb Z[x]/(x^2+1)$"}
{"_id": "1966953", "title": "", "text": "$P(E)=\\frac12$"}
{"_id": "3830", "title": "", "text": "$n!\\sim\\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n$"}
{"_id": "5452443", "title": "", "text": "$P[X_n\\geq1\\space\\forall n\\geq1\\mid X_0=0]=\\dfrac{6}{\\pi^2}$"}
{"_id": "7174509", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y}= \\frac{1}{13}$"}
{"_id": "8108442", "title": "", "text": "$φ_a (z)=\\dfrac{z-a}{1-\\overline{a} z}  ,a∈\\mathbb{D}$"}
{"_id": "3259941", "title": "", "text": "$\\pi (a,n) > m e^{ \\gamma } \\prod_{5 \\leqslant p \\leqslant n} (1 - 1/p)$"}
{"_id": "5049609", "title": "", "text": "$\\cos 2 \\varphi = 2 \\cos^2 \\varphi - 1 \\\\ \\sin 2 \\varphi = 2 \\sin \\varphi \\cos \\varphi,$"}
{"_id": "7574244", "title": "", "text": "$\\sum_{k>n}\\Vert a_k-b_k\\Vert^2<1$"}
{"_id": "4962350", "title": "", "text": "$\\lim_{x \\to 0} f(x)g(x) = \\lim_{x \\to 0} f(x) \\cdot \\lim_{x \\to 0} g(x).$"}
{"_id": "3659777", "title": "", "text": "$\\int_0^\\infty f^2(x)~dx=1$"}
{"_id": "5755827", "title": "", "text": "$||\\hat{y}|| = \\frac{1}{||f||}$"}
{"_id": "6515061", "title": "", "text": "$f(ax+by)=f(c).$"}
{"_id": "3145693", "title": "", "text": "$(40-x_4)$"}
{"_id": "2858517", "title": "", "text": "$d(f,g) = \\frac{1}{\\sup_x |h(x)|} - 1.$"}
{"_id": "7352517", "title": "", "text": "$\\tan(z)=\\frac{\\pi}{2}$"}
{"_id": "7924281", "title": "", "text": "$ \\begin{array}{c} A^\\top \\gamma\\,P + \\gamma\\,P\\,A \\leq -\\gamma\\,I < 0, \\\\ \\gamma\\,P \\geq \\gamma\\,I > 0. \\end{array} $"}
{"_id": "4618428", "title": "", "text": "$ P(m), m<n \\implies P(n) $"}
{"_id": "6318393", "title": "", "text": "$\\frac{x^2 y^2 z^2+x^2 y^2 z-x^2 y z^2-x y z-x z^2+y-z+1}{(x y+y+1) (x z+x+1) (y z+y+1)}$"}
{"_id": "7545300", "title": "", "text": "$L:\\begin{cases} x=1+2t \\\\ y=1+2t \\\\ z=2-t \\end{cases}$"}
{"_id": "827725", "title": "", "text": "$\\tag 1 J = (a, \\gamma) \\sqcup  (\\gamma, b)$"}
{"_id": "8909593", "title": "", "text": "$[T_X^{0,1},T_X^{0,1}]\\subset T_X^{0,1}$"}
{"_id": "4029415", "title": "", "text": "$Z = \\{\\{x\\}\\}$"}
{"_id": "1289403", "title": "", "text": "$ds = \\sqrt{1+\\left ( \\frac{dy}{dx} \\right )^2} dx = \\frac{dx}{\\sqrt{1-x^2}}$"}
{"_id": "7576957", "title": "", "text": "$ p + e = S(p)$"}
{"_id": "6291339", "title": "", "text": "$\\mid\\mid\\dot\\gamma\\times\\ddot\\gamma\\mid\\mid = \\mid\\mid\\dot\\gamma\\mid\\mid\\ddot\\gamma\\mid\\mid = \\mid\\mid\\ddot\\gamma\\mid\\mid=\\kappa$"}
{"_id": "6562431", "title": "", "text": "$\\{u_n\\} \\subset W_0^{1,1}((0,1))$"}
{"_id": "6371108", "title": "", "text": "$\\gamma(s) = (\\gamma_x(s), \\gamma_y(s), \\gamma_z(s))$"}
{"_id": "5900111", "title": "", "text": "$f(n+1)+f(n+2)+...+f(2n)=n^2$"}
{"_id": "6559397", "title": "", "text": "$\\lim_{n \\to \\infty} \\dfrac{\\sqrt 1 + \\sqrt 2 + \\dots + \\sqrt{n}}{n\\sqrt{n}}=\\lim_{n \\to \\infty} \\dfrac{1}{n}\\sum_{r=1}^n\\sqrt{\\frac{r}{n}}=\\int_0^1\\sqrt{x}=\\frac{2}{3}$"}
{"_id": "2733235", "title": "", "text": "$\\frac{1}{2^{n/2}}$"}
{"_id": "3574716", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} f(x_n)\\ge f(x)$"}
{"_id": "7494132", "title": "", "text": "$ \\sum_{n=1}^{\\infty}\\sum_{m=1}^{\\infty} \\frac{1}{{\\rm lcm}(m,n)}. $"}
{"_id": "4678001", "title": "", "text": "$\\frac{2^x}{2^x-8}>2$"}
{"_id": "2663294", "title": "", "text": "$ \\sqrt{\\cos^2{u}} = |\\cos(u)| $"}
{"_id": "4209180", "title": "", "text": "$m!+2, m!+3,\\dots,m!+m$"}
{"_id": "5112371", "title": "", "text": "$\\frac4x + \\frac9y + \\frac{16}z = D$"}
{"_id": "3638905", "title": "", "text": "$\\int_0^x f(t)g(t)\\,dt = \\int_0^x f(t+a)g(t+a) \\,dt$"}
{"_id": "6980334", "title": "", "text": "$G_1:=\\{x\\in H_1: y\\in H_2, x=2y\\}$"}
{"_id": "580467", "title": "", "text": "$\\gamma(t) = (\\gamma_x(t), \\gamma_y(t)) = (\\gamma_x, \\gamma_y)$"}
{"_id": "3663633", "title": "", "text": "$P(X_{n+1}=x'|X_t=x)=0$"}
{"_id": "3286573", "title": "", "text": "$40/10$"}
{"_id": "673821", "title": "", "text": "$a^{log_a(z)}=z$"}
{"_id": "8075271", "title": "", "text": "$\\sum_n\\|f_n-g_n\\|^2<1$"}
{"_id": "1189259", "title": "", "text": "$\\mathbb{P}[X_n=1|X_0=1]$"}
{"_id": "1731024", "title": "", "text": "$0\\in V\\subseteq\\bar V\\subseteq W\\subseteq \\bar W\\subseteq U$"}
{"_id": "8159067", "title": "", "text": "$x=∑_{i=1}^∞⟨x,e_i⟩e_i$"}
{"_id": "5612100", "title": "", "text": "$x_{1}\\cdot...\\cdot x_{n}$"}
{"_id": "328359", "title": "", "text": "$(k+1)^3=k^3\\cdot\\frac{(k+1)^3}{k^3}=k^3\\left(\\frac{k+1}k\\right)^3\\;.$"}
{"_id": "2052409", "title": "", "text": "$ \\sum_{n\\geq 1}\\arctan\\frac{1}{2n^2}\\leq \\sum_{n\\geq 1}\\frac{1}{2n^2}=\\frac{\\pi^2}{12}<1\\tag{1}$"}
{"_id": "3140354", "title": "", "text": "$f\\text{ is continuous at }c\\iff (\\forall\\epsilon>0,\\exists\\delta>0:|x-c|<\\delta\\implies|f(x)-f(c)|<\\epsilon)\\tag{1}$"}
{"_id": "14404", "title": "", "text": "$\\begin{bmatrix} a & b \\\\ -b & a\\end{bmatrix}$"}
{"_id": "2690680", "title": "", "text": "$6^x + 4^x =9^x$"}
{"_id": "6138303", "title": "", "text": "$\\text{Li}_{s,N}(z) = \\frac{z^{N+1}}{\\Gamma(s)}\\int_0^{+\\infty}\\frac{x^{s-1}e^{-Nx}}{e^x-z}dx$"}
{"_id": "520034", "title": "", "text": "$\\binom{n}{2}-(n-1) = \\binom{n-1}{2}$"}
{"_id": "2290966", "title": "", "text": "$f(x)=\\frac{x^2}{1+x^3}$"}
{"_id": "2268668", "title": "", "text": "$A := \\{ \\omega \\in \\Omega: \\lim_{n \\rightarrow \\infty} X_n(\\omega) = X(\\omega)\\}$"}
{"_id": "6384269", "title": "", "text": "$3|14^{2(k+1)}-1 \\implies 3|14^2*14^{k+1}-1$"}
{"_id": "1736804", "title": "", "text": "$\\int_0^\\infty  {xf(x)\\,{\\rm d}x}$"}
{"_id": "1034826", "title": "", "text": "$\\gamma\\in[\\gamma_1,\\gamma_2]$"}
{"_id": "5582295", "title": "", "text": "$\\frac{(1+x^2)\\cos(x\\sqrt{2})-1)}{x^4(1+x^2)}$"}
{"_id": "6713121", "title": "", "text": "$\\zeta(s)\\int_{0}^\\infty \\frac{x^{s-1}}{e^x}dx=\\int_{0}^\\infty \\frac{x^{s-1}}{e^x-1}dx$"}
{"_id": "3880369", "title": "", "text": "$cov(X) - cov(X,Y) cov(Y,Y)^{-1} cov(Y,X)$"}
{"_id": "3790832", "title": "", "text": "$\\left \\| A \\right \\|_2 = \\sqrt{\\lambda_{max} \\left ( A^H A \\right )}$"}
{"_id": "7843692", "title": "", "text": "$3^x+4^x+5^x-6^x=0$"}
{"_id": "5763760", "title": "", "text": "$|G| = p\\cdot q\\cdot r$"}
{"_id": "5649192", "title": "", "text": "$C_{n} = n^2 - n + 2$"}
{"_id": "6430059", "title": "", "text": "$a^{\\log_a b} + a^{-\\log_a b} > a^{\\log_a b} = b$"}
{"_id": "1611126", "title": "", "text": "$ \\int_a^b \\sqrt{1 + (\\frac{dy}{dx})^2} $"}
{"_id": "8754671", "title": "", "text": "$\\int \\kappa ds = 2\\pi$"}
{"_id": "7446862", "title": "", "text": "$|ab| = 35,$"}
{"_id": "2426822", "title": "", "text": "$a^n = (1 + \\delta)^n = 1 + n\\delta + \\cdots + \\delta^n > 1 + n\\delta.$"}
{"_id": "2496278", "title": "", "text": "$\\sum_{k=1}^n \\cos(\\frac{2 \\pi k}{n}) = 0$"}
{"_id": "6727066", "title": "", "text": "$d(x,M)=d(x,P(x))$"}
{"_id": "3463879", "title": "", "text": "$ \\zeta(s)=\\frac{1}{\\Gamma(s)}\\left(1-\\frac{2}{2^s}\\right)^{-1}\\int_{0}^{+\\infty}\\frac{x^{s-1}}{e^x+1}\\,dx \\tag{2}$"}
{"_id": "6189693", "title": "", "text": "$\\frac{\\partial l}{\\partial \\theta} = -(x-\\theta) = 0 \\implies \\theta = x$"}
{"_id": "1932208", "title": "", "text": "$f(x) = \\frac{4}{x}+2x+10+\\frac{3+x}{4x^2+1}$"}
{"_id": "7394037", "title": "", "text": "$\\forall\\varepsilon\\gt0,\\exists \\delta\\gt 0, \\text{ such that } |x| \\lt \\delta ,|f(x)| \\lt \\varepsilon \\tag{1}$"}
{"_id": "34160", "title": "", "text": "$P(m)$"}
{"_id": "7655523", "title": "", "text": "$\\langle x^3-Q(x), 1 \\rangle =0$"}
{"_id": "7309370", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta_{f,\\epsilon} >0, |x| \\leq \\delta \\implies |f(x)-f(0)| \\leq \\epsilon$"}
{"_id": "3807118", "title": "", "text": "$v(x)=\\int_a^x f(t)dt$"}
{"_id": "2231402", "title": "", "text": "$\\sum_{n} ||a_n - b_n ||^2 < 1$"}
{"_id": "6216240", "title": "", "text": "$-\\sum_{n=1}^{\\infty}\\sum_{k=1}^{\\infty}\\frac{\\cos(\\frac{\\pi k}{3})\\cos(\\frac{\\pi n}{3})}{n^{2}(k+n)}$"}
{"_id": "6194592", "title": "", "text": "$f^{-1}(x) = \\frac{3+2x}{1-x}$"}
{"_id": "3898699", "title": "", "text": "$\\int_{0}^{1} \\frac{(\\sin(x) )}{ x} \\,\\mathrm dx$"}
{"_id": "8997367", "title": "", "text": "$f(a_1) > a_1.$"}
{"_id": "225834", "title": "", "text": "$\\overline{V} \\subseteq U$"}
{"_id": "1084914", "title": "", "text": "$\\sum e_n x_n$"}
{"_id": "1856074", "title": "", "text": "$=\\int_{y=\\frac{1}{2}}^{y=1} 2 \\cdot \\frac{1}{2} \\pi (1-y^2) dy $"}
{"_id": "1480611", "title": "", "text": "$[X,Y]=XY-YX.$"}
{"_id": "6062082", "title": "", "text": "$ \\sum_{n=1}^{\\infty} \\sum_{m\\geq n^{\\mu-1+\\delta}} \\frac{\\sin(\\sin(nm))}{n^2+m^2}. $"}
{"_id": "8003934", "title": "", "text": "$p(e) = p'(e')$"}
{"_id": "9200067", "title": "", "text": "$(\\det{\\alpha}) (\\det{\\gamma}) = (\\det{\\alpha})(\\det{\\beta}) = dD$"}
{"_id": "4784375", "title": "", "text": "$M=J \\oplus T$"}
{"_id": "15863", "title": "", "text": "$f_3$"}
{"_id": "1278017", "title": "", "text": "$\\int \\frac{\\cos(x)}{(1+\\cos(x))^3} \\, dx$"}
{"_id": "3646214", "title": "", "text": "$AB=BA=A^*B^*=B^*A^*=B^*A=A^*B=0\\quad \\text{and}\\quad AB^*x=0$"}
{"_id": "3447384", "title": "", "text": "$\\frac{w^n-w^{n+1}-w^nz^n+w^{n+1}z^n + z^n-z^{n+1}-w^nz^n+z^{n+1}w^n } {1-w-z+zw^n+wz^n - w^n z^n}.$"}
{"_id": "3654711", "title": "", "text": "$F(a+b) \\leq F(a) + F(b)$"}
{"_id": "5798532", "title": "", "text": "$y=\\frac{a+bx}{x+c}$"}
{"_id": "4449271", "title": "", "text": "$d(x,b)\\leqslant d(x,y)+d(y,b)$"}
{"_id": "4290112", "title": "", "text": "$f(a+b)=f(a)+'f(b)$"}
{"_id": "4482056", "title": "", "text": "$\\mathbb A = \\{ \\{(a,-b),(-c,d) \\} | \\forall a,b,c,d \\in \\mathbb{R} \\}$"}
{"_id": "6379488", "title": "", "text": "$\\log(2)=\\lim_{N\\to \\infty}\\sum_{n=1}^N\\frac{1}{n+N}$"}
{"_id": "8198529", "title": "", "text": "$ \\frac{x}{2} = \\tan u$"}
{"_id": "3996692", "title": "", "text": "$U\\subseteq V\\subseteq X\\times X$"}
{"_id": "5119305", "title": "", "text": "$\\lfloor\\alpha\\rfloor+\\lfloor\\beta\\rfloor+\\lfloor\\gamma\\rfloor$"}
{"_id": "2839656", "title": "", "text": "$t(a, b) = (a + b, a - b)$"}
{"_id": "4778654", "title": "", "text": "$ \\begin{align} \\left(\\frac{n!}{n^{n/2}}\\right)^2 &=\\overbrace{\\frac{1(n-0)}{n}\\frac{2(n-1)}{n}}^{\\ge1}\\overbrace{\\frac{3(n-2)}{n}\\cdots\\frac{(n-2)3}{n}}^{\\ge\\left(\\frac32\\right)^{n-4}}\\overbrace{\\frac{(n-1)2}{n}\\frac{(n-0)1}{n}}^{\\ge1}\\\\ &\\ge\\left(\\frac32\\right)^{n-4} \\end{align} $"}
{"_id": "2663929", "title": "", "text": "$1+2+4+8+\\cdots = -1$"}
{"_id": "3078456", "title": "", "text": "$F_{X_n}(x)=P(X_n\\leq x)$"}
{"_id": "3271338", "title": "", "text": "$ \\int_0^\\infty f(t)\\ dt = \\int_0^3 f(t)\\ dt = 3.5 $"}
{"_id": "6576912", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\sum\\limits_{r=0}^{\\lfloor\\frac{n}{2}\\rfloor}\\frac{1}{n}f(\\frac{r}{n})$"}
{"_id": "3621595", "title": "", "text": "$ -x^2 y^2 z^2-2 x y z (x+y+z)^3+(x y+x z+y z)^2 (x+y+z)^2+4 x y z (x y+x z+y z) (x+y+z)-2 (x y+x z+y z)^3= $"}
{"_id": "3829903", "title": "", "text": "$f(1)+2^2 f(2)+3^2 f(3)+..+(n-1)^2 f(n-1)+n^2 f(n)=n^3 f(n)$"}
{"_id": "69029", "title": "", "text": "$\\gamma(st)=\\gamma(s+...+s)=\\gamma(s)+...+\\gamma(s)$"}
{"_id": "4777629", "title": "", "text": "$ M=\\begin{pmatrix}A&B\\\\ -B &A\\end{pmatrix}, $"}
{"_id": "4250199", "title": "", "text": "$\\ds{\\int_0^\\infty f(x)\\, dx}.$"}
{"_id": "1628450", "title": "", "text": "$I_n = \\frac12 \\int_{-\\pi}^\\pi \\frac{\\cos (nx)}{a+\\cos x}\\,dx,$"}
{"_id": "6295407", "title": "", "text": "$\\mathsf P(n)\\implies\\mathsf P(n+1)$"}
{"_id": "1503073", "title": "", "text": "$\\begin{pmatrix} 1 & x &x^2\\\\1&y&y^2\\\\1&z&z^2\\end{pmatrix} \\begin{pmatrix} a_0\\\\a_1 \\\\ a_2\\end{pmatrix} = \\begin{pmatrix} c_0\\\\c_1 \\\\ c_2\\end{pmatrix}$"}
{"_id": "702897", "title": "", "text": "$x = a^{\\log_a(x)}$"}
{"_id": "6189867", "title": "", "text": "$\\int\\frac{\\tan{x}}{(x^2 +1)^2}dx$"}
{"_id": "6142894", "title": "", "text": "$\\int{\\frac{1}{(x^2 + a^2)^n}}dx, n,a>0$"}
{"_id": "1238596", "title": "", "text": "$\\sin(\\theta)=\\frac{\\tan(\\theta)}{\\sqrt{1+\\tan^2(\\theta)}}$"}
{"_id": "3663632", "title": "", "text": "$P(X_{n+1}=x'|X_t=x)\\geq \\epsilon>0$"}
{"_id": "195991", "title": "", "text": "$=2\\pi\\lim_{n\\to\\infty}\\sum_{k=1}^n \\left(\\frac{f(x_k^*)-mx_k^*-b}{\\sqrt{m^2+1}}\\right)\\sqrt{1+\\left[f'(x_k*)\\right]^2}\\left(\\frac{1+mf'(x_k^*)}{\\sqrt{m^2+1}}\\right)\\Delta{x}$"}
{"_id": "7522824", "title": "", "text": "$x_n \\to x \\implies \\dfrac{x_1+x_2+\\ldots +x_n}{n} \\to x$"}
{"_id": "6139595", "title": "", "text": "$dm=(1)|ab|=|ab|$"}
{"_id": "104687", "title": "", "text": "$\\lambda^+$"}
{"_id": "1624565", "title": "", "text": "$S= \\int 2 \\pi y ds$"}
{"_id": "812282", "title": "", "text": "$1+2+3+\\cdots=-\\frac{1}{12}$"}
{"_id": "8347253", "title": "", "text": "$ \\begin{align} \\int_0^1\\frac{\\sin(x)}{x^a}\\,\\mathrm{d}x &\\le\\int_0^1x^{1-\\alpha}\\,\\mathrm{d}x \\end{align} $"}
{"_id": "9178264", "title": "", "text": "$ \\bigl(\\gamma(a_1)\\,\\gamma(a_2)\\,\\dots\\,\\gamma(a_k)\\bigr). $"}
{"_id": "3316222", "title": "", "text": "$\\star x=\\{x,\\top\\}$"}
{"_id": "478094", "title": "", "text": "$f(x) = \\frac{x^3}{1+x^3}$"}
{"_id": "6357471", "title": "", "text": "$B = A_1 \\times A_2 \\times \\cdots \\times A_n \\times \\cdots$"}
{"_id": "8263912", "title": "", "text": "$ \\lnot (\\exists c_0) (\\forall x) (x = c_0) $"}
{"_id": "1820535", "title": "", "text": "$ \\int_0^{\\infty} \\int_0^{\\infty} f(x,y) dydx $"}
{"_id": "1377976", "title": "", "text": "$\\forall \\epsilon>0, \\exists\\delta>0,|x-7|<\\delta\\implies|f(x)-f(7)|<\\epsilon.$"}
{"_id": "344587", "title": "", "text": "$\\lim_{x\\to a} f(x) = L \\iff \\forall \\varepsilon \\gt 0 \\ \\exists \\delta  \\gt 0 \\ni \\left| x-a \\right| \\le \\delta \\Rightarrow  \\left| f(x)-L \\right| \\le \\varepsilon$"}
{"_id": "3680412", "title": "", "text": "$B(x, R-||x||_2)\\cap F = \\{x, e_2, e_3, \\dots, e_n\\}$"}
{"_id": "3128841", "title": "", "text": "$d(x,y)\\leq d(x,x_{n})+d(x_{n},y_{n})+d(y_{n},y).$"}
{"_id": "4176365", "title": "", "text": "$0\\le e^x-\\left(1+\\frac xn\\right)^n\\le\\dfrac{x^2e^x}{2n}$"}
{"_id": "7724889", "title": "", "text": "$\\displaystyle P\\bigg(\\int_0^1e^{as}dB_s\\le\\int_0^2e^{as}dB_s\\bigg)=P\\bigg(\\int_1^2e^{as}dB_s\\ge 0\\bigg)=1/2$"}
{"_id": "8733009", "title": "", "text": "$(p^{2})^{2} = p\\sin(\\phi) \\cos(\\theta)$"}
{"_id": "5343167", "title": "", "text": "$\\lim_{n\\to\\infty} \\sum_{r=1}^n \\frac 1 {2^r}.$"}
{"_id": "371236", "title": "", "text": "$(78,14) =2$"}
{"_id": "2488458", "title": "", "text": "$(x^n - y^n) = (x+y)^{n-1}(x-y)^{n-1}$"}
{"_id": "1297700", "title": "", "text": "$p_1 p_2\\mid a$"}
{"_id": "3104465", "title": "", "text": "$f(x,y) = \\frac{1}{2\\pi\\sqrt{\\sigma^2(1-p^2)}} \\exp\\left[\\frac{-(x^2-2py+y^2)}{2\\sigma^2(1-p^2)}\\right]$"}
{"_id": "7067918", "title": "", "text": "$d(x',a)-d(x,a)\\leqslant d(x',x)$"}
{"_id": "7758300", "title": "", "text": "$I_n = \\int_{-\\pi/2}^{\\pi/2} (\\cos{x})^n$"}
{"_id": "3562483", "title": "", "text": "$6^{x+1} - 6^x = 3^{x+4} - 3^x$"}
{"_id": "6231824", "title": "", "text": "$ g^1 \\gamma -\\gamma = \\partial(H\\gamma) - H(\\partial \\gamma).$"}
{"_id": "230620", "title": "", "text": "$|f(z)| = 2|g(z)|$"}
{"_id": "1499488", "title": "", "text": "$dl=\\sqrt{r^2+\\left(\\frac{dr}{d\\phi}\\right)^2}$"}
{"_id": "4349535", "title": "", "text": "$S=\\{\\{a\\},\\{b\\}\\}$"}
{"_id": "3705472", "title": "", "text": "$tan\\theta=\\cfrac{x}{3}$"}
{"_id": "2436444", "title": "", "text": "$\\|\\psi_i^n-e_i\\|\\le 1/n$"}
{"_id": "5476513", "title": "", "text": "$2J_n^{(0, -1)}(x)= J_n^{(0, 0)}(x)+J_{n-1}^{(0, 0)}(x) $"}
{"_id": "8248640", "title": "", "text": "$d(x,A) \\leq d(x,y) + d(y,p)$"}
{"_id": "506097", "title": "", "text": "$\\operatorname{dVol}(e_1,e_2,e_3,\\ldots,e_n) = 1$"}
{"_id": "844372", "title": "", "text": "$\\text{Spec } \\mathbb{R}[x]/(x^2 + 1)$"}
{"_id": "1763442", "title": "", "text": "$\\{ x^2, xy, y^2, xz, yz, z^2 \\}$"}
{"_id": "4037410", "title": "", "text": "$xRy=yRx$"}
{"_id": "6285434", "title": "", "text": "$(\\frac{1}{2^2})^{10}$"}
{"_id": "2886152", "title": "", "text": "$S_n=\\frac{n(n+3)}{2}$"}
{"_id": "8566602", "title": "", "text": "$\\sum_{k=1}^\\infty \\dfrac{1}{n}\\cos\\left(\\dfrac{k\\pi}{2n}\\right)$"}
{"_id": "3145288", "title": "", "text": "$H(x) = \\int_a^{x}f(t)g(t) dt$"}
{"_id": "8813858", "title": "", "text": "$\\|A\\| = \\sqrt{\\mbox{tr}(A^H A)}$"}
{"_id": "8456949", "title": "", "text": "$(a^{1/n} - \\epsilon')^n < a - \\delta < x < a + \\delta < (a^{1/n} + \\epsilon')^n$"}
{"_id": "7029404", "title": "", "text": "$\\displaystyle{d(a, H)=\\frac{|f(a)|}{\\|f\\|}}$"}
{"_id": "4552544", "title": "", "text": "$Pr(A) = 1/6$"}
{"_id": "4768677", "title": "", "text": "$\\lim_{N\\to\\infty}\\sum_{n=1}^{N}\\frac{a_n}{n^2}$"}
{"_id": "7597763", "title": "", "text": "$ \\begin{align} &A_x=\\{(a,b) \\in\\mathbb{N}^2 : a+b=x,\\,\\,1<a\\le b\\} \\\\ &B_x=\\{(a,b) \\in\\mathbb{N}^2  : ab=x,\\,\\,1<a\\le b\\} \\\\ &C_x=\\{ab : (a,b)\\in A_x\\} \\\\ &D_x=\\{a+b : (a,b)\\in B_x\\} \\\\ &R=\\{p,\\,pq,\\,p^3 \\,:\\, p,q\\,\\text{  are prime}. \\} \\\\ &T=\\{x : C_x \\cap R=\\emptyset\\} \\end{align} $"}
{"_id": "4029236", "title": "", "text": "$ \\zeta(s)=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^s}=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{t^{s-1}}{e^t+1}\\,dt $"}
{"_id": "167935", "title": "", "text": "$U(A)\\subseteq V$"}
{"_id": "2182193", "title": "", "text": "$\\frac{8}{a}+\\frac{2}{b}=1$"}
{"_id": "4649302", "title": "", "text": "$(a,a)_K=\\{\\{a\\}\\}$"}
{"_id": "6963531", "title": "", "text": "$\\|x_n-x_m\\|<c$"}
{"_id": "9121765", "title": "", "text": "$\\boldsymbol{x} = (x,y)$"}
{"_id": "3926481", "title": "", "text": "$log(2^x + 3^x - 4^x + 6^x - 9^x) = log (1296^x) = x log(1296)$"}
{"_id": "1797014", "title": "", "text": "$a^{\\log_ab}=b.$"}
{"_id": "1065474", "title": "", "text": "$(6^k + 3) - (4^{k-1} + 2)$"}
{"_id": "839206", "title": "", "text": "$\\{a,a+d,a+2d,......\\}$"}
{"_id": "1834632", "title": "", "text": "$\\begin{cases} x=1+16t\\\\ y=7t\\\\z=-1-19t \\end{cases}$"}
{"_id": "2522312", "title": "", "text": "$\\delta^{(n)}(-x) = (-1)^n\\delta^{(n)}(x)$"}
{"_id": "4981536", "title": "", "text": "$x^2(x-y)(x-z)+y^2(y-z)(y-x)+z^2(z-x)(z-y)\\ge0$"}
{"_id": "7960766", "title": "", "text": "$ f(0) = 0\\\\ f(\\pi) = 0\\\\ f(2\\pi) = 0\\\\ $"}
{"_id": "1973676", "title": "", "text": "$={9\\over 9+3}$"}
{"_id": "7307529", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{n^7}{e^{n\\pi}-1}-16^2\\sum_{n=1}^{\\infty}\\frac{n^7}{e^{4n\\pi}-1}=\\frac{17}{32}$"}
{"_id": "8022391", "title": "", "text": "$\\pi^{-(1-s)/2}\\Gamma(\\frac{1-s}{2})\\zeta(1-s)=\\pi^{-s/2}\\Gamma(\\frac{s}{2})\\zeta(s)$"}
{"_id": "769540", "title": "", "text": "$\\frac{(1-p)p}{1-(1-p)^2} = \\frac{(1-p)p}{1-1+2p-p^2}=\\frac{1-p}{2-p}$"}
{"_id": "8815270", "title": "", "text": "$\\|A\\|=\\sqrt{\\lambda_\\max(A^\\ast A)}$"}
{"_id": "8352321", "title": "", "text": "$\\|x\\| \\geq \\sup_{y \\neq 0} \\frac{|\\langle x,y \\rangle|}{\\|y\\|}$"}
{"_id": "5028547", "title": "", "text": "$\\sum_{k=1}^{n}\\frac{1}{1+ak} = \\frac{1}{a}\\sum_{k=1}^{n}\\frac{1}{k+\\frac{1}{a}}=\\frac{H_n}{a}-\\frac{1}{a^2}\\sum_{k=1}^{n}\\frac{1}{k\\left(k+\\frac{1}{a}\\right)}=\\frac{1}{a}H_n+O\\left(\\frac{1}{a^2}\\right). $"}
{"_id": "978880", "title": "", "text": "$U^{\\bot\\bot} \\cap V^{\\bot\\bot} \\subseteq(U^{\\bot\\bot}\\cap V)\\subseteq (U\\cap V)^{\\bot\\bot\\bot\\bot}$"}
{"_id": "7571756", "title": "", "text": "$s(x)=\\sum\\limits_{n=0}^k \\alpha_n\\chi_{A_n}(x)$"}
{"_id": "5526552", "title": "", "text": "$(A_1\\times ... \\times A_{n+1})$"}
{"_id": "654418", "title": "", "text": "$F(a+b)=F(a)+F(b)$"}
{"_id": "4576994", "title": "", "text": "$\\gcd(a, p_1 p_2)\\ne 1$"}
{"_id": "1520875", "title": "", "text": "$F_a(x) = \\left\\{ \\begin{array}{lr}   f_a & 0 \\le x \\le \\gamma \\\\ 0 &  x > \\gamma \\\\ \\end{array} \\right.$"}
{"_id": "7417546", "title": "", "text": "$\\frac{\\sum\\limits_{r=1}^{\\infty}r^0}{\\sum\\limits_{r=1}^{\\infty}r}$"}
{"_id": "1276603", "title": "", "text": "$\\log_a B$"}
{"_id": "4001691", "title": "", "text": "$\\overrightarrow{RP}=\\pmatrix{2\\\\-6\\\\-1}$"}
{"_id": "8067858", "title": "", "text": "$\\forall \\delta \\gt 0, \\exists\\epsilon \\gt 0 : |x-a| \\lt \\delta \\implies |f(x)-f(a)|\\lt \\epsilon$"}
{"_id": "5017491", "title": "", "text": "$\\left[\\begin{array}{rrr|r}1&a&0 & 1\\\\0&1&b & 1\\\\c&0&1 & 1\\end{array}\\right]$"}
{"_id": "8223477", "title": "", "text": "$ \\begin{vmatrix}     1  & 1  & 1 \\\\     x  & y  & z \\\\     yz & xz & xy  \\end{vmatrix}  $"}
{"_id": "2582156", "title": "", "text": "$\\sum ^\\infty_1 \\| e_n - f_n \\| < 1$"}
{"_id": "7856512", "title": "", "text": "$\\lfloor\\frac{\\lfloor\\frac{x}{n}\\rfloor}{m}\\rfloor = \\lfloor\\frac{x}{mn}\\rfloor$"}
{"_id": "3010713", "title": "", "text": "$ \\begin{array}{ccc} \\left(\\alpha\\beta\\gamma\\right)^{2}-a_{0} & = & 0\\\\ -2\\alpha\\beta\\gamma\\left(\\alpha\\beta+\\gamma\\beta+\\alpha\\gamma\\right)-a_{1} & = & 0\\\\ (\\alpha\\beta+\\gamma\\beta+\\alpha\\gamma)^{2}+2\\alpha\\beta\\gamma(\\alpha+\\beta+\\gamma)-a_{2} & = & 0\\\\ -2\\alpha\\beta\\gamma-2(\\alpha+\\beta+\\gamma)(\\alpha\\beta+\\gamma\\beta+\\alpha\\gamma)-a_{3} & = & 0\\\\ (\\alpha+\\beta+\\gamma)^{2}+2(\\alpha\\beta+\\gamma\\beta+\\alpha\\gamma)-a_{4} & = & 0\\\\ -2(\\alpha+\\beta+\\gamma)-a_{5} & = & 0 \\end{array} $"}
{"_id": "8157026", "title": "", "text": "$|F(z)|\\leq |f(z)||f(−z)|\\leq 2\\cdot 1.$"}
{"_id": "2215424", "title": "", "text": "$\\frac{1}{2^{n-1-1}} = \\frac{1}{2^{n-2}}$"}
{"_id": "4292720", "title": "", "text": "$3^x + 4^x + 5 ^x = 6^x$"}
{"_id": "7694049", "title": "", "text": "$\\lim_{x\\to c^{+}} ~ f(x) =l$"}
{"_id": "7183813", "title": "", "text": "$\\frac{\\sqrt[n]{x}}{(1+\\frac{x}{n})^n}\\sim \\sqrt[n]{x}e^{-x}$"}
{"_id": "2152817", "title": "", "text": "$\\lfloor \\bar{.9}\\rfloor = \\left\\lfloor \\sum_{i=1}^{\\infty}\\frac{9}{10^{i}}\\right\\rfloor=\\lfloor 1 \\rfloor = 1.$"}
{"_id": "1174421", "title": "", "text": "$\\mathbf P[X_1 < X_0 \\, | \\, X_0 = p] = p.$"}
{"_id": "6969433", "title": "", "text": "$\\leq \\lim_{n\\rightarrow \\infty}\\inf\\int_{E_n} f d\\mu = \\lim_{n\\rightarrow\\infty}\\inf\\int f 1_{E_n} d\\mu$"}
{"_id": "5663411", "title": "", "text": "$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\\;\\forall x,y\\in [0,+\\infty)$"}
{"_id": "9276805", "title": "", "text": "$f^{-1}(x)f(x)=f(x)f^{-1}(x)=1$"}
{"_id": "9300389", "title": "", "text": "$\\gamma^3+\\gamma(\\gamma+1)=\\gamma^3\\gamma^4+\\gamma^5+\\gamma^4=(\\gamma^2+\\gamma)(\\gamma^3+\\gamma^2)=\\varphi(\\alpha)\\cdot\\varphi(\\alpha\\beta)$"}
{"_id": "4815998", "title": "", "text": "$f_Y(x) = \\frac{1}{\\sqrt{2\\pi}} e^\\frac{-x^2}{2}$"}
{"_id": "5454031", "title": "", "text": "$ F(u) = \\int _{-\\infty}^{\\infty}f(x) m\\left(\\frac{u}{x}\\right) \\mathrm d x$"}
{"_id": "1406276", "title": "", "text": "${C_R}: |z| = R$"}
{"_id": "5939453", "title": "", "text": "$a_n = \\frac{1}{2}n(n+1) + 1.$"}
{"_id": "655352", "title": "", "text": "$V \\subseteq \\overline{V} \\subseteq U$"}
{"_id": "1214889", "title": "", "text": "$\\{(x,y):a<x\\le b, c<y\\le d\\}.$"}
{"_id": "5996195", "title": "", "text": "$S_n=\\frac{n(n+1)(2n+1)}{6}$"}
{"_id": "5896104", "title": "", "text": "$(-2b-2a,a,b,a)$"}
{"_id": "7084273", "title": "", "text": "$f(x)=x^2-x+2$"}
{"_id": "4825200", "title": "", "text": "$\\tan(\\theta) = x/2$"}
{"_id": "7175661", "title": "", "text": "$\\sum_{n\\in \\mathbb Z} e^{-n^2\\pi x} = \\frac{1}{\\sqrt{x}}\\sum_{n\\in \\mathbb Z}e^{\\frac{-n^2\\pi}{x}} $"}
{"_id": "8515193", "title": "", "text": "$\\frac{\\pi^2}{6}=\\sum_{n=1}^\\infty\\frac{1}{n^2}=\\sum_{n=1}^\\infty \\frac{1}{(2n-1)^2}+\\sum_{n=1}^\\infty\\frac{1}{(2n)^2}$"}
{"_id": "6751067", "title": "", "text": "$\\phi(n)=|\\mathbb{Z}_n^*|$"}
{"_id": "971822", "title": "", "text": "$\\mathbb{R}[x]/( x^2 + x + 1 )$"}
{"_id": "1828657", "title": "", "text": "$\\frac 1 3 < x < \\frac 1 2 \\implies 2\\pi < \\frac{\\pi}{x} < 3 \\pi \\implies 1 < 1+ \\sin \\frac{\\pi}x <2$"}
{"_id": "6592493", "title": "", "text": "$ \\det A=c^{n(p-1)}(c-p)^{n-1}(c+p(n-1))\\;. $"}
{"_id": "4717549", "title": "", "text": "$k^3-1 = (k-1)(k^2 + k +1) , (a-d)/(b-c) = (k^2 +k+1)/k >= 3$"}
{"_id": "2518410", "title": "", "text": "$\\Bbb{P}[X_t \\in B | X_s] = \\Bbb{P}[X_{t-s} \\in B | X_0 = x]_{x= X_s}$"}
{"_id": "3639440", "title": "", "text": "$K=\\Bbb R[x]/(x^2+1)$"}
{"_id": "4498222", "title": "", "text": "$ f(2n)=f(2)f(n)-f(n+1)=3(n+1)-(n+2)=2n+1 $"}
{"_id": "1632120", "title": "", "text": "$\\displaystyle \\int_0^\\infty f_X(x)\\,dx=1$"}
{"_id": "5327332", "title": "", "text": "$\\phi\\circ\\gamma(a+b) = \\phi(\\gamma(a+b)) = \\phi(\\gamma(a)+\\gamma(b)) = \\phi(\\gamma(a)) + \\phi(\\gamma(a)) = \\phi\\circ\\gamma(a)+\\phi\\circ\\gamma(b)$"}
{"_id": "7623812", "title": "", "text": "$t = \\frac{1+2x}{1-x}$"}
{"_id": "9121197", "title": "", "text": "$\\sum_{n\\geq 1}\\frac{4}{(2n+2)^2-4}=\\sum_{n\\geq 1}\\frac{1}{n(n+2)}$"}
{"_id": "3720714", "title": "", "text": "$\\sigma\\gamma = \\gamma\\tau\\gamma^{-1}\\gamma = \\gamma\\tau\\circ e = \\gamma\\tau$"}
{"_id": "7532612", "title": "", "text": "$f(x^2+y)=xf(x)+f(y)$"}
{"_id": "5777947", "title": "", "text": "$[X,Y]=X*Y-Y*X$"}
{"_id": "4756724", "title": "", "text": "$\\gamma^{4}+\\gamma+1=(\\gamma +b)(c\\gamma^{3}+d\\gamma^{2}+e\\gamma+\\gamma)$"}
{"_id": "6384271", "title": "", "text": "$3|14^2*14^{k+1}-1$"}
{"_id": "4824055", "title": "", "text": "$f(a+bx)=f(a)+f(b)f(x)$"}
{"_id": "433666", "title": "", "text": "$p_1+p_2-p_1p_2$"}
{"_id": "4541230", "title": "", "text": "$\\gamma \\cap \\{a\\} = \\emptyset$"}
{"_id": "5284686", "title": "", "text": "$z=\\sqrt[4]{3}e^{\\frac{\\pi ki}{2}}\\Longleftrightarrow$"}
{"_id": "2552312", "title": "", "text": "$\\lfloor-p^\\gamma\\rfloor-\\lfloor-(p+1)^\\gamma\\rfloor$"}
{"_id": "7816882", "title": "", "text": "$f(x)=\\sqrt{2\\pi x}\\left(\\frac{3}{e}\\right)^x$"}
{"_id": "7025463", "title": "", "text": "$\\left(1+\\frac{1}{2^n}\\right)^{2^n}$"}
{"_id": "4307623", "title": "", "text": "$\\left(1+\\frac{x}{N}\\right)^N \\approx e^x$"}
{"_id": "3072521", "title": "", "text": "$[A \\mid b] = \\begin{bmatrix}-3 & 6 & -1 & 1 & -7\\\\ 1 & -2 & 2 & 3 & -1\\\\ 2 & -4 & 5 & 8 & -4 \\end{bmatrix}$"}
{"_id": "3781099", "title": "", "text": "$ \\mathbb E(|X+\\mathbb E(Y)|)\\leqslant\\mathbb E(|X+Y|). $"}
{"_id": "1016097", "title": "", "text": "$f(x)=\\frac{2}{\\theta^2}x$"}
{"_id": "2296443", "title": "", "text": "$ \\zeta(s)=\\sum_{n\\geq 1}\\frac{1}{n^s}=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^s}\\stackrel{\\mathcal{L}^{-1}}{=}\\frac{1}{\\Gamma(s)}\\left(1-\\frac{2}{2^s}\\right)^{-1}\\int_{0}^{+\\infty}\\frac{t^{s-1}}{e^t+1}\\,dt $"}
{"_id": "6146142", "title": "", "text": "$\\prod\\limits_{k = 1}^{n} \\sin \\dfrac{k\\pi}{n} = 0$"}
{"_id": "4288069", "title": "", "text": "$ \\frac{d}{dx}[f'(x)^2 + f(x)^2] = 2f''(x)f'(x) + 2f'(x)f(x) = 2f'(x)[f(x) + f''(x)] $"}
{"_id": "2116982", "title": "", "text": "$\\displaystyle \\sum_{k=-9}^{9} e^{-ikx} =  e^{9ix} \\sum_{k=0}^{18} e^{-ikx} $"}
{"_id": "9333016", "title": "", "text": "$=-2\\int \\frac{d(1-\\tan x)}{\\left(1-\\tan x\\right)^2}$"}
{"_id": "8288447", "title": "", "text": "$ b = \\min\\{s \\in [a,1] \\mid \\gamma(s) = -\\gamma(a)\\} $"}
{"_id": "3576803", "title": "", "text": "$\\det\\begin{pmatrix}A&B\\\\0&D\\end{pmatrix}=\\det(A)\\det(D)$"}
{"_id": "2957287", "title": "", "text": "$\\tilde{\\nabla}_\\dot{\\gamma}\\dot{\\gamma}=(\\nabla_\\dot{\\gamma}\\dot{\\gamma})^\\top=(\\partial_\\dot{\\gamma}\\dot{\\gamma})^\\top=\\ddot{\\gamma}^\\top,$"}
{"_id": "4238180", "title": "", "text": "$\\mathbb{R}^n \\sqcup \\mathbb{R}^n$"}
{"_id": "1754380", "title": "", "text": "$k!+2, k!+3, \\dots, k!+k$"}
{"_id": "3709395", "title": "", "text": "$K \\subseteq V \\subseteq \\operatorname{Cl}(V) \\subseteq U$"}
{"_id": "8936546", "title": "", "text": "$V = \\lbrace v_1, v_2, v_3,....,v_n\\rbrace$"}
{"_id": "362873", "title": "", "text": "$(3, 7) = 1$"}
{"_id": "7993997", "title": "", "text": "$A = A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq ...$"}
{"_id": "2746552", "title": "", "text": "$\\left(1+\\frac xn\\right)^n<e^x<\\left(1-\\frac xn\\right)^{-n}$"}
{"_id": "1650838", "title": "", "text": "$x_0, x_1, x_1 \\in \\Bbb N$"}
{"_id": "3290163", "title": "", "text": "$x! \\approx \\sqrt{2\\pi x}\\left(\\frac{x}{e}\\right)^x, \\text{ for } x \\to \\infty$"}
{"_id": "492483", "title": "", "text": "$\\begin{bmatrix} a & -\\overline b \\\\ b & \\overline a \\end{bmatrix}$"}
{"_id": "3317205", "title": "", "text": "$ !Z= \\begin{bmatrix} A & -B\\\\B&A \\end{bmatrix} \\ , $"}
{"_id": "1711148", "title": "", "text": "$\\det M(A,B)=\\det M(A,-B)$"}
{"_id": "799126", "title": "", "text": "$ \\underline{Lim} A_n = \\{ x \\in X : \\exists n_0 \\in \\omega \\forall n \\geq n_0, x \\in A_n \\} $"}
{"_id": "4417261", "title": "", "text": "$\\displaystyle i\\sum_{n=-\\infty}^\\infty \\sum_{m=1}^\\infty \\frac 1{n+n^3}e^{inx}$"}
{"_id": "6895569", "title": "", "text": "$\\deg(a) = 4, \\ \\deg(b) =3,\\ \\deg(c) = 3, \\ \\deg(d) = 3,\\ \\deg(e) = 3, \\ \\deg(f) = 3, \\ \\deg(g) = 3, \\ \\deg(h) = 2, \\ \\deg(i) = 2.$"}
{"_id": "33092", "title": "", "text": "$\\forall \\epsilon\\gt0, \\exists \\delta\\gt0$"}
{"_id": "6587266", "title": "", "text": "$f_X(0;\\theta) = \\theta$"}
{"_id": "4401133", "title": "", "text": "$B = \\{ \\{d\\} ,\\{s\\}\\}$"}
{"_id": "3774630", "title": "", "text": "$\\|e_n - e_m\\| = 1$"}
{"_id": "102203", "title": "", "text": "$A\\vec v = \\left[\\matrix{Ry_1+Qy_1'+Py_1''\\cr Ry_2+Qy_2'+Py_2''\\cr Ry_3+Qy_3'+Py_3''\\cr }\\right]=\\left[\\matrix{0\\cr 0\\cr 0\\cr}\\right],$"}
{"_id": "7636237", "title": "", "text": "$(x+y,x-y)=1$"}
{"_id": "668457", "title": "", "text": "$t=1 \\implies \\frac x 2=\\frac{\\pi}4+k\\pi \\implies x=\\frac{\\pi}2+2k\\pi $"}
{"_id": "7754828", "title": "", "text": "$[x, y] = \\{x, y\\}$"}
{"_id": "5956318", "title": "", "text": "$ [F_n, F_n ] \\subseteq F_n $"}
{"_id": "5289689", "title": "", "text": "$\\bigg\\lfloor\\frac{\\lfloor \\frac{x}{m}\\rfloor}{n}\\bigg\\rfloor=\\bigg\\lfloor \\frac{x}{mn}\\bigg\\rfloor$"}
{"_id": "4029877", "title": "", "text": "$y=\\frac{bx}{1+bx}$"}
{"_id": "5521815", "title": "", "text": "$\\frac{x}{2}=\\sin \\theta.$"}
{"_id": "2501357", "title": "", "text": "$R = A_1 \\times \\dotsc \\times A_s$"}
{"_id": "4969849", "title": "", "text": "$g(a_1)>a_1$"}
{"_id": "3442206", "title": "", "text": "$\\displaystyle\\sum_{k = -(n-1)}^{n-1}e^{-k^2\\tfrac{\\pi}{n}} = \\sum_{k = -\\infty}^{\\infty}\\sqrt{n}e^{-n\\pi k^2} - \\displaystyle\\sum_{|k| \\ge n}e^{-k^2\\tfrac{\\pi}{n}}$"}
{"_id": "4332489", "title": "", "text": "$f(x) = \\limsup_n \\frac{x_1+x_2+\\cdots+x_n}{n}$"}
{"_id": "6712063", "title": "", "text": "$E(|X+Y|^r)\\leq c_r[E(|X|^r)+E(|Y|^r)]$"}
{"_id": "152917", "title": "", "text": "$\\lim_{k\\to \\infty}\\mu(A_k)=\\mu(A)$"}
{"_id": "5180081", "title": "", "text": "$f_2(3n,\\ 3k) = \\binom{n}{k}$"}
{"_id": "3168277", "title": "", "text": "$f(x) = \\frac{a+bx}{1+cx}$"}
{"_id": "8674128", "title": "", "text": "$ \\|A^TA\\|_2=\\rho(A^TA)=\\|A\\|_2^2. $"}
{"_id": "6166754", "title": "", "text": "$P(t|F) = p, P(f|T)=q, P(t|T)=r, \\text{ and } P(T) = P(T|f)P(f) + P(T|t)P(t)$"}
{"_id": "6848944", "title": "", "text": "$L\\{f(t)*g(t)\\}=\\int_0^t f(t-\\tau)g(\\tau)d\\tau=\\int_0^t f(\\tau)g(t-\\tau)dt=\\int_0^tf(t)dt$"}
{"_id": "6420528", "title": "", "text": "$d(\\gamma(a),\\gamma(b)) \\le L(\\alpha)$"}
{"_id": "6625482", "title": "", "text": "$\\lim_{n \\to \\infty} \\inf s_n \\le \\lim_{n \\to \\infty} \\inf t_n.$"}
{"_id": "6531237", "title": "", "text": "$ \\lim_{n\\rightarrow\\infty}[(1+i)^n+(1-i)^n]. $"}
{"_id": "5578879", "title": "", "text": "$W^*(N) = W^*(C^*(N,I))$"}
{"_id": "320099", "title": "", "text": "$F(x)=\\int_a^xf(t)\\,dt$"}
{"_id": "995485", "title": "", "text": "$\\lim_{n\\to\\infty} \\mu(A_n)=0$"}
{"_id": "2384339", "title": "", "text": "$J(y)=\\int_{0}^{1}{ \\sqrt{1+\\frac{dy}{dx}  ^{2}}  dx}$"}
{"_id": "3702188", "title": "", "text": "$f(x)+f(x^2)=x$"}
{"_id": "9059172", "title": "", "text": "$ f(f(x)^2y)=x^3f(xy)\\space\\forall x,y\\in\\mathbb{Q^+} $"}
{"_id": "5397783", "title": "", "text": "$[x,y] = [-1,i]$"}
{"_id": "5196205", "title": "", "text": "$P(\\{3\\})=1/6$"}
{"_id": "6461144", "title": "", "text": "$\\text{Spec}(\\mathbb{C}[s]/(s^2 - 1)) = \\mathbb{Z}/2$"}
{"_id": "8226193", "title": "", "text": "$f(xf(y)+y^2)=f((x+y)^2)-xf(x)$"}
{"_id": "3331755", "title": "", "text": "$\\   M=   \\left[ {\\begin{array}{ccccc}    1 & \\alpha & -1 & 2 \\\\    2 & -1 & \\alpha & 5 \\\\    1 & 10 & -6 & 1 \\\\   \\end{array} } \\right]$"}
{"_id": "3833971", "title": "", "text": "$D=\\{\\langle x,y\\rangle\\in E^2:x^2+y^2\\le 1\\}$"}
{"_id": "7067917", "title": "", "text": "$d(x,a)-d(x',a)\\leqslant d(x,x')$"}
{"_id": "2285003", "title": "", "text": "$|X+Y|^2 \\le |X|^2 + |Y|^2$"}
{"_id": "2542549", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\sum_{k=n}^{\\infty}\\frac{\\left(-1\\right)^{k+1}}{k^{2}+x^{2}}$"}
{"_id": "4665979", "title": "", "text": "$\\lim \\limits _{n \\to \\infty} \\sum _{i=0} ^{n} \\frac{1}{2^i}$"}
{"_id": "6932450", "title": "", "text": "$F(x) = \\int_a^x f(y) \\textrm{d}\\mu$"}
{"_id": "423598", "title": "", "text": "$x^3 - x^2 - x - 1$"}
{"_id": "27555", "title": "", "text": "$(\\frac 12)^2$"}
{"_id": "4170292", "title": "", "text": "$z={\\sqrt{2}}e^{5{\\pi}i/12}$"}
{"_id": "13881", "title": "", "text": "$Ax + By = C$"}
{"_id": "9171579", "title": "", "text": "$\\displaystyle \\int |u|^p dμ \\le C^p(\\int |u|^r dμ)^{p/r} $"}
{"_id": "1764920", "title": "", "text": "$\\operatorname{support}(\\widehat{f}) \\subseteq [-\\gamma, \\gamma]$"}
{"_id": "5474214", "title": "", "text": "$\\sum_{n=1}^\\infty \\|e_n - f_n\\| \\leqslant \\frac{1}{8M\\|P\\|},$"}
{"_id": "6499384", "title": "", "text": "$\\int_a^b f(t) \\sin(t) \\mathrm{d}t=\\int_a^b f(t) \\cos(t) \\mathrm{d} t=0. $"}
{"_id": "2729539", "title": "", "text": "$P \\subseteq V$"}
{"_id": "4034405", "title": "", "text": "$\\begin{bmatrix}a & -b\\\\b & a\\end{bmatrix},$"}
{"_id": "258170", "title": "", "text": "$Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)$"}
{"_id": "2474635", "title": "", "text": "$J_n=\\int_{-\\pi}^{\\pi}\\frac{\\cos(nt)}{1+r^2-2r\\cos t }dt$"}
{"_id": "9205932", "title": "", "text": "$P = (C+E)\\dfrac{r(1+r)^N}{(1+r)^N-1}$"}
{"_id": "7281823", "title": "", "text": "$f'(x)=2 (x-1)-\\frac{2}{5} e^{-\\frac{x^2}{10}} \\left(2+e^{-\\frac{x^2}{10}}\\right) x$"}
{"_id": "2435513", "title": "", "text": "$[x,y] = x^2 = (a\\ b\\ c)$"}
{"_id": "3250950", "title": "", "text": "$n!+2, \\dots n!+n$"}
{"_id": "2168", "title": "", "text": "$\\mathrm{P}[X\\geq x]=0$"}
{"_id": "6665608", "title": "", "text": "$ 1 + 2 + 3 + 4 + \\cdots = \\sum_{k=1}^\\infty k = -\\frac{1}{12}. $"}
{"_id": "3288785", "title": "", "text": "$[ 2 (k + 1)^3 + 3 (k + 1)^2 + (k + 1) ] / 6 $"}
{"_id": "534910", "title": "", "text": "$x! \\approx. \\sqrt{2\\pi x} \\left(\\frac{x}{e}\\right)^x$"}
{"_id": "3410386", "title": "", "text": "$\\lim \\limits_{n\\to +\\infty } \\left(1+\\frac{x}{n}\\right)^n=\\text{e}^x$"}
{"_id": "474503", "title": "", "text": "$|f(x_n)| \\leq 2^{-n} |f(x)|$"}
{"_id": "658026", "title": "", "text": "$\\lim_{x\\to 0^+} f'(x)=\\lim_{x\\to 0^-} f'(x)=0$"}
{"_id": "8352319", "title": "", "text": "$\\|x \\| = \\sup_{y \\neq 0} \\frac{|\\langle x,y \\rangle|}{\\|y\\|}$"}
{"_id": "4205241", "title": "", "text": "$x > \\frac{1}{\\pi} \\implies \\frac{1}{x} < \\pi \\implies 0<\\sin \\frac{1}{x} < 1 \\implies \\left\\lfloor \\sin \\frac{1}{x} \\right\\rfloor = 0$"}
{"_id": "4250435", "title": "", "text": "$=\\sum \\limits_{n=1}^\\infty (-1)^{n+1}\\frac{1}{ \\left\\lfloor(n+1)/2\\right\\rfloor}$"}
{"_id": "8583991", "title": "", "text": "$\\frac{\\alpha}{||F||}$"}
{"_id": "3130424", "title": "", "text": "$\\mathrm{dist}(x,\\ker f)=\\frac{|f(x)|}{\\|f\\|}$"}
{"_id": "7529544", "title": "", "text": "$\\Bbb R^{n+m}=\\Bbb R^n$"}
{"_id": "7884816", "title": "", "text": "$\\Bbb E[|XY|] \\leq \\sqrt{\\Bbb E[|X|^2] \\Bbb E[|Y|^2]}$"}
{"_id": "3476198", "title": "", "text": "$[x,1] = x$"}
{"_id": "7258434", "title": "", "text": "$ (x, y) + [(x', y') + (x'', y'')] = (x, y) + (x' + x'', y' + y'') $"}
{"_id": "4607249", "title": "", "text": "$\\log_a b+\\log_a c =\\log_a (bc)$"}
{"_id": "3626784", "title": "", "text": "$(X^4-X^2+1)$"}
{"_id": "1747932", "title": "", "text": "$\\begin{align*}\n \\lim_{n\\to\\infty}\\frac{\\ln\\left(\\frac{1}{2} + \\frac{1}{\\pi}\\arctan\\frac{nx}{t}\\right)}{\\frac{1}{n}} &\\stackrel{\\mathrm{L'H}}{=} \\lim_{n\\to\\infty}\\left(\\frac{1}{\\frac{1}{2} + \\frac{1}{\\pi}\\arctan\\frac{nx}{t}}\\right)\\left(\\frac{x}{t\\pi}\\right)\\left(\\frac{-t^2n^2}{t^2+n^2x^2}\\right)\\\\\n &=\\frac{1}{1}\\times\\frac{x}{t\\pi}\\times\\frac{-t^2}{x^2}\\\\\n &= -\\frac{t}{x\\pi}.\n \\end{align*}$"}
{"_id": "267923", "title": "", "text": "$x_i ==> \\infty$"}
{"_id": "3371218", "title": "", "text": "$\\mbox{denote }a=\\frac{x_1+x_2+\\ldots+x_n}{n}$"}
{"_id": "1207893", "title": "", "text": "$f(a \\cdot b)=f(a)+f(b)$"}
{"_id": "5096146", "title": "", "text": "$\\sum_{k=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{n^2k^2(n+k)^2}$"}
{"_id": "3297759", "title": "", "text": "$a, a+d, a+2d,...,$"}
{"_id": "8226928", "title": "", "text": "$\\{a+b,a-b\\}$"}
{"_id": "6824615", "title": "", "text": "$ f(x+\\delta)=(x+\\delta,A(x+\\delta))=(x,Ax)+(\\delta,Ax)+(x,A\\delta)+(\\delta,A\\delta). $"}
{"_id": "4318415", "title": "", "text": "$\\displaystyle \\sum_{x=1}^n \\frac{1}{x^2} = \\sum_{x=1}^\\infty \\frac{1}{x^2} + O\\left(\\frac{1}{n}\\right) = \\frac16 \\pi^2 + O\\left(\\frac{1}{n}\\right)$"}
{"_id": "8940904", "title": "", "text": "$\\sin(x)\\int_x^{x+2\\pi}f(t)\\cos(t)dt-\\cos(x)\\int_x^{x+2\\pi}f(t)\\sin(t)dt=0$"}
{"_id": "6737831", "title": "", "text": "$\\bigoplus\\limits_P E_n:=\\{(x_n):x_n\\in E_n,\\sum\\limits_n\\lVert x_n\\rVert^pw_n<\\infty\\}$"}
{"_id": "9229503", "title": "", "text": "$f(b)<k<f(a)$"}
{"_id": "3277230", "title": "", "text": "$ \\frac { \\left( 1- \\left( \\tan \\left( \\frac x 2 \\right)  \\right)^2 \\right)^2}{ \\left( 1+ \\left( \\tan \\left( \\frac x 2 \\right)  \\right)^2 \\right)^2 (1+\\tan(x)) } $"}
{"_id": "3484115", "title": "", "text": "$\\int f(x) = \\frac14 \\left [-4e^\\frac{-x}{2}(-\\frac{x}{2} - 1) + C \\right] $"}
{"_id": "117461", "title": "", "text": "$\\;1+2+3+\\ldots=-\\frac1{12}\\;$"}
{"_id": "6760194", "title": "", "text": "$I_n=(-(2\\pi n + \\pi), 2 \\pi n).$"}
{"_id": "5500767", "title": "", "text": "$\\tan\\theta = \\displaystyle\\frac{x}{2}$"}
{"_id": "2050166", "title": "", "text": "$\\Gamma(s) \\zeta(s) = \\lim_{N \\to \\infty} \\Gamma(s) \\sum_{n=1}^N n^{-s} = \\lim_{N \\to \\infty} \\int_0^\\infty x^{s-1} \\frac{1-e^{-Nx}}{e^x-1}dx = \\int_0^\\infty \\frac{x^{s-1}}{e^{x}-1} dx$"}
{"_id": "1374655", "title": "", "text": "$\\gcd(r,s) = 1, z = rs$"}
{"_id": "7786471", "title": "", "text": "$P[X=x|X+Y=k]$"}
{"_id": "7681513", "title": "", "text": "$xRy, yRz \\Rightarrow xRz$"}
{"_id": "3577591", "title": "", "text": "$\\int_0^\\infty f_n(x)\\,dx = 1$"}
{"_id": "6883002", "title": "", "text": "$f[x,y]=xy$"}
{"_id": "698153", "title": "", "text": "$(k+1)^3 - (k-1)^3 = 6 k^2 + 2$"}
{"_id": "1439736", "title": "", "text": "$z=e^{i\\vartheta}$"}
{"_id": "4399168", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\sum_{k=n}^\\infty 1$"}
{"_id": "4149552", "title": "", "text": "$a^n - a^{n-1} > \\delta$"}
{"_id": "8312539", "title": "", "text": "$x=y=z=\\frac{1}{\\lambda+2}$"}
{"_id": "6858713", "title": "", "text": "$P(X_n\\leq x) = x$"}
{"_id": "2812861", "title": "", "text": "$[x,y]=h$"}
{"_id": "6985281", "title": "", "text": "$Cov(X,Y)=Cov(X,g(Y)),$"}
{"_id": "3428405", "title": "", "text": "$A=\\{(x,y)\\in \\mathbb{R}^2:a\\leq x\\leq b, f(x)\\leq y\\leq g(x)\\}$"}
{"_id": "6861799", "title": "", "text": "$C_r=r-1$"}
{"_id": "1001882", "title": "", "text": "$\\int_0^\\infty f^2(x)dx$"}
{"_id": "7252739", "title": "", "text": "$f^{-1}(f(x) + f(e)) = x$"}
{"_id": "9017207", "title": "", "text": "$\\vartheta(u+\\frac 1\\tau,-i\\tau)=e^{\\pi/q+2\\pi\\tau}\\vartheta(u,-i\\tau)$"}
{"_id": "3998270", "title": "", "text": "$\\begin{align} \\int\\frac1{(1+\\sqrt x)^4}\\,dx&=2\\int \\frac{u}{(1+u)^4}\\,du\\\\\\\\ &=2\\int \\frac{t-1}{t^4}\\,dt\\\\\\\\ &=\\frac{2}{3t^3}-\\frac{1}{t^2}+C\\\\\\\\ &=\\frac{2}{3(1+u)^3}-\\frac{1}{(1+u)^2}+C\\\\\\\\ &=\\frac{2}{3(1+\\sqrt x)^3}-\\frac{1}{(1+\\sqrt x)^2}+C \\end{align}$"}
{"_id": "6441953", "title": "", "text": "$p\\mid n_1n_2$"}
{"_id": "4151606", "title": "", "text": "$\\left\\lfloor\\frac{a}{b}\\right\\rfloor \\geq \\left\\lfloor\\frac{a}{b+1}\\right\\rfloor$"}
{"_id": "6930994", "title": "", "text": "$E[(X+Y)^p] \\le 2^{p-1} (E[X^p] + E[Y^p])$"}
{"_id": "218310", "title": "", "text": "$\\frac{n}{{2^{n - 1} }}$"}
{"_id": "6033825", "title": "", "text": "$(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$"}
{"_id": "1531864", "title": "", "text": "$ |E_A|\\leq \\gamma_2|A|, \\quad |E_B|\\leq \\gamma_2|B|. $"}
{"_id": "6549584", "title": "", "text": "$\\lVert A \\rVert_F=\\sqrt{\\operatorname{tr}{(A^TA)}}$"}
{"_id": "2602007", "title": "", "text": "$d(x,y)\\le d(x,a) + d(a,y)$"}
{"_id": "2483640", "title": "", "text": "$\\lim\\limits_{n \\to ∞} \\frac{{\\sqrt 1}+\\cdots +{\\sqrt n}}{n{\\sqrt n}}$"}
{"_id": "2197948", "title": "", "text": "$f(x)=x^2+x+2$"}
{"_id": "1018884", "title": "", "text": "$\\zeta\\left(s\\right)=\\frac{1}{\\Gamma\\left(s\\right)}\\int_{0}^{\\infty}\\frac{x^{s-1}}{e^{x}-1}dx,\\, s>1.$"}
{"_id": "1132814", "title": "", "text": "$1/|x|^\\lambda$"}
{"_id": "7476262", "title": "", "text": "$\\int^{x}_{0}\\frac{\\sin(x)}{x}$"}
{"_id": "8281322", "title": "", "text": "$\\int_0^{\\infty} |f(x)|$"}
{"_id": "6603815", "title": "", "text": "$u=\\left(ac^{\\frac{1-\\gamma}{\\theta}}+b d^{\\frac{1-\\gamma}{\\theta}}\\right)^{\\frac{\\theta}{1-\\gamma}}\\Rightarrow u^{\\frac{1-\\gamma}{\\theta}}=ac^{\\frac{1-\\gamma}{\\theta}}+b d^{\\frac{1-\\gamma}{\\theta}}$"}
{"_id": "7104320", "title": "", "text": "$L^{AB}_1 \\subseteq L^{AB}_2\\subseteq \\cdots \\subseteq L^{AB}_{k-1}\\subseteq L^{AB}_{k}\\subseteq \\cdots\\subseteq L^{AB}_{n}$"}
{"_id": "8799281", "title": "", "text": "$C_R\\cap \\mathbb{R}$"}
{"_id": "1164876", "title": "", "text": "$M=N\\oplus K$"}
{"_id": "463355", "title": "", "text": "$\\lim_{n\\to\\infty} \\mu(A(n)) = \\mu(A(\\infty))$"}
{"_id": "5942173", "title": "", "text": "$ r^n e^{rx} = \\pm k^2 e^{rx} \\quad \\Longrightarrow \\quad r^n = \\pm k^2 \\quad \\Longrightarrow \\quad r = \\sqrt[n]{\\pm k^2} e^{\\frac{2\\pi i j}n}, \\quad 0 \\le j \\le n-1. $"}
{"_id": "4107910", "title": "", "text": "$\\lim_{n\\to \\infty} \\sum_{r=1}^n {\\frac{r^4}{4r^2-1}}$"}
{"_id": "6802372", "title": "", "text": "$ax_0+by_0=d$"}
{"_id": "7955152", "title": "", "text": "$\\int_0^\\infty x dF(x),$"}
{"_id": "3691801", "title": "", "text": "$(e_i)_{1\\le i\\le n}$"}
{"_id": "5519433", "title": "", "text": "$f(t)=\\sum_{j=1}^{n}\\alpha_{j}\\chi_{S_{j}}(t)$"}
{"_id": "8430513", "title": "", "text": "$\\varphi(x) = \\sum_{n = 1}^{N} c_{n} \\chi_{E_{n}}(x)$"}
{"_id": "8904538", "title": "", "text": "$ D = \\{(x, y) | a \\leq x \\leq b, \\ c \\leq y \\leq d\\}.$"}
{"_id": "2760574", "title": "", "text": "$2^x+3^x+4^x=5^x+6^x$"}
{"_id": "3027161", "title": "", "text": "$\\ker A:=\\{\\vec{x} \\in \\mathbb R^n \\mid A\\vec{x}=0\\}$"}
{"_id": "318876", "title": "", "text": "$i < \\alpha^{+}$"}
{"_id": "55302", "title": "", "text": "$P_{m - 1} \\implies P_m$"}
{"_id": "3847907", "title": "", "text": "$\\int_0^\\infty f(x) x \\ dx$"}
{"_id": "5425157", "title": "", "text": "$A_1\\supset A_2\\supset\\cdots A_n\\supset A_{n+1}\\supset\\cdots$"}
{"_id": "6934201", "title": "", "text": "$m_1m_2\\mid n$"}
{"_id": "1587414", "title": "", "text": "$\\int\\limits_{0}^{1200} \\int\\limits_0^{1200-x}(\\frac{1}{1200})^2 e^{\\frac{-x}{1200}} e^{\\frac{-y}{1200}} dydx = \\int\\limits_{0}^{1200}(\\frac{1}{1200})e^{\\frac{-x}{1200}} \\int\\limits_0^{1200-x}(\\frac{1}{1200}) e^{\\frac{-y}{1200}} dydx = \\int\\limits_{0}^{1200}(\\frac{1}{1200})e^{\\frac{-x}{1200}} [-e^{\\frac{-y}{1200}}]^{1200-x}_0 dydx = \\int\\limits_{0}^{1200}(\\frac{1}{1200})e^{\\frac{-x}{1200}}[1-e^{-(1-\\frac{x}{1200})}]dx = \\int\\limits_{0}^{1200}(\\frac{1}{1200})e^{\\frac{-x}{1200}}-\\frac{1}{1200}e^{-1}dx = [1-e^{-1}]-e^{-1} =1-2e^{-1} = Poisson(N\\geq2, \\lambda = 1)$"}
{"_id": "7777398", "title": "", "text": "$y+\\frac{1}{z}=10$"}
{"_id": "5915685", "title": "", "text": "$Cov(Y, X) = Cov(Y', X)$"}
{"_id": "4678411", "title": "", "text": "$T_x\\subset[0,1]$"}
{"_id": "3464243", "title": "", "text": "$f(x)=4^x+6^{x^2}-5^x-5^{x^2}$"}
{"_id": "672258", "title": "", "text": "$\\lim_{k \\rightarrow \\alpha^+} e^{ik}$"}
{"_id": "7437245", "title": "", "text": "$ \\lim_{n\\to\\infty}   \\frac{\\dfrac{\\sqrt{n}+1}{n\\sqrt{n}+\\sqrt{n}+1}}{\\dfrac{1}{n}}= \\lim_{n\\to\\infty}\\frac{n\\sqrt{n}+n}{n\\sqrt{n}+\\sqrt{n}+1}= \\lim_{n\\to\\infty}   \\frac{n\\sqrt{n}}{n\\sqrt{n}}   \\frac{1+\\dfrac{1}{\\sqrt{n}}+\\dfrac{1}{n\\sqrt{n}}}        {1+\\dfrac{1}{n}+\\dfrac{1}{n\\sqrt{n}}}=1 $"}
{"_id": "621712", "title": "", "text": "$ \\begin{align}  &\\coth(x \\pi)=\\frac{x}{\\pi}\\sum_{n=-\\infty}^\\infty\\frac{1}{x^2+n^2}   & \\cot(x \\pi)= \\frac{x}{\\pi}\\sum_{n=-\\infty}^\\infty\\frac{1}{x^2-n^2}         \\\\  & \\text{csch}(x \\pi )=\\frac{x}{\\pi} \\sum_{n=-\\infty}^\\infty{\\frac{(-1)^n }{x^2+n^2}}   &\\csc(x \\pi)= \\frac{x}{\\pi} \\sum_{n=-\\infty}^\\infty{\\frac{(-1)^n }{x^2-n^2}} \\\\ & \\tanh(x \\pi)=\\frac{4x}{\\pi}\\sum_{n=-\\infty}^\\infty{\\frac{1}{(2n+1)^2+4x^2}} &\\tan(x \\pi) = \\frac{4x}{\\pi}\\sum_{n=-\\infty}^\\infty{\\frac{1}{(2n+1)^2-4x^2}} \\end{align} $"}
{"_id": "5301832", "title": "", "text": "$\\hat{\\beta}=\\frac{X_1+X_2+\\ldots+X_n}{n}$"}
{"_id": "3581341", "title": "", "text": "$\\ ax + by = j$"}
{"_id": "1500098", "title": "", "text": "$f(x)=\\frac{e^x}{1+x}$"}
{"_id": "3441181", "title": "", "text": "$df_\\lambda(x,y,z)=\\left[\\matrix{2x &2y &-2z\\cr 1&1&\\lambda\\cr}\\right]\\ .$"}
{"_id": "7210619", "title": "", "text": "$\\beta^2(s)=||\\gamma(s)-\\gamma(0)||^2=\\langle\\gamma(s)-\\gamma(0),\\gamma(s)-\\gamma(0)\\rangle\\\\=\\langle\\gamma(s),\\gamma(s)\\rangle-2\\langle\\gamma(s),\\gamma(0)\\rangle+\\langle\\gamma(0),\\gamma(0)\\rangle$"}
{"_id": "7506995", "title": "", "text": "$f(x)= \\frac{1}{1000}e^{\\frac{-x}{1000}}$"}
{"_id": "6610259", "title": "", "text": "$f(x) = x^2 \\frac{1-\\frac{1}{(1+x^2)^{n+1}}}{1-\\frac{1}{1+x^2}}=x^2\\frac{(1+x^2)^{n+1}-1}{x^2(1+x^2)^{n}} = \\frac{(1+x^2)^{n+1}-1}{(1+x^2)^{n}}$"}
{"_id": "3079106", "title": "", "text": "$\\displaystyle \\int_0^\\infty f(t)g(t)dt\\le 0$"}
{"_id": "5375925", "title": "", "text": "$P_1P_2E$"}
{"_id": "7320087", "title": "", "text": "$g : \\overline{\\mathbb R}\\to \\overline{\\mathbb R}$"}
{"_id": "1898417", "title": "", "text": "$\\sum_{n=1}^\\infty a_n\\leq\\sum_{n=1}^\\infty\\frac{1}{n^2}+\\sum_{n=1}^\\infty \\frac{1}{n^{4/3}}=\\frac{\\pi^2}{6}+\\zeta(4/3)<\\infty.$"}
{"_id": "5714632", "title": "", "text": "$\\color{green}{\\left\\{ \\begin{array}{rcl} s = x + 1 \\\\ t = y + 3 \\end{array}\\right.} \\iff \\left\\{ \\begin{array}{rcl} x = s - 1 \\\\ y = t - 3 \\end{array}\\right. $"}
{"_id": "714097", "title": "", "text": "$P(\\bar{A} \\cap B) = 1/6$"}
{"_id": "7295450", "title": "", "text": "$ds=\\sqrt{(dx)^2+(dy)^2}=\\sqrt{\\left(-\\frac12\\sin t\\right)^2+\\left(\\frac12\\cos t\\right)^2}\\,dt=\\frac12dt$"}
{"_id": "8757988", "title": "", "text": "$\\sum_{n=1}^{\\infty} n-\\sum_{n=1}^{\\infty}n=\\sum_{n=1}^{\\infty} (n-n)=\\sum_{n=1}^{\\infty} 0=0.$"}
{"_id": "6738836", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac{1}{n^2(n+1)} = \\sum_{n=1}^\\infty \\frac{1}{n^2} - \\sum_{n=1}^\\infty \\frac{1}{n(n+1)} = \\frac{\\pi^2}{6} - 1$"}
{"_id": "8801446", "title": "", "text": "$\\mathbb{R}^{n+1} = \\mathbb{R}^{m+1} \\times \\mathbb{R}^{n-m}$"}
{"_id": "2407773", "title": "", "text": "$\\int_{-\\infty}^{\\infty}\\frac{\\sin^2{x}}{x^2}dx$"}
{"_id": "25969", "title": "", "text": "$R^+$"}
{"_id": "2232156", "title": "", "text": "$|f(z^2)| ≤ 2|f(z)|$"}
{"_id": "727880", "title": "", "text": "$\\frac{n+1}{2}C_n, \\frac{n+3}{2}C_n, \\ldots, (n-2)C_n, (n-1)C_n$"}
{"_id": "4583236", "title": "", "text": "$ \\frac {4}{n}+\\frac {10}{n}=$"}
{"_id": "8446487", "title": "", "text": "$ |f(re^{i\\gamma})|=r^ae^{-r\\sin\\gamma}\\le \\Bigl(\\frac{a}{\\sin\\gamma}\\Bigr)^ae^{-a},\\quad r>0. $"}
{"_id": "8280260", "title": "", "text": "$\\frac{-B_1}{1+r}+\\frac{B_1}{(1+r)^2}$"}
{"_id": "6522679", "title": "", "text": "$\\frac{1}{n^2}(n+2)\\cdot 2^{n-1}$"}
{"_id": "439049", "title": "", "text": "$x^2+3x-10+\\frac{3}{x}+\\frac{1}{x^2}=0.$"}
{"_id": "1365989", "title": "", "text": "$\\bar \\partial \\pi :T^{0,1}M\\rightarrow T^{0,1}N$"}
{"_id": "8001509", "title": "", "text": "$[x,y]=[c,c^2]$"}
{"_id": "4744271", "title": "", "text": "$A\\simeq A_1\\times\\cdots\\times A_n$"}
{"_id": "9245608", "title": "", "text": "$E[X] = \\int P[X \\geq x] dx$"}
{"_id": "4173744", "title": "", "text": "$\\forall \\epsilon>0,\\;\\exists \\delta>0,\\;|x-a|<\\delta\\implies |f(x)-g(x)| <\\epsilon|g(x)| $"}
{"_id": "1359824", "title": "", "text": "$\\int_{0}^{1} \\frac{\\sin(x)}{x^{1.5}} \\ dx $"}
{"_id": "2645518", "title": "", "text": "$(k+1)!+2,\\ (k+1)!+3,\\ ...\\ ,\\ (k+1)!+(k+1)$"}
{"_id": "89224", "title": "", "text": "$M^+$"}
{"_id": "3076531", "title": "", "text": "$I_n=|\\int_{(n-1)\\pi}^{n\\pi}f(x)\\;dx|.$"}
{"_id": "183037", "title": "", "text": "$5|n^5-n$"}
{"_id": "979912", "title": "", "text": "$ \\lim_{n\\to\\infty} \\left(1 + \\frac{x}{n}\\right)^{n} = e^{x}, $"}
{"_id": "7982495", "title": "", "text": "$n! = \\sqrt{2 \\cdot \\pi \\cdot n} \\cdot (\\frac{n}{e})^n  $"}
{"_id": "6721349", "title": "", "text": "$\\frac{3}{9^{3/2}} = \\frac{3}{27} = \\frac{1}{9}$"}
{"_id": "4776144", "title": "", "text": "$T=\\frac{r(nr^{n+1}-(n+1)r^n+1)}{(1-r)^2}$"}
{"_id": "1593438", "title": "", "text": "$(f(z))^2=f(z)f(z)=f(zz)=f(z^2)=z$"}
{"_id": "4637914", "title": "", "text": "$\\int_0^1\\int_0^1 \\frac{\\sin^{-1}(xy)}{xy} dx dy$"}
{"_id": "4480756", "title": "", "text": "$A_1 \\times A_2 \\times ...$"}
{"_id": "2409287", "title": "", "text": "$\\begin{align*} \\int_0^\\infty \\dfrac{\\sin^3x}xdx &=\\dfrac14 \\int_0^\\infty\\dfrac{3\\sin x}xdx-\\int_0^\\infty\\dfrac{\\sin3x}xdx\\\\&=\\dfrac{3\\pi}8-\\int_0^\\infty \\dfrac{\\sin y}ydy\\\\&=\\dfrac\\pi4\\end{align*}$"}
{"_id": "5549200", "title": "", "text": "$f(x) = \\frac{2^{\\frac{1}{2}-n} n e^{-\\frac{x^2}{2}} \\left(1+\\text{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)\\right)^{n-1}}{\\sqrt{\\pi }}$"}
{"_id": "2910089", "title": "", "text": "$R=\\{(t,y)\\in \\mathbb{R}^{2}: \\alpha<t<\\beta,\\, \\delta <y<\\gamma\\},$"}
{"_id": "1987847", "title": "", "text": "$p_1p_2\\cdots p_k\\mid x$"}
{"_id": "4793704", "title": "", "text": "$\\int_a^xf(t)dt-[F(x)-F(a)]=0$"}
{"_id": "622519", "title": "", "text": "$ax+by=k$"}
{"_id": "6676768", "title": "", "text": "$\\lfloor a\\rfloor+\\lfloor b\\rfloor=\\lfloor c\\rfloor$"}
{"_id": "4998496", "title": "", "text": "$10+15+40+60=125$"}
{"_id": "8643912", "title": "", "text": "$I = \\int\\limits_0^\\pi \\frac{(\\pi-x)\\sin x}{1+\\cos^2 x}dx\\tag 2$"}
{"_id": "1824582", "title": "", "text": "$p(\\lambda)=(\\lambda+1)(\\lambda-1)^{n-1}$"}
{"_id": "1924367", "title": "", "text": "$S_n=\\frac{n(n+1)}2$"}
{"_id": "87547", "title": "", "text": "$p_1p_2p_3$"}
{"_id": "8729128", "title": "", "text": "$f(x)=\\sum_{m\\in M}\\langle x,m\\rangle f(m)$"}
{"_id": "1520307", "title": "", "text": "$P(n)=n^2-n$"}
{"_id": "8632397", "title": "", "text": "$ds=\\sqrt{1^2+\\left(\\frac{dy}{dx}\\right)^2}dx$"}
{"_id": "6112165", "title": "", "text": "$\\lim_{n \\to \\infty} \\sum_{k=1}^n\\frac{1}{k\\cdot 2^k}$"}
{"_id": "607657", "title": "", "text": "$ \\frac{1}{2^1} + \\frac{2}{2^2} + ... +\\frac{n-1}{2^{n-1}} + \\frac{n}{2^n} <2 $"}
{"_id": "3806174", "title": "", "text": "$a_n=n(n+1)/3$"}
{"_id": "310035", "title": "", "text": "$|ab|=|a||b|,$"}
{"_id": "1661341", "title": "", "text": "$\\int\\frac{1}{(x^3+1)^2}\\mathrm dx \\cdots$"}
{"_id": "4149223", "title": "", "text": "$g = \\sum_{j=1}^{N} c_j \\chi_{Q_j}$"}
{"_id": "8879809", "title": "", "text": "$\\det \\left(\\begin{array}{cc} 0 & A\\\\ -B & I \\end{array}\\right) = \\det(AB) $"}
{"_id": "7998190", "title": "", "text": "$\\int_0^T u(t) \\phi'(t) dt = \\int_0^T u'(t) \\phi(t) dt$"}
{"_id": "6156262", "title": "", "text": "$\\bigwedge^{m}(T^{1,0})\\otimes \\bigwedge^{l}(T^{0,1})$"}
{"_id": "5857387", "title": "", "text": "$\\sum\\limits_{n=1}^\\infty\\left|\\frac{1}{n^{1/p}}\\right|^r=\\sum\\limits_{n=1}^\\infty\\left|\\frac{1}{n^{r/p}}\\right|,$"}
{"_id": "3757020", "title": "", "text": "$Cov(X,Y') = Cov(X,Y) + Cov(X,Z) = Cov(X,Y)$"}
{"_id": "7610327", "title": "", "text": "$\\sum_{j=1}^{\\infty}\\sum_{i=1}^{\\infty} \\cfrac{(-1)^{(i+j)}}{i^2+2nij+j^2} $"}
{"_id": "5403190", "title": "", "text": "$\\frac {12}{y} =1$"}
{"_id": "358959", "title": "", "text": "$R \\, : \\, \\mathbb{R}^{n+1} \\, \\rightarrow \\, \\mathbb{R}^{n+1}$"}
{"_id": "4378699", "title": "", "text": "$J_n=\\int_0^{\\pi}\\left(\\frac{\\sin nx}{\\sin x}\\right)^2dx$"}
{"_id": "4035958", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} \\sum_{i=1}^{\\infty} h_n(i) = \\sum_{i=1}^{\\infty} h(i) $"}
{"_id": "3738549", "title": "", "text": "$\\kappa = \\langle dn\\gamma', d\\phi \\gamma'\\rangle = - \\mathrm {I\\!I}(\\gamma',\\gamma')$"}
{"_id": "4203591", "title": "", "text": "$T_0[0,1],T_1[0,1]$"}
{"_id": "5568283", "title": "", "text": "$\\sum\\limits_{k=1}^{n}k \\cos{\\frac{2k\\pi}{n}} =-\\frac {n}2$"}
{"_id": "2353846", "title": "", "text": "$a_n=\\int_0^1 \\frac{f(x)}{(1+x^n)(1+x^{n+k})}dx$"}
{"_id": "3439381", "title": "", "text": "$1/|x|^p$"}
{"_id": "1442455", "title": "", "text": "$f(x)=\\sum_{j=1}^{n}\\chi_{E_{j}}(x)y_{j}$"}
{"_id": "3946604", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\frac{3+\\sqrt{3}+\\sqrt[3]{3}+\\dots+\\sqrt[n]{3}-n}{\\ln n}$"}
{"_id": "1528630", "title": "", "text": "$\\implies \\lfloor\\frac{a}{b}\\rfloor \\lfloor\\frac{c}{d}\\rfloor + \\lfloor\\frac{a}{b}\\rfloor y + \\lfloor\\frac{c}{d}\\rfloor x + xy$"}
{"_id": "4684105", "title": "", "text": "$C_1\\subseteq V\\subseteq\\overline{V}\\subseteq U$"}
{"_id": "3720968", "title": "", "text": "$\\sqrt{1!\\sqrt{2!\\sqrt{3!\\sqrt{\\cdots\\sqrt{n!}}}}} <3$"}
{"_id": "1417580", "title": "", "text": "$ \\bbx{\\zeta\\pars{1 \\over 2}  = \\lim_{N \\to \\infty}\\pars{\\sum_{k = 1}^{N}{1 \\over \\root{k}} - 2\\root{N}}} $"}
{"_id": "287865", "title": "", "text": "$C_R' \\subseteq C_R$"}
{"_id": "7352183", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac{1}{n(n!)} = \\int_0^1\\frac{e^x-1}{x}dx$"}
{"_id": "5256911", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       k&1&1&-2\\\\       k&k^2&k&k\\\\       k&k&k^2&k     \\end{array} \\right] $"}
{"_id": "71947", "title": "", "text": "$\\mathbb{Q}^{+}$"}
{"_id": "1407518", "title": "", "text": "$\\displaystyle \\omega^2 = e^{\\frac{4\\pi i}{3}}$"}
{"_id": "3534853", "title": "", "text": "$x!\\approx \\sqrt{2\\pi x}~x^xe^{-x}$"}
{"_id": "9286783", "title": "", "text": "$ax+by=d\\mid n$"}
{"_id": "6263260", "title": "", "text": "$a_j, i_j$"}
{"_id": "6411655", "title": "", "text": "$v \\in T_p \\mathbb{R}^{n+1} = \\mathbb{R}^{n+1}$"}
{"_id": "5201940", "title": "", "text": "$S_n = \\frac{X_1 + X_2 + \\dots + X_n }{n}$"}
{"_id": "8056154", "title": "", "text": "$\\mathbb{var}[\\overline{x}]= \\mathbb{E}[(\\overline{x}-\\mathbb{E}[\\overline{x}])^2]= \\mathbb{E}[\\overline{x}^2]-\\mathbb{E}[\\overline{x}]^2= \\mathbb{E}[\\overline{x}^2]-m_1^2$"}
{"_id": "1999417", "title": "", "text": "$G = \\{ z \\mid |z| >2, 0 < arg(z) \\leq \\pi\\}$"}
{"_id": "6139515", "title": "", "text": "$X\\subseteq\\overline{V}\\subseteq X$"}
{"_id": "3121929", "title": "", "text": "$\\sum_{n=0}^{\\infty}\\frac{2^{n}}{(2n)!\\cdot2^{n}} = \\sum_{n=0}^{\\infty}\\frac{1}{(2n)!}$"}
{"_id": "2102595", "title": "", "text": "$ \\limsup \\frac{x_1+x_2+...+x_n+x_{n+1}}{x_n} \\geq 4.$"}
{"_id": "1231099", "title": "", "text": "$ \\begin{align} \\sum_{k=0}^{n-1}\\cos\\left(\\frac{2\\pi k}{n}+\\phi\\right) &=\\mathrm{Re}\\left(\\sum_{k=0}^{n-1}e^{\\frac{2\\pi ik}n+i\\phi}\\right)\\\\ &=\\mathrm{Re}\\left(e^{i\\phi}\\sum_{k=0}^{n-1}e^{\\frac{2\\pi ik}n}\\right)\\\\ &=\\mathrm{Re}\\left(e^{i\\phi}\\frac{1-e^{\\frac{2\\pi in}n}}{1-e^{\\frac{2\\pi i}n}}\\right)\\\\ &=\\mathrm{Re}\\left(e^{i\\phi}\\frac{1-e^{2\\pi i}}{1-e^{\\frac{2\\pi i}n}}\\right)\\\\[6pt] &=\\mathrm{Re}(0)\\\\[12pt] &=0 \\end{align} $"}
{"_id": "3605106", "title": "", "text": "$P(m, n) \\implies P(m, n + 1)$"}
{"_id": "1057696", "title": "", "text": "$R_{n}\\left( x\\right) =\\dfrac {f^{n+1}\\left( a\\right) \\left( x-a\\right) ^{n+1}} {\\left( n+1\\right) !} $"}
{"_id": "4189838", "title": "", "text": "$\\theta^a=-\\frac{1}{2}(\\gamma^{-1})^{ab}\\rho(G_b)_c^{\\phantom{a} d}(u^{-1})_d^{\\phantom{a} e}\\mathrm du_e^{\\phantom{a} c}$"}
{"_id": "849813", "title": "", "text": "$2\\pi\\int_0^{\\sqrt{1-m^2/3}} (1-r^2)\\rho\\mathrm d\\rho= 2\\pi\\int_h^1(1-r^2)r\\mathrm dr=2\\pi\\left(\\frac{1-h^2}2-\\frac{1-h^4}4\\right).$"}
{"_id": "6853533", "title": "", "text": "$\\mathbb{E}\\vert g(X_1,\\ldots, X_m)\\vert^r < \\infty$"}
{"_id": "4644083", "title": "", "text": "$-\\sum |x_n|\\le\\sum x_n \\le \\sum |x_n|$"}
{"_id": "4374903", "title": "", "text": "$f(a_1)=f(a_1)$"}
{"_id": "3723673", "title": "", "text": "$\\frac{\\partial^2}{\\partial x^2}f(x,t,a) = -t(1+t)(1+x)^{-2-t}-2a \\frac{(1+x)^{2+t}-(2+t) x \\big(1+\\frac{1+t}2x\\big)-1}{x^3(1+x)^{2+t}}. $"}
{"_id": "5026964", "title": "", "text": "$\\begin{pmatrix} x & y  \\end{pmatrix} \\begin{pmatrix}  1& 1 &\\frac{-z}{x} \\\\   2& \\frac{z}{y} &1  \\end{pmatrix}=\\begin{pmatrix}  x+2y&x+z &y-z  \\end{pmatrix}$"}
{"_id": "7634675", "title": "", "text": "$Ax + By = D - Cz$"}
{"_id": "9111333", "title": "", "text": "$c=\\frac {1- \\gamma a}{\\gamma} $"}
{"_id": "2949785", "title": "", "text": "$(\\Bbb R[X] / (X^2)) \\setminus (X)$"}
{"_id": "6988952", "title": "", "text": "$f(x)=\\frac{1}{\\sqrt{2\\pi t}}e^{\\frac{-x^2}{2t}}$"}
{"_id": "6806560", "title": "", "text": "$ M(t)=\\|p-f(t)\\|^2=(p-f(t),p-f(t)) $"}
{"_id": "3040050", "title": "", "text": "$9^{x}=6^x +4^x$"}
{"_id": "752162", "title": "", "text": "$1 + 2 + 3 + \\cdots = -\\frac{1}{12}.$"}
{"_id": "3924240", "title": "", "text": "$(x,y) = (a+b, a-b)$"}
{"_id": "6349987", "title": "", "text": "$\\lim_{x \\to c^+}f'(x) = f'_+(c) = f'(c)  = f'_-(c) = \\lim_{x \\to c^-}f'(x)$"}
{"_id": "85713", "title": "", "text": "$\\{A, -B\\}$"}
{"_id": "5923281", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       1 & 1 & 1&4\\\\         0 & 0 & 1&2 \\\\         0 & 0&a-4&a-2\\\\     \\end{array} \\right] $"}
{"_id": "8812222", "title": "", "text": "$g(x) = (x^3+x^4+x^5+x^6+x^7+x^8+x^9)(1+x+...+x^9)^4$"}
{"_id": "6912211", "title": "", "text": "$f(x)=\\frac{1}{1+\\frac{27}{9^x}}$"}
{"_id": "9073066", "title": "", "text": "$\\sum_{k=0}^{n-1}\\left\\lceil \\frac{\\left\\lceil y\\right\\rceil+k}{n}\\right\\rceil=\\sum_{k=0}^{n-1}\\left\\lceil \\frac {y+k}{n}\\right\\rceil$"}
{"_id": "8273803", "title": "", "text": "$N = (n+1)^2 - n^2 = 2n+1$"}
{"_id": "2861171", "title": "", "text": "$P(m− 1)\\to P(m)$"}
{"_id": "5313627", "title": "", "text": "$Cov(X,Y^2)=0$"}
{"_id": "2573111", "title": "", "text": "$3|a^3-a$"}
{"_id": "785576", "title": "", "text": "$Cov(Y,c)=0$"}
{"_id": "101615", "title": "", "text": "$\\frac{27!}{3!^9}$"}
{"_id": "9178261", "title": "", "text": "$ \\gamma\\xi\\gamma^{-1}(\\gamma(a_1))= \\gamma\\xi(a_1)=\\gamma(a_2). $"}
{"_id": "8388876", "title": "", "text": "$X=\\lbrace 0,e_0,e_1,\\ldots,e_n \\rbrace$"}
{"_id": "2759604", "title": "", "text": "$a^{log_n(b)}=n^{log_n(a)log_n(b)}$"}
{"_id": "1790216", "title": "", "text": "$C_k(n)=\\binom{n}{k}$"}
{"_id": "6331790", "title": "", "text": "$ \\left\\{    \\begin{array}{l l}      x=1 + \\frac{4t}{\\sqrt{32}}  \\\\     y=2 - \\frac{4t}{\\sqrt{32}}\\\\   \\end{array} \\right.   $"}
{"_id": "1759296", "title": "", "text": "$\\left(\\begin{array}{r}-B\\\\A\\end{array}\\right)$"}
{"_id": "4067197", "title": "", "text": "$ \\int_0^\\infty f(x) \\, dx = C,$"}
{"_id": "4013249", "title": "", "text": "$\\mathcal{P}(k)\\implies\\mathcal{P}(k+1)$"}
{"_id": "3543630", "title": "", "text": "$ (\\star) \\quad {[{\\gamma^{*}}(\\nabla)]_{\\frac{\\partial}{\\partial x}}}({\\gamma^{*}}(X)) = {\\gamma^{*}} \\left( {\\nabla_{{d \\gamma} \\left( \\frac{\\partial}{\\partial x} \\right)}}(X) \\right) = \\gamma^{*} \\left( {\\nabla_{\\gamma'}}(X) \\right). $"}
{"_id": "4102548", "title": "", "text": "$F_0 = P\\oplus T$"}
{"_id": "4948552", "title": "", "text": "$S=\\{z \\in \\mathbb C \\: | (z+i)^n  = (z-i)^n\\}; \\: n \\in \\mathbb N, n \\ge 2$"}
{"_id": "9222065", "title": "", "text": "$ \\cases{ x=st \\cr \\displaystyle y={1\\over2}s(t^2-1)\\cr \\displaystyle z={1\\over2}s(t^2+1)\\cr } $"}
{"_id": "4565629", "title": "", "text": "$f(x)=\\frac{1}{4\\pi}e^{\\frac{-x^2}{4}}$"}
{"_id": "4090480", "title": "", "text": "$\\gamma=nq\\gamma_1+n\\gamma_2 \\implies \\frac{\\gamma}{n}=q\\gamma_1+\\gamma_2$"}
{"_id": "8931533", "title": "", "text": "$\\therefore f(a+b)=f(a)+f(b)$"}
{"_id": "4257083", "title": "", "text": "$\\#_A(n)$"}
{"_id": "2557376", "title": "", "text": "$\\overline {\\mathbb{R}}$"}
{"_id": "541433", "title": "", "text": "$\\,[x,y] = {\\rm lcm}(x,y)$"}
{"_id": "3729220", "title": "", "text": "$ A \\subseteq V \\subseteq \\overline{V} \\subseteq U $"}
{"_id": "7644897", "title": "", "text": "$ \\int_{0}^{1} \\frac{\\tan^{-1}(x)}{\\sqrt{1-x^2}} dx$"}
{"_id": "4650474", "title": "", "text": "$N= \\{\\omega: \\lim_{n \\rightarrow \\infty} X_n(\\omega) \\ne 0\\}$"}
{"_id": "4349571", "title": "", "text": "$|ab|=n$"}
{"_id": "2515551", "title": "", "text": "$\\begin{cases} x_1=1+1s+2t \\\\ x_2=0+1s+0t \\\\ x_3=0+0s+1t \\end{cases}$"}
{"_id": "628353", "title": "", "text": "$\\{a\\} \\ne \\{\\{a\\}\\}$"}
{"_id": "1620260", "title": "", "text": "$ \\begin{cases} r\\equiv 1\\mod p-1\\\\ r\\equiv 1\\mod q-1 \\end{cases} $"}
{"_id": "967645", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty} e^{-\\frac{1}{2} An^2+iBn} = \\sqrt{\\frac{2\\pi}{A}} \\sum_{l=-\\infty}^{\\infty} e^{-\\frac{1}{2A}(B-2\\pi l)^2}$"}
{"_id": "926726", "title": "", "text": "$R_\\varphi(0,1) = (\\cos(\\varphi+\\frac\\pi 2),\\sin(\\varphi+\\frac\\pi 2)) = (-\\sin\\varphi,\\cos\\varphi)$"}
{"_id": "120729", "title": "", "text": "$\\textrm{cov}(X,Y) = 0$"}
{"_id": "4773878", "title": "", "text": "$\\liminf \\limits_{n \\to \\infty} \\, \\mu_n(G) \\ge \\mu(G)$"}
{"_id": "7805850", "title": "", "text": "$x! \\approx \\sqrt{2 \\pi x} \\cdot \\left(\\dfrac{x}{e}\\right)^x $"}
{"_id": "6733991", "title": "", "text": "$ \\begin{eqnarray*} P\\left(\\dfrac{Y^2}{X^2}<z\\right){}={}P\\left(Y^2<z\\,X^2\\right)&{}={}&P\\left(Y<\\sqrt{z}\\,X\\right)\\,,\\,\\,\\,\\,\\,\\,\\mbox{ since }Y\\,,\\,X\\,,z>0\\newline &{}={}&\\int\\limits_{0}^{\\infty}P\\left(Y<\\sqrt{z}\\,x\\,\\bigg|\\,X=x\\right)f_{X}(x)\\,\\mathrm{d}x\\newline &{}={}&\\int\\limits_{0}^{\\infty}P\\bigg(Y<\\sqrt{z}\\,x\\bigg)f_{X}(x)\\,\\mathrm{d}x\\newline &{}={}&\\int\\limits_{0}^{\\infty}\\bigg(1-e^{-\\sqrt{z}\\,x}\\bigg)e^{-x}\\,\\mathrm{d}x\\newline &{}={}&\\left(1-\\dfrac{1}{1+\\sqrt{z}}\\right){\\bf{1}}_{\\left\\{z>0\\right\\}}\\,. \\end{eqnarray*} $"}
{"_id": "6719315", "title": "", "text": "$P(1)=1/6$"}
{"_id": "3763619", "title": "", "text": "$\\frac{x}{20}=\\tan\\theta.$"}
{"_id": "4944752", "title": "", "text": "$ \\left|^\\mathbb{N} \\mathbb{R}\\right| = |\\mathbb R| < |P(\\mathbb R)| \\leq \\left|^\\mathbb R \\mathbb N\\right|. $"}
{"_id": "85266", "title": "", "text": "$\\overline{ \\mathbb{R}}$"}
{"_id": "8731692", "title": "", "text": "$a_n = n^2 -n +2.$"}
{"_id": "4238170", "title": "", "text": "$p_j \\mid p_1 p_2 \\cdots p_k$"}
{"_id": "5473900", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\frac {\\sin nx}{(1+2^x) \\sin x} $"}
{"_id": "3485100", "title": "", "text": "$(a-b,b-c)=(0,0)$"}
{"_id": "1087841", "title": "", "text": "$H = [a,b]\\times [c,d] = \\{(x,y)| a \\le x \\le b; c \\le y \\le d\\}$"}
{"_id": "1871781", "title": "", "text": "$Cov(X,Y)=-5$"}
{"_id": "5304156", "title": "", "text": "$\\mathcal B=\\{v_1,v_2,....,v_n \\}$"}
{"_id": "3869906", "title": "", "text": "$x^6-6 x^4+9 x^2-x-2 = 0,$"}
{"_id": "6994615", "title": "", "text": "$f(x)=\\min\\{1,1/|x|\\}$"}
{"_id": "2638433", "title": "", "text": "$\\int_{0}^{\\pi }f(x+t)dt=\\int_{0}^{\\left [ n\\pi \\right ]}f(x+t)dt=0$"}
{"_id": "5643352", "title": "", "text": "$\\left | \\left | A \\right | \\right |_{2}=\\sqrt{tr(A^{t}A)}$"}
{"_id": "5687664", "title": "", "text": "$|\\theta-\\sin \\theta| < |2\\tan\\frac\\theta 2 - \\sin\\theta| < 4|\\sin^3\\theta|<4|\\theta|^3$"}
{"_id": "9162942", "title": "", "text": "$P(16)\\implies P(20)$"}
{"_id": "1767245", "title": "", "text": "$X_1\\subset X_2\\subset \\dots$"}
{"_id": "4029668", "title": "", "text": "$\\text{Max}[f(x)+f(y)]=\\text{Max}f(x)=\\text{Max}f(y).$"}
{"_id": "4830371", "title": "", "text": "$\\forall \\epsilon>0,\\exists\\delta>0: |x-c|<\\delta \\implies |f(x)-f(c)|<\\epsilon.$"}
{"_id": "1647983", "title": "", "text": "$\\langle a,b\\rangle=\\det(\\langle a_i,b_i\\rangle)$"}
{"_id": "8373540", "title": "", "text": "$f_X(x)=\\frac{2x}{r^2}$"}
{"_id": "1762245", "title": "", "text": "$ -\\frac1{12}=1+2+3+4+\\dots\\tag{5} $"}
{"_id": "9051008", "title": "", "text": "$|\\mathbb{E}|X_n|^p-\\mathbb{E}|X|^p|\\le \\mathbb{E}|X_n-X|^p\\to 0$"}
{"_id": "8108769", "title": "", "text": "$f(x)=\\frac{x^2}{x^2+3}$"}
{"_id": "1342741", "title": "", "text": "$P(X ≤ 10) = 1/6$"}
{"_id": "7803009", "title": "", "text": "$y=\\frac{10}{4}$"}
{"_id": "9136697", "title": "", "text": "$ \\int_{-\\pi}^{\\pi}{\\cos\\left(\\, ax\\,\\right) \\over 1-bx^{2}}\\,{\\rm d}x $"}
{"_id": "4918146", "title": "", "text": "$\\tan \\theta=\\frac{3}{2}$"}
{"_id": "6912165", "title": "", "text": "$ \\begin{pmatrix}   1 & 1 & 1 \\\\   0 & y-x & z-x \\\\   0 & y^2-x^2 &z^2-x^2\\\\ \\end{pmatrix} = \\begin{pmatrix}   1 & 1 & 1 \\\\   0 & y-x & z-x \\\\   0 & (y-x)(y+x) &z^2-x^2\\\\ \\end{pmatrix}  $"}
{"_id": "6044041", "title": "", "text": "$|xy| = pq$"}
{"_id": "2232985", "title": "", "text": "$e^x-\\left(1+\\frac{x}{n}\\right)^n$"}
{"_id": "5984961", "title": "", "text": "$T_n \\gamma =b.(\\gamma -Sd \\gamma -T_{n-1}\\partial_n \\gamma)$"}
{"_id": "196608", "title": "", "text": "$\\{0\\}\\subset V_0\\subset V_1\\subset \\cdots \\subset V_r\\subset k^n,$"}
{"_id": "3599183", "title": "", "text": "$S=\\{e_{1}, e_{2}, e_{3}, e_{4}\\}$"}
{"_id": "4827369", "title": "", "text": "$\\mathbb R^n → \\mathbb R^n$"}
{"_id": "9163344", "title": "", "text": "$P(1) \\implies P(2) \\implies  P(3) \\implies \\dots \\implies P(k) \\implies P(k+1) \\implies \\dots $"}
{"_id": "7214825", "title": "", "text": "$\\begin{bmatrix}     1 & -2 \\\\     2 & 3 \\\\     \\end{bmatrix}$"}
{"_id": "315894", "title": "", "text": "$(\\overline{S} \\cap \\overline{T})^\\perp \\subset \\overline{S \\cap T}^\\perp$"}
{"_id": "4878953", "title": "", "text": "$2(n-1) = {n\\choose2} = \\frac{n(n-1)}{2}$"}
{"_id": "668986", "title": "", "text": "$\\lim_{x\\to 0} f(x)g(x) = \\lim_{x\\to0} f(x)\\cdot \\lim_{x\\to0}g(x)$"}
{"_id": "4684113", "title": "", "text": "$A\\subseteq V_1\\subseteq \\overline{V_1}\\subseteq U$"}
{"_id": "5807520", "title": "", "text": "$p(n)=\\binom{n}k$"}
{"_id": "61119", "title": "", "text": "$\\begin{bmatrix}a&b\\\\b&-a\\end{bmatrix}$"}
{"_id": "5037505", "title": "", "text": "$f(...f(a_1)...)=a_1$"}
{"_id": "143081", "title": "", "text": "$F = x - y$"}
{"_id": "4745339", "title": "", "text": "$3|x^3-x$"}
{"_id": "1106770", "title": "", "text": "$\\left(1+\\frac{x}{n}\\right)^n\\leq e^{x}.$"}
{"_id": "466920", "title": "", "text": "$  k^2 = 2 \\; \\left( \\begin{array}{c} k \\\\ 2 \\end{array} \\right) + \\left( \\begin{array}{c} k \\\\ 1 \\end{array} \\right),  $"}
{"_id": "6268532", "title": "", "text": "$x^2 + y^2 + z^2 = x y + y z + x z$"}
{"_id": "2091707", "title": "", "text": "$ax+by=m$"}
{"_id": "1878631", "title": "", "text": "$\\cdots \\subseteq A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\cdots \\subseteq A_n \\subseteq \\cdots .$"}
{"_id": "1644448", "title": "", "text": "$\\frac{sr}{1-s} = \\frac{(1-p)^N}{1-(1-p)^N}$"}
{"_id": "2802779", "title": "", "text": "$\\mathbb{R}^{n+1}\\times \\mathbb{R}^{n+1}$"}
{"_id": "3429972", "title": "", "text": "$F_3=(f_3(n))_n$"}
{"_id": "5049225", "title": "", "text": "$M=\\begin{pmatrix} 1 & 1 & 1\\\\ x & y & z\\\\ x^2 & y^2 & z^2 \\end{pmatrix}$"}
{"_id": "180303", "title": "", "text": "$f(x + 2x f(x)^2) = y f(x) + f(f(x) + 1)$"}
{"_id": "1450182", "title": "", "text": "$\\{e_i: 1 \\leq i \\leq n\\}$"}
{"_id": "439995", "title": "", "text": "$\\left| \n \\begin{array}{cccc}1 & 1 & 1 & 1\\\\\n x & y & 1 & 0\\\\\n x^2 & y^2 & 1 &0\\\\\n x^3 & y^3 & 1&0\n \\end{array} \n \\right|$"}
{"_id": "2900983", "title": "", "text": "$z=a e^{i \\pi/4}$"}
{"_id": "1553514", "title": "", "text": "$\\lim_{n \\to \\infty} \\inf s_n \\le \\lim_{n \\to \\infty} \\inf t_n$"}
{"_id": "5080639", "title": "", "text": "$ xf'(x)+f(x)=(x-y)f'(x+y)+f(x+y). $"}
{"_id": "3534386", "title": "", "text": "$\\mathbf{B}=\\left[ \\matrix{ \\mathbf{A}_1&\\mathbf{A}_2&\\ldots&\\mathbf{A}_{p-1}&\\mathbf{A}_p \\\\  \\mathbf{I}_M&\\mathbf{I}_M&\\ldots&\\mathbf{I}_M&\\mathbf{O}_M \\\\  \\vdots &\\vdots &\\ddots& \\vdots& \\vdots\\\\  \\mathbf{I}_M&\\mathbf{I}_M&\\ldots&\\mathbf{I}_M&\\mathbf{O}_M \\\\ } \\right]:(pM \\times pM)$"}
{"_id": "8482664", "title": "", "text": "$C-\\sum_{p\\leq x}\\frac{1}{p^{2}}=\\sum_{p>x}\\frac{1}{p^{2}}$"}
{"_id": "5547811", "title": "", "text": "$1+2+\\cdots+(n-1)={n\\choose2}$"}
{"_id": "7312733", "title": "", "text": "$ Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)$"}
{"_id": "5250728", "title": "", "text": "$f(a,b,c,\\gamma)=\\int\\left(\\gamma -a-b\\cos \\gamma -c\\sin \\gamma\\right)^2\\,d\\gamma$"}
{"_id": "2347875", "title": "", "text": "$(y-x)z+(y-x)(y+x)=(y-x)(z+y+x)$"}
{"_id": "1370731", "title": "", "text": "$ \\lfloor \\frac{\\lfloor\\frac{a}{b}\\rfloor}{c} \\rfloor = \\lfloor \\frac{a}{b \\times c} \\rfloor $"}
{"_id": "1122579", "title": "", "text": "$d(x,z)>d(x,y)+d(y,z)$"}
{"_id": "8659626", "title": "", "text": "$\\mathbb{E}|X|^r<\\infty$"}
{"_id": "688915", "title": "", "text": "$Cov(Y,Z) =0$"}
{"_id": "8742269", "title": "", "text": "$\\operatorname{ord}_p(n) = k$"}
{"_id": "4716057", "title": "", "text": "$\\forall \\epsilon>0 ,\\exists \\delta > 0 :|x-a|<\\delta \\implies |f(x) - L|<\\epsilon$"}
{"_id": "4078323", "title": "", "text": "$(-1)^n\\bigl(x - (n-1)\\bigr)(x+1)^{n-1}$"}
{"_id": "5256700", "title": "", "text": "$∇^ 2u=X''(x)Y(y)Z(z)+X(x)Y''(y)Z(z)+X(x)Y(y)Z''(z)=0$"}
{"_id": "2961660", "title": "", "text": "$x^{\\log_x(a)}=a$"}
{"_id": "7214231", "title": "", "text": "$\\phi(n) = n + I$"}
{"_id": "6098018", "title": "", "text": "$\\det \\begin{pmatrix}A & 0 \\\\ B & C\\end{pmatrix} = \\det (A) \\det (C)$"}
{"_id": "1993616", "title": "", "text": "$\\|A\\| = \\sqrt{\\lambda_{\\max} (A^T A)}$"}
{"_id": "3796696", "title": "", "text": "$E(X)=\\frac{n(n-1)}{N}.$"}
{"_id": "8941435", "title": "", "text": "$\\lfloor c^{a+b} \\rfloor$"}
{"_id": "1992858", "title": "", "text": "$|f'(z)| = \\frac \\pi 2 |f(z)|$"}
{"_id": "3465303", "title": "", "text": "$\\sum_{k=1}^n\\cos\\frac{k\\pi}{n+1}$"}
{"_id": "6276221", "title": "", "text": "$F(x)=\\int_a^{x}xf(t)dt$"}
{"_id": "4908596", "title": "", "text": "$ \\lvert \\gamma \\times \\gamma' \\rvert^2 = (\\gamma \\cdot \\gamma)(\\gamma' \\cdot \\gamma') - (\\gamma \\cdot \\gamma')^2 = (\\gamma \\cdot \\gamma)(\\gamma' \\cdot \\gamma')  \\neq 0. $"}
{"_id": "7163599", "title": "", "text": "$101!+1,101!+2,101!+3,\\cdots,101!+101$"}
{"_id": "199715", "title": "", "text": "$\\int_0^{\\infty} \\frac{\\tan^{-1}(x^2)}{x^2+a^2} \\ dx$"}
{"_id": "6239763", "title": "", "text": "$\\frac{y+4}{5+4} = \\frac{x+10}{8+10}$"}
{"_id": "1512077", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{\\cos(nx)\\cos(my)}{n^2+m^2},$"}
{"_id": "150764", "title": "", "text": "$\\in \\mathbb{R^+}$"}
{"_id": "4637917", "title": "", "text": "$I = \\int_0^1 \\frac{\\sin^{-1}(t)}{t} A'(t) dt = - \\int_0^1 \\frac{\\sin^{-1}(t)}{t}\\log t dt = -\\frac12 \\int_0^1 \\sin^{-1}(t) (\\log^2(t))' dt$"}
{"_id": "3491758", "title": "", "text": "$x^3<1,x\\in\\mathbb{R}$"}
{"_id": "480437", "title": "", "text": "$9^x + 12^x = 16^x$"}
{"_id": "5626788", "title": "", "text": "$ \\gamma(a) + \\gamma(b) = \\gamma(a+b). $"}
{"_id": "4152470", "title": "", "text": "$\\frac8x+\\frac1y=1$"}
{"_id": "6817196", "title": "", "text": "$M = F' \\oplus T(M)$"}
{"_id": "5529074", "title": "", "text": "$f(n)=3n^2-n+4$"}
{"_id": "5362346", "title": "", "text": "$\\lim\\limits_{x \\to a^+} f(x)=\\lim\\limits_{x\\to a^-}f(x)=L$"}
{"_id": "1862750", "title": "", "text": "$L_\\alpha = [H_i,L_\\alpha] \\subseteq [L_i,L_\\alpha] \\subseteq L_i.$"}
{"_id": "3452923", "title": "", "text": "$\\frac{f(f(x)y)}{x^2} =  \\frac{f(f(x) + f(y))}{x + y }\\tag{1}$"}
{"_id": "4021663", "title": "", "text": "$\\lim_{x\\to1^+}f'(x)\\neq\\lim_{x\\to1^-}f'(x)$"}
{"_id": "5790231", "title": "", "text": "$p|m \\implies p |n$"}
{"_id": "714963", "title": "", "text": "$ \\ \\tan \\theta = \\frac{x}{a} \\ $"}
{"_id": "876277", "title": "", "text": "$ \\frac1{\\binom{n+a}{a}}=\\frac{a}{a-1}\\left[\\frac1{\\binom{n+a-1}{a-1}}-\\frac1{\\binom{n+a}{a-1}}\\right]\\tag{1} $"}
{"_id": "3351519", "title": "", "text": "$f''(x) = n(n-1)(x+1)^{n-1}$"}
{"_id": "3936209", "title": "", "text": "$\\,p_1 \\mid q_{2}\\cdots q_{n}\\,$"}
{"_id": "1676023", "title": "", "text": "$f(x)=1/|x|$"}
{"_id": "5072791", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty}e^{-a|n|}=\\sum_{m=-\\infty}^{\\infty}\\dfrac{2a}{a^2+(2\\pi m)^2}$"}
{"_id": "7706444", "title": "", "text": "$z^{1/n} = r^{\\frac 1n} e^{\\frac{\\theta}ni + \\frac{2k}n \\pi i} = [r^{\\frac 1n} e^{\\frac{\\theta}ni}]*e^{\\frac{2k}n \\pi i}$"}
{"_id": "7044406", "title": "", "text": "$M_n=\\frac{x_1+x_2+\\dots+x_n}{n}.$"}
{"_id": "4327057", "title": "", "text": "$\\zeta(s)=\\frac{1}{\\Gamma(s)}\\int_0^{\\infty}\\frac{x^{s-1}dx}{e^x-1},$"}
{"_id": "3485607", "title": "", "text": "$\\begin {array}{ll} x_1 = a\\lambda /(\\lambda+1)\\\\ y_1 = b\\lambda /(\\lambda+1)\\\\ (a-x_1)^2+(b-y_1)^2-r^2 =0 \\\\ \\end{array}$"}
{"_id": "7384236", "title": "", "text": "$z_1=3e^{\\frac{i\\pi}{6}}$"}
{"_id": "2979139", "title": "", "text": "$\\int_\\gamma \\! \\frac{1}{z+1} \\, \\mathrm{d}z = \\int_0^1 \\! \\frac{\\gamma'(t)}{\\gamma(t)+1} \\, \\mathrm{d}t$"}
{"_id": "498743", "title": "", "text": "$x-a,\\:\\:x\\in \\left[a,\\:\\infty \\right]$"}
{"_id": "6601875", "title": "", "text": "$ (a_1) \\subseteq (a_2) \\subseteq \\dots \\subseteq (a_i) \\subseteq (a_{i+1}) \\subseteq \\dots\\tag{seq} $"}
{"_id": "8614091", "title": "", "text": "$[\\mathbb{x}]_d^t:=[1,x,y,x^2,xy,y^2,x^3,x^2y,xy^2,y^3,\\dots,y^d]$"}
{"_id": "3662734", "title": "", "text": "$xRy \\rightarrow yRx?$"}
{"_id": "8981236", "title": "", "text": "$|f(z)|\\le N|g(z)|$"}
{"_id": "352241", "title": "", "text": "$COV(X,Y) = COV(X, E(Y|X))$"}
{"_id": "4417", "title": "", "text": "$1/|x|$"}
{"_id": "6171862", "title": "", "text": "$\\Vert \\cdot \\Vert_2:=\\sqrt{\\lambda_{\\max}(A^TA)}$"}
{"_id": "401407", "title": "", "text": "$f_3(n)=\\binom{n+3}{4}$"}
{"_id": "5093340", "title": "", "text": "$ \\sum_{k = -\\infty}^{\\infty} e^{-\\pi k n^2} = \\frac{1}{\\sqrt{k}} \\sum_{k=-\\infty}^{\\infty} e^{-\\pi n^2/k} $"}
{"_id": "9107257", "title": "", "text": "$t = f(x) = f(lim_{n \\rightarrow \\infty }x_n) = lim_{n \\rightarrow \\infty }f(x_n) = lim_{n \\rightarrow \\infty } t_n$"}
{"_id": "2859409", "title": "", "text": "$\\mathcal{X} = \\mathcal{A}$"}
{"_id": "2072737", "title": "", "text": "$P(k + 1): k + 1 = (k + 1 – 3) + 3$"}
{"_id": "8685396", "title": "", "text": "$log_a(x)>0$"}
{"_id": "221451", "title": "", "text": "$M^*M =MM^*$"}
{"_id": "5269438", "title": "", "text": "$\\lim_{n\\to\\infty}J(v_n)=x=\\inf_{v\\in K}J(v),$"}
{"_id": "4826058", "title": "", "text": "$\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}+\\frac{\\boxed{}}{\\boxed{}}$"}
{"_id": "60563", "title": "", "text": "$\\int {1 \\over (x^2-1)^2}dx$"}
{"_id": "8819506", "title": "", "text": "$\\lim_{n\\to \\infty} \\sum_{r=1}^n \\frac {1}{2^r}\\tan \\left(\\frac {1}{2^r}\\right)$"}
{"_id": "4603318", "title": "", "text": "$             \\|x\\| \\ge \\frac{|\\langle x,y\\rangle|}{\\|y\\|} \\\\             |\\langle x,y\\rangle| \\le \\|x\\|\\|y\\| $"}
{"_id": "6760375", "title": "", "text": "$f(f(x)+f(y))=af(y)+f(a)+f(y)-f(x)y.$"}
{"_id": "4825212", "title": "", "text": "$ \\tan \\theta = \\dfrac{x}{2}.$"}
{"_id": "9080309", "title": "", "text": "$\\displaystyle\\sum_{k=1}^{n-1}\\cos\\frac{2k\\pi }n=$"}
{"_id": "1471720", "title": "", "text": "$f(f(x)^2 + xf(x)) = xf(x) + x^2$"}
{"_id": "3414592", "title": "", "text": "$\\frac{P}{M} = \\frac{\\left(1 + J\\right)^N - 1}{J(1+J)^N}$"}
{"_id": "8668382", "title": "", "text": "$\\forall x(xRb\\lor bRx)$"}
{"_id": "6753348", "title": "", "text": "$f(-\\pi)=f(\\pi)=0$"}
{"_id": "5365185", "title": "", "text": "$z^{\\frac{1}{n}}=\\sqrt[n]{r}e^{i \\varphi /n}$"}
{"_id": "3364857", "title": "", "text": "$F: \\Bbb R^{n + 1} = \\Bbb R \\times \\Bbb R^n \\to \\Bbb R^n \\tag 2$"}
{"_id": "6231452", "title": "", "text": "$|f(z^{2^n})| \\le |f(z)| \\to |f(0)| \\le |f(z)|$"}
{"_id": "8087314", "title": "", "text": "$\\ddot x = x^3 -x$"}
{"_id": "6290951", "title": "", "text": "$\\zeta(s)=2^{s} \\pi^{s-1} \\sin \\frac{\\pi s}{2} \\Gamma(1-s)\\zeta(1-s).$"}
{"_id": "6516542", "title": "", "text": "$A=\\{(a,b,c,d)|a,b,c,d\\in\\mathbb{R}_{\\ge0}; a+b+c+d=m\\}$"}
{"_id": "1647424", "title": "", "text": "$(1 + \\tfrac{x}{n})^n < \\mathrm{e}^x$"}
{"_id": "5536514", "title": "", "text": "$r:\\\\x=\\lambda\\\\y=\\lambda\\\\z=1-\\lambda$"}
{"_id": "2269153", "title": "", "text": "$\\left(|a|^r+|b|^r\\right)(1+1)^{r-1}\\geq\\left(\\left(|a|^r\\cdot1^{r-1}\\right)^{\\frac{1}{1+r-1}}+\\left(|b|^r\\cdot1^{r-1}\\right)^{\\frac{1}{1+r-1}}\\right)^{1+r-1}=\\left(|a|+|b|\\right)^r$"}
{"_id": "6489613", "title": "", "text": "$f(k)\\leq k^2+k+2$"}
{"_id": "5327411", "title": "", "text": "$ \\Gamma(x)\\approx\\sqrt{\\frac{2\\pi}{x}}\\left(\\frac{x}{e}\\right)^x $"}
{"_id": "4741188", "title": "", "text": "$R=\\{\\langle x,y\\rangle\\in X\\times X:\\exists i\\in I:\\{x,y\\}\\subseteq U_i\\}$"}
{"_id": "2328920", "title": "", "text": "$\\intop\\left(1-x^{2/3}\\right)^{3/2}dx=3\\intop\\cos^{4}\\left(y\\right)\\sin^{2}\\left(y\\right)dy.$"}
{"_id": "2578909", "title": "", "text": "$\\lim_{n\\to\\infty}\\|x_{n}-y_{n}\\|=0$"}
{"_id": "4859383", "title": "", "text": "$F_n\\neq \\cfrac{n(n+1)}{2}$"}
{"_id": "4102552", "title": "", "text": "$P\\oplus T$"}
{"_id": "177140", "title": "", "text": "$b^{log_b(n)} = log_b(b^n) = n$"}
{"_id": "1425564", "title": "", "text": "$a=\\frac{n(n+1)}{6}$"}
{"_id": "5736691", "title": "", "text": "$t^2=(t_x^2,t_y^2)$"}
{"_id": "1019245", "title": "", "text": "$\\|g\\|_{L^1} = \\sum_{n=-\\infty}^\\infty \\int_{-\\infty}^\\infty e^{-n^4 (x-n)^2} dx =  \\sum_{n=-\\infty, n \\ne 0}^\\infty \\frac{1}{n^2}\\int_{-\\infty}^\\infty e^{-x^2} dx = 2\\frac{\\pi^2}{6}\\sqrt{2\\pi}$"}
{"_id": "742359", "title": "", "text": "$e^x > \\frac{x^{n+1}}{(n+1)!}$"}
{"_id": "595546", "title": "", "text": "$p_1p_2...p_k - 1$"}
{"_id": "2304795", "title": "", "text": "$u = 1-\\displaystyle\\sum_{k=1}^n\\frac{1}{2^k}$"}
{"_id": "1162626", "title": "", "text": "$|x+a||x-a|=(x+a)(x-a)$"}
{"_id": "1251691", "title": "", "text": "$t^{0,1}_7=8$"}
{"_id": "1809334", "title": "", "text": "$\\begin{cases} 2x \\equiv 1 (29) \\\\ 2x \\equiv -1 (29) \\end{cases} $"}
{"_id": "4256472", "title": "", "text": "$n^2-\\left(\\frac{n-1}{2}\\right)^2-\\left(\\frac{n+1}{2}\\right)^2+\\frac{n-1}{2}=\\frac 1 2 (n^2+n-2),$"}
{"_id": "6878252", "title": "", "text": "$df: T^{0,1}D \\to f^*(T^{0,1}D)$"}
{"_id": "1854708", "title": "", "text": "$d(x) = \\inf_{m \\in M} d(x,m)$"}
{"_id": "5287356", "title": "", "text": "$ \\begin{cases} y=5\\\\ x=3\\\\ z=7 \\end{cases}\\Longleftrightarrow $"}
{"_id": "8687393", "title": "", "text": "$P_{\\Theta|X}(\\theta=0|x) = \\frac{P(\\theta=0)f_{X|\\Theta}(x|\\theta=0)}{f_X(x)} = \\frac{1}{f_X(x)}((1-p)*1)$"}
{"_id": "7092520", "title": "", "text": "$cov(x,y)=E(xy)-E(x)E(y)$"}
{"_id": "99561", "title": "", "text": "$W^{\\bot}=\\{\\frac{1}{2}(1,-1,-1,1), \\frac{1}{2}(1,-1,1,-1)\\}$"}
{"_id": "3970899", "title": "", "text": "$\\big\\{\\langle x,y \\rangle | \\langle x,y \\rangle \\in \\mathbb{R}^2,\\quad\\exists\\;a,b,c,d \\in \\mathbb{R}^2 such \\; that \\; a<x<b \\quad c<y<d\\big\\}$"}
{"_id": "3021952", "title": "", "text": "$ \\mathbb{P}\\{X > (1+\\gamma)P\\} \\leq \\left(\\frac{e^\\gamma}{(1+\\gamma)^{1+\\gamma}}\\right)^P \\tag{1} $"}
{"_id": "409916", "title": "", "text": "$\\mathbb{P}(X_n \\geq K \\;| \\;n \\geq N) = 1.] $"}
{"_id": "3930596", "title": "", "text": "$4^x+6^x=9^x\\implies 1+u=u^2$"}
{"_id": "8177109", "title": "", "text": "$\\lim_{n \\to \\infty}\\dfrac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\cdots+\\sqrt{n-1}}{n\\sqrt{n}}.$"}
{"_id": "5498614", "title": "", "text": "$9^x=5^x+2\\sqrt{20^x}+4^x$"}
{"_id": "4678436", "title": "", "text": "$\\alpha^+\\times \\alpha^+$"}
{"_id": "6708421", "title": "", "text": "$X \\text{ ind } Y \\implies Cov(X,Y)=0$"}
{"_id": "1544885", "title": "", "text": "$ \\left[\\begin{array}{ccc|c} 1 & 4 & 5 & 2\\\\ 2 & 2 & 4 & 10\\\\ 2 & 1 & 1 & 1\\\\ \\end{array}\\right]. $"}
{"_id": "3080457", "title": "", "text": "$\\frac{s(n)}{2}=\\frac{1}{2^2}+\\frac{1}{2^3}+...+\\frac{1}{2^{n}}-\\frac{n}{2^{n+1}}\\\\ s(n)=\\left[\\frac{1}{2}+\\frac{1}{2^2}+...+\\frac{1}{2^{n-1}}\\right]-\\frac{n}{2^{n}}$"}
{"_id": "7839779", "title": "", "text": "$ \\begin{split} &\\hat{\\nabla}_{(\\gamma \\circ \\tau)'(s)} (\\gamma \\circ \\tau)' \\\\&=\\nabla_{(\\gamma \\circ \\tau)'(s)} (\\gamma \\circ \\tau)' + 2 d \\sigma((\\gamma \\circ \\tau)'(s)) \\cdot (\\gamma \\circ \\tau)'(s) - \\| (\\gamma \\circ \\tau)'(s) \\| grad \\sigma \\\\&=\\nabla_{(\\gamma \\circ \\tau)'(s)} (\\gamma \\circ \\tau)' + 2 d \\sigma((\\gamma \\circ \\tau)'(s)) \\cdot (\\gamma \\circ \\tau)'(s) \\\\&=\\frac{\\tau ''(s)}{\\tau'(s)} \\cdot (\\gamma \\circ \\tau)'(s) + 2 d \\sigma((\\gamma \\circ \\tau)'(s)) \\cdot (\\gamma \\circ \\tau)'(s). \\end{split} $"}
{"_id": "1591787", "title": "", "text": "$ \\varphi_0(f)=\\sum_{j=1}^hc_jf(n_j) $"}
{"_id": "723738", "title": "", "text": "$P(A_{k})=1/6$"}
{"_id": "2854765", "title": "", "text": "$|y-y'| < |y| = -y$"}
{"_id": "3153890", "title": "", "text": "$\\phi_1(t,t_{0,1},x_{0,1})=e^{A_1(t-t_{0,1})}x_{0,1},$"}
{"_id": "3676764", "title": "", "text": "$f(f(x)^2+y)=x^2+f(y)$"}
{"_id": "4667064", "title": "", "text": "$E_n = X_n$"}
{"_id": "3890340", "title": "", "text": "$y',y'',y''',y''''$"}
{"_id": "579454", "title": "", "text": "$\\cot(\\sin^{-1}(z))=\\frac{\\sqrt{1-z^2}}{z}$"}
{"_id": "1389613", "title": "", "text": "$\\vert f(z) \\vert \\leq \\vert g(z) \\vert$"}
{"_id": "6617383", "title": "", "text": "$S = \\{\\{a\\}, \\{a\\}\\}$"}
{"_id": "1268895", "title": "", "text": "$A=\\{b_1,b_2,...,b_n,...\\}$"}
{"_id": "3117195", "title": "", "text": "$x_\\phi=e^{2\\pi i/\\phi}\\sqrt[k]a$"}
{"_id": "8575574", "title": "", "text": "$ \\gamma_R=\\gamma_{R,1}+\\gamma_{R,2}-\\gamma_{R,3} $"}
{"_id": "403364", "title": "", "text": "$L:ax+by=c$"}
{"_id": "1508609", "title": "", "text": "$N=rs+r+s$"}
{"_id": "5275524", "title": "", "text": "$ \\frac{dy}{dx}=\\frac{\\sqrt{1-x^2}}{x} $"}
{"_id": "5454645", "title": "", "text": "$g(\\pi) = f(\\pi) - f(0) = - (f(0) - f(\\pi))$"}
{"_id": "2119570", "title": "", "text": "$p_1p_2...p_k \\mid m$"}
{"_id": "7266644", "title": "", "text": "$2\\int_0^1\\frac{\\sin(x)}{\\arcsin(x)} \\, dx$"}
{"_id": "2392593", "title": "", "text": "$0\\leq \\sum_{n\\geq 1}\\frac{1}{\\sqrt{n}}\\,\\sin\\frac{1}{n}\\leq \\sum_{n\\geq 1}\\frac{1}{n\\sqrt{n}}=\\zeta(3/2).$"}
{"_id": "1002346", "title": "", "text": "$e^x> \\frac{x^{n+1}}{(n+1)!}.$"}
{"_id": "7039728", "title": "", "text": "$ \\zeta(s) = \\pi^{s - \\frac12} \\frac{\\Gamma(\\frac{1-s}{2})}{\\Gamma (\\frac s2)} \\zeta(1-s) $"}
{"_id": "5133442", "title": "", "text": "$res_{z_0}f = \\text{lim}_{z\\rightarrow z_0}{1 \\over (n-1)!}\\bigg({d \\over dz}\\bigg)^{n-1}(z-z_0)^n[f(z)-G(z)].$"}
{"_id": "2962102", "title": "", "text": "$y= c_1\\left | x \\right |^r+ c_2\\left | x \\right |^rln(\\left | x \\right |)$"}
{"_id": "2370058", "title": "", "text": "$x=(\\frac{1}{2})^2$"}
{"_id": "4676837", "title": "", "text": "$\\sqrt{m}\\|A\\|_\\infty=\\|A\\|_2$"}
{"_id": "8052479", "title": "", "text": "$\\sum_{k = 0}^{n - 1} \\sin \\frac{2\\pi k}{n} = 0$"}
{"_id": "5196206", "title": "", "text": "$P(E_1)=3/6$"}
{"_id": "1727404", "title": "", "text": "$f(x) = \\frac{1}{\\sqrt{2\\pi}} e^{\\frac{-x^2}{2}},$"}
{"_id": "7319093", "title": "", "text": "$\\int_{(i)}f(z)dz=\\int_{(ii)}f(z)dz=\\int_{-1}^{1}\\frac{k}{cos(k)-1}dz$"}
{"_id": "2268677", "title": "", "text": "$\\Omega - \\{ \\omega \\in \\Omega: \\lim_{n \\rightarrow \\infty} X_n(\\omega) = X(\\omega)\\}$"}
{"_id": "290056", "title": "", "text": "$\\begin{array}{11} x=u+1 \\\\ v = y\\\\ J(u,v) = 1 \\\\ u=r\\cos\\theta\\\\v=r\\sin\\theta\\\\0 \\leq r \\leq 1 \\\\ 0 \\leq \\theta \\leq \\pi  \\end{array}$"}
{"_id": "4989820", "title": "", "text": "$d(x,T)=d(S,T)$"}
{"_id": "4921254", "title": "", "text": "$P(X \\leq x \\mid H_{0})$"}
{"_id": "7382793", "title": "", "text": "$\\lim_{x\\rightarrow a}f(x)g(x)=\\lim_{x\\rightarrow a}f(x)\\times\\lim_{x\\rightarrow a}g(x)$"}
{"_id": "6553699", "title": "", "text": "$ 1/x + 4/y + 9/z $"}
{"_id": "2036792", "title": "", "text": "$K\\subset X-Â$"}
{"_id": "1953912", "title": "", "text": "$\\lim_{x \\rightarrow c} f(x) := L$"}
{"_id": "7926167", "title": "", "text": "$7|n^7-n$"}
{"_id": "390121", "title": "", "text": "$ F(x) = \\int_1^x f(t)dt $"}
{"_id": "961258", "title": "", "text": "$\\lfloor \\frac{\\lfloor x \\rfloor}{n} \\rfloor =  \\lfloor \\frac{ bn + r_2}{n} \\rfloor = b$"}
{"_id": "7402009", "title": "", "text": "$\\textrm{coefficient of x^2 in } \\frac{(1+x)^n}{x} + \\frac{((1+x)^{n-3}-1).(1+x)^n}{x^2(1+x)^{n-3}} $"}
{"_id": "4975863", "title": "", "text": "$\\gamma(Tx) = \\gamma(ax) = \\gamma(a)\\gamma(x) = \\gamma(x)$"}
{"_id": "2826207", "title": "", "text": "$c=(a+b,a-b)$"}
{"_id": "8945661", "title": "", "text": "$\\tan t = \\frac{x}{3} $"}
{"_id": "7039727", "title": "", "text": "$ \\pi^{-\\frac s2} \\Gamma (\\frac s2) \\zeta(s) = \\pi^{-\\frac{1-s}{2}} \\Gamma(\\frac{1-s}{2}) \\zeta(1-s)$"}
{"_id": "2502083", "title": "", "text": "$A_1 \\times A_2\\times A_3\\times \\cdots $"}
{"_id": "1360393", "title": "", "text": "$\\mathbb{R}[x]/(x^2+k)$"}
{"_id": "1767281", "title": "", "text": "$E^{(p)}=E^p$"}
{"_id": "784363", "title": "", "text": "$ \\begin{align} \\color{#C00000}{\\frac{(1+x)^{n+1}-1}{x}} &=\\sum_{j=0}^n\\color{#00A000}{\\binom{n+1}{j+1}}x^j \\end{align} $"}
{"_id": "5995689", "title": "", "text": "$B(p,q)=\\Big\\{\\langle x,y\\rangle\\in\\Bbb R^2:a<x\\le b\\text{ and }c<y\\le d\\Big\\}\\;,$"}
{"_id": "6402757", "title": "", "text": "$\\lim_{\\gamma\\to0}\\frac{x^\\gamma-1}\\gamma=\\left.\\frac{\\mathrm d}{\\mathrm d \\gamma}x^\\gamma\\right|_{\\gamma=0}=\\ln x.\\tag{III}$"}
{"_id": "7284443", "title": "", "text": "$(x*y)*z = x+y+z-[x+y]-[x+y+z-[x+y]]$"}
{"_id": "8940905", "title": "", "text": "$\\int_x^{x+2\\pi}f(t)\\cos(t)dt=\\int_x^{x+2\\pi}f(t)\\sin(t)dt=0$"}
{"_id": "6112360", "title": "", "text": "$\\lim_{x \\to a} f(x) = L \\iff \\lim_{x \\to a+}f(x) = L = \\lim_{x \\to  a-}f(x)$"}
{"_id": "6243137", "title": "", "text": "$\\max (C_r)=r$"}
{"_id": "8566734", "title": "", "text": "$f(x)=\\frac{16}{x^2+4}$"}
{"_id": "4416589", "title": "", "text": "$r=\\frac{1+\\sqrt{1-4p^2}}{2p},\\qquad s=\\frac{1-\\sqrt{1-4p^2}}{2p}.$"}
{"_id": "2200322", "title": "", "text": "$d=ax+by.$"}
{"_id": "2209863", "title": "", "text": "$|(f^3(x))'| =  3|f'(x)\\cdot f'(f^2)(x)| \\le \\frac 34<1$"}
{"_id": "170986", "title": "", "text": "$|f(y')|\\leq\\frac{1}{2}|f(a)|$"}
{"_id": "376258", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} b_n = \\lim\\limits_{n\\to\\infty} (\\sqrt{n+\\sqrt{2n}}-(n+\\frac{\\sqrt2}2) ) - $"}
{"_id": "4209665", "title": "", "text": "$\\int_\\gamma df(\\partial/\\partial\\gamma)=f(\\gamma(b))-f(\\gamma(a)),$"}
{"_id": "4060888", "title": "", "text": "$E(m)\\implies E(m+1)$"}
{"_id": "3429634", "title": "", "text": "$\\Gamma(x+1)={\\sqrt {2\\pi x}}\\left({\\frac {x}{e}}\\right)^{x}\\left(1+\\mathcal O\\left({\\frac {1}{x}}\\right)\\right)$"}
{"_id": "4999816", "title": "", "text": "$\\sum_{i=0}^x M(1+r)^{x-i} = M\\sum_{i=0}^x (1+r)^{i} = M\\frac{1-(1+r)^{x+1}}{1-(1+r)}.$"}
{"_id": "1733486", "title": "", "text": "$ (x, y, z) \\mapsto (1, x, y, z, \\frac{1}{2} ( x^2 + y^2 + z^2 ) )$"}
{"_id": "6145190", "title": "", "text": "$|\\mathbb EX^b|\\leq\\mathbb E|X^b|=\\mathbb E|X|^b\\leq \\mathbb E(1+|X|^a)=1+\\mathbb E|X|^a<\\infty$"}
{"_id": "3349619", "title": "", "text": "$(\\frac 1{2^n})$"}
{"_id": "577426", "title": "", "text": "$n^2-(n-1)=n^2-n+1$"}
{"_id": "7748286", "title": "", "text": "$\\varepsilon = \\frac{a + bi \\sqrt{3}}{2}$"}
{"_id": "1768549", "title": "", "text": "$\\alpha^+ = \\alpha +1$"}
{"_id": "754186", "title": "", "text": "$\\lim_{x\\to+\\infty}e^x-(1+\\frac{x}{n})^n=+\\infty$"}
{"_id": "2832593", "title": "", "text": "$\\lim_{x \\to 0} f(x) \\cdot \\lim_{x \\to 0} f(x) = \\lim_{x \\to 0} f(x)f(x) = \\lim_{x \\to 0} f(x + x)$"}
{"_id": "7914456", "title": "", "text": "$A\\in \\mathbb{R}^{m,n} \\Rightarrow \\ \\|A\\|_2 = \\sqrt{\\rho(A^TA)}$"}
{"_id": "3442212", "title": "", "text": "$ \\sum_{m=-\\infty}^\\infty e^{-\\pi m^2 x} = x^{-1/2} \\sum_{n=-\\infty}^\\infty e^{-\\pi n^2/x} $"}
{"_id": "1670513", "title": "", "text": "$\\lim_{n \\to \\infty}n\\mu(E)=\\infty$"}
{"_id": "6915706", "title": "", "text": "$C_r+C_e+(-R,-e)+(R,e)$"}
{"_id": "2391270", "title": "", "text": "$\\|\\frac{\\langle u,v\\rangle}{\\|v\\|^2}v\\|^2=|\\frac{\\langle u,v\\rangle}{\\|v\\|^2}\\times \\|v\\||^2=\\frac{|\\langle u,v\\rangle|^2}{\\|v\\|^2}$"}
{"_id": "5214208", "title": "", "text": "$y = \\frac{d x+k}{x+d}$"}
{"_id": "7238713", "title": "", "text": "$\\int_{-\\infty}^{\\infty} \\frac{1}{(1+x^2)^n}dx$"}
{"_id": "511025", "title": "", "text": "$\\frac{100}{a} + \\frac{100}{b} = 1$"}
{"_id": "6145192", "title": "", "text": "$\\mathbb E|X|^a=\\mathbb E|X^a|<\\infty$"}
{"_id": "144400", "title": "", "text": "$\\lim_{n\\to\\infty} T_n=\\infty$"}
{"_id": "4502785", "title": "", "text": "$p\\circ t$"}
{"_id": "2582457", "title": "", "text": "$(\\mathbb Z[x]/(x^2 - 3))/(1+2√3)$"}
{"_id": "191529", "title": "", "text": "$(1,x,y,x^2,xy,y^2)$"}
{"_id": "4998441", "title": "", "text": "$f_3=g_3=\\frac n2$"}
{"_id": "122557", "title": "", "text": "$\\overline{p_1p_2}$"}
{"_id": "6957199", "title": "", "text": "$[x,y]=h,[h,x]=x$"}
{"_id": "956631", "title": "", "text": "$I(a, b) = \\int_0^{2 \\pi} \\frac{1 -  a \\sin(\\theta) -  b \\cos(\\theta)}{(1 +a^2 +b^2 - 2 a \\sin(\\theta) - 2 b \\cos(\\theta) )^{3/2}} \\mathrm{d} \\theta, \\quad 0 \\le a, b < 1$"}
{"_id": "8750138", "title": "", "text": "$A=\\textrm{Spec} (\\mathbf{R}[x]/(x^2+1))$"}
{"_id": "3130385", "title": "", "text": "$f(n^2-n)=f(n)^2-f(n)$"}
{"_id": "1661812", "title": "", "text": "$ \\int_0^1\\frac{\\tan^{-1}(x)}{1+x}\\,\\mathrm{d}x=\\frac\\pi8\\log(2) $"}
{"_id": "4085553", "title": "", "text": "$\\int_{-\\infty}^\\infty  \\sin(ax) \\log\\Big(\\frac{x+1}{x-1} \\Big) \\,dx=\\frac{2 \\pi  (\\sin (a)-i \\cos (a))}{a}$"}
{"_id": "4893463", "title": "", "text": "$\\{a,a+d,a+2d, \\cdots, a+(n-1)d \\}$"}
{"_id": "7553746", "title": "", "text": "$f(x+f(y)) = f(x)+f(y)$"}
{"_id": "6190023", "title": "", "text": "$x^2+y^2=4 \\Bigl(1-2\\sin^2 (t)\\cos^2 (t)\\Bigr)$"}
{"_id": "909978", "title": "", "text": "$\\bigg\\lfloor\\frac{\\lfloor 4.5 \\rfloor }{5}\\bigg\\rfloor \\overset{?}=\\bigg\\lfloor\\frac{4.5  }{5}\\bigg\\rfloor $"}
{"_id": "8942724", "title": "", "text": "$σ(x,t)=σ·x$"}
{"_id": "312337", "title": "", "text": "$\\dfrac 2a+\\dfrac 2b=1$"}
{"_id": "2333914", "title": "", "text": "$A=\\{\\{1\\}\\}$"}
{"_id": "7522466", "title": "", "text": "$\\limsup P[X_n< x]\\leq P[X<x]$"}
{"_id": "4791182", "title": "", "text": "$V=P\\oplus L\\oplus W$"}
{"_id": "3682403", "title": "", "text": "$(1+\\frac{x}{n})^n=e^x(1+O(n^{-1}))$"}
{"_id": "6531232", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}[(1+i)^n+(1-i)^n]$"}
{"_id": "4356297", "title": "", "text": "$H = P \\ltimes T$"}
{"_id": "5049722", "title": "", "text": "$\\displaystyle \\sum_{n\\geq1}\\frac1{n^2}.$"}
{"_id": "537020", "title": "", "text": "$F(n) = F(n-2) + F(n-3) + 2n$"}
{"_id": "1700044", "title": "", "text": "$\\neg\\alpha\\lor \\neg\\neg\\alpha$"}
{"_id": "6990865", "title": "", "text": "$\\gamma_0 \\subseteq \\gamma_1 \\subseteq \\gamma_2...$"}
{"_id": "2486538", "title": "", "text": "$ \\lim_{n\\to\\infty}F[u_{l_n}]=\\liminf_{n\\to\\infty}F[u_n] $"}
{"_id": "7367234", "title": "", "text": "$\\mathcal{X}(\\mathcal{A}) = \\mathcal{X}(\\mathcal{B})$"}
{"_id": "9265591", "title": "", "text": "$G/K = \\mathbb{Z}_2^m \\implies G \\simeq \\mathbb{Z}_2^{m+1}$"}
{"_id": "5054005", "title": "", "text": "$16/\\!/2 = 8$"}
{"_id": "7743387", "title": "", "text": "$Cov(S,D)=0$"}
{"_id": "5580242", "title": "", "text": "$A = \\{v_1,v_2,....,v_n\\}$"}
{"_id": "1248970", "title": "", "text": "$\\mathbb{E}|\\zeta_n|\\le c_k[\\mathbb{E}|X_n-X|^k+\\mathbb{E}|\\zeta|]$"}
{"_id": "5473903", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx = \\int_{0}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx + \\int_{0}^{\\pi}\\frac {\\sin ny}{(1+2^{-y})\\sin y}dy$"}
{"_id": "2256974", "title": "", "text": "$d(x,a)\\leqslant d(x,y)+d(y,a)<d(x,y)+r$"}
{"_id": "1148786", "title": "", "text": "$\\Delta^+$"}
{"_id": "4235389", "title": "", "text": "$2\\sin^2\\varphi = 1 - \\cos(2\\varphi)$"}
{"_id": "2320543", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} X_n(\\omega)$"}
{"_id": "7596772", "title": "", "text": "$\\mathbb{R}^{n+1}/\\mathbb{R}_+$"}
{"_id": "6755202", "title": "", "text": "$ \\mathbb E(|X_n^p-X^p|)\\leqslant p\\mathbb E(|X_n-X|\\,|X_n|^{p-1})+p\\mathbb E(|X_n-X|\\,|X|^{p-1}). $"}
{"_id": "8634760", "title": "", "text": "$ z_1 = e^{\\frac{i \\pi}{4}} $"}
{"_id": "8976386", "title": "", "text": "$\\cos k\\frac{2\\pi}{n}\\cos (k+1)\\frac{2\\pi}{n}+\\sin k\\frac{2\\pi}{n}\\sin (k+1)\\frac{2\\pi}{n}=\\cos[(k+1)-k]\\left(\\frac{2\\pi}{n}\\right)=\\cos\\frac{2\\pi}{n}$"}
{"_id": "3038309", "title": "", "text": "$\\lim_{N\\to \\infty}(-\\sqrt1 - \\sqrt2 - ... -\\sqrt{N} + \\sqrt2 + \\sqrt3 +...+\\sqrt{N} + \\sqrt{N+1})$"}
{"_id": "4001103", "title": "", "text": "$\\color{red}{\\det \\left(\\begin{matrix}  A& -B\\\\B&A \\end{matrix}\\right)= \\bigg|\\det(A+iB)\\bigg|^2 }$"}
{"_id": "5564544", "title": "", "text": "$\\sum_{n=1}^\\infty\\|f_n-e_n\\|^2<1$"}
{"_id": "4102550", "title": "", "text": "$ P \\oplus F = P \\oplus(\\underset{i \\in \\mathbb{N}}{\\oplus}F_0) = P \\underset{i \\in \\mathbb{N}} \\oplus (P\\oplus T ) =$"}
{"_id": "6743767", "title": "", "text": "$P(m/p) \\implies P(m)$"}
{"_id": "2074886", "title": "", "text": "$(a+b, a-b) = d$"}
{"_id": "1754867", "title": "", "text": "$|\\alpha|^+$"}
{"_id": "6097228", "title": "", "text": "$x(f(x+a) - f(x)) = nf(x)$"}
{"_id": "2288951", "title": "", "text": "$\\log_a (BC) = \\log_a (B) + \\log_a (C)$"}
{"_id": "7292206", "title": "", "text": "$\\log_a b + \\log_a c = \\log_a bc$"}
{"_id": "4684112", "title": "", "text": "$\\overline{V_1}\\subseteq V_2^c\\subseteq U$"}
{"_id": "2948073", "title": "", "text": "$x \\notin y \\lor y \\notin x$"}
{"_id": "7212421", "title": "", "text": "$ \\zeta(s)=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\eta(s)= \\frac{2^s}{\\Gamma(s)(2^s-2)}\\int_{0}^{+\\infty}\\frac{x^{s-1}}{e^x+1}\\,dx\\tag{1}$"}
{"_id": "5343184", "title": "", "text": "$\\displaystyle L = \\lim_{n\\rightarrow \\infty}\\sum_{r=1}^{n}\\frac{1}{2^r} = \\lim_{n\\rightarrow \\infty}\\frac{1}{2}+\\frac{1}{2^2}+\\cdots+\\frac{1}{2^n}$"}
{"_id": "1283321", "title": "", "text": "$\\|f\\|_K^2=\\int_0^K f(t)^Tf(t) \\, dt\\leq \\int_0^\\infty f(t)^Tf(t) \\, dt$"}
{"_id": "7663376", "title": "", "text": "$\\{1,-1,-2-n,1+n,1+2+3(k-l),-1-2-2(k-l-1)..\\},l<k$"}
{"_id": "6862406", "title": "", "text": "$(y-|y|,y+|y|)$"}
{"_id": "2867529", "title": "", "text": "$ F_n(x) = \\int_a^x f_n(t)dt, \\hspace{2mm}  F(x) = \\int_a^x f(t)dt $"}
{"_id": "1611487", "title": "", "text": "$c_n=(-1)^n\\frac{1}{n!}<b_n=\\frac{1}{2^n} \\text{ for $n>3$},$"}
{"_id": "8262949", "title": "", "text": "$B \\sim \\left( \\begin{array}{ccc} 1 & x & x^{2} \\\\ 1 & y & y^{2} \\\\ 1 & z & z^{2} \\end{array} \\right) \\sim \\left( \\begin{array}{ccc} 1 & x & x^{2} \\\\ 0 & y - x & y^{2} - x^{2} \\\\ 0 & z - x & z^{2} - x^{2} \\end{array} \\right)$"}
{"_id": "54112", "title": "", "text": "$d(x,z)\\leq d(x,y)+d(y,z)$"}
{"_id": "3253850", "title": "", "text": "$ \\sum_{k=1}^{\\infty}\\mu(A_{k}):=\\lim_{n\\to\\infty}\\sum_{k=1}^{n}\\mu(A_{k})\\leq \\mu(A). $"}
{"_id": "5085449", "title": "", "text": "$\\frac{1}{2^{x+2}}$"}
{"_id": "5997747", "title": "", "text": "$ \\tag{2} f(x)^2+f(y)^2 = f(x+y)(f(x)+f(f(y))) $"}
{"_id": "7719233", "title": "", "text": "$2^x + 3^x = 6^x + 6$"}
{"_id": "6528942", "title": "", "text": "$\\overline f(a) \\stackrel{\\cdot}{+} \\overline f(b) = \\overline f(a+b)$"}
{"_id": "1971885", "title": "", "text": "$A_0 \\subset A_1 \\subset \\ldots \\subset A_n$"}
{"_id": "4591274", "title": "", "text": "$ \\sum_{j=1}^\\infty\\sum_{k=1, k\\neq j}^\\infty \\frac{1}{j^2-k^2}$"}
{"_id": "4855980", "title": "", "text": "$\\{f \\in \\mathbb{Z}^{\\mathbb{N}}: \\forall n \\in \\mathbb{N} \\ \\ f(2n) + f(2n+1) = 0\\}$"}
{"_id": "4772040", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} \\sum_{k=1}^{n}\\cos\\left( \\frac{2\\pi k}{2n+1} \\right)$"}
{"_id": "5343907", "title": "", "text": "$\\begin{matrix} a & b \\\\ -b & a\\end{matrix}$"}
{"_id": "8374373", "title": "", "text": "$P^\\mu[X_{S+t}\\in\\Gamma|X_S]=0$"}
{"_id": "5023887", "title": "", "text": "$\\lim_{n\\to\\infty} inf A_n=\\{0\\}$"}
{"_id": "8779604", "title": "", "text": "$f(x) =\\sum_{j=1}^n a_j\\chi_{E_j},$"}
{"_id": "6299485", "title": "", "text": "$\\sum_{n \\geq 1} \\frac{1}{p_n} < \\sum_{n \\geq 1} \\frac{1}{n^2} = \\frac{\\pi^2}{6} < \\infty.$"}
{"_id": "1958022", "title": "", "text": "$ (x^a-y^a)(x^b-y^b)=x^{a+b}+y^{a+b} -x^{a}y^{b} -x^{b}y^a$"}
{"_id": "6079100", "title": "", "text": "$|ab|=ab \\tag{eq1}$"}
{"_id": "8079432", "title": "", "text": "$(x_2,y_2,z_2) = (1,1,1)$"}
{"_id": "8194947", "title": "", "text": "$\\begin{align} \\int_{0}^{\\infty}f(x)dx \\end{align}$"}
{"_id": "6197796", "title": "", "text": "$\\sum_{n \\leq x} \\frac{1}{n} \\left\\lfloor \\frac{x}{n} \\right\\rfloor = \\sum_{n \\leq x} \\frac{x}{n^2}  + O(1)$"}
{"_id": "5858016", "title": "", "text": "$\\int_0^\\infty f_n(x)dx=\\infty$"}
{"_id": "4111324", "title": "", "text": "$A_1^c \\cap A_2^c \\ldots A_{n-1}^c \\cap A_n \\subseteq A_n$"}
{"_id": "4714840", "title": "", "text": "$\\frac{2}{2^n} = \\frac{1}{2^{n-1}}$"}
{"_id": "4141619", "title": "", "text": "$P(X = k) = \\binom {3}{k} p^k(1-p)^{3-k}$"}
{"_id": "5239459", "title": "", "text": "$\\frac{\\sqrt{1+\\frac xa}}{\\sqrt{1}+\\sqrt{1+\\frac xa}}=\\frac{\\sqrt{1-\\frac xa}}{\\sqrt{1}-\\sqrt{1-\\frac xa}}$"}
{"_id": "8801903", "title": "", "text": "$1\\le n \\leq e_i,$"}
{"_id": "7823728", "title": "", "text": "$(a, b)+_\\mathbb{C}(c, d)=(a+_\\mathbb{R}b, c+_\\mathbb{R}d)$"}
{"_id": "125748", "title": "", "text": "$K=\\frac{1}{1980}+\\frac{1}{1981}+\\frac{1}{1982}+........+\\frac{1}{2012}<\\frac{1}{1980}+\\frac{1}{1980}+\\frac{1}{1980}+........+\\frac{1}{1980}=\\frac{33}{1980}=\\frac{1}{60}$"}
{"_id": "3178851", "title": "", "text": "$\\left(\\text{a}+\\text{b}\\right)^2=\\left(\\text{a}+\\text{b}\\right)\\left(\\text{a}+\\text{b}\\right)=\\text{a}\\text{a}+\\text{a}\\text{b}+\\text{a}\\text{b}+\\text{b}\\text{b}=\\text{a}^2+\\text{b}^2+2\\text{a}\\text{b}$"}
{"_id": "4323991", "title": "", "text": "$\\overline{V_1} \\subseteq {V_2}^c \\subseteq U$"}
{"_id": "3779430", "title": "", "text": "$ \\mathcal A'=\\{a\\setminus\\{\\gamma\\}\\mid a\\in\\mathcal A\\mbox{ and }\\min(a)=\\gamma\\}, $"}
{"_id": "167002", "title": "", "text": "$\\int^1_0\\frac{\\sin^2(x)}{x^2}dx$"}
{"_id": "1471142", "title": "", "text": "$f(f(x)+2f(y))=f(x)+2y \\,.$"}
{"_id": "5064729", "title": "", "text": "$|x^3-x_0^3|\\le\\delta(\\delta^2+3\\delta|x_0|+2|x_0|^2)$"}
{"_id": "865521", "title": "", "text": "$\\lim_{k\\rightarrow \\infty}  \\frac{2}{3 (\\sqrt{k+1} - \\sqrt{k-1})(\\sqrt{k+2} + \\sqrt{k})} = \\frac{1}{3}$"}
{"_id": "7155018", "title": "", "text": "$ \\frac{|\\langle u,v\\rangle |}{\\|u\\|} $"}
{"_id": "7892640", "title": "", "text": "$\\ds{\\int_{\\gamma -\\ic\\infty}^{\\gamma + \\ic\\infty}      {s\\expo{st} \\over \\root{\\pars{s + a}^3}}\\,{\\dd s \\over 2\\pi\\ic}      =\\int_{\\gamma -\\ic\\infty}^{\\gamma + \\ic\\infty}      s\\pars{s + a}^{-3/2}\\expo{st}\\,{\\dd s \\over 2\\pi\\ic}\\,,\\quad\\gamma > \\verts{a}\\ \\mbox{and}\\ t > 0.\\quad}$"}
{"_id": "2208420", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 2 & 2 & 0 &2 \\\\ 0 &k &1 &1 \\\\ 1 &2 &k&2 \\end{array}\\right]$"}
{"_id": "7107013", "title": "", "text": "$f(x) = \\displaystyle \\frac{2x}{(x^2+1)^2}$"}
{"_id": "2328892", "title": "", "text": "$[-B, A]$"}
{"_id": "3130484", "title": "", "text": "$ \\left( \\begin{array}{ccc|c} 1&1&b&1\\\\b&3&-1&-2\\\\3&4&1&c \\end{array} \\right) $"}
{"_id": "7469348", "title": "", "text": "$v_n=\\frac{u_1+u_2+...+u_n}{n}$"}
{"_id": "1213432", "title": "", "text": "$\\binom{k}{2} + \\binom{n-k}{2} + k(n-k) = \\binom{n}{2}$"}
{"_id": "1862749", "title": "", "text": "$[L_i,L_\\alpha]\\subseteq L_i$"}
{"_id": "199312", "title": "", "text": "$T^{2, 0} M \\otimes T^{0, 2} M \\to T^{1, 1} \\cong \\operatorname{End} TM$"}
{"_id": "2343743", "title": "", "text": "$\\lim_{x \\to 0^-} f'(x) = 0 = \\lim_{x \\to 0^+} f'(x) = (e^x - 1)\\big |_{x =0}.$"}
{"_id": "3058640", "title": "", "text": "$\\int_0^1\\frac1{(1+x^3)^{n}}dx$"}
{"_id": "7511164", "title": "", "text": "$\\int_{0}^{\\infty}f(x)\\text dx $"}
{"_id": "1250100", "title": "", "text": "$g[x1] = x2 + zg$"}
{"_id": "8858506", "title": "", "text": "$\\left(\\begin{smallmatrix}2&1&0&5\\\\-1&1&1&6\\\\5&1&-1&4\\\\5&1&3&0\\end{smallmatrix}\\right)$"}
{"_id": "6974927", "title": "", "text": "$A_0 \\times\\dotsc\\times A_k$"}
{"_id": "961158", "title": "", "text": "$\\vartheta(z)$"}
{"_id": "3465305", "title": "", "text": "$S_n:=\\sum_{k=1}^n\\cos\\frac{k\\pi}{n+1},$"}
{"_id": "6180517", "title": "", "text": "$\\Delta''= \\begin{vmatrix} x&y&z\\\\ x^2&y^2&z^2\\\\ x^3&y^3&z^3\\\\ \\end{vmatrix}$"}
{"_id": "1127920", "title": "", "text": "$T_xX+T_xY=T_xZ$"}
{"_id": "8379616", "title": "", "text": "$\\mathfrak{g}=\\mathfrak{p} \\oplus \\mathfrak{t}$"}
{"_id": "6000379", "title": "", "text": "$\\sigma_1^2 = \\frac{t}{2}$"}
{"_id": "3411345", "title": "", "text": "$ \\log \\mathbb{E} |X|-\\mathbb{E}\\log |X|\\leq\\mathbb{E}\\left[\\frac{1}{2}\\left(1+|X|\\right)(\\log|X|)^2\\right]$"}
{"_id": "4579009", "title": "", "text": "$\\begin{pmatrix}a&-\\bar{b}\\\\b&\\bar{a}\\end{pmatrix}$"}
{"_id": "5807402", "title": "", "text": "$\\epsilon \\equiv \\frac{1}{4!} \\epsilon^{abcd} \\gamma_a \\gamma_b \\gamma_c \\gamma_d = \\gamma_0 \\gamma_1 \\gamma_2 \\gamma_3 \\implies \\epsilon^{abcd} = \\epsilon \\cdot (\\gamma^d \\wedge \\gamma^c \\wedge \\gamma^b \\wedge \\gamma^a)$"}
{"_id": "7631000", "title": "", "text": "$i\\in\\{a,a+1,a+2,\\cdots,b\\}$"}
{"_id": "5100256", "title": "", "text": "$\\begin{align} \\lim_{N\\to \\infty}\\prod_{k=0}^N(1+z^{2^k})&=\\lim_{N\\to \\infty}\\sum_{k=0}^{2^{N+1}-1}z^k\\\\\\\\ &=\\lim_{N\\to \\infty}\\frac{1-z^{2^{N+1}}}{1-z}\\\\\\\\ &=\\frac{1}{1-z} \\end{align}$"}
{"_id": "925640", "title": "", "text": "$4^x+9^x=2\\cdot 6^x$"}
{"_id": "6333123", "title": "", "text": "$\\int_0^ag(x)\\cos(x)dx=\\int_0^\\infty g(x)\\cos(x)dx=0$"}
{"_id": "5984118", "title": "", "text": "$log_a (b)=c$"}
{"_id": "2955330", "title": "", "text": "$\\sum_{n\\geq1} u_n=\\frac12\\sum_{n\\geq1} u_{n-1}+\\sum_{n\\geq1}\\frac1{2^n}$"}
{"_id": "142231", "title": "", "text": "$ f_x(x) = \\frac {2x}{10^{(2)}} $"}
{"_id": "8157848", "title": "", "text": "$A=A_0 \\times ... \\times A_n$"}
{"_id": "8649266", "title": "", "text": "$E(x_n e_n) = E(x_n)E(e_n)=0$"}
{"_id": "6216239", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\sum_{k=1}^{\\infty}\\frac{\\cos(\\frac{\\pi k}{3})\\cos(\\frac{\\pi n}{3})}{kn^{2}}$"}
{"_id": "8870420", "title": "", "text": "$\\lim_{n\\to \\infty} \\mathbb{P}(A_n)=1$"}
{"_id": "9116442", "title": "", "text": "$(a,\\gamma) (b,\\delta) = (ab^{\\gamma_\\alpha},\\gamma \\delta)$"}
{"_id": "3242451", "title": "", "text": "$A_1\\times...\\times A_m$"}
{"_id": "6504995", "title": "", "text": "$\\left[Q[x]/(x^2+1)^2\\right]^{n_1}\\oplus \\left[Q[x]/(x^2+1)\\right]^{n_2}$"}
{"_id": "3786087", "title": "", "text": "$|ab|=0$"}
{"_id": "9133387", "title": "", "text": "$\\|x_n-x_m\\| \\leq 1$"}
{"_id": "5064075", "title": "", "text": "$S = \\{\\{a\\}, \\{a,b\\}\\}$"}
{"_id": "8753643", "title": "", "text": "$\\int 2\\pi y dx $"}
{"_id": "6402664", "title": "", "text": "$\\mathbf{x}:=(x,y)$"}
{"_id": "8881679", "title": "", "text": "$\\tan u = \\frac{x}{1}.$"}
{"_id": "3836203", "title": "", "text": "$x \\in V_{i,x} \\subseteq \\overline{V_{i,x}} \\subseteq U_i$"}
{"_id": "4680530", "title": "", "text": "$ = \\frac {1}{n} (\\frac{log_{c}x}{log_{c} b})$"}
{"_id": "2525189", "title": "", "text": "$\\hat f_n(t)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}\\frac{\\sin x\\sin nx}{x^2}e^{-ixt}dx$"}
{"_id": "8364915", "title": "", "text": "$\\sum\\limits_{r=1}^{n} (r^2+1)(r!)$"}
{"_id": "6018710", "title": "", "text": "$\\tan \\theta = \\frac{x-y}2$"}
{"_id": "7625502", "title": "", "text": "$\\left\\lfloor\\frac{c}{ab}\\right\\rfloor$"}
{"_id": "2474239", "title": "", "text": "$\\sum_{n=1}^\\infty \\|f_n\\|_\\infty = \\sum_{n=1}^\\infty 1/n =\\infty.  $"}
{"_id": "2775491", "title": "", "text": "$\\begin{cases}x=2\\mp1,\\\\ y=2\\pm1,\\\\ z=2,\\end{cases}$"}
{"_id": "6207387", "title": "", "text": "$P_1(-.6\\mid 1.2),P_2(1\\mid 2)$"}
{"_id": "8096316", "title": "", "text": "$ 2a < a+b \\\\ a+b < 2b $"}
{"_id": "829279", "title": "", "text": "$ \\lim_{n\\rightarrow \\infty} \\frac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+.....+\\sqrt{n}-\\frac23n\\sqrt{n}}{\\sqrt{n}} = \\frac12. $"}
{"_id": "5704662", "title": "", "text": "$\\|x_n - y_n \\|< 2\\epsilon_n$"}
{"_id": "8722697", "title": "", "text": "$f(x)=1\\wedge x$"}
{"_id": "3433303", "title": "", "text": "$\\lim_{N \\to +\\infty}\\frac{1}{N}\\sum_{k=1}^{N}\\frac{\\phi(k)}{k}=\\frac{6}{\\pi^2}$"}
{"_id": "6893955", "title": "", "text": "$u_\\epsilon(x) = \\frac{2x}{x^2+\\epsilon^2}$"}
{"_id": "8776634", "title": "", "text": "$\\|A\\| = \\sqrt{\\mathrm{tr}(A^* A)}. $"}
{"_id": "1532480", "title": "", "text": "$B\\subset V\\subset\\overline{V}\\subset U$"}
{"_id": "8958432", "title": "", "text": "$dist(v,U)=\\frac {\\lvert \\left\\langle v,u \\right\\rangle\\rvert}{\\lVert u\\rVert}$"}
{"_id": "8403648", "title": "", "text": "$\\begin{align}\\int\\frac{1}{\\sqrt{x^2+1}}\\,dx&=\\int\\frac{\\sec^2 t}{\\sec t}\\,dt \\\\&=\\int\\sec t\\,dt \\\\&=\\ln(\\tan t+\\sec t)+C \\\\&=\\ln\\left(x+\\frac{1}{\\sqrt{x^2+1}}\\right)+C\\end{align}$"}
{"_id": "5477702", "title": "", "text": "$\\int\\int z^2\\,dS,$"}
{"_id": "2074888", "title": "", "text": "$(a+b,a-b) = (a+b,2b)$"}
{"_id": "9114053", "title": "", "text": "$f(a + 1) = f(a) + f(1)$"}
{"_id": "4420045", "title": "", "text": "$(a,b)_\\mathbb{Q}=(a,c)_\\mathbb{Q} \\cup (c,b)_\\mathbb{Q}.$"}
{"_id": "5054006", "title": "", "text": "$2/\\!/16 = 1$"}
{"_id": "8265485", "title": "", "text": "$b^y=a^{y\\log_a(b)}=x$"}
{"_id": "5214727", "title": "", "text": "$\\alpha^{\\sup_{\\gamma<\\beta}\\{\\gamma\\}}=\\sup_{\\gamma<\\beta}\\{\\alpha^\\gamma\\}$"}
{"_id": "5979329", "title": "", "text": "$G(x)=P[T\\geq x|H_{0}]$"}
{"_id": "67986", "title": "", "text": "$S(\\lambda_1)=7\\sqrt{\\frac{2}{15}}$"}
{"_id": "107742", "title": "", "text": "$\\gamma^q-\\gamma=0 \\implies \\gamma^{-1}(\\gamma^q-\\gamma)=\\gamma^{q-1}-\\gamma^0=\\gamma^{q-1}-1=0\\implies \\gamma^{q-1}=1 \\implies (\\gamma^{q-1})^{1/2}=\\gamma^{(q-1)/2}=1^{1/2}=1$"}
{"_id": "6312926", "title": "", "text": "$1/|\\log x|$"}
{"_id": "5237666", "title": "", "text": "$\\lim_{n\\to \\infty}{\\inf[a_n]}\\le\\lim_{n\\to \\infty}{\\inf[b_n]}$"}
{"_id": "196326", "title": "", "text": "$a\\in\\mathbb{F}$"}
{"_id": "5186104", "title": "", "text": "$(\\mathrm{pr}_1(\\tilde K)\\cup \\mathrm{pr}_2(\\tilde K)) \\subseteq V \\subseteq \\overline V \\subseteq  \\Omega$"}
{"_id": "2136200", "title": "", "text": "$ (x+y)+z=x+y+z=y+z+x=(y+z)+x=x+(y+z) $"}
{"_id": "4637546", "title": "", "text": "$\\sum_{n=1}^\\infty \\frac{\\cos \\pi n}{2^n}$"}
{"_id": "5666423", "title": "", "text": "$ \\ \\pi \\ \\int \\ y^2 \\ dx \\ $"}
{"_id": "316151", "title": "", "text": "$D^{n+1}\\times S^{m-n}$"}
{"_id": "2535627", "title": "", "text": "$T^{0,1}X$"}
{"_id": "7366584", "title": "", "text": "$s(x)= \\int \\sqrt{1+ \\Big(\\frac{dy}{dx}\\Big)^2} dx$"}
{"_id": "8455287", "title": "", "text": "$a = \\lim a_n = \\sup a_n = \\sup_{n}\\inf X_n\\\\b=\\lim b_n = \\inf b_n = \\inf_{n} \\sup X_n$"}
{"_id": "8382256", "title": "", "text": "$x̃=\\frac{x_1+x_2+\\dots+x_n}{n}$"}
{"_id": "4613352", "title": "", "text": "$ \\left[ \\begin{array}{ccc|c}   1&5&-6&2\\\\   k&1&-1&3\\\\   5&-k&3&7 \\end{array} \\right] $"}
{"_id": "2842859", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} \\inf f(x_n) \\geq f(x).$"}
{"_id": "4803994", "title": "", "text": "$\\lim \\limits_{n \\to \\infty}\\left( 1 + \\sqrt{2} + \\sqrt[3]{3} + \\dots \\sqrt[n]{n} \\right) \\ln{2n+1 \\over n}$"}
{"_id": "5123607", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\frac{\\sum_{i=1}^{n}x_{i}}{n}=\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}\\frac{x_{i}}{n}=\\sum_{i=1}^{n}\\lim_{n\\rightarrow\\infty}\\frac{x_{i}}{n}=\\sum_{i=1}^{n}0=0$"}
{"_id": "712115", "title": "", "text": "$\\{e_1, e_2, \\ldots, e_{n+1} \\}$"}
{"_id": "7688244", "title": "", "text": "$g(c_r)=r$"}
{"_id": "2759605", "title": "", "text": "$b^{log_n(a)}=n^{log_n(b)log_n(a)}$"}
{"_id": "5876921", "title": "", "text": "$T^n = T^{n+1} = T^{(n+1) + 1} = T^{n+2} = \\dotsc = T^{2n} = (T^n)^2$"}
{"_id": "1866488", "title": "", "text": "$\\frac{z}{xy}\\,\\frac{\\displaystyle\\prod_{p\\text{ prime}}\\,\\left(1-\\frac{1}{p^2}\\right)}{\\displaystyle \\prod_{\\substack{{p\\text{ prime}}\\\\{p\\mid \\gcd(xy,z)}}}\\,\\left(1+\\frac{1}{p}\\right)}\\gtrsim \\frac{z}{\\zeta(2)\\,xy\\,\\ln(xy)}\\gtrsim 1\\,.$"}
{"_id": "6301241", "title": "", "text": "$\\log_aa^b=b,\\log_a(ab)=\\log_aa+\\log_ab=1+\\log_ab$"}
{"_id": "1844678", "title": "", "text": "$|e_i| \\le 1$"}
{"_id": "672556", "title": "", "text": "$-\\log_a(b)$"}
{"_id": "5679221", "title": "", "text": "$\\sum_{k=1}^n \\sin(2 \\pi j k/n) = 0$"}
{"_id": "5678898", "title": "", "text": "$f^{-1}(f(x) + f(y)) = \\frac{x + y}{1 + xy}$"}
{"_id": "2176254", "title": "", "text": "$\\begin{gathered}   \\frac{{113}}{{50}} = \\frac{{100}}{{50}} + \\frac{{13}}{{50}} = 2 + \\frac{1}{{\\frac{{39}}{{13}} + \\frac{{11}}{{13}}}} = 2 + \\frac{1}{{3 + \\frac{{11}}{{13}}}} \\hfill \\\\    = 2 + \\frac{1}{{3 + \\frac{{11}}{{11 + 2}}}} = 2 + \\frac{1}{{3 + \\frac{1}{{1 + \\frac{2}{{10 + 1}}}}}} \\hfill \\\\    = 2 + \\frac{1}{{3 + \\frac{1}{{1 + \\frac{1}{{5 + \\frac{1}{2}}}}}}} = [2;3,1,5,2] \\hfill \\\\  \\end{gathered}$"}
{"_id": "8754399", "title": "", "text": "$p_1 p_2 p$"}
{"_id": "8853330", "title": "", "text": "$\\langle Ax,x\\rangle=\\langle(-m+in)x,x\\rangle= (-m+in)\\langle x,x\\rangle=(-m+in)\\|x\\|^2$"}
{"_id": "5480282", "title": "", "text": "$\\left\\lfloor \\frac{\\left\\lfloor\\frac ab \\right\\rfloor}c \\right\\rfloor =\\left\\lfloor \\frac{\\left\\lfloor\\frac ac \\right\\rfloor}b \\right\\rfloor  $"}
{"_id": "899862", "title": "", "text": "$\\displaystyle \\begin{vmatrix} a &a^3  &1 \\\\  b &b^3  &1 \\\\  c &c^3  &1 \\end{vmatrix}$"}
{"_id": "1276168", "title": "", "text": "$D = \\{(x,y)\\in\\mathbb{R}^2: 0\\leq x \\leq 1, 0\\leq y \\leq x\\}$"}
{"_id": "2515548", "title": "", "text": "$\\begin{cases} x_1=1+x_2+2x_3 \\\\ x_2=s \\\\ x_3=t \\end{cases}$"}
{"_id": "7838900", "title": "", "text": "$S=\\{e_1,e_2,e_3\\}$"}
{"_id": "3659836", "title": "", "text": "$\\zeta(-s) = -2^{-s}\\pi^{-s-1}\\sin (\\frac{\\pi s}{2}) \\Gamma(1+s)\\zeta(1+s)$"}
{"_id": "3588118", "title": "", "text": "$(a + b, b - (a + b)) = (a + b, -a)$"}
{"_id": "1204120", "title": "", "text": "$P(AB)=1/6$"}
{"_id": "5737190", "title": "", "text": "${N \\choose r} \\frac{1}{2^N}$"}
{"_id": "3909393", "title": "", "text": "$\\mu (\\lim\\inf_n A_n) \\le \\lim\\inf_n \\mu( A_n) \\le  \\lim\\sup_n \\mu( A_n) \\le  \\mu (\\lim\\sup_n A_n)$"}
{"_id": "4498346", "title": "", "text": "$f(n)=(n-2)$"}
{"_id": "6963026", "title": "", "text": "$\\delta(n,a) = \\delta(n+1,a),$"}
{"_id": "7913961", "title": "", "text": "$|ab|=\\alpha$"}
{"_id": "6997773", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{n \\left (1-2^{-1-n} \\right )}{n+1} \\sum_{k=1}^n \\frac{1}{2^k} = 1$"}
{"_id": "4504982", "title": "", "text": "$\\sum_{i=0}^{n-1} \\cos\\left(\\frac{2\\pi i}{n}\\right) = 0 \\tag{1} \\label{1} $"}
{"_id": "7022245", "title": "", "text": "$\\left| A \\right| = \\left| {\\matrix{    {x + y} & {xy} & 0 &  \\cdots  &  \\cdots  & 0  \\cr     1 & {x + y} & {xy} &  \\cdots  &  \\cdots  & 0  \\cr     0 & 1 & {x + y} &  \\cdots  &  \\cdots  & 0  \\cr      \\cdots  &  \\cdots  &  \\cdots  &  \\cdots  &  \\cdots  &  \\vdots   \\cr     0 &  \\cdots  & 0 & 1 & {x + y} & {xy}  \\cr     0 &  \\cdots  & 0 & 0 & 1 & {x + y}  \\cr   } } \\right|$"}
{"_id": "766513", "title": "", "text": "$f(x)= f(a)+\\int_a^x f'(t)dt$"}
{"_id": "3075609", "title": "", "text": "$r \\equiv s$"}
{"_id": "9150615", "title": "", "text": "$A_1, A_2, \\ldots,A_{\\ell-1}, A_{\\ell+1}\\ldots \\ldots, A_{m} $"}
{"_id": "12728", "title": "", "text": "$\\mathbb{Z}^{+}$"}
{"_id": "7097689", "title": "", "text": "$\\begin{pmatrix}r&p\\\\p&-r\\end{pmatrix}$"}
{"_id": "1826412", "title": "", "text": "$ \\begin{bmatrix} a & -b \\\\ b & a \\end{bmatrix}. $"}
{"_id": "4012766", "title": "", "text": "${\\|A\\|}_2 = \\infty$"}
{"_id": "6653897", "title": "", "text": "$|ab| = |ba|.$"}
{"_id": "1932203", "title": "", "text": "$\\frac{4}{x}+2x+10+\\frac{3+x}{4x^2+1}$"}
{"_id": "3745912", "title": "", "text": "$0 for (x,y) = 0$"}
{"_id": "4889224", "title": "", "text": "$ \\mathcal{R} = \\begin{cases} 0\\leq x \\leq 60\\\\ 0 \\leq y \\leq 60 \\\\ y \\leq x+15\\\\ y \\geq x-15 \\end{cases} $"}
{"_id": "88404", "title": "", "text": "$ax+by=N$"}
{"_id": "722769", "title": "", "text": "$\\mathbb{R}^{n+1}\\to \\mathbb{R}$"}
{"_id": "6643085", "title": "", "text": "$I=-\\int_{\\pi}^{-\\pi}\\dfrac{\\sin n(-x)}{(1+(\\pi)^{-x})\\sin (-x)} dx$"}
{"_id": "4344337", "title": "", "text": "$(\\Bbb Z[i]/(1+i)^3)^*$"}
{"_id": "3165017", "title": "", "text": "$\\mathscr{C} = \\{\\{a\\},\\{b\\}\\}$"}
{"_id": "3617836", "title": "", "text": "$M_n = \\dfrac{n(n^2+1)}{2}$"}
{"_id": "4998440", "title": "", "text": "$f_3+g_3=n$"}
{"_id": "7980827", "title": "", "text": "$a=\\frac{\\langle x,y\\rangle}{\\lVert y \\rVert^2}\\overset{\\color{pink} \\heartsuit}{\\implies}\\lvert a \\rvert=\\frac{\\lVert x \\rVert}{\\lVert y \\rVert}$"}
{"_id": "7941753", "title": "", "text": "$n=4: \\dfrac{k(k^2 + 3 k + 14)}{6}$"}
{"_id": "4597813", "title": "", "text": "$G = A_1 \\times \\cdots \\times A_n$"}
{"_id": "754756", "title": "", "text": "$\\rho\\sigma\\rho^{-1} = \\left(\\begin{array}{cccccccc} 1 & 2 & 3 & 4 & \\dots & r-2 & r-1 & r\\\\ r & r-1 & r-2 & r-3 & \\dots & 3 & 2 & 1\\\\ 1 & r & r-1 & r-2 & \\dots & 4 & 3 & 2\\\\ r & 1 & 2 & 3 & \\dots & r-3 & r-2 & r-1 \\end{array}\\right)=\\sigma^{-1}  $"}
{"_id": "4975238", "title": "", "text": "$\\forall \\epsilon >0, \\exists \\ \\delta>0, \\ s.t. \\ |x-a|<\\delta\\implies|f(x)-f(a)|<\\epsilon$"}
{"_id": "7777399", "title": "", "text": "$z+\\frac{1}{x}=10$"}
{"_id": "6019125", "title": "", "text": "$\\int_{x_0}^\\infty f(x)\\,dx$"}
{"_id": "3761163", "title": "", "text": "$\\left[  \\begin{array}{cccc}    1 & -8 & 10 & 15\\\\    0 & 2 & -3 & -3\\\\    1 & -4 & 4 & d\\\\   \\end{array} \\right]$"}
{"_id": "435855", "title": "", "text": "$n^n=(s+(n-s))^n = \\sum_{i=0}^n \\binom{n}{i}s^i(n-s)^{n-i}\\ge \\binom{n}{s}s^s(n-s)^{n-s}\\ge \\binom{n}{s}$"}
{"_id": "9033718", "title": "", "text": "$\\frac{1}{|x|^n}=(1+\\delta)^n>1+\\delta^n+n\\delta>n\\delta>K$"}
{"_id": "5616835", "title": "", "text": "$\\frac{1-p^2}{1-2p\\cos x+p^2}-\\frac{1-2p\\cos x+p^2}{1-2p\\cos x+p^2}$"}
{"_id": "105143", "title": "", "text": "$\\sin x + \\cos x=t$"}
{"_id": "2315611", "title": "", "text": "$||A||_2 \\le \\sqrt {m}||A||_\\infty$"}
{"_id": "7635896", "title": "", "text": "$M = \\int_0^\\infty f(x)\\, dx$"}
{"_id": "373913", "title": "", "text": "$\\zeta(s)=\\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{x^{s-1}}{e^x-1}dx$"}
{"_id": "6119177", "title": "", "text": "$f_Y(y)=\\frac{2y}{\\theta^2}$"}
{"_id": "4022285", "title": "", "text": "$A:=\\{\\langle x,y\\rangle\\mid0\\leq x\\leq1, y^2\\leq x\\}\\subseteq\\mathbb R^2$"}
{"_id": "4346153", "title": "", "text": "$f(n) = {n \\choose 4} $"}
{"_id": "7613127", "title": "", "text": "$z=\\sqrt{2}e^{i\\pi/4}$"}
{"_id": "350912", "title": "", "text": "$p_n = \\frac{n(3n - 1)}{2}$"}
{"_id": "6391567", "title": "", "text": "$ \\lim_{n \\to \\infty} \\frac{\\sqrt{n}(\\sqrt{1} + \\sqrt{2} + ... + \\sqrt{n})}{n^2} $"}
{"_id": "1715944", "title": "", "text": "$n^+ = 1 \\Rightarrow n^+ = ∅^+ \\Rightarrow n^+ = ∅^+ = \\{∅\\}$"}
{"_id": "79536", "title": "", "text": "$(X,T_X)$"}
{"_id": "6776418", "title": "", "text": "$G_1 = A_1 \\times \\ldots \\times A_t$"}
{"_id": "3590883", "title": "", "text": "$ \\ x \\ \\ne \\ y \\ , \\ z \\ = \\ - 2 \\lambda \\ $"}
{"_id": "1199370", "title": "", "text": "$ S_1=(e_1,e_2,..,e_n)$"}
{"_id": "8769856", "title": "", "text": "$(a_{k},b_k)\\subset I_k$"}
{"_id": "1287275", "title": "", "text": "$T_k = \\pmatrix{k^2 \\sin(1/k) & k^2 \\cos(1/k)\\cr 0 & 1\\cr}$"}
{"_id": "3715810", "title": "", "text": "$\\sqrt{1!\\sqrt{2!\\sqrt{3!\\sqrt{\\cdots\\sqrt{n!}}}}} <2$"}
{"_id": "2638468", "title": "", "text": "$f(\\det\\gamma(t),0)=\\det\\gamma(t),\\qquad f(\\det\\gamma(t),1)=\\det\\gamma(0).$"}
{"_id": "462978", "title": "", "text": "$P(m+1) \\Rightarrow P(m)$"}
{"_id": "7659849", "title": "", "text": "$1/2^{k+2}$"}
{"_id": "6655351", "title": "", "text": "$\\pi^{-\\frac{s}{2}}\\Gamma(\\frac{s}{2})\\zeta(s)=\\pi^{-\\frac{1-s}{2}}\\Gamma(\\frac{1-s}{2})\\zeta(1-s)=\\Lambda(s)$"}
{"_id": "4564928", "title": "", "text": "$H=\\{h_0,h_1,h_2,....,h_{m-1}\\}$"}
{"_id": "6565087", "title": "", "text": "$E[N | X_0=x] = 1/x$"}
{"_id": "6118580", "title": "", "text": "$\\mathfrak o^*/(1+2\\mathfrak o)$"}
{"_id": "446410", "title": "", "text": "$B_{\\text{alt}} = \\{ \\gamma(\\epsilon) \\mid \\gamma \\text{ is a geodesic}, \\gamma(0)=p, \\text{ and } \\langle \\gamma'(0), \\gamma'(0) \\rangle  \\lt 1 \\} .$"}
{"_id": "3701840", "title": "", "text": "$x\\in V\\subset\\overline{V}\\subset U$"}
{"_id": "9086070", "title": "", "text": "$(x+y)^r-x^r=ryc^{r-1}\\leq ryx^{r-1}\\leq ry^r$"}
{"_id": "5424387", "title": "", "text": "$\\phi(x) =\\sum_{i=1}^n a_i \\Large {\\chi}_{A_i}$"}
{"_id": "7054448", "title": "", "text": "$C \\rightarrow \\left( \\begin{smallmatrix} A&-\\bar B\\\\  B&\\bar A \\end{smallmatrix} \\right)$"}
{"_id": "1527214", "title": "", "text": "$1/|x_n|$"}
{"_id": "5074557", "title": "", "text": "$s(t) = \\sum_{j=1}^n c_j 1_{(a_j,b_j]}(t),$"}
{"_id": "1624568", "title": "", "text": "$ds= \\int \\sqrt{1+(\\frac{dx}{dy})^2} dy$"}
{"_id": "6643086", "title": "", "text": "$I= \\int_{-\\pi}^{\\pi}\\dfrac{\\sin n(-x)}{(1+(\\pi)^{-x})\\sin (-x)} dx$"}
{"_id": "5612739", "title": "", "text": "$\\sum_{q=0}^{n-x} p^{x+q}  (1-p)^{n-(x+q)}$"}
{"_id": "5237777", "title": "", "text": "$P(X\\leq x) = P(X_1\\leq x,\\ldots,X_n\\leq x) = P(X_1\\leq x)^n.$"}
{"_id": "8151017", "title": "", "text": "$g(a_1)=a_1$"}
{"_id": "7140314", "title": "", "text": "$\\wedge^{0,1}F=(T_a^*X)^{0,1}$"}
{"_id": "7104258", "title": "", "text": "$ \\lim_{N\\to\\infty} \\sum_{n=1}^N \\frac{1}{n} $"}
{"_id": "7754183", "title": "", "text": "$\\frac {(k+1)(k+2)(k+3)}{(k+3)!}-\\frac{1 }{(k+3)!} $"}
{"_id": "2450770", "title": "", "text": "$ f(m) = {n \\choose m}$"}
{"_id": "6269253", "title": "", "text": "$\\int_0^{\\infty}\\frac{x^n}{(x^2+1)^n}\\,\\mathrm dx$"}
{"_id": "6507449", "title": "", "text": "$rs \\mid r^2+s^2$"}
{"_id": "4389992", "title": "", "text": "$k\\equiv a+sm \\equiv a \\pmod m$"}
{"_id": "6104577", "title": "", "text": "$A:=A_1\\times\\ldots\\times A_n\\in\\bigotimes_{i=1}^{n}\\mathcal{A}_i$"}
{"_id": "2759553", "title": "", "text": "$P : \\mathbb{R}^{n+1} \\to  \\mathbb{R}^n$"}
{"_id": "8898358", "title": "", "text": "$I(m)=\\frac12\\int_0^1u^{\\frac{m-1}2}(1-u)^{\\frac{m-1}2}du$"}
{"_id": "6263472", "title": "", "text": "$\\frac{\\left(2\\pi i\\right)\\cdot\\left(e^{\\frac{x\\pi i}{3}}\\right)}{\\left(1-e^{\\frac{i2\\pi x}{3}}\\right)\\cdot\\left(3e^{\\frac{2\\pi i}{3}}\\right)}=$"}
{"_id": "292058", "title": "", "text": "$(1+\\frac{x}{n})^n < \\frac1{1-x}$"}
{"_id": "4386769", "title": "", "text": "$\\exists! x(Px \\mid Qx)$"}
{"_id": "6246498", "title": "", "text": "$a^{log_b(c)}=c^{log_b(a)}$"}
{"_id": "2057756", "title": "", "text": "$f(x) = \\sum_{m=1}^{M}a_m \\chi_{A_m}(x)$"}
{"_id": "1378432", "title": "", "text": "$\\sum||f_n-g_n||^2<1$"}
{"_id": "4176096", "title": "", "text": "$\\rho^+$"}
{"_id": "4761878", "title": "", "text": "$ab = rk$"}
{"_id": "5845567", "title": "", "text": "$P(x) \\implies P(x+1)$"}
{"_id": "3827149", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}{\\dfrac{1}{x_n}}=\\dfrac{1}{\\lim\\limits_{n\\to\\infty}{x_n}}.$"}
{"_id": "7956922", "title": "", "text": "$\\displaystyle \\int_{-\\pi/2}^{\\pi/2} \\frac{\\sin^{2012}{x}}{\\left(1+ \\alpha^x\\right)\\left(\\sin^{2012}  {x}+\\cos^{2012}{x}\\right)}\\;{dx} $"}
{"_id": "7918212", "title": "", "text": "$ T^* = (VP)^* = (PV)^* = V^*P^* = V^*P, $"}
{"_id": "2114154", "title": "", "text": "$\\frac{n(n-1)}2 + 1 = \\frac12(n^2 - n + 2)$"}
{"_id": "5985420", "title": "", "text": "$v_2^2-v_1^2 = \\sum_{k=1}^d\\left(\\cos^2\\frac{(k+1)\\pi}d-\\sin^2\\frac{(k+1)\\pi}d\\right) = \\sum_{k=1}^d\\cos\\frac{(k+1)2\\pi}{d} = 0,$"}
{"_id": "8840868", "title": "", "text": "$(ab)^{rs}=a^{rs}b^{rs} =(a^r)^s (b^s)^r \\equiv 1 \\mod p $"}
{"_id": "7439045", "title": "", "text": "$[x,y]=xy$"}
{"_id": "902349", "title": "", "text": "$(\\frac{1}{2n+1})^2$"}
{"_id": "792016", "title": "", "text": "$ Z(\\phi,s)  Z(\\phi^*,s)  = \\sum_{n,m} \\frac{\\phi(n) \\phi^*(m)}{n^s m^s}$"}
{"_id": "8613548", "title": "", "text": "$\\left(y + \\dfrac{dy}{dx} x\\right)$"}
{"_id": "309325", "title": "", "text": "$A(t) = A^*(t) = A^{**}(t) = B^{**}(t) = B^*(t) = B(t)$"}
{"_id": "5067070", "title": "", "text": "$\\binom{n+1}{3} * \\frac{(n-1)! + (n-2)!}{(n+1)!} = \\frac{(n+1)!}{3!(n+1-3)!} * \\frac{(n-1)! + (n-2)!}{(n+1)!} = \\frac{(n+1)!}{3!(n-2)!} * \\frac{(n-1)! + (n-2)!}{(n+1)!} = \\frac{(n-1)! + (n-2)!}{3!(n-2)!} = \\frac{(n-2)!((n-1) + 1)}{3!(n-2)!}=\\frac{n}{3!} = \\frac{n}{6}$"}
{"_id": "4139757", "title": "", "text": "$1/|X|$"}
{"_id": "5319133", "title": "", "text": "$ \\begin{align} \\int_0^1\\frac{\\sin^{-1}\\left(\\sqrt{x}\\right)}{x^2-x+1}\\,\\mathrm{d}x &=\\int_0^1\\frac{\\frac12\\cos^{-1}(1-2x)}{x^2-x+1}\\,\\mathrm{d}x\\tag{1}\\\\ &=\\int_{-1}^1\\frac{\\cos^{-1}(x)}{3+x^2}\\,\\mathrm{d}x\\tag{2}\\\\ &=\\frac12\\int_{-1}^1\\frac{\\cos^{-1}(x)+\\cos^{-1}(-x)}{3+x^2}\\,\\mathrm{d}x\\tag{3}\\\\ &=\\frac\\pi2\\int_{-1}^1\\frac1{3+x^2}\\,\\mathrm{d}x\\tag{4}\\\\ &=\\frac\\pi{2\\sqrt3}\\int_{-1/\\sqrt3}^{1/\\sqrt3}\\frac1{1+x^2}\\,\\mathrm{d}x\\tag{5}\\\\ &=\\frac\\pi{\\sqrt3}\\tan^{-1}\\left(\\frac1{\\sqrt3}\\right)\\tag{6}\\\\ &=\\frac{\\pi^2}{6\\sqrt3}\\tag{7} \\end{align} $"}
{"_id": "7820882", "title": "", "text": "$f(x)-f(x+1)-1 < f(x-1)-f(x+1) \\le f(x)-f(x+1)$"}
{"_id": "890134", "title": "", "text": "$\\sum\\limits_{n = 1}^\\infty \\sum\\limits_{m = 1}^\\infty \\frac{1}{(n^2 + m^2)^3}$"}
{"_id": "5930096", "title": "", "text": "$\\lim_{n\\to\\infty}\\inf a_n =0$"}
{"_id": "438473", "title": "", "text": "$\n d(x,y) \\geq \\|p(x)\\| = \\left\\|\\frac{\\langle x,a\\rangle}{\\|a\\|^2} a\\right \\| = \\frac{|\\langle x,a\\rangle|}{\\|a\\|^2} \\|a\\| = \\frac{|\\langle x,a\\rangle|}{\\|a\\|}.\n $"}
{"_id": "7897401", "title": "", "text": "$\\lim_{N\\rightarrow\\infty}\\sum_{n=-\\infty}^\\infty f_N(n)=\\sum_{n=-\\infty}^\\infty f(n).$"}
{"_id": "8003632", "title": "", "text": "$1\\neq P_1\\mid P_2$"}
{"_id": "2465976", "title": "", "text": "$A_N=\\frac{n(n-1)}{2}$"}
{"_id": "1439255", "title": "", "text": "$\\int_{0}^{\\infty}f(x)dx, $"}
{"_id": "1320419", "title": "", "text": "$9^x = 6^x + 2.4^x$"}
{"_id": "6526676", "title": "", "text": "$|AB|=|CD|=|EF|=r$"}
{"_id": "2999904", "title": "", "text": "$P[X \\ge (i+j) | X \\ge j] = P[X \\ge i]$"}
{"_id": "4550022", "title": "", "text": "$ \\Vert proj_{W}\\mathbf{v}\\Vert=\\Vert\\frac{\\langle v,u\\rangle}{\\langle u,u\\rangle}u\\Vert=\\frac{|\\langle v,u\\rangle|}{|\\langle u,u\\rangle|}\\Vert u\\Vert=\\frac{|\\langle v,u\\rangle|}{||u||^{2}}\\Vert u\\Vert=\\frac{|\\langle u,v\\rangle|}{\\Vert u\\Vert} $"}
{"_id": "4965967", "title": "", "text": "$Cov (U,X) = 0$"}
{"_id": "3314979", "title": "", "text": "$[T_n^0,T_n^n]$"}
{"_id": "9060548", "title": "", "text": "$a\\cdot v= \\det(a,b,c),\\ \\forall a$"}
{"_id": "4959534", "title": "", "text": "$ \\sum_{j=1}^{M/2}\\sum_{k=1}^{M-1}\\cos{\\left(\\frac{2\\pi}{M}kj\\right)}=-M/2 $"}
{"_id": "3691351", "title": "", "text": "$f(n) = {n\\choose 2}+{11-n\\choose 2} = n^2-11n+55$"}
{"_id": "1493821", "title": "", "text": "$(-1)^{n-1}*2(\\frac{1}{2^{n-1}})$"}
{"_id": "6594741", "title": "", "text": "$s_n =\\frac{(a_1 + a_2 + ...+ a_n)}{n}$"}
{"_id": "5566263", "title": "", "text": "$P(E(X|{\\mathcal F}_0)\\in C)=1$"}
{"_id": "1573346", "title": "", "text": "$N!+2, N!+3,...,N!+N$"}
{"_id": "8479174", "title": "", "text": "$ \\begin{array}{r} &&1&1&2\\\\\\hline 1&0&1&-1&3\\\\ 0&1&-1&2&-5\\\\ 5&3&2&1&0\\\\ \\end{array}\\tag{2} $"}
{"_id": "447299", "title": "", "text": "$x,y\\in \\mathbb{R}, f(x)+f(y)=f(x+y)$"}
{"_id": "5392972", "title": "", "text": "$(n+1)!+2, (n+1)!+3, \\ldots ,(n+1)!+(n+1)$"}
{"_id": "5137240", "title": "", "text": "$\\gamma(v,w)=\\gamma(v+0,w)=\\gamma_{|u}(v,w)+0(0,w)=\\gamma_{|u}(v,w)+0=\\gamma_{|u}(v,w)$"}
{"_id": "8534263", "title": "", "text": "$y'=-y,y'''=-y$"}
{"_id": "361294", "title": "", "text": "$\\lim_{x \\to0^{-}} f(x) = \\lim_{x \\to 0^{+}} f(x) = \\lim_{x \\to 0} f(x) = C,$"}
{"_id": "5731792", "title": "", "text": "$t = |y|/|x| < 1$"}
{"_id": "2987640", "title": "", "text": "$\\begin{array}\\\\ \\gamma\\\\ \\gamma\\ \\cup\\{\\delta\\}\\implies \\lnot\\ \\gamma\\\\ \\gamma\\ \\cup\\{\\delta\\}\\implies \\gamma\\\\ \\delta\\\\ \\hline \\gamma\\ \\land \\lnot\\ \\gamma \\end{array}$"}
{"_id": "6215617", "title": "", "text": "$\\frac{\\cot(x+110^\\circ)}{\\cot x} = \\cot(x+60^\\circ)\\cdot \\cot(x-60^\\circ)$"}
{"_id": "90258", "title": "", "text": "$p_1p_2p_3\\cdots p_n+1$"}
{"_id": "6183679", "title": "", "text": "$c_1\\geq c_0$"}
{"_id": "144157", "title": "", "text": "$(a+b,a+2b).$"}
{"_id": "5078955", "title": "", "text": "$\\,f_2(n) = \\dfrac{n(3n\\!-\\!1)}2\\,$"}
{"_id": "6714301", "title": "", "text": "$\\lim_{x\\to0} f(c(x))=L$"}
{"_id": "6246430", "title": "", "text": "$\\sum_{n=2}^\\infty (\\frac1{n}-\\frac1{n^2})$"}
{"_id": "4490022", "title": "", "text": "$\\lim\\limits_{x\\to a^-}h(x)=\\lim\\limits_{x\\to a^-}f(x)=f(a)=h(a)=g(a)=\\lim\\limits_{x\\to a^+}g(x)=\\lim\\limits_{x\\to a^+}h(x)$"}
{"_id": "6755435", "title": "", "text": "$\\lfloor \\frac{a-1}{d}\\rfloor = \\lfloor \\frac{(a-1)\\gcd (ab,c)}{ab} \\rfloor$"}
{"_id": "4680523", "title": "", "text": "$Log_{a} (x^{n})=c $"}
{"_id": "3090031", "title": "", "text": "$f(x+2)=f(1)-f(x+1)=f(1)-(f(1)-f(x))=f(x)$"}
{"_id": "8593834", "title": "", "text": "$ S=2\\pi\\int x\\ ds \\\\ V=\\pi\\int xy\\ dx  $"}
{"_id": "544362", "title": "", "text": "$a=b^{\\log_b(a)}$"}
{"_id": "7461176", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\frac{1}{n^3} =\\sum_{n=1}^k \\frac{1}{n^3} + \\sum_{n=k+1}^{\\infty} \\frac{1}{n^3}< \\sum_{n=1}^k \\frac{1}{n^3}+\\int_{k}^{\\infty} \\frac{1}{x^3} dx =\\sum_{n=1}^k \\frac{1}{n^3}+\\frac{1}{2 k^2}$"}
{"_id": "3888719", "title": "", "text": "$log_a(b)$"}
{"_id": "5976124", "title": "", "text": "$0\\le M-m+m^*-M^*=(M-M^*)+(m^*-m)\\,,$"}
{"_id": "5963176", "title": "", "text": "$r=\\lambda-1$"}
{"_id": "6695489", "title": "", "text": "$\\forall{x,y} \\;\\; (xRy \\lor yRx)$"}
{"_id": "4754088", "title": "", "text": "$S^a := \\{f \\in X^* : f(x) = 0 \\ \\ \\forall x \\in S\\}.$"}
{"_id": "172053", "title": "", "text": "$|a+b|=s+t$"}
{"_id": "1630068", "title": "", "text": "$c_0 = c_1$"}
{"_id": "8828548", "title": "", "text": "$\\alpha\\alpha^\\omega=\\alpha^{1+\\omega}=\\alpha^\\omega.$"}
{"_id": "4001942", "title": "", "text": "$\\cdot \\frac{r}{1-(1+r)^{-n}}$"}
{"_id": "8451826", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1&1&k&1\\\\0&k-1&1-k&0\\\\0&1-k&1-k^2&1-k\\end{array}\\right]$"}
{"_id": "1002281", "title": "", "text": "$[x,y]=t$"}
{"_id": "3361018", "title": "", "text": "$\\to^+$"}
{"_id": "4590097", "title": "", "text": "$ \\operatorname{Ext}_\\gamma=\\{a\\not \\in \\gamma : \\operatorname{Ind}_\\gamma a=0\\} $"}
{"_id": "2388362", "title": "", "text": "$\\lim_{n \\to ∞} \\frac{\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+\\sqrt{4}+...+\\sqrt{n}}{n\\sqrt{n}}$"}
{"_id": "3403710", "title": "", "text": "$-\\frac12\\int_0^1\\frac{\\ln(1-x^2)}x\\ dx=\\int_0^1\\sum_{n=0}^\\infty\\frac{x^{2n+1}}{2n+1}\\ dx=\\sum_{n=0}^\\infty\\frac1{(2n+1)^2}$"}
{"_id": "5319062", "title": "", "text": "$[x,y]=-x$"}
{"_id": "6524518", "title": "", "text": "$ \\small{\\sum_{k=1}^N \\! \\left( \\cos \\frac{\\pi}{2k}-\\cos \\frac{\\pi}{2(k+2)} \\right)=\\!\\!\\left(\\cos \\frac{\\pi}{2}-\\cos \\frac{\\pi}{2(N+1)}\\right)\\!+\\!\\left(\\cos \\frac{\\pi}{4}-\\cos \\frac{\\pi}{2(N+2)}\\right)} $"}
{"_id": "4432813", "title": "", "text": "$ F(x) := \\int_a^x f(t) \\, \\mathrm{d}t. $"}
{"_id": "4786466", "title": "", "text": "$f (x) =\\sum_{k=1}^\\infty r_k \\chi_{A_k}(x)$"}
{"_id": "1110850", "title": "", "text": "$b_n = \\frac{a_1 + a_2 + \\dots + a_n}{n}$"}
{"_id": "4295633", "title": "", "text": "$\\dfrac{\\sqrt{1-t^2}}{2\\pi}\\displaystyle\\int_{-\\pi}^{\\pi}\\dfrac{g(\\theta)}{1-t\\cos\\theta}\\,d\\theta$"}
{"_id": "634139", "title": "", "text": "$ \\sqrt{ \\frac{dx}{dt}^2 + \\frac{dy}{dt}^2 }$"}
{"_id": "3055769", "title": "", "text": "$\\mathbf x = (x_1,x_2,....,x_{n-1},x_n)$"}
{"_id": "8940289", "title": "", "text": "$\\lim_{x\\to 0}{\\frac{\\sin^{-1}(x)}{x}} =1$"}
{"_id": "582898", "title": "", "text": "$\\forall \\epsilon>0, \\exists r>0:|u-0| <r \\implies |f(u) - f(0)| < \\epsilon$"}
{"_id": "4629416", "title": "", "text": "$x^3,x^3-x,x^3+x$"}
{"_id": "9246161", "title": "", "text": "$\\lim_{N \\to \\infty}\\sum\\limits_{n=1}^{N} \\frac{1}{2^n}$"}
{"_id": "7296094", "title": "", "text": "$n \\uparrow\\uparrow 5 = n^{n^{n^{n^n}}} < (m^m)^{(m^m)^{(m^m)^{m^{m^m - m^{m-1}}}}} = m^{m^{m^{m^{m^m - m^{m-1} + 1} + 1} + 1}} < m^{m^{m^{m^{m^m - m^{m-1} + 1} + 2}}} < m^{m^{m^{m^{m^m - m^{m-1} + 2}}}}< m^{m^{m^{m^{m^m}}}}.$"}
{"_id": "3002076", "title": "", "text": "$q^n-q^{l+2}$"}
{"_id": "768682", "title": "", "text": "$f_1(n) = (-1)^n$"}
{"_id": "7405289", "title": "", "text": "$\\mathbb{E}(Z) = \\frac{1}{2}(\\mathbb{E}(X)+\\mathbb{E}(Y))$"}
{"_id": "8155451", "title": "", "text": "$\\sum_{k=1}^\\infty\\frac1{k+k^3}\\text{ and }\\sum_{k=1}^\\infty\\frac1{1+k^2}$"}
{"_id": "5504047", "title": "", "text": "$10^{40}=2^{40}\\cdot5^{40}$"}
{"_id": "1858014", "title": "", "text": "$cov(X+Y,Z+W)=cov(X,Z)+cov(X,W)+cov(Y,Z)+cov(Y,W)$"}
{"_id": "95762", "title": "", "text": "$\\left [\\begin{array}{cc} 1 & -1\\\\0 & 2\\end{array}\\right]$"}
{"_id": "6371959", "title": "", "text": "$\\sqrt{1}+\\sqrt{2}+\\sqrt{3}+...+\\sqrt{k}\\ge\\frac{2}{3}k\\sqrt{k}$"}
{"_id": "588612", "title": "", "text": "$\\mid p_1 - p_2\\mid \\rightarrow 0$"}
{"_id": "856653", "title": "", "text": "$ \\begin{pmatrix} 1 & 3 & 2 & 1 \\\\ 0 & 1 & 4 & -4 \\\\ 0 & -1 & -6 & 7 \\\\ 0 & 1 & 3 & 1 \\\\ \\end{pmatrix} $"}
{"_id": "3141892", "title": "", "text": "$e_n=x_n/|x_n|\\in G$"}
{"_id": "6247159", "title": "", "text": "$f(x) = \\frac2 {\\sqrt{2 \\pi}}e^\\frac{-x^2}{2} $"}
{"_id": "7607544", "title": "", "text": "$\\int_{-\\pi}^\\pi\\frac{\\sin(13x)}{\\sin x}\\cdot\\frac1{1+2^x}\\mathrm dx?$"}
{"_id": "1884063", "title": "", "text": "$1+2+3+4+5+...=-\\frac{1}{12}\\Longleftrightarrow$"}
{"_id": "27183", "title": "", "text": "$\\mathbb{R}^+$"}
{"_id": "4102553", "title": "", "text": "$F=\\bigoplus_{i\\in\\mathbb{N}}(P\\oplus T)$"}
{"_id": "6736379", "title": "", "text": "$k!+2,\\dots, k!+k$"}
{"_id": "4669649", "title": "", "text": "$d(x,q)\\leqslant d(x,y)+d(y,q)$"}
{"_id": "4867401", "title": "", "text": "$ \\langle a\\wedge b,c\\wedge d\\rangle=\\det\\begin{pmatrix} \\langle a,c\\rangle & \\langle a,d\\rangle \\\\ \\langle b,c\\rangle & \\langle b,d\\rangle \\end{pmatrix}. $"}
{"_id": "6093857", "title": "", "text": "$ n! + 2, .... n! + n$"}
{"_id": "8395719", "title": "", "text": "$I_n = \\int_{-\\pi}^{\\pi} \\frac{\\sin nx}{(1+2^x)\\sin x}dx = \\int_{0}^{\\pi} \\frac{\\sin nx}{(1+2^x)\\sin x}dx + \\int_{-\\pi}^{0} \\frac{\\sin nx}{(1+2^x)\\sin x}dx$"}
{"_id": "3602690", "title": "", "text": "$\\Gamma(s) \\zeta(s) =  \\int_{0}^{\\infty} \\frac{u^{s-1}}{e^{u}-1} du$"}
{"_id": "1379416", "title": "", "text": "$p_1 \\mid p_2^n$"}
{"_id": "1374964", "title": "", "text": "$f(x) =\\dfrac{3x+5}{x^2+2} $"}
{"_id": "5073055", "title": "", "text": "$d(x,v) \\le d(x,y)+d(y,v)$"}
{"_id": "5807401", "title": "", "text": "$F = \\frac{1}{2} F_{ab} \\gamma^a \\gamma^b \\implies F_{ab} = F \\cdot (\\gamma_b \\wedge \\gamma_a)$"}
{"_id": "5501124", "title": "", "text": "$ \\bar{A}=\\{x \\in \\mathbb{R^n}\\mid \\forall r \\in \\mathbb{R^+}:B_r(x)\\ \\cap A \\neq \\phi \\} $"}
{"_id": "848031", "title": "", "text": "$P(E_i) = 1/2$"}
{"_id": "3206642", "title": "", "text": "$\\left( 1 + \\frac{x}{n} \\right)^n \\approx e^x$"}
{"_id": "3624986", "title": "", "text": "$E=1,6$"}
{"_id": "3057478", "title": "", "text": "$\\delta = \\gamma + 1 =\\gamma \\cup \\{\\gamma\\}$"}
{"_id": "6574464", "title": "", "text": "$C = (p-nT) \\sum_\\gamma (\\partial_\\gamma u_\\gamma) + \\sum_{\\alpha,\\beta} (A_{\\alpha\\beta} \\partial_\\alpha u_\\beta) - (\\zeta \\sum_\\gamma \\partial_\\gamma u_\\gamma)\\sum_\\alpha(\\partial_\\alpha u_\\alpha) + n^2 \\sum_\\gamma (\\partial_\\gamma u_\\gamma) + K \\left( -\\frac12 (\\sum_\\gamma \\partial_\\gamma n) (\\sum_\\gamma \\partial_\\gamma n) (\\sum_\\alpha \\partial_\\alpha u_\\alpha) - n(\\sum_\\gamma \\partial_\\gamma n)(\\sum_{\\alpha,\\gamma} \\partial_\\gamma \\partial_\\alpha u_\\alpha) - (\\sum_\\gamma \\partial_\\gamma n)(\\sum_{\\alpha,\\gamma} \\partial_\\gamma u_\\alpha)(\\sum_\\alpha \\partial_\\alpha n) \\right)$"}
{"_id": "331387", "title": "", "text": "$\\overline{p_1 p_2\\ldots p_{12}}$"}
{"_id": "9050711", "title": "", "text": "$Cov(\\epsilon, \\theta)=0$"}
{"_id": "5986476", "title": "", "text": "$R^{n+1}_{n+1}=\\mathbb{R}^{n+1}$"}
{"_id": "4279619", "title": "", "text": "$ det(A) = (\\alpha - \\beta)^{n-1}(\\alpha + (n-1)\\beta). $"}
{"_id": "7745656", "title": "", "text": "$ \\begin{cases}x=1.3\\\\ y=0\\\\ z=0.8912 \\end{cases} $"}
{"_id": "106582", "title": "", "text": "$A=\\{a_1,a_2,a_3,\\ldots\\}$"}
{"_id": "5134433", "title": "", "text": "$f_R(r) = 2r/R^2_{\\max}$"}
{"_id": "1909682", "title": "", "text": "$\\log_A(B)$"}
{"_id": "5671736", "title": "", "text": "$\\lim\\limits_{n\\rightarrow \\infty}\\sum\\limits_{r=1}^{n}\\frac{1}{T_r}$"}
{"_id": "3866341", "title": "", "text": "$S = A ( (1 + R)^{n + 1} - (1 + R) ) / R$"}
{"_id": "2937745", "title": "", "text": "$\\frac {2^{n-2}-1}{2^{n-2}} + \\frac 13*\\frac 1{2^{n-2}}(2 + 1)= \\frac {2^{n-2}-1}{2^{n-2}}+\\frac 1{2^{n-2}} = \\frac {2^{n-2}}{2^{n-2}} = 1$"}
{"_id": "1517072", "title": "", "text": "$\\left\\{\\begin{array}{cc}x=3t \\\\ y=-5(t+1) \\\\ z=2t+3\\end{array}\\right.,\\space t\\in\\Bbb{R}$"}
{"_id": "7754799", "title": "", "text": "$s(x)=\\sum_{k=1}^n a_n \\chi_{A_n}(x)$"}
{"_id": "8122328", "title": "", "text": "$||x|| = ||x − y + y|| ≤ ||x − y + y|| ⇒ ||x−y||≤||x − y||$"}
{"_id": "8722621", "title": "", "text": "$\\mathbb{P}\\{\\sup_{\\{t \\geq 0 \\}} M_t > x |\\mathcal{F}_0\\}=1 \\wedge \\frac{M_0}{x}.$"}
{"_id": "6510649", "title": "", "text": "$f_X(x|Z=z^*)$"}
{"_id": "2909859", "title": "", "text": "$f^+,$"}
{"_id": "7272061", "title": "", "text": "$\\lim_{y\\to 1^-} \\frac{\\sqrt {1-y^2}}{y-1}$"}
{"_id": "1656493", "title": "", "text": "$V \\subseteq P$"}
{"_id": "1483032", "title": "", "text": "$M_n=\\#_{i=1}^{n}T^2$"}
{"_id": "7232565", "title": "", "text": "$[\\frac{p}{3^k}, \\frac{p+1}{3^k}]$"}
{"_id": "7277696", "title": "", "text": "$\\forall \\omega\\in\\Omega\\setminus N:\\lim_{n\\rightarrow\\infty}X_n(\\omega)=X(\\omega)$"}
{"_id": "3620181", "title": "", "text": "$\\textbf{P}_{\\gamma} = \\exp(\\textbf G \\gamma)$"}
{"_id": "1607602", "title": "", "text": "$\\left | e^x - \\left( 1 + \\frac{x}{n} \\right) ^ n \\right|$"}
{"_id": "6408754", "title": "", "text": "$\\gamma(0)=\\gamma_1(0)=\\gamma_2(1)=\\gamma(2)$"}
{"_id": "3643930", "title": "", "text": "$\\varphi'(x)=\\frac{d}{dx}\\int_0^1f(t)\\cos(xt)dt =\\int_0^1\\frac{\\partial}{\\partial x}f(t)\\cos(xt)dt =-\\int_0^1t f(t)\\sin(xt)dt$"}
{"_id": "2542376", "title": "", "text": "$\\displaystyle\\sum_{n}\\mu(A_{n})\\geq\\mu(A_{K})=\\sqrt{3}=\\mu\\left(\\displaystyle\\bigcup_{n}A_{n}\\right)$"}
{"_id": "9250689", "title": "", "text": "$\\forall a,b,c\\in\\mathbb{R},d(a,c)=0\\leq d(a,b)+d(a,c)=0$"}
{"_id": "5063792", "title": "", "text": "$(\\mathbb{Q}[x]/(x^2-5))[y]/(y^2+3)$"}
{"_id": "4259912", "title": "", "text": "$(\\mathbb{R}^{\\mathbb{N}}, \\mathcal{R}^{\\mathbb{N}})$"}
{"_id": "4604883", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}x_n \\lim_{n \\rightarrow \\infty}(1/t_n)= x.$"}
{"_id": "4793858", "title": "", "text": "$\\vartheta(z)+\\vartheta(z+1)=2\\vartheta(4z)$"}
{"_id": "2130654", "title": "", "text": "$\\det\\begin{pmatrix}A & C \\\\ 0 & B\\\\ \\end{pmatrix}= \\det(A)\\operatorname{det}(B),$"}
{"_id": "8550024", "title": "", "text": "$x/t_x$"}
{"_id": "7104152", "title": "", "text": "$\\mathcal F=\\{\\{x\\},\\{y\\}\\}$"}
{"_id": "3003953", "title": "", "text": "$y=\\frac{-(-2)±\\sqrt{(-2)^{2}-8(-220)}}{4}$"}
{"_id": "8047470", "title": "", "text": "$ X \\in \\mathcal{F}_0, X \\subseteq Y \\Rightarrow Y \\in \\mathcal{F}_0.$"}
{"_id": "1109167", "title": "", "text": "$(\\mathbb Q^+,.)$"}
{"_id": "7477279", "title": "", "text": "$P_0=\\{\\xi\\colon \\xi=x-x_0,\\,x\\in P\\}$"}
{"_id": "8690885", "title": "", "text": "$ dm = \\left|ab\\right| ,$"}
{"_id": "3963178", "title": "", "text": "$f(x\\mid\\theta) = \\dfrac{2x}{\\theta^2}\\mathbb{1}_{(0,\\theta)}(x)$"}
{"_id": "6832111", "title": "", "text": "$\\phi = \\tan ^{-1}\\frac{\\sqrt{1-\\delta^2}}{\\delta}$"}
{"_id": "1477850", "title": "", "text": "$\\forall n\\in \\mathbb{N}^*, \\cases{ p_{2n}=9\\times 10^{n-1} \\\\ p_{2n+1}=10\\times p_{2n}=9\\times10^n }$"}
{"_id": "4571357", "title": "", "text": "$\\lim_{x\\to a^+}f(x)=\\lim_{x\\to a^-}f(x)=\\lim_{x\\to a}f(x)=f(a)$"}
{"_id": "4186114", "title": "", "text": "$\\Phi(f) = \\lim_{n\\to\\infty} \\Phi(f_n) = \\sup \\{\\Phi(g) : g \\in \\mathcal{F}\\},$"}
{"_id": "6387534", "title": "", "text": "$h(x) = \\sum_{j=1}^n a_j 1_{C_j}(x).$"}
{"_id": "3016910", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-\\pi\\alpha n^2} = \\frac{1}{\\sqrt{\\alpha}}\\sum_{n=-\\infty}^\\infty e^{-\\pi n^2/\\alpha}.$"}
{"_id": "4469085", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor x\\rfloor+n}m\\right\\rfloor = \\left\\lfloor\\frac{x+n}m\\right\\rfloor$"}
{"_id": "7694788", "title": "", "text": "$\\int \\frac{x+1}{(x^2+x+1)^2}dx$"}
{"_id": "789629", "title": "", "text": "$x \\in V \\subseteq \\overline{V} \\subseteq U$"}
{"_id": "6299085", "title": "", "text": "$cov(X, Y) = 0,$"}
{"_id": "4971864", "title": "", "text": "$\\displaystyle\\sum_{n\\ge1}\\frac{n\\cos(nx)}{n^3}=\\sum_{n\\ge1}\\frac{\\cos(nx)}{n^2}$"}
{"_id": "9114054", "title": "", "text": "$f(a + (b + 1)) = f(a) + f(b + 1)$"}
{"_id": "1791347", "title": "", "text": "$p\\in V\\subseteq U\\subseteq X$"}
{"_id": "8260288", "title": "", "text": "$f(v)=f(a)+f(b)=f(b)$"}
{"_id": "3418299", "title": "", "text": "$\\gamma(1,2,3,4,5)\\gamma^{-1}=(1,\\gamma(2),\\gamma(3),\\gamma(4),\\gamma(5))$"}
{"_id": "8281047", "title": "", "text": "$\\lim_{n\\to \\infty}\\int_{1}^{n}\\frac{1}{(x^{2}+1)^{n}}dx\\sim \\frac{1}{n\\cdot 2^{n}}?.$"}
{"_id": "2011686", "title": "", "text": "$\\left(1+\\frac xn\\right)^n=\\mathrm e^x\\left(1+O\\left(\\frac1n\\right)\\right)$"}
{"_id": "8281076", "title": "", "text": "$\\int_{1}^{n} \\frac{1}{(1+x^2)^n} dx$"}
{"_id": "2796369", "title": "", "text": "$\\mathrm{cov}(X,Z) = 0$"}
{"_id": "4190015", "title": "", "text": "$\\lim_{n\\rightarrow\\infty} X_n(\\omega) = \\liminf_{n\\rightarrow\\infty} X_n(\\omega)$"}
{"_id": "4468700", "title": "", "text": "$\\displaystyle \\int_0^{\\pi/2}\\frac{\\sin(2n+1)x}{\\sin x}dx=\\frac\\pi2$"}
{"_id": "296601", "title": "", "text": "$a\\mathbf{u} + b\\mathbf{v} = \\mathbf{0}$"}
{"_id": "9113948", "title": "", "text": "$p(x)=x^2-x+2.$"}
{"_id": "6068380", "title": "", "text": "$ \\begin{vmatrix} 0 & x & y & z \\\\ x & 0 & z & y \\\\ y & z & 0 & x \\\\ z & y & x & 0 \\\\ \\end{vmatrix} = \\begin{vmatrix} 0 & 1 & 1 & 1 \\\\ 1 & 0 & z^2 & y^2 \\\\ 1 & z^2 & 0 & x^2 \\\\ 1 & y^2 & x^2 & 0 \\\\ \\end{vmatrix} $"}
{"_id": "2661697", "title": "", "text": "$f (z/2) =f (z) /2$"}
{"_id": "7293082", "title": "", "text": "$ \\begin{align}    \\int_{-1}^1\\frac1{(x^2+1)^3}\\:dx&=2\\int_0^1\\frac1{(x^2+1)^3}\\:dx    \\\\\\\\&=2\\int_0^{\\pi/4}\\frac{(\\tan^2\\theta+1)\\:d\\theta}{(\\tan^2\\theta+1)^3}    \\\\\\\\&=2\\int_0^{\\pi/4}\\cos^4\\!\\theta \\:d\\theta \\\\\\\\&=\\frac12\\int_0^{\\pi/4}(1+\\cos    (2\\theta))^2 \\:d\\theta \\\\\\\\&=\\frac12\\int_0^{\\pi/4}(1+2\\cos    (2\\theta)+\\cos^2(2\\theta)) \\:d\\theta \\end{align} $"}
{"_id": "8224610", "title": "", "text": "$J=\\int_0^a \\frac{\\sin ^{-1}(x)}{x}\\,dx$"}
{"_id": "9034801", "title": "", "text": "$ \\begin{pmatrix} A & -B \\\\ -B^T & D \\end{pmatrix} $"}
{"_id": "1106781", "title": "", "text": "$e^{-x}\\ge \\left(1-\\frac xn\\right)^n$"}
{"_id": "2779789", "title": "", "text": "$\\text{ Deoes } \\lim\\limits_{n\\to\\infty}y_nx_n\\geq\\alpha?$"}
{"_id": "7719840", "title": "", "text": "$\\,f=\\big\\{(\\{a\\},a)\\big\\}$"}
{"_id": "7043713", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       1&-2&3&-4\\\\       2&1&1&2\\\\       0&a&-1&b     \\end{array} \\right] $"}
{"_id": "4373216", "title": "", "text": "$z+I_j=a_j+I_j$"}
{"_id": "3343673", "title": "", "text": "$f'(n)=xn^{x-1}-x(n-1)^{x-1}=x(n-1)^{x-1}[(1+\\frac{1}{n-1})^{x-1}-1]>0$"}
{"_id": "3573210", "title": "", "text": "$\\mid P_1P_2\\mid $"}
{"_id": "4234004", "title": "", "text": "$ \\left(\\begin{array}{ccc}         x_1^2 & x_1 & 1\\\\         x_2^2 & x_2 & 1\\\\         x_3^2 & x_3 & 1\\\\ \\end{array}\\right) % \\left(\\begin{array}{c}          a\\\\          b\\\\          c\\\\ \\end{array}\\right) % = \\left(\\begin{array}{c}          y_1\\\\          y_2\\\\          y_3\\\\ \\end{array}\\right) $"}
{"_id": "9155259", "title": "", "text": "$M'=M+(M'-M)$"}
{"_id": "6778435", "title": "", "text": "$P(X_{1}=k|X_{0}=0)$"}
{"_id": "8794129", "title": "", "text": "$\\sqrt{z}=\\sqrt{r}\\: \\mathrm{e}^{\\frac{\\psi \\mathrm{i}}{2}}$"}
{"_id": "4497854", "title": "", "text": "$\\lVert XY\\rVert_r \\leq \\lVert |X|^r\\rVert_p^{1/r} \\lVert|Y|^r\\rVert_q^{1/r} ,$"}
{"_id": "2729217", "title": "", "text": "$\\lim\\limits_{n\\to\\infty} \\sqrt n a_n = \\sqrt3$"}
{"_id": "2388366", "title": "", "text": "$\\mathcal{O}_X[T, T^{-1}] \\to \\mathcal{O}_X[T, T^{-1}] \\otimes \\mathcal{O}_X[T, T^{-1}]$"}
{"_id": "5287357", "title": "", "text": "$ \\begin{cases} x=3\\\\ y=5\\\\ z=7 \\end{cases} $"}
{"_id": "5696261", "title": "", "text": "$\\tag{1}\n \\sup_{\\|w^\\ast\\|_{W^\\ast} \\leq 1}{\n   \\left|\n     \\langle w^{\\ast\\ast}, w^\\ast\\rangle_{W^{\\ast\\ast},W^\\ast}\n   \\right|\n } =\n \\|w^{\\ast\\ast}\\|_{W^{\\ast\\ast}}\n \\stackrel{\\color{red}{(!!)}}{=}\n \\|i^{\\ast\\ast}w^{\\ast\\ast}\\|_{V^{\\ast\\ast}} = \n \\sup_{\\|v^{\\ast}\\|_{V^{\\ast}} \\leq 1}{\n   \\left| \n     \\langle i^{\\ast\\ast}w^{\\ast\\ast},v^{\\ast}\\rangle_{V^{\\ast\\ast},V^{\\ast}}\n   \\right|\n }. \\quad\n $"}
{"_id": "2649461", "title": "", "text": "$\\sum_{n=101}^\\infty\\frac1{n^2} >\\sum_{n=101}^\\infty\\frac1{n^2+n}=\\frac1{101}. $"}
{"_id": "3009824", "title": "", "text": "$C = \\{(x,y)\\ |\\ a \\leq x \\leq b , c \\leq y \\leq d \\}$"}
{"_id": "4467552", "title": "", "text": "$u(0) = \\frac{1}{2\\pi}\\int_0^{2\\pi} u(e^{i\\varphi})\\,d\\varphi = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\cos(\\varphi)e^{\\cos\\varphi}\\cos (\\sin\\varphi) - \\sin(\\varphi)e^{\\cos\\varphi}\\sin(\\sin\\varphi)\\,d\\varphi.$"}
{"_id": "3480037", "title": "", "text": "$S:=\\{a,a+b,a+2b,\\ldots,a+nb,\\ldots\\}$"}
{"_id": "2977000", "title": "", "text": "$=\\sqrt{1+\\frac{\\partial f}{\\partial x}^2+\\frac{\\partial f}{\\partial y}^2}$"}
{"_id": "5647902", "title": "", "text": "$\\bigcup\\mathscr A = X$"}
{"_id": "4967481", "title": "", "text": "$n^a - (n-3)^a= \\frac{1}{\\left(\\frac{1}{n}\\right)^a} -\\frac{\\left(1-\\frac{3}{n}\\right)^a}{\\left(\\frac{1}{n}\\right)^a}\\\\ =\\frac{1}{\\left(\\frac{1}{n}\\right)^a}\\left(1-\\left(1-\\frac{3}{n}\\right)^a\\right) $"}
{"_id": "4823591", "title": "", "text": "$\\left(1 + \\frac{x}{n}\\right)^n \\leqslant e^x \\leqslant \\left(1 - \\frac{x}{n}\\right)^{-n}.$"}
{"_id": "7308955", "title": "", "text": "$m : M ⊗ M → M$"}
{"_id": "4647290", "title": "", "text": "$B^{log_B(C)}=C$"}
{"_id": "745299", "title": "", "text": "$=\\frac{(1-p)^{n-2}\\cdot p}{1-(1-p)}=(1-p)^{n-2}$"}
{"_id": "6620621", "title": "", "text": "$a_1\\times\\cdots\\times a_n$"}
{"_id": "7111770", "title": "", "text": "$ \\vartheta(z + \\alpha + \\beta \\tau, \\tau ) = \\vartheta(z, \\tau) e^{- \\pi i \\beta^2 \\tau - 2 \\pi i \\beta z} $"}
{"_id": "8045265", "title": "", "text": "$\\displaystyle\\;\\frac{\\sqrt{1-z^2}}{1+z^2} \\sim \\frac{i}{z}$"}
{"_id": "4273075", "title": "", "text": "$a=\\{\\{\\{x\\}\\}\\}$"}
{"_id": "7043085", "title": "", "text": "$c_r(x)=r$"}
{"_id": "566757", "title": "", "text": "$f(x,y,z) = (x^2, y^2, z^2)$"}
{"_id": "117394", "title": "", "text": "$f(x)=\\frac{4^x}{4^x+2}$"}
{"_id": "8842549", "title": "", "text": "$x=y=z=-\\lambda$"}
{"_id": "6555430", "title": "", "text": "$\\sum_{s=2}^\\infty \\frac{\\zeta(s)-1}{s}=1-\\gamma$"}
{"_id": "5070026", "title": "", "text": "$\\prod_{k=0}^{n-1} \\cos \\frac{2\\pi k}{n} = \\frac{1}{2^{n-1}}$"}
{"_id": "5342443", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       3&2&1&0\\\\       -2&1&-1&2\\\\       2&-1&2&-1     \\end{array} \\right] $"}
{"_id": "4997617", "title": "", "text": "$a_1....a_n$"}
{"_id": "6073019", "title": "", "text": "$\\phi(n) = i^n$"}
{"_id": "1366402", "title": "", "text": "$c_1(T^{0,1}X)=0$"}
{"_id": "6191732", "title": "", "text": "$\\left(\\begin{array}{l}1&x&x^2\\\\1&y&y^2\\\\1&z&z^2\\end{array}\\right)$"}
{"_id": "6444813", "title": "", "text": "$(x_n, e_n) \\to 0$"}
{"_id": "1605036", "title": "", "text": "$\\int_0^\\infty\\frac{ \\sin^2 x} x\\,dx$"}
{"_id": "44260", "title": "", "text": "$\\int_{0}^{2} f(t) dt=0$"}
{"_id": "2269281", "title": "", "text": "$\\{\\{A\\}\\}$"}
{"_id": "3949349", "title": "", "text": "$\\int _0 ^{2 \\pi} \\sin x \\cos x dx = 2 \\int _0 ^\\pi \\sin x \\cos x dx = 0,$"}
{"_id": "415492", "title": "", "text": "$\\det(A)=(a+(n-1)b)\\begin{vmatrix}\n       1 & 1 & 1 & \\cdots & 1 \\\\\n       0 & a-b & 0 &\\cdots & 0 \\\\\n      0 & 0 & a-b &\\cdots & 0 \\\\\n       \\vdots & \\vdots & \\vdots  & \\ddots & \\vdots \\\\\n       0 & 0 & 0 & \\ldots & a-b \\\\\n     \\end{vmatrix}=(a+(n-1)b)(a-b)^{n-1}.$"}
{"_id": "6123576", "title": "", "text": "$\\ f(a) = \\lim_{x\\to a^+}f(x) = \\lim_{x\\to a^-}f(x)$"}
{"_id": "3381013", "title": "", "text": "$\\implies \\frac{\\pi}{x} < \\frac{\\pi}{2}$"}
{"_id": "958883", "title": "", "text": "$d=|f(Yn)-f(a)| \\le ||f||*||Yn-a||$"}
{"_id": "7054814", "title": "", "text": "$|\\gamma| \\ge d(\\gamma(a),\\gamma(b))$"}
{"_id": "5242936", "title": "", "text": "$P(D) = 2/3$"}
{"_id": "4202873", "title": "", "text": "$(2a,b),  (a + b, 2b + a),  (2b,2a)$"}
{"_id": "8263607", "title": "", "text": "$\\int_{0}^{\\infty}f(x^2)dx$"}
{"_id": "5623946", "title": "", "text": "$ (1-r-\\lambda)^{n-1}(1+(n-1)r-\\lambda)=0 $"}
{"_id": "3215409", "title": "", "text": "$\\lim_{x \\rightarrow a^+}f(x)=\\lim_{x \\rightarrow a^-}f(x)=\\lim_{x \\rightarrow a}f(x)$"}
{"_id": "4903326", "title": "", "text": "$\\bigg\\lfloor\\frac{q_1}{a_2}\\bigg\\rfloor=\\Bigg\\lfloor\\frac{\\bigg\\lfloor\\frac{n}{a_1}\\bigg\\rfloor}{a_2}\\Bigg\\rfloor=\\bigg\\lfloor\\frac{n}{a_1a_2}\\bigg\\rfloor$"}
{"_id": "9088962", "title": "", "text": "$\\lim_{N\\to\\infty}\\sum_{n=1}^{N}\\frac{1}{(n+1)}\\sum_{i=1}^{n}\\frac{1}{i(n+1-i)}$"}
{"_id": "159702", "title": "", "text": "$(1 + {x \\over n})^n$"}
{"_id": "3777546", "title": "", "text": "$ \\begin{align} P(N \\geq k \\mid X_0 = x) &= \\int_0^1 P(N \\geq k \\mid X_0=x,X_1=y)\\,dy \\\\ & = \\int_x^1P(N \\geq k \\mid X_1 = y)\\,dy\\\\ & = \\int_x^1P(N \\geq k-1\\mid X_0=y)\\,dy \\end{align} $"}
{"_id": "4661522", "title": "", "text": "$\\int_{-\\infty}^\\infty \\frac{x^{2n}}{(x^2 + 1)^{n + 1}}\\ dx = 2\\pi i \\operatorname{res}_i f.$"}
{"_id": "4473436", "title": "", "text": "$f(n)=3n^2-n+2$"}
{"_id": "3993944", "title": "", "text": "$\\displaystyle\\int_0^2 F(\\gamma)\\gamma ' +\\displaystyle\\int_2^3 F(\\gamma)\\gamma ' +\\displaystyle\\int_3^4 F(\\gamma)\\gamma '$"}
{"_id": "2195797", "title": "", "text": "$\\begin{cases} x+2y=0 \\\\ z=0 \\\\ 0=0\\end{cases}$"}
{"_id": "1584019", "title": "", "text": "$Cov(x,y)$"}
{"_id": "656492", "title": "", "text": "$a_1 \\ldots a_n$"}
{"_id": "6848766", "title": "", "text": "$Tr(-B)$"}
{"_id": "8202856", "title": "", "text": "$r_2\\mid r_1n$"}
{"_id": "7130002", "title": "", "text": "$\\dot x = x-x^3, x \\in \\mathbb{R}$"}
{"_id": "8909594", "title": "", "text": "$T_N^{0,1}\\subset T_M^{0,1}$"}
{"_id": "510695", "title": "", "text": "$(y'-y)' = y'' - y' = y - y' = -(y'-y)$"}
{"_id": "285930", "title": "", "text": "$\\{a+b,ab,a^b,\\dots\\}$"}
{"_id": "351827", "title": "", "text": "$\\int \\limits_{0}^{\\infty}\\frac {\\sin (x^n)} {x^n}dx$"}
{"_id": "4510848", "title": "", "text": "$\\frac{1}{\\gamma}\\log\\left(\\mathbb{E}[e^X]\\right)^{-\\gamma}\\leq\\lim_{k\\rightarrow \\infty}\\frac{2^k}{\\gamma}\\log\\mathbb{E}[e^{-\\gamma \\frac{X}{2^k}}]\\leq \\frac{1}{\\gamma}\\log \\mathbb{E}[e^{-\\gamma X}]$"}
{"_id": "6051036", "title": "", "text": "$(a+bi,a-b)=(\\alpha,\\beta)$"}
{"_id": "3120648", "title": "", "text": "$\\{a,a+m,a+2m,\\cdots\\}$"}
{"_id": "8428336", "title": "", "text": "$x(1+x)^n = (1+x)^{n+1}-(1+x)^n$"}
{"_id": "5078508", "title": "", "text": "$   \\int_0^1 dx = \\lim_{n \\rightarrow \\infty} \\sum_{n=1}^n 1/n               = \\lim_{n \\rightarrow \\infty} n.1/n               = \\lim_{n \\rightarrow \\infty} 1 = 1 $"}
{"_id": "968071", "title": "", "text": "$\\forall x,y \\in \\mathbb R f(xy+f(x))=xf(y)+f(x)$"}
{"_id": "2532026", "title": "", "text": "$f_n(x)=\\frac{g(x)}{n^2}$"}
{"_id": "7370314", "title": "", "text": "$\\displaystyle\\int_{-\\infty}^\\infty \\frac{1}{(1+x^2)^m} \\, dx$"}
{"_id": "3776589", "title": "", "text": "$ \\int_0^\\pi f(x)\\sin(2\\,n\\,x)\\,dx=\\int_0^\\pi f(\\pi-x)\\sin(2\\,n\\,\\pi-2\\,n\\,x)\\,dx=-\\int_0^\\pi f(x)\\sin(n\\,x)\\,dx. $"}
{"_id": "7042675", "title": "", "text": "$ XRY + YRX = 0$"}
{"_id": "7858948", "title": "", "text": "$\\sum\\limits_{n=0}^\\infty n^{2m}e^{-an^2}  =\\left(-\\frac{\\partial}{\\partial a}\\right)^m \\sum\\limits_{n=0}^\\infty e^{-an^2}$"}
{"_id": "910107", "title": "", "text": "$q(x)=1/\\|x\\|_2$"}
{"_id": "1230772", "title": "", "text": "$ \\iiint_T 2x \\,\\,\\, dxdydz=2\\int_{0}^{1}\\int_{-\\pi}^{\\pi}\\int_{0}^{\\pi/2} \\rho^2 \\sin(\\theta)\\cos(\\sigma) d\\theta d\\sigma d\\rho=\\frac{2}{3}\\int_{-\\pi}^{\\pi}\\cos(\\sigma)d\\sigma $"}
{"_id": "224371", "title": "", "text": "$H^+$"}
{"_id": "9150397", "title": "", "text": "$1+8^x+27^x = 2^x+12^x+9^x$"}
{"_id": "8235778", "title": "", "text": "$\\forall N >0, \\; \\exists M >0: \\; x > M \\Rightarrow f(x) > N$"}
{"_id": "6607224", "title": "", "text": "$ r\\colon \\; \\begin{cases} x=at\\\\ y=bt\\\\ z=ct \\end{cases} $"}
{"_id": "3143990", "title": "", "text": "$\\sigma(a)<\\sigma(b)$"}
{"_id": "7678618", "title": "", "text": "${1 \\over 6^{n-2}}$"}
{"_id": "4260713", "title": "", "text": "$\\frac{dw}{dx} = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}$"}
{"_id": "2987887", "title": "", "text": "$P[Y_n=1|\\ Y_m=1]=1/2$"}
{"_id": "2700839", "title": "", "text": "$ e_1 = \\alpha+\\beta+\\gamma = -\\frac{b}{a} \\\\ e_2 = \\alpha\\beta+\\beta\\gamma+\\gamma\\alpha = +\\frac{c}{a} \\\\ e_3 = \\alpha\\beta\\gamma = -\\frac{d}{a} $"}
{"_id": "3790831", "title": "", "text": "$\\left \\| A \\right \\|_F = \\sqrt{\\mathrm{trace} \\left ( A^H A \\right )}$"}
{"_id": "3231657", "title": "", "text": "$F(x,y)= \\sum_{n=-\\infty}^{\\infty} f(|n|x) e^{-2 i \\pi |n|y}= \\frac{1}{x} \\sum_{n=-\\infty}^{\\infty} \\hat{f}(\\frac{n+y}{x})$"}
{"_id": "8730011", "title": "", "text": "$log(b)/log(a) = n$"}
{"_id": "7396965", "title": "", "text": "$\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin^2(x)}{\\sin(x)+\\cos(x)}$"}
{"_id": "2108806", "title": "", "text": "$K\\cong \\Bbb Q(\\gamma)=\\{a+b\\gamma+c\\gamma^2: a,b,c\\in\\Bbb Q\\}$"}
{"_id": "8539657", "title": "", "text": "$ dS=\\sqrt{\\left(\\frac{dx}{d\\phi}\\right)^2 + \\left(\\frac{dy}{d\\phi}\\right)^2} $"}
{"_id": "7749337", "title": "", "text": "$\\int_0^{\\pi/2} \\cos^n(t) \\sin(nt) \\,dt = \\int_0^{\\epsilon/n} \\cos^n(t) \\sin(nt) \\,dt + \\int_{\\epsilon/n}^{\\pi/2} \\cos^n(t) \\sin(nt) \\,dt$"}
{"_id": "3604506", "title": "", "text": "$f(x)=\\frac{2x}{R^2}$"}
{"_id": "1412522", "title": "", "text": "$ \\int_{0}^{ \\infty} f(x) dx $"}
{"_id": "1171330", "title": "", "text": "$d(x,Y)=||x-y||$"}
{"_id": "6264624", "title": "", "text": "${A_{ik}} = \\left\\{ \\matrix{   1 - {1 \\over P}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,i = k\\, \\hfill \\cr   - {1 \\over P}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,i \\ne k\\, \\hfill \\cr}  \\right.\\tag{4}$"}
{"_id": "8798850", "title": "", "text": "$\\gamma = (\\gamma_1, \\gamma_2, \\ldots \\gamma_n)$"}
{"_id": "4949595", "title": "", "text": "$\\underset{n=1}{\\overset{\\infty}{\\sum}}\\parallel z_{n}-x_{n}\\parallel_{n}^{2}<\\epsilon^{2}$"}
{"_id": "8217065", "title": "", "text": "$rn = r(1)n$"}
{"_id": "24778", "title": "", "text": "$M=\\pmatrix{a&b\\\\c&d}$"}
{"_id": "8266850", "title": "", "text": "$4+\\lim_{n \\rightarrow \\infty} \\sum_{i=1}^n 2=\\infty$"}
{"_id": "7877432", "title": "", "text": "$(a(x)b(y))'=a'(x)b(y)+a(x)b'(y)$"}
{"_id": "1470178", "title": "", "text": "$\\rm {\\bf Lemma}\\quad\\ \\lfloor x/(mn)\\rfloor = \\lfloor{\\lfloor x/m\\rfloor}/n\\rfloor\\quad for\\ \\ \\ n > 0 $"}
{"_id": "2230514", "title": "", "text": "$\\det \\begin{pmatrix} A & -B \\\\ B & A \\end{pmatrix} = \\det{(A+iB)}\\det{(A-iB)}$"}
{"_id": "2915684", "title": "", "text": "$f'(x) =\\frac{f(\\pi) - f(0) }{ \\pi - 0 }$"}
{"_id": "1716", "title": "", "text": "$[x,y]$"}
{"_id": "1811303", "title": "", "text": "$f(x) = 1/|x|^{d-2}$"}
{"_id": "5619706", "title": "", "text": "$(2^{2^{n-2}})^2$"}
{"_id": "8090814", "title": "", "text": "$6a(x)[(3-x)a(x)-3a(x)^2]=-xa(x)+x(2-x)a'(x)-x^2a''(x)$"}
{"_id": "1991047", "title": "", "text": "$\\frac{x_1+x_2+\\ldots+x_n}n\\;,$"}
{"_id": "8768581", "title": "", "text": "$y+1=y+(y+(-y))=(y+y)+(-y)=y+(-y)=1$"}
{"_id": "728148", "title": "", "text": "$\\sigma \\subseteq \\gamma \\subseteq \\tau.$"}
{"_id": "1990361", "title": "", "text": "$\\frac {d}{dt} \\int_0^t (E i(x)-R i(x)^2)dx=E i(t)-R i(t)^2=(E-R i(t)) i(t).$"}
{"_id": "611162", "title": "", "text": "$p(x)=x^2+x+2$"}
{"_id": "4687648", "title": "", "text": "$y_n=\\dfrac{x_1+x_2+x_3+\\cdots+x_n}{n}$"}
{"_id": "4549497", "title": "", "text": "$(5,-2,-2,2,0) = x(3,-2,-1,1,-1) + y(1,0,-1,1,1)$"}
{"_id": "1136546", "title": "", "text": "$\\mathbb{R}[x]/(x^3)$"}
{"_id": "2301387", "title": "", "text": "$\\beta = \\{A_1,A_2,A_3,...,A_n\\}$"}
{"_id": "5517296", "title": "", "text": "$\\begin{align}\\frac{ds}{dt}= \\sqrt{(\\frac{dy}{dt})^2+(\\frac{dx}{dt})^2}\\end{align}$"}
{"_id": "649605", "title": "", "text": "$P(2) \\implies P(3),$"}
{"_id": "3984530", "title": "", "text": "$e^x\\geq \\left(1+\\frac{x}{6}\\right)^6$"}
{"_id": "3518022", "title": "", "text": "$  \\int_{0}^{1} \\frac{\\sin(x)dx}{x} $"}
{"_id": "3893377", "title": "", "text": "$(\\alpha\\beta)^{+}=\\beta^{+}\\alpha^{+}$"}
{"_id": "2032816", "title": "", "text": "$3q^2p \\equiv \\begin{cases} 1 & p\\equiv q \\equiv 1\\\\                               0 & \\text{otherwise} \\end{cases}$"}
{"_id": "6362231", "title": "", "text": "$p\\oplus t$"}
{"_id": "4089466", "title": "", "text": "$\\int_a^b f(t)\\sin(t)dt=0=\\int_a^b f(t)\\cos(t)dt$"}
{"_id": "6337881", "title": "", "text": "$A_1 \\subseteq \\dots \\subseteq A_r$"}
{"_id": "5267382", "title": "", "text": "$X=\\frac{\\frac{c(1+r)^{T+1}+(1-c)r^2}{r(1+r)}}{\\frac{(c-r)(1+r)^{T}+(1-c)r}1}$"}
{"_id": "4800285", "title": "", "text": "$E(X^n) = E((X + 1)^{n−1})$"}
{"_id": "722238", "title": "", "text": "$\\operatorname E|X_n|^r\\to\\operatorname E|X|^r$"}
{"_id": "6791012", "title": "", "text": "$(x,y) = \\mathbf{x}$"}
{"_id": "408848", "title": "", "text": "$(1 + x/n)^n \\approx e^x$"}
{"_id": "3128486", "title": "", "text": "$\\left[\\begin{array}{cccc|c} 1&-1&2&1 & 2\\\\   2&-3&2&0 & 3\\\\ -1&1&2&3 & 6\\\\ -3&2&0&3 & 9\\\\ \\end{array}\\right]$"}
{"_id": "5329227", "title": "", "text": "$\\det(A_{x,i,j})=cofactor(a_{i,j})x+\\det(A)$"}
{"_id": "4512661", "title": "", "text": "$M = \\{\\{\\emptyset\\}\\}$"}
{"_id": "4329084", "title": "", "text": "$\\mathbb{R}^2: (a,b) + (c,d) = (a+b, c+d)$"}
{"_id": "1887332", "title": "", "text": "$\\frac1{x}+\\frac1{y}=1$"}
{"_id": "1487628", "title": "", "text": "$\\sum_{n=1}^r \\frac{1}{1+nQ}\\leq\\sum_{t=1}^\\infty\\left(\\sum_{n=k+1}^\\infty\\frac{1}{p_n}\\right)^t$"}
{"_id": "9191042", "title": "", "text": "$\\rho(A)=\\|A\\|_\\infty=1$"}
{"_id": "3876726", "title": "", "text": "$ \\gamma+\\beta\\gamma^{-1}=\\frac{\\gamma^2+\\beta}{\\gamma}=\\alpha\\gamma^{-1} $"}
{"_id": "7831996", "title": "", "text": "$(x-a)^{m-1}\\,(x-b)^{n-1}$"}
{"_id": "6702254", "title": "", "text": "$\\lim_{a\\rightarrow\\infty}\\frac{a^2-(a-1-1-1/a)^2}{a^2-(a-1+1+1/a)^2} = \\lim_{a\\rightarrow\\infty}\\frac{a^2-(a-2-1/a)^2}{a^2-(a+1/a)^2} = -\\infty$"}
{"_id": "9176489", "title": "", "text": "$\\mathscr{A}_0 =\\{a\\}$"}
{"_id": "5492935", "title": "", "text": "$\\lim_{x \\to c} f(x) = \\lim_{x \\to c} f'(x) = 0$"}
{"_id": "6304043", "title": "", "text": "$P(X\\geq 1 \\vert X\\geq 2) = 1$"}
{"_id": "5267383", "title": "", "text": "$X=\\frac{(c(1+r)^{T+1}+(1-c)r^2)1}{r(1+r)((c-r)(1+r)^{T}+(1-c)r)}$"}
{"_id": "8714368", "title": "", "text": "$\\frac{n(n+1)(n+2)(n+3)}{24}+\\frac{(n+1)(n+2)(n+3)}6=\\frac{(n+1)(n+2)(n+3)(n+4)}{24}$"}
{"_id": "311585", "title": "", "text": "$\\begin{align*} && P(0)\\implies P(1) &&&&\\text{is true and}\\\\ && P(1)\\implies P(2) &&&&\\text{is true and}\\\\ && P(2)\\implies P(3) &&&&\\text{is true and}\\\\ && &&\\vdots && \\end{align*}$"}
{"_id": "307763", "title": "", "text": "$\\left \\lfloor \\sum_{i=1}^{n}\\{x_i\\} \\right \\rfloor = L$"}
{"_id": "7333284", "title": "", "text": "$I_n=\\frac{1}{(x^2+1)^n}$"}
{"_id": "7935580", "title": "", "text": "$\\int_0^1 x^2\\sqrt{1-x^2}\\,dx = \\int_0^{\\pi/2} \\sin^2\\varphi \\sqrt{\\cos^2\\varphi}\\cos\\varphi\\,d\\varphi = \\int_0^{\\pi/2}\\sin^2\\varphi\\cos^2\\varphi\\,d\\varphi.$"}
{"_id": "7706451", "title": "", "text": "$\\sqrt[n]{z} = \\sqrt[n]{r}e^{\\theta i/n}$"}
{"_id": "4920769", "title": "", "text": "$(y-z)(y+z)+(z-x)(z+x)+(x-y)(x+y)=y^2-z^2+z^2-x^2+x^2-y^2=0$"}
{"_id": "8595668", "title": "", "text": "$\\tan u = \\frac{x}{2}$"}
{"_id": "4156519", "title": "", "text": "$\\,p_1p_2\\,\\,\\mid\\,\\,p_1p_2\\,=>\\,p_1p_2\\,\\,\\mid\\,\\,p_1\\,\\,\\,\\vee \\,\\,\\,p_1p_2\\,\\,\\mid\\,\\,p_2$"}
{"_id": "2768722", "title": "", "text": "$T_k[f](x_n+e_n) \\approx f(x_n + e_n) = 0$"}
{"_id": "7888235", "title": "", "text": "$P(i)=\\frac{1}{6}$"}
{"_id": "2469507", "title": "", "text": "$ \\gamma \\in K :\\gamma B \\subset R$"}
{"_id": "1122880", "title": "", "text": "$L=\\lim\\limits_{x\\to c}f(x)$"}
{"_id": "8100708", "title": "", "text": "$\\left(\\begin{array}{ccc|c} 1 & 1 & 1 & 4\\\\ 0 & 5 & 4 & 2\\\\ 0 & 5 & 4 & 2\\end{array}\\right)\\begin{array}{c} \\\\ R_2+R_3\\\\ R_3+R_2\\end{array} \\Leftrightarrow\\left(\\begin{array}{ccc|c} 1 & 0 & 1/5 & 18/5\\\\ 0 & 1 & 4/5 & 2/5\\\\ 0 & 0 & 0 & 0\\end{array}\\right)\\begin{array}{c} R_1-R_2/5\\\\ R_2/5\\\\ R_2-R_3\\end{array}$"}
{"_id": "4894399", "title": "", "text": "$\\begin{align*} \\lim\\limits_{n\\to \\infty}\\dfrac{1}{\\ln(n)}\\sum\\limits_{r=1}^{n}\\dfrac{a_r}{r} &=\\lim\\limits_{n\\to \\infty}\\dfrac{1}{n}\\cdot\\dfrac{n}{\\ln(n)}\\sum\\limits_{r=1}^{n}\\dfrac{a_r}{r}\\\\ &=\\lim\\limits_{n\\to \\infty}\\dfrac{1}{n}\\sum\\limits_{r=1}^{n}\\dfrac{a_r}{r/n}\\cdot\\dfrac{1}{\\ln(n)} \\end{align*}$"}
{"_id": "3267597", "title": "", "text": "$\\int_0^{\\infty}\\frac{1}{(x^2+y)^n}dx$"}
{"_id": "789964", "title": "", "text": "$f(n)-f(n-1) = f(n-1) + f(n-2) - f(n-2) - f(n-3)$"}
{"_id": "2798054", "title": "", "text": "$\\lim_{x \\to c}f(x)k(x) = \\lim_{x \\to c}h(x)g(x)$"}
{"_id": "5151599", "title": "", "text": "$ \\lfloor \\frac{a+b}{c}\\rfloor +\\lfloor \\frac{b+c}{a} \\rfloor+\\lfloor \\frac{c+a}{b} \\rfloor $"}
{"_id": "7173772", "title": "", "text": "$ d=\\frac{a+bc}{a-1} $"}
{"_id": "550138", "title": "", "text": "$\\{a\\}\\in\\{\\{a\\}\\}$"}
{"_id": "7266391", "title": "", "text": "$\\sum_{k \\in A} |x_k + y_k|^2 \\leq 4\\left( \\sum_{k \\in A} |x_k|^2 + \\sum_{k\\in A} |y_k|^2 \\right) < \\infty,$"}
{"_id": "4694139", "title": "", "text": "$\\int |fg|^r\\leq \\left(\\int |f|^{r\\frac{p}{r}}\\right)^{r/p}\\left(\\int |g|^{r\\frac{q}{r}}\\right)^{r/q}.$"}
{"_id": "6251794", "title": "", "text": "$[y,x]=y$"}
{"_id": "3183850", "title": "", "text": "$e\\in V\\subseteq \\overline{V}\\subseteq U$"}
{"_id": "1058736", "title": "", "text": "$\\int_{0}^{1}\\frac{\\sin(x^a)}{x^b}$"}
{"_id": "6843887", "title": "", "text": "$V=\\ker(P)\\oplus U$"}
{"_id": "714678", "title": "", "text": "$P(E)P(E')=P(E\\cap E')$"}
{"_id": "7810569", "title": "", "text": "$\\vartheta(x):=\\sum_{n=-\\infty}^\\infty e^{-n^2\\pi x}={1\\over\\sqrt{x}}\\vartheta\\left({1\\over x}\\right)\\ .$"}
{"_id": "6847148", "title": "", "text": "$\\sum_{k=1}^{n-1}\\cos\\frac{2\\pi k}{n}=\\frac{\\sum\\limits_{k=1}^{n-1}2\\sin\\frac{\\pi}{n}\\cos\\frac{2\\pi k}{n}}{2\\sin\\frac{\\pi}{n}}=\\frac{\\sum\\limits_{k=1}^{n-1}\\left(\\sin\\frac{(2k+1)\\pi}{n}-\\sin\\frac{(2k-1)\\pi }{n}\\right)}{2\\sin\\frac{\\pi}{n}}=$"}
{"_id": "6696306", "title": "", "text": "$ Var(X^*_t | X^*_0 = x^*_0) = \\frac{\\sigma^2}{2 \\mu},\\quad \\forall t > 0 $"}
{"_id": "1268847", "title": "", "text": "$u(x)=1 - 1/|x|$"}
{"_id": "5809864", "title": "", "text": "$V(a)\\subseteq V(a)$"}
{"_id": "2030126", "title": "", "text": "$4 \\times2^x+3^x=5^x$"}
{"_id": "3976620", "title": "", "text": "$\\mu\\left(\\liminf A_{n}\\right)=\\lim_{n\\to\\infty}\\mu\\left(B_{n}\\right)\\leq\\lim_{n\\to\\infty}\\inf_{k\\geq n} \\mu A_{k}  =\\liminf\\mu A_{n}$"}
{"_id": "7036984", "title": "", "text": "$\\forall \\epsilon >0, \\exists \\delta >0, \\forall x \\in \\mathbb R, s.t. |f(x)-f(a)|< \\epsilon, |x-a|< \\delta$"}
{"_id": "1075817", "title": "", "text": "$\\begin{cases} x=t \\\\ y=2-t \\\\ z= 5\\end{cases}$"}
{"_id": "3697526", "title": "", "text": "$a+b,a+c,a+d \\in \\mathbb{Q}$"}
{"_id": "1889514", "title": "", "text": "$(x,y) = x$"}
{"_id": "3494877", "title": "", "text": "$\\mathop {\\lim }\\limits_{n \\to \\infty } \\int_{ - \\pi }^\\pi  {{x^2}} \\frac{{\\sin \\left( {2nx} \\right)}}{{\\sin x}}xdx =  - 8\\pi \\sum\\limits_{k \\geqslant 1} {\\frac{1}{{{{\\left( {2k - 1} \\right)}^2}}}}  =  - 8\\pi \\frac{{{\\pi ^2}}}{8} =  - {\\pi ^3}$"}
{"_id": "900577", "title": "", "text": "$\\lim_{n\\rightarrow \\infty}{1^n}=\\lim_{n\\rightarrow \\infty}{1}=1$"}
{"_id": "2130683", "title": "", "text": "$\\det\\begin{pmatrix}A' & C' \\\\ 0 & B\\\\ \\end{pmatrix} = \\det(A')\\det(B) $"}
{"_id": "444334", "title": "", "text": "$(y,y,x,*),(y,y,*,x),(y,x,y,*),(y,*,y,x)$"}
{"_id": "8242550", "title": "", "text": "$\\frac{1}{|D_n|} f_n(2) = {n+2 \\choose 2}$"}
{"_id": "4153239", "title": "", "text": "$ \\gamma'(t) = \\left(\\frac{\\partial \\gamma_1}{\\partial t},\\frac{\\partial \\gamma_2}{\\partial t}\\right)\\approx\\frac{\\gamma(t+h)-\\gamma(t)}{h}$"}
{"_id": "9119003", "title": "", "text": "$ \\begin{vmatrix} x & y & 1  \\\\  x1 & y1 & 1 \\\\ x2 & y2 & 1  \\end{vmatrix} $"}
{"_id": "4777630", "title": "", "text": "$ det\\begin{pmatrix}A & -B\\\\ B & A \\end{pmatrix}=|det(A+iB)|^2, $"}
{"_id": "3190656", "title": "", "text": "$6^{2x-3}=9x$"}
{"_id": "1573873", "title": "", "text": "$x_3,x_2\\in \\mathbb R$"}
{"_id": "8426774", "title": "", "text": "$\\lim_{n \\to \\infty} \\sum\\limits_{k=1}^n \\frac{1}{k 2^k} $"}
{"_id": "2850438", "title": "", "text": "$I_j^{(A)},I_j^{(B)}$"}
{"_id": "5585257", "title": "", "text": "$\\sum_{n=0}^\\infty {(x-1)^n\\over (n+2)!}={1 \\over (x-1)^2} \\sum_{n=0}^\\infty {(x-1)^{n+2} \\over (n+2)!}$"}
{"_id": "218430", "title": "", "text": "$\\Bigl(1+\\dfrac1{n+1}\\Bigr)^{\\!n+1}-\\Bigl(1+\\dfrac1n\\Bigr)^{\\!n}<\\mathrm e-2.$"}
{"_id": "898081", "title": "", "text": "$=\\begin{vmatrix} x &x^2  &1 \\\\  y &y^2  &1 \\\\  z &z^2  &1  \\end{vmatrix}+xyz\\begin{vmatrix} 1 &x  &x^2 \\\\  1 &y  &y^2 \\\\  1 &z  &z^2  \\end{vmatrix}=\\ldots$"}
{"_id": "466222", "title": "", "text": "$ax+by=d$"}
{"_id": "6317937", "title": "", "text": "$\\text{for}\\;\\;1\\le k\\le n\\;,\\;\\;\\;i_k=\\begin{cases}0\\;,\\;\\;\\text{if}\\;\\;k\\notin X\\\\{}\\\\1\\;,\\;\\;\\text{if}\\;\\;k\\in X\\end{cases}$"}
{"_id": "4372724", "title": "", "text": "$\\cos(n\\theta) = \\sum_{p=0}^{\\lfloor \\frac{n}{2} \\rfloor} (-1)^{p} \\begin{pmatrix} n \\\\ 2p \\end{pmatrix} \\sin^{2p}(\\theta)\\cos^{n-2p}(\\theta)$"}
{"_id": "4743076", "title": "", "text": "$3\\mid k^3-k$"}
{"_id": "9026725", "title": "", "text": "$\\psi = \\sum_{i=0}^n a_i \\chi_{A_i}$"}
{"_id": "5149765", "title": "", "text": "$\\mathcal{R} = \\{\\{0\\}\\}$"}
{"_id": "5150334", "title": "", "text": "$\\log_a(a)+\\log_a(b)=\\log_a(ab)$"}
{"_id": "529265", "title": "", "text": "$d(x,y)=\\frac1n$"}
{"_id": "6586715", "title": "", "text": "$ e^x = e^{\\frac{x}{n} \\cdot n} \\geq \\left(1 + \\frac{x}{n} \\right)^n $"}
{"_id": "4172676", "title": "", "text": "$\\sum_{k=1}^{n}\\cos(\\frac{k\\pi}{2n})\\frac{ \\pi}{2n}$"}
{"_id": "8783449", "title": "", "text": "$|xy|=rs$"}
{"_id": "1564466", "title": "", "text": "$z^n = |z|^n (\\cos(n\\phi) + i\\sin(n\\phi))$"}
{"_id": "4110976", "title": "", "text": "$S(\\log_2n)=5n+2$"}
{"_id": "1355552", "title": "", "text": "$\\;\\mathrm e^{i\\tfrac{(n+1)x}2}$"}
{"_id": "4573902", "title": "", "text": "$e^{2 i \\pi/(p-1)}$"}
{"_id": "6902509", "title": "", "text": "$A_n=a_1+\\ldots+a_n = \\frac{n(n+1)}{2}$"}
{"_id": "3582027", "title": "", "text": "$\\sum_{n=1}^{\\infty}(1/n)=\\infty.$"}
{"_id": "9126563", "title": "", "text": "$(a+2b)a = - (b+2a)b,$"}
{"_id": "8434160", "title": "", "text": "$f_\\theta(x)=\\dfrac{2x}{\\theta^2}$"}
{"_id": "5905183", "title": "", "text": "$ \\cos^4 \\varphi - 6 \\cos^2 \\varphi \\sin^2 \\varphi + \\sin^4 \\varphi = (\\cos^2 \\varphi - \\sin^2 \\varphi)^2 - 4 \\cos^2 \\varphi \\sin^2 \\varphi = \\cos^2 2\\varphi -  \\sin^2 2\\varphi  $"}
{"_id": "1662123", "title": "", "text": "$\\frac 1r[(1+r)^n -1] = \\sum_1^T (1+r)^{T-1}$"}
{"_id": "4719504", "title": "", "text": "$ \\frac{\\langle x,v \\rangle}{\\vert\\vert v \\vert\\vert} \\leq \\frac{\\vert \\langle x,v \\rangle \\vert}{\\vert\\vert v \\vert\\vert} \\leq \\vert\\vert x \\vert\\vert $"}
{"_id": "8981183", "title": "", "text": "$d_{g}(x,y) = \\inf\\{\\ell(\\gamma)\\mid\\gamma:[0,1]\\to M,\\gamma\\text{ is a curve},\\gamma(0)=x,\\gamma(1)=y\\}$"}
{"_id": "8946218", "title": "", "text": "$y,y',y'',y'''$"}
{"_id": "2116686", "title": "", "text": "$ P(k+1) : (k+1)^3 < 3^{k+1} $"}
{"_id": "6650492", "title": "", "text": "$\\forall m P(n,m) \\Rightarrow \\forall m P(n++,m)$"}
{"_id": "9171952", "title": "", "text": "$\\int_0^\\infty\\int_t^\\infty f(x)dxdt = \\int_0^\\infty f(x) \\int_0^xdt dx$"}
{"_id": "7710981", "title": "", "text": "$ \\left[\\begin{array}{ccc|c} 1 & -1 & 1 & 1 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & -6 & 4 & 5 \\end{array}\\right] $"}
{"_id": "5790136", "title": "", "text": "$m=rs, a^m=a^{rs}=(a^r)^s\\implies (a^r-b^r)\\mid\\{(a^r)^s- (b^r)^s\\}$"}
{"_id": "3776135", "title": "", "text": "$ax+by=f(a,b)$"}
{"_id": "9326203", "title": "", "text": "${\\sqrt[3]{1} + \\sqrt[3]{2} + \\sqrt[3]{3} + \\dots + \\sqrt[3]{n} \\over \\sqrt[3]{n^4}}$"}
{"_id": "536394", "title": "", "text": "$\\frac{||Tf||}{||f||}\\leq c <\\infty$"}
{"_id": "5825900", "title": "", "text": "$\\left\\lfloor\\frac{x/a}b\\right\\rfloor = \\left\\lfloor\\frac{\\lfloor x/a\\rfloor}b\\right \\rfloor$"}
{"_id": "4073478", "title": "", "text": "$A_1 \\subseteq A_2 \\supseteq A_3 \\subseteq A_4 \\supseteq ...$"}
{"_id": "5264861", "title": "", "text": "$=1+\\dfrac{a_1}{1-a_1} \\left(1+\\dfrac{a_2}{1-a_2}+\\dfrac{a_3}{1-a_3}\\right)+\\dfrac{a_2}{1-a_2} \\left(1+\\dfrac{a_1}{1-a_1}+\\dfrac{a_3}{1-a_3}\\right)+\\dfrac{a_3}{1-a_3} \\left(1+\\dfrac{a_1}{1-a_1}+\\dfrac{a_2}{1-a_2}\\right)$"}
{"_id": "1339087", "title": "", "text": "$\\widehat{X}=\\frac{X_1+X_2+...+X_n}{n}$"}
{"_id": "1678910", "title": "", "text": "$R/2R\\cong \\mathbb Z_2[x]/(x^4 + 1)$"}
{"_id": "6098604", "title": "", "text": "$\\begin{bmatrix}1 & 3 & -1\\\\4 & -1 & 2\\\\2 & -1 & -3\\end{bmatrix}$"}
{"_id": "8341196", "title": "", "text": "$D = \\lbrace (x,y) \\in \\Bbb{R}^2 | 0\\le x \\le 1, 0 \\le y \\le x \\rbrace$"}
{"_id": "3317468", "title": "", "text": "$P(s) = 1/6$"}
{"_id": "8472832", "title": "", "text": "$\\|A\\|_2^2=\\rho(A^TA)\\leq\\|A^TA\\|_\\infty$"}
{"_id": "6770639", "title": "", "text": "$\\gamma(a)\\le\\max\\{\\gamma(x_1),\\gamma(x_2)\\}$"}
{"_id": "7730615", "title": "", "text": "$         \\begin{pmatrix}         1 & 1 & 1 \\\\         a & b & c \\\\         a^2 & b^2 & c^2 \\\\         \\end{pmatrix}^{-1}         \\begin{pmatrix}         1  \\\\         t  \\\\         t^2 \\\\         \\end{pmatrix} =        \\begin{pmatrix}         x  \\\\         y  \\\\         z \\\\         \\end{pmatrix} $"}
{"_id": "8113984", "title": "", "text": "$\\displaystyle \\left\\{(1+x)^{r}+(1+x)^{r+1}+(1+x)^{r+1}\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot+(1+x)^{n}\\right\\} = \\frac{(1+x)^{n+1}-(1+x)^r}{(1+x)-1}=\\frac{(1+x)^{n+1}-(1+x)^r}{x}$"}
{"_id": "926811", "title": "", "text": "$\\cos(\\pi/2^{n+2})$"}
{"_id": "8344632", "title": "", "text": "$\\{f, 0_{[0,1]},f, 0_{[0,1]},\\dots \\}$"}
{"_id": "4996264", "title": "", "text": "$(x∘y)∘z = x⊗y⊗z + y⊗x⊗z + z⊗x⊗y + z⊗y⊗x$"}
{"_id": "8396750", "title": "", "text": "$xHx^{-1}, x\\in G$"}
{"_id": "1044740", "title": "", "text": "$ \\sum_{n=2}^\\infty x^{\\ln n}=\\sum_{n=2}^\\infty \\frac1{n^{-\\ln x}} $"}
{"_id": "726541", "title": "", "text": "$Cov(X,T)=0$"}
{"_id": "6892003", "title": "", "text": "$\\frac{3}{27}=\\frac{1}{9}$"}
{"_id": "2399210", "title": "", "text": "$b=as$"}
{"_id": "9258416", "title": "", "text": "$R^{n+1} -  $"}
{"_id": "1594811", "title": "", "text": "$9+9x+3x^3+6x^4+3x^5+x^6$"}
{"_id": "2396713", "title": "", "text": "$0=B_0(1+r)^N - C\\frac{(1+r)^N-1}{r}$"}
{"_id": "7184543", "title": "", "text": "$ \\begin{align} \\left(1+\\frac{x}n\\right)^n &\\le\\lim_{n\\to\\infty}\\left(1+\\frac{x}n\\right)^n\\\\[6pt] &=e^x\\tag4 \\end{align} $"}
{"_id": "60069", "title": "", "text": "$\\|A\\|_{2} = \\sqrt{\\rho(A^{T}A)}$"}
{"_id": "5082659", "title": "", "text": "$ f(m+a)=f(m)+b$"}
{"_id": "7679696", "title": "", "text": "$A_3 \\subset A_2 \\subset A_1$"}
{"_id": "3385931", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^n p_i(1-p_i)=\\infty$"}
{"_id": "961245", "title": "", "text": "$\\left\\lfloor\\frac{\\lfloor x\\rfloor}{n}\\right\\rfloor\\le\\left\\lfloor\\frac{x}{n}\\right\\rfloor$"}
{"_id": "5012931", "title": "", "text": "$ \\pi\\,f=(f^*\\,\\pi^*)^*=(f\\,\\pi)^*=(\\pi\\,f\\,\\pi)^*=\\pi^*\\,f^*\\,\\pi^*=\\pi\\,f\\,\\pi=f\\,\\pi. $"}
{"_id": "3698800", "title": "", "text": "$g(x)=1/|x|$"}
{"_id": "318867", "title": "", "text": "$\\alpha^{+}$"}
{"_id": "4900761", "title": "", "text": "$P_{1}P_{2}=14$"}
{"_id": "6059028", "title": "", "text": "$AC\\lor \\bar BC\\lor A\\bar B$"}
{"_id": "6045704", "title": "", "text": "$ \\frac{2}{T}\\frac{1}{\\dot{\\theta}(\\theta)}=\\frac{\\sqrt{\\left(1-e^2\\right)^3}}{\\pi\\left(1+e\\cos{\\theta}\\right)^2} $"}
{"_id": "8975910", "title": "", "text": "$\\sum_{t=1}^{n}\\frac{t^2}{(1+r)^t}$"}
{"_id": "1711549", "title": "", "text": "$A=\\lim_{n\\to \\infty}\\sqrt{1+\\sqrt{\\frac{1}{2}+\\sqrt{\\frac{1}{3}+\\cdots+\\sqrt{\\frac{1}{n}}}}}$"}
{"_id": "72059", "title": "", "text": "${x}^{2}-z\\,x-y\\,x+{z}^{2}-y\\,z+{y}^{2}=1$"}
{"_id": "2592939", "title": "", "text": "$Cov(X,Y)=\\rho$"}
{"_id": "5208318", "title": "", "text": "$\\begin{align} \\int \\tan^n x\\,dx &= \\int \\tan^{n-2}x\\,(\\sec^2 x - 1)\\,dx \\\\ &= \\int \\tan^{n-2}x\\,\\sec^2 x\\,dx - \\int \\tan^{n-2}x\\,dx \\\\ &= \\int u^{n-2}\\,\\,du - \\int \\tan^{n-2}x\\,dx \\\\ &= \\frac{1}{n-1}u^{n-1} - \\int \\tan^{n-2}x\\,dx \\\\ &= \\frac{1}{n-1}\\tan^{n-1} x - \\int \\tan^{n-2}x\\,dx \\end{align}$"}
{"_id": "6197932", "title": "", "text": "$ n = m+1\\\\ \\text{and}\\\\ P(n) \\implies P(m+1) $"}
{"_id": "3787807", "title": "", "text": "$1+\\frac{1}{4}\\left((e^{x/20})^2-2+(e^{-x/10})^2\\right).$"}
{"_id": "2234279", "title": "", "text": "$\\mathcal{A}+\\mathcal{X}=\\mathcal{A}+\\mathcal{Y}$"}
{"_id": "1011051", "title": "", "text": "$\\frac{X_1+X_2+\\ldots+X_N}{N}=1$"}
{"_id": "6070268", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{x+y}.$"}
{"_id": "6911609", "title": "", "text": "$x = \\frac{u}{\\gamma} - \\frac{ay}{2\\gamma^{2}}$"}
{"_id": "6701871", "title": "", "text": "$\\left(\\frac12\\right)^{n-1}.$"}
{"_id": "3083620", "title": "", "text": "$\\int_{0}^{\\infty}\\frac{\\sin^2(x)}{x^3}\\,dx$"}
{"_id": "2687917", "title": "", "text": "$\\int_{0}^{\\infty} f(x)^2dx$"}
{"_id": "5511064", "title": "", "text": "$\\displaystyle\\sum_{m=0}^{n-1}\\cos\\frac{2m\\pi}n=\\frac0{2^{n-1}}$"}
{"_id": "6674614", "title": "", "text": "$P(X_2=1|X_1=1)=1/2$"}
{"_id": "2642265", "title": "", "text": "$\\Bbb{E}(|X-Y|) \\le \\Bbb{E}(|X+Y|)$"}
{"_id": "2084714", "title": "", "text": "$ \\lim_{x\\rightarrow a^-}f(x)=\\lim_{x\\rightarrow a^+}f(x)=\\lim_{x\\rightarrow a}f(x)=f(a) $"}
{"_id": "324947", "title": "", "text": "$|x-a|<\\delta\\implies |f(x)-f(a)|<\\epsilon$"}
{"_id": "2326495", "title": "", "text": "$\\int_C \\cos z\\ dz=\\int_0^{\\pi/2}f(z(t))\\dot{z}(t)\\ dt=\\int_0^{\\pi/2}\\cos(\\sqrt{2}\\cos t+i\\sin t)\\cdot(-\\sqrt{2}\\sin t+i\\cot t)\\ dt.$"}
{"_id": "3879035", "title": "", "text": "$Cov_3(x,y)>0$"}
{"_id": "5511370", "title": "", "text": "$z=\\sqrt[3]{4}e^{\\frac{i\\pi}{k}}$"}
{"_id": "1701504", "title": "", "text": "$\\tan\\theta=\\dfrac{x}{100}$"}
{"_id": "4798535", "title": "", "text": "$f(x)=x^2\\sum_{n\\ge0}{\\frac{(-1)^n}{x^n}}=\\sum_{n\\ge0}{\\frac{(-1)^nx^2}{x^n}}=\\sum_{n\\ge0}{\\frac{(-1)^n}{x^{n-2}}}=\\sum_{n\\ge0}{\\frac{1}{(-x)^{n-2}}}$"}
{"_id": "7322989", "title": "", "text": "$=\\frac{1}{\\gamma(a\\gamma-1)}\\int \\frac{dt}{t+1}-\\frac{1}{\\gamma(a\\gamma-1)}\\int \\frac{dt}{t+a\\gamma}$"}
{"_id": "4405160", "title": "", "text": "$ \\liminf_{x\\to 0}f(x)=\\liminf_{|y|\\to\\infty}f\\Big(\\frac{1}{y}\\Big)=\\liminf_{y\\to\\pm\\infty}f\\Big(\\frac{1}{y}\\Big). $"}
{"_id": "3618291", "title": "", "text": "$S = \\sqrt{1+\\left(\\frac{dz}{dx}\\right)^2 + \\left(\\frac{dz}{dy}\\right)^2}$"}
{"_id": "980886", "title": "", "text": "$n = \\frac{m(m+1)}{2}$"}
{"_id": "131425", "title": "", "text": "$\\begin{pmatrix} a^2 & a & 1 \\\\ b^2 & b & 1  \\\\ c^2 & c & 1\\\\ \\end{pmatrix} $"}
{"_id": "5265854", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\mu^*(A_n) \\leq \\mu^*(A)$"}
{"_id": "5020439", "title": "", "text": "$\\frac {a^6 - (a^3-a) - (a^2-a) - a}{6} + \\frac  {a^3-a}{3}+ \\frac  {a^2-a}{2} + a$"}
{"_id": "7857759", "title": "", "text": "$ \\Vert A \\Vert_2 =\\sqrt{\\lambda_\\max (A^TA)}=\\sqrt{\\lambda_\\max (D^2)}$"}
{"_id": "1251373", "title": "", "text": "$\\lim_{n\\to\\infty} \\inf s_n \\leq \\lim_{n\\to\\infty} \\inf t_n,$"}
{"_id": "1348967", "title": "", "text": "${{\\vartheta }_{3}}\\left( z|\\tau  \\right)=\\sum\\limits_{n=-\\infty }^{\\infty }{{{e}^{i\\pi {{n}^{2}}\\tau }}{{e}^{2niz}}}$"}
{"_id": "1758279", "title": "", "text": "$\\frac{n}{n+1} = \\sum\\limits_{r=1}^n \\frac{(-1)^{r+1}}{r+1}\\binom{n}{r} .$"}
{"_id": "85014", "title": "", "text": "$1/\\sqrt{x}$"}
{"_id": "5014308", "title": "", "text": "$ \\sum_{n=1}^\\infty \\frac1{n^2} = \\sum_{n=1}^\\infty \\frac1{(2n)^2}+\\sum_{n=1}^\\infty \\frac1{(2n-1)^2}.$"}
{"_id": "4512258", "title": "", "text": "$(a,x) *(b,y)=(a,x)$"}
{"_id": "1087842", "title": "", "text": "$H = [a,b]\\times[c,d] := \\{(x,y)| a \\le x \\le b; c \\le y \\le d\\}$"}
{"_id": "8850087", "title": "", "text": "$f(n)=n^2-7n+12.$"}
{"_id": "2460936", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty}\\cos\\frac{2n\\pi}{3} = 0$"}
{"_id": "684736", "title": "", "text": "$M=\\begin{pmatrix}A&B\\\\0&C\\end{pmatrix}$"}
{"_id": "1390195", "title": "", "text": "$F(x) = \\int\\limits_a^{b(x)} {f(t)dt} $"}
{"_id": "6297873", "title": "", "text": "$ s(x) = \\sum \\limits_{i = 1}^{n} \\alpha_{i} \\chi_{A_{i}}(x)$"}
{"_id": "9162202", "title": "", "text": "$T_X X$"}
{"_id": "1451951", "title": "", "text": "$P(E_2) = 3/10.$"}
{"_id": "4108269", "title": "", "text": "$K = \\{k_1, k_2, k_3,....k_m \\}$"}
{"_id": "2083545", "title": "", "text": "$\\frac{a+bx}{1+x}$"}
{"_id": "9017212", "title": "", "text": "$\\vartheta(u,-i\\tau)=\\vartheta(u+1,-i\\tau)$"}
{"_id": "5709893", "title": "", "text": "$A\\in\\{A_1, A_2, \\ldots\\}$"}
{"_id": "927414", "title": "", "text": "$P(m) \\implies P(m + 1)$"}
{"_id": "7276834", "title": "", "text": "$\\mathbb{E}[X|X_{0} = x_{0}]$"}
{"_id": "3933220", "title": "", "text": "$P(x)=\\frac{1}{6}$"}
{"_id": "6517948", "title": "", "text": "$I_{n,a}=\\int_{-\\infty} ^{\\infty} \\frac{dx}{(1+\\frac{x^2}{a})^n}$"}
{"_id": "3706385", "title": "", "text": "$A_1 ... A_k$"}
{"_id": "7269251", "title": "", "text": "$\\sum_{n=0}^\\infty \\left \\lvert \\frac{(-1)^n}{n+x^2}\\right \\rvert = \\sum^\\infty_{n=0} \\frac{1}{n+x^2} = \\sum_{0\\le n < x^2} \\frac{1}{n+x^2} + \\sum_{n \\ge x^2} \\frac{1}{n+x^2} \\ge \\sum_{0\\le n < x^2} \\frac{1}{n+x^2} + \\sum_{n\\ge x^2} \\frac{1}{2n}.$"}
{"_id": "9110287", "title": "", "text": "$\\{a, aa, bbb, aab, z, Z\\}$"}
{"_id": "4912843", "title": "", "text": "$xRy, yRh$"}
{"_id": "917857", "title": "", "text": "$f(z)=f(-z)=(-z)^2=z^2$"}
{"_id": "3980849", "title": "", "text": "$  p(k)=(\\frac{2}{3})^{k-1}(\\frac{1}{3}), $"}
{"_id": "526904", "title": "", "text": "$x_4 = 40$"}
{"_id": "5914540", "title": "", "text": "$x(n(x+y)^{n-1} + x n(n-1)(x+y)^{n-2}) = \\sum_{k=0}^n k^2 \\binom{n}{k} x^k y^{n-k}$"}
{"_id": "712667", "title": "", "text": "$\\tan u = x/2$"}
{"_id": "9260485", "title": "", "text": "$S(m) \\Rightarrow S(m+1)$"}
{"_id": "9081388", "title": "", "text": "$ X_n \\leq x \\Rightarrow X \\leq x + \\epsilon$"}
{"_id": "4410998", "title": "", "text": "$1 + 3 + 9 + 27 + ... = -\\frac{1}{2}$"}
{"_id": "7767322", "title": "", "text": "$\\int_{0}^{\\infty }\\frac{x^{n-1}}{e^{x}-1}\\, \\mathrm{d}x=\\sum_{k=1}^{\\infty }\\int_{0}^{\\infty }x^{n-1}e^{-kx}\\, \\mathrm{d}x\\overset{[1]}{=}\\sum_{k=1}^{\\infty }\\frac{1}{k^{n}}\\int_{0}^{\\infty }x^{n-1}e^{-x}\\, \\mathrm{d}x=\\zeta \\left ( n \\right )\\Gamma \\left ( n \\right )$"}
{"_id": "6998661", "title": "", "text": "$f'(c) = L = \\lim_{x \\to c}f'(x)$"}
{"_id": "6723659", "title": "", "text": "$xRy \\wedge yRx \\Longrightarrow y=x$"}
{"_id": "138787", "title": "", "text": "$\\dfrac{10!}{3!\\cdot 7!}=120 \\text{ ways.}$"}
{"_id": "3063808", "title": "", "text": "$\\{\\gamma \\in W(\\omega_1): \\gamma \\leq a\\}$"}
{"_id": "417982", "title": "", "text": "$\n \\|M\\|_2 = \\sqrt{ \\rho(M^T M) } \n $"}
{"_id": "8451808", "title": "", "text": "$\\left[\\begin{array}{ccc|c}1&1&k&1\\\\1&k&1&1\\\\k&1&1&1\\end{array}\\right]$"}
{"_id": "3824910", "title": "", "text": "$\\gamma_n(t) = \\gamma(\\frac{1}{n}\\lfloor n t \\rfloor)$"}
{"_id": "514345", "title": "", "text": "$P_1P_2P_3$"}
{"_id": "8023422", "title": "", "text": "$||A||_2 = \\sqrt{\\rho(A^HA)}$"}
{"_id": "5456377", "title": "", "text": "$\\gamma = \\gamma_+ - \\gamma_-$"}
{"_id": "2181608", "title": "", "text": "$\\gamma*(\\gamma*\\gamma) = (\\gamma*\\gamma)*\\gamma$"}
{"_id": "3437764", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{3}{2014}$"}
{"_id": "4158951", "title": "", "text": "$K\\subset V\\subset \\overline{V}\\subset U$"}
{"_id": "2005566", "title": "", "text": "$\\sum\\limits_{n=1}^{\\infty}{(\\frac{1}{n^2}-\\frac{1}{n})}:=\\lim\\limits_{M\\to\\infty}{\\sum\\limits_{n=1}^{M}{(\\frac{1}{n^2}-\\frac{1}{n})}}=\\lim\\limits_{M\\to\\infty}{\\sum\\limits_{n=1}^{M}{\\frac{1}{n^2}}}-\\lim\\limits_{M\\to\\infty}{\\sum\\limits_{n=1}^{M}{\\frac{1}{n}}}=:\\sum\\limits_{n=1}^{\\infty}{\\frac{1}{n^2}}-\\sum\\limits_{n=1}^{\\infty}{\\frac{1}{n}}$"}
{"_id": "6391019", "title": "", "text": "$\\mathbb{E}|X-Y|^\\alpha \\leq 1/2$"}
{"_id": "5729077", "title": "", "text": "$\\mathbb{P}(X_1 = 1 | X_0 =n) > 0$"}
{"_id": "4109741", "title": "", "text": "$\\operatorname{det}Df(\\varphi)=-\\sin(\\varphi)\\cos(\\varphi)$"}
{"_id": "6967162", "title": "", "text": "$f[(0,\\pi)]=(0,\\pi)$"}
{"_id": "2039250", "title": "", "text": "$\\frac{m}{d} \\in [x,y]$"}
{"_id": "1830549", "title": "", "text": "$\\tan(E) = \\frac{\\sqrt{1-e^2} \\sin(\\theta)}{e + \\cos(\\theta)}$"}
{"_id": "5617467", "title": "", "text": "$y=\\frac{a+b x}{c+ x}$"}
{"_id": "5260080", "title": "", "text": "$\\displaystyle \\frac {\\partial} {\\partial x} f(x,y(x))=\\frac {\\partial} {\\partial x}(x^2+y(x)x+y(x)^2)=2x+y'(x)x+y(x)+2y'(x)y(x)$"}
{"_id": "4895647", "title": "", "text": "$f'(b) < k < f'(a)$"}
{"_id": "7067919", "title": "", "text": "$d(x,A)-d(x',A)\\leqslant d(x,x')$"}
{"_id": "3636928", "title": "", "text": "$\\displaystyle I = \\int_{0}^{\\pi}\\frac{\\sin (884 x)\\sin (1122x)}{2\\sin x}dx$"}
{"_id": "1037385", "title": "", "text": "$|ab|=2$"}
{"_id": "1213623", "title": "", "text": "$F(x)=\\int_a^xf(t)dt=\\int_a^x \\tilde f(t) dt=\\tilde F(x)$"}
{"_id": "5914541", "title": "", "text": "$xn(x+y)^{n-1} + x^2 n(n-1)(x+y)^{n-2} = \\sum_{k=0}^n k^2 \\binom{n}{k} x^k y^{n-k}$"}
{"_id": "6667731", "title": "", "text": "$\\{a, a + b, \\ldots, a + (N − 1)b\\}$"}
{"_id": "3893782", "title": "", "text": "$16^x = 12^x + 9^x$"}
{"_id": "2205996", "title": "", "text": "$(1-r)(1+r+r^{2}+\\ldots + r^{n})=1 - r^{n+1}$"}
{"_id": "7081449", "title": "", "text": "$d{\\bf g}(0,0)=\\left[\\matrix{1&0\\cr 0&1\\cr}\\right]\\ .$"}
{"_id": "6793011", "title": "", "text": "$\\text{Li}_s(x)=\\sum_{n=1}^\\infty\\frac{x^n}{n^s} ,\\text{    }\\text{ } \\text{ } \\zeta(s)=\\sum_{n=1}^\\infty\\frac{1}{n^s}$"}
{"_id": "1289023", "title": "", "text": "$\\frac{\\mathbb{R}[x]}{x^4-1}$"}
{"_id": "704717", "title": "", "text": "$q!-q+1,\\ldots, q!-2$"}
{"_id": "2382551", "title": "", "text": "$d(x,A)=inf\\{d(x,y):y \\in A\\}$"}
{"_id": "4324878", "title": "", "text": "$\\circ: \\mathcal{C}'(\\mathbb{R}^n,\\mathbb{R}^n) \\times \\mathcal{C}'(\\mathbb{R}^n,\\mathbb{R}^n) \\longrightarrow \\mathcal{C}'(\\mathbb{R}^n,\\mathbb{R}^n)$"}
{"_id": "9280978", "title": "", "text": "$\\frac{t(1+r)^{t-1}}{(\\sum_{k=1}^T (1+r)^{k-t})^2}$"}
{"_id": "1661342", "title": "", "text": "$\\displaystyle \\int\\frac{1}{(x^3+1)^2} \\mathrm dx$"}
{"_id": "1379077", "title": "", "text": "$10[(1+r)^{n-1}+(1+r)^{n-2}+\\ldots (1+r)^1+1]=10\\frac {(1+r)^n-1}{1+r}$"}
{"_id": "3695868", "title": "", "text": "$\\int_{-\\pi}^\\pi\\frac{\\sin(x)}{1+x^2}{\\rm~d}x=0$"}
{"_id": "3743460", "title": "", "text": "$\\gcd(a/\\gcd(a,b)$"}
{"_id": "7734356", "title": "", "text": "$S=-\\sum_{k=0}^nk\\cos\\dfrac{2\\pi k}n$"}
{"_id": "1996833", "title": "", "text": "$f_Z(x)=\\frac{1}{x^2}$"}
{"_id": "6166239", "title": "", "text": "$g_n(x) = (\\frac{k-1}{2^{n-1}})^2.$"}
{"_id": "3143851", "title": "", "text": "$ f(x)=x^2-x+5 $"}
{"_id": "191528", "title": "", "text": "$(1,x,y,x^2,xy,y^2,x^3,\\ldots,y^4)$"}
{"_id": "1865759", "title": "", "text": "$ \\int \\frac{\\sin^{-1}(x)}{\\sqrt{1+x}} \\; dx$"}
{"_id": "7558102", "title": "", "text": "$T_x M = T_x X$"}
{"_id": "4965966", "title": "", "text": "$Cov (Z,X) = 0$"}
{"_id": "5821906", "title": "", "text": "$E=\\{\\langle x,y\\rangle\\in\\Bbb R^2:x=y\\}$"}
{"_id": "6252513", "title": "", "text": "$ a, a + d, a + 2d, \\cdots $"}
{"_id": "7165305", "title": "", "text": "$\\sum_{n=1}^\\infty \\sum_{m=1}^\\infty \\frac{1}{mn^2+m^2n+2mn}$"}
{"_id": "6716136", "title": "", "text": "$B = \\{A\\} = \\{\\{a\\}\\}$"}
{"_id": "7031188", "title": "", "text": "$Cov(X,Y) = 1$"}
{"_id": "7881551", "title": "", "text": "$\\frac{dx}{dt} = \\frac{t+3x}{t-x}$"}
{"_id": "4519929", "title": "", "text": "$\\int_{0}^{1/4}\\frac{\\sin^{-1}(2x)}{2x}dx$"}
{"_id": "474842", "title": "", "text": "$p = (a, a+k, a+2k,...)$"}
{"_id": "3675309", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{x_1+x_2+...+x_n}{n}=\\lim_{n\\to\\infty}x_n$"}
{"_id": "2011919", "title": "", "text": "$F_{X_{(2)}|X_{(3)}}(x;y)=\\frac{2x}{y^2}$"}
{"_id": "5345851", "title": "", "text": "$=\\int\\frac{x\\left(1+\\tan^2\\frac{x}{2}\\right)}{1+\\tan^2\\frac{x}{2}+1-\\tan^2\\frac{x}{2}}dx$"}
{"_id": "2517163", "title": "", "text": "$\\sum_{k=0}^{n-1}\\cos\\left(\\frac{2\\pi k}{n}\\right)=0=\\sum_{k=0}^{n-1}\\sin\\left(\\frac{2\\pi k}{n}\\right)$"}
{"_id": "53222", "title": "", "text": "$F_3$"}
{"_id": "6062064", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\sum_{m=1}^{\\infty} \\frac{\\sin(\\sin(nm))}{n^2+m^2}$"}
{"_id": "9085327", "title": "", "text": "$P(F|T)=\\frac{P(T|F)P(F)}{P(T|F)P(F)+P(T|F^c)P(F^c)}$"}
{"_id": "8207586", "title": "", "text": "$\\begin{cases}a_0=0\\\\b_0=1\\\\c_0=0\\end{cases}\\quad$"}
{"_id": "4820537", "title": "", "text": "$x\\rho y \\: \\wedge \\: y \\rho x$"}
{"_id": "1090622", "title": "", "text": "$e^{+}$"}
{"_id": "142918", "title": "", "text": "$\\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix}$"}
{"_id": "3740375", "title": "", "text": "$ a = b^{\\log_b(a)} \\tag{a}.$"}
{"_id": "1818864", "title": "", "text": "$n! \\approx \\sqrt{2 \\pi n} \\cdot \\left( \\frac{n}{e} \\right)^n$"}
{"_id": "4415024", "title": "", "text": "$ \\left[       \\begin{array}{ccc|c}         1&0&-3&-2\\\\         3&1&-2&5 \\\\         2&2&1&4       \\end{array}     \\right]$"}
{"_id": "1019895", "title": "", "text": "$ A = \\int_{0}^{1}(1-x^{2/3})^{3/2}\\,dx = \\frac{3}{2}\\int_{0}^{1}z^{1/2}(1-z)^{3/2}\\,dz=\\frac{3}{2}\\,B\\left(\\frac{3}{2},\\frac{5}{2}\\right)\\tag{1}$"}
{"_id": "212045", "title": "", "text": "$\\int\\frac{\\partial^2{\\bar{u}}}{\\partial\\sigma\\partial\\gamma}d\\sigma=g(\\gamma)\\implies \\int\\frac{\\partial{\\bar{u}}}{\\partial\\gamma}=\\int g(\\gamma)d\\gamma$"}
{"_id": "4170615", "title": "", "text": "$v(x)= ae^{\\frac{x}{-A}}+be^\\frac{-x}{-A}$"}
{"_id": "1359192", "title": "", "text": "$1-i =\\sqrt{2}e^{i\\pi\\over 4}$"}
{"_id": "9087386", "title": "", "text": "$\\lim_{n\\to\\infty}\\dfrac{1+\\sqrt[n]{2}+\\sqrt[n]{3}+\\cdots+\\sqrt[n]{n}}{n}=1$"}
{"_id": "7718359", "title": "", "text": "$antilog_a (y) = x$"}
{"_id": "6122184", "title": "", "text": "$(F_2[x]/(x^3+x+1))/(x^2+x+1) $"}
{"_id": "2308446", "title": "", "text": "$F(x) = \\int_b^x f(t)\\,dt$"}
{"_id": "1752933", "title": "", "text": "$\\lim \\limits_{x \\to \\alpha^+}{f(x)}$"}
{"_id": "532426", "title": "", "text": "$F(x) = \\int_{a}^x f(t) \\ dt = \\frac{x-a}{b-a}$"}
{"_id": "5478977", "title": "", "text": "$f(x) = \\sum_{j=1}^n c_j 1_{A_j}(x).$"}
{"_id": "1574615", "title": "", "text": "$\\pi_1(\\mathrm{Spec}(\\mathbb{Z}[x]/(x^6-1))) \\cong \\widehat{\\mathbb{Z}}$"}
{"_id": "3309204", "title": "", "text": "$(\\mathbb{Z}[x]/(x^2))^\\times.$"}
{"_id": "2458023", "title": "", "text": "$y(x)=1+a(x−1)+…$"}
{"_id": "5187515", "title": "", "text": "$\\lfloor\\gamma\\rfloor$"}
{"_id": "968072", "title": "", "text": "$f(xy+f(x))=xf(y)+f(x)$"}
{"_id": "4466006", "title": "", "text": "$Cov(X,Y)=4$"}
{"_id": "7218443", "title": "", "text": "$x^n-x^{n-1}$"}
{"_id": "5963829", "title": "", "text": "$f(x)\\equiv -1$"}
{"_id": "3619376", "title": "", "text": "$I_n:=\\int_{0}^{\\pi}sin(x)^ndx$"}
{"_id": "4427643", "title": "", "text": "$p_{1}p_{2} = a^2$"}
{"_id": "3303368", "title": "", "text": "$(1+\\frac xn)^n=e^x$"}
{"_id": "6280841", "title": "", "text": "$\\alpha=\\sqrt 2e^\\frac{2\\pi i}{3}$"}
{"_id": "2659248", "title": "", "text": "$A_1\\times  A_2\\times \\cdots\\times A_n \\times A_{n+1}$"}
{"_id": "4382385", "title": "", "text": "$f = \\{\\{a\\}\\}$"}
{"_id": "4973547", "title": "", "text": "$A_i=\\{a_i, a_i+d_i, a_i+2d_i, \\ldots\\}$"}
{"_id": "1866680", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-n^2\\pi x}= \\frac{1}{\\sqrt{x}}\\sum_{n=-\\infty}^\\infty e^{-n^2\\pi / x}$"}
{"_id": "1866683", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty e^{-n^2\\pi / x}=\\vartheta _3\\left(0,e^{-\\frac{\\pi }{x}}\\right)$"}
{"_id": "6298053", "title": "", "text": "$ \\sum_{k=0}^{n}kr^k = r\\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $"}
{"_id": "21306", "title": "", "text": "$\\|u\\|_{L^\\infty(0, 1)} \\le \\epsilon\\|u'\\|_{L^p(0, 1)} + C\\|u\\|_{L^1(0, 1)}$"}
{"_id": "2031337", "title": "", "text": "$a^2 b+ b^2 a= a^2 + b^2$"}
{"_id": "6678701", "title": "", "text": "$f(K)=[f(a),f(b)]$"}
{"_id": "2742666", "title": "", "text": "$-9(x + 9)^{\\frac{2}{3}} +27(x +9)^{\\frac{1}{3}} =-9-9 +27 $"}
{"_id": "8172885", "title": "", "text": "$\\sum\\limits_{r=0}^{n+1} \\frac{{(n+1)}!}{r!(n+1-r)!} =\\sum\\limits_{r=0}^{n+1} \\Bigl(\\frac{n!}{r!(n-r)!}+\\frac{n!}{(r-1)!(n-r+1)!}\\Bigr),$"}
{"_id": "5180082", "title": "", "text": "$f_3(5n,\\ 5k) = \\binom{n}{k}$"}
{"_id": "6896599", "title": "", "text": "$\\lim_{\\text x \\rightarrow c}c(x)= L$"}
{"_id": "1743764", "title": "", "text": "$\\tan(\\vartheta)=-\\rho/\\tau$"}
{"_id": "2371710", "title": "", "text": "$(n+1)! + 2, (n + 1)! + 3, ..., (n + 1)! + n + 1$"}
{"_id": "6035172", "title": "", "text": "$\\cos(\\frac{\\pi}{k}) + \\cos(\\frac{2\\pi}{k}) + \\cos(\\frac{3\\pi}{k}) + ... + \\cos(\\frac{n\\pi}{k})$"}
{"_id": "582334", "title": "", "text": "$(A^2＋B^2)(A-B)＝A^3-B^3＋B^2A-A^2B＝B^2A-A^2B$"}
{"_id": "8020108", "title": "", "text": "$yb + xa = d$"}
{"_id": "1614070", "title": "", "text": "$\\mathbb{P}(X_1=1\\vert X_0=2) = \\mathbb{P}(X_1=2\\vert X_0=3)=1.$"}
{"_id": "7194727", "title": "", "text": "$\\left(\\sum^\\infty_{i_j=1} \\frac{x^{i_j}}{i_j!} \\right)$"}
{"_id": "6547691", "title": "", "text": "$ \\tan (\\gamma+a)=\\frac{k-\\cos(\\gamma)}{\\sin(\\gamma)}$"}
{"_id": "5819374", "title": "", "text": "$ \\mathcal E = \\{ e_1, e_2, \\dots, e_n, \\dots \\} $"}
{"_id": "5536947", "title": "", "text": "$xy-1,x\\in P$"}
{"_id": "4225637", "title": "", "text": "$|AB|=b$"}
{"_id": "5053088", "title": "", "text": "$ \\begin{align} 2^{n-1}\\prod_{k=1}^{n-1}\\cos\\left(\\frac{2k\\pi}n\\right) &=\\prod_{k=1}^{n-1}\\left(1+e^{4\\pi ik/n}\\right)e^{-2\\pi ik/n}\\\\ &=(-1)^{n-1}\\left[\\prod_{k=1}^{n-1}\\left(-z^2+e^{4\\pi ik/n}\\right)\\right]_{\\large z=i}\\\\ &=\\left[\\prod_{k=1}^{n-1}\\left(z^2-e^{4\\pi ik/n}\\right)\\right]_{\\large z=i}\\\\ &=\\left[\\prod_{k=1}^{n-1}\\left(z-e^{2\\pi ik/n}\\right)\\prod_{k=1}^{n-1}\\left(z+e^{2\\pi ik/n}\\right)\\right]_{\\large z=i}\\\\[3pt] &=\\left[\\frac{z^n-1}{z-1}\\frac{z^n-(-1)^n}{z+1}\\right]_{\\large z=i}\\\\[6pt] &=\\frac{(i^n-1)(i^n-(-1)^n)}{-2}\\\\[9pt] &=\\frac{2(-1)^n-i^n(1+(-1)^n)}{-2}\\\\[12pt] &=\\cos(n\\pi/2)-(-1)^n \\end{align} $"}
{"_id": "6159929", "title": "", "text": "$(p_1)∩(p_2)=(p_1p_2)$"}
{"_id": "9044825", "title": "", "text": "$w = e^{\\frac{2\\pi\\iota}{n}}$"}
{"_id": "9105872", "title": "", "text": "$f_X(x;\\theta)=g(\\theta)h(x)\\theta^x$"}
{"_id": "3305952", "title": "", "text": "$\\bigg\\lfloor\\frac{I+F}{c}\\bigg\\rfloor=\\bigg\\lfloor\\frac{I}{c}\\bigg\\rfloor$"}
{"_id": "6339964", "title": "", "text": "$[x+y]=[[x]+\\{x\\}+[y]+\\{y\\}]=[x]+[y]+[\\{x\\}+\\{y\\}]$"}
{"_id": "3417272", "title": "", "text": "$t^2=\\frac{\\sqrt{2r^2(1-\\cos(x))}+\\sqrt{2R^2(1-\\cos(x))}}{g\\sin(x)}$"}
{"_id": "3342386", "title": "", "text": "$\\neg P(s(m))\\implies \\neg P(m)$"}
{"_id": "8764110", "title": "", "text": "$ A_i \\times \\cdots \\times A_n$"}
{"_id": "5499061", "title": "", "text": "$(t^{\\frac{j-1}{2}}-1)^2 \\equiv 0 \\pmod{p^2}$"}
{"_id": "6678616", "title": "", "text": "$S=\\sum_{n\\geqslant1}\\frac{1}{n^2}-\\sum_{n\\geqslant1}\\frac{2}{(2n)^2}=\\frac12\\zeta(2).$"}
{"_id": "2844689", "title": "", "text": "$\\frac{36}{19}=1+\\frac{17}{19}=1+\\frac1{\\frac{19}{17}}=1+\\frac1{1+\\frac2{17}}=1+\\frac1{1+\\frac1{\\frac{17}2}}=1+\\frac1{1+\\frac1{8+\\frac12}}$"}
{"_id": "1182369", "title": "", "text": "$||x||_{E} \\geq \\frac{1}{||f||}$"}
{"_id": "7848765", "title": "", "text": "$ \\begin{cases} x = 2 \\alpha \\\\ y = -\\alpha \\\\ z = -\\alpha \\end{cases} $"}
{"_id": "6996945", "title": "", "text": "$Pr\\left(X_{n+1}=\\frac{X_{n}}{2}|\\mathcal{F}_{n}\\right)=1-X_{n}$"}
{"_id": "3060894", "title": "", "text": "$d(x,F)=\\parallel y-x\\parallel$"}
{"_id": "4875135", "title": "", "text": "$\\mathbb{E}|X - Y|^2 \\leq 1/2$"}
{"_id": "8907403", "title": "", "text": "$p(x|a) = \\frac{2x}{a^2}$"}
{"_id": "3108030", "title": "", "text": "$L_2 = \\begin{cases} x= 1+t \\\\ y=2-t \\\\ z=3+2t \\end{cases}$"}
{"_id": "5919466", "title": "", "text": "$T =T_1\\oplus T'$"}
{"_id": "3206716", "title": "", "text": "$\\lVert\\gamma(s)-\\gamma(t)\\rVert\\lt\\frac{\\varepsilon}{4n}$"}
{"_id": "1805405", "title": "", "text": "$\\mathbb{E}[f(X_T)^2]\\le\\mathbb{E}[(2d)^{2T}]f(0)^2$"}
{"_id": "6056802", "title": "", "text": "$\\frac{(\\gamma)_k}{(\\gamma+1)_k} = \\frac{\\gamma(\\gamma+1)\\cdots(\\gamma+k-1)}{(\\gamma+1)(\\gamma+2)\\cdots(\\gamma+k)} = \\frac{\\gamma}{\\gamma+k}$"}
{"_id": "5491197", "title": "", "text": "$\\operatorname{dist}(x,A) \\leqslant d(x,y) + \\operatorname{dist}(y,A),\\tag{2'}$"}
{"_id": "7306870", "title": "", "text": "$\\left[\\frac{10}4,\\frac{10}4,\\frac{10}4,\\frac{10}4\\right].$"}
{"_id": "6319844", "title": "", "text": "$1/x - 1/|x|$"}
{"_id": "3670779", "title": "", "text": "$D_n=a(a-x)^{n-1}+x\\left(\\sum_{k=0}^{n-1}(a-x)^{k}(a-y)^{n-1-k}-(a-x)^{n-1}\\right)$"}
{"_id": "7019644", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{1}{m^2+n^2}=\\sum_{N=1}^\\infty \\sum_{n+m=N}^\\infty  \\frac{1}{m^2+n^2}=\\sum_{N=1}^\\infty \\sum_{m=1}^{N-1}  \\frac{1}{m^2+(N-m)^2}$"}
{"_id": "7207841", "title": "", "text": "$\\begin{array}{c|c|c|c} & \\begin{pmatrix}        0&1\\\\       1&0 \\end{pmatrix} & \\begin{pmatrix}        0&-i\\\\       i&0 \\end{pmatrix} & \\begin{pmatrix}       1&0\\\\       0&-1 \\end{pmatrix} \\\\\\hline \\begin{pmatrix}         0&1\\\\       1&0 \\end{pmatrix} & \\begin{pmatrix}        1&0\\\\       0&1 \\end{pmatrix} & \\begin{pmatrix}       i&0\\\\       0&-i \\end{pmatrix} & \\begin{pmatrix}       0&-1\\\\       1&0 \\end{pmatrix} \\\\\\hline \\begin{pmatrix}        0&-i\\\\       i&0 \\end{pmatrix} & \\begin{pmatrix}        -i&0\\\\       0&i \\end{pmatrix} & \\begin{pmatrix}       1&0\\\\       0&1 \\end{pmatrix} & \\begin{pmatrix}       0&i\\\\       i&0 \\end{pmatrix} \\\\\\hline \\begin{pmatrix}        1&0\\\\       0&-1 \\end{pmatrix} & \\begin{pmatrix}        0&1\\\\       -1&0 \\end{pmatrix} & \\begin{pmatrix}       0&-i\\\\       -i&0 \\end{pmatrix} & \\begin{pmatrix}       1&0\\\\       0&1 \\end{pmatrix} \\\\ \\end{array}$"}
{"_id": "6485562", "title": "", "text": "$\\sup_{0\\le |z|\\le 1/2} |f(z)|\\le\\sup_{0\\le |z|\\le 1/2} |f(z^2)|=\\sup_{0\\le |z|\\le 1/4} |f(z)|.$"}
{"_id": "582895", "title": "", "text": "$\\forall \\epsilon>0, \\exists r>0:|x-y| <r \\implies |f(x) - f(y)| < \\epsilon$"}
{"_id": "4870020", "title": "", "text": "$ f(x) = \\frac{e^x}{e^{2x} + 2e^x + 1}$"}
{"_id": "2656572", "title": "", "text": "$f(x) = \\frac{x^{2}}{1+2x}$"}
{"_id": "117851", "title": "", "text": "$\\mathbb{E}|\\xi| \\leq \\lim_{p \\downarrow 1} (\\mathbb{E} |\\xi|^p)^{1/p}$"}
{"_id": "7888207", "title": "", "text": "$\\sum_{k=1}^n\\cos(2\\phi_k)=0$"}
{"_id": "2369639", "title": "", "text": "$a_1+a_1+a_1+a_1+a_1+a_1$"}
{"_id": "8017096", "title": "", "text": "$a^{y/x}=b-a^x$"}
{"_id": "8653292", "title": "", "text": "$P(\\lambda_0)=-\\frac{|\\langle x,y\\rangle|^2}{\\|y\\|^2}$"}
{"_id": "5875065", "title": "", "text": "$ \\int_{-\\pi}^\\pi \\frac{\\sin(x)}{1-2a\\cos(x)+a^2}\\sin(nx)\\,dx=\\frac{\\pi}{a^{n+1}}? $"}
{"_id": "2160271", "title": "", "text": "$ \\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} \\frac{x^{s-1}}{e^x-1} \\, dx, $"}
{"_id": "3772884", "title": "", "text": "$k= a^{log_b c}$"}
{"_id": "3857591", "title": "", "text": "$ k! + 2, k! + 3, \\ldots k! + k, $"}
{"_id": "4609761", "title": "", "text": "$\\,2\\mid n^2+n,\\,$"}
{"_id": "779017", "title": "", "text": "$f(A+B)=f(A)+f(B)$"}
{"_id": "7108727", "title": "", "text": "$\\zeta\\left(s\\right)=\\frac{1}{\\Gamma\\left(s\\right)}\\int_{0}^{\\infty}\\frac{u^{s-1}}{e^{u}-1}du,\\,\\mathrm{Re}\\left(s\\right)>1$"}
{"_id": "9250160", "title": "", "text": "$K\\subseteq G\\subseteq \\overline{G}\\subseteq V\\subseteq \\overline{V}\\subseteq U.$"}
{"_id": "981153", "title": "", "text": "$\\begin{pmatrix}A&0\\\\0&B\\end{pmatrix}$"}
{"_id": "5284920", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\frac{\\cos {\\frac{n \\pi}3}}n$"}
{"_id": "5854925", "title": "", "text": "$ \\int_0^\\infty \\frac{\\sin^{2n}x}{x^2}dx$"}
{"_id": "9040353", "title": "", "text": "$ds = \\sqrt{1 + \\left(\\frac{3x^2}{y}\\right)^2}$"}
{"_id": "7781484", "title": "", "text": "$\\forall_{\\epsilon > 0} \\exists{\\delta > 0}: |x-a| < \\delta \\implies |f(x) - f(a)| < \\epsilon$"}
{"_id": "2862436", "title": "", "text": "$[a_n,d_n) \\subseteq U_n$"}
{"_id": "9262531", "title": "", "text": "$(X+y+z)(x+Y+z)(x+y+Z)(X+y+z)(X+y+z)$"}
{"_id": "6119859", "title": "", "text": "$f\\big(xy + f(y)\\big) = y\\,f(x).$"}
{"_id": "9246970", "title": "", "text": "$1\\equiv a^{n-1} \\equiv a^{sd+r} \\equiv a^r\\mod{n}$"}
{"_id": "1661805", "title": "", "text": "$\\int_0^1 {\\tan^{-1}\\left(x\\right) \\over {1+x}}\\,{\\rm d}x$"}
{"_id": "5032447", "title": "", "text": "$3^x + 6^x = 5^x + 4^x$"}
{"_id": "5020913", "title": "", "text": "$ (1+r/n)^{rn} $"}
{"_id": "2384321", "title": "", "text": "$ -(b^n + b^{n-1}) $"}
{"_id": "787445", "title": "", "text": "$\\begin{cases} x=2,\\\\ y=2, \\\\z=\\pm 1,-1. \\end{cases}$"}
{"_id": "915041", "title": "", "text": "$\\begin{pmatrix}     0.8 & -0.6 \\\\     0.6 & 0.8 \\end{pmatrix}$"}
{"_id": "1942160", "title": "", "text": "$d(x,c) \\leq d(x,y) + d(y,c)$"}
{"_id": "8825635", "title": "", "text": "$ \\frac{1}{2^n-2}-\\frac{1}{2^n-1}=\\frac{1}{(2^n-2)(2^n-1)} $"}
{"_id": "7822638", "title": "", "text": "$\\rm{dist}(f,Y)=\\frac{|T(f)|}{||T||}.$"}
{"_id": "9205931", "title": "", "text": "$C+E = \\dfrac{P((1+r)^N-1)}{r(1+r)^N}$"}
{"_id": "154893", "title": "", "text": "$S_1\\subset S_2\\subset...\\subset S_i\\subset...$"}
{"_id": "6277557", "title": "", "text": "$ E = \\left( \\begin{matrix} A & -B \\\\ B & A \\end{matrix} \\right) $"}
{"_id": "5778305", "title": "", "text": "$\\dim(M+H) = \\dim M + \\dim H = \\dim M + \\dim K = \\dim(M+K)$"}
{"_id": "4725019", "title": "", "text": "$f_n(x) = {(nx) \\over n^2}$"}
{"_id": "3237454", "title": "", "text": "$\\mathbb R^n=\\mathbb R^{n-1}\\times \\mathbb R$"}
{"_id": "2465158", "title": "", "text": "$P[M_n \\leq x \\mid X_0](\\omega) = I_{[X_0 \\leq x]}(\\omega)x^n$"}
{"_id": "239100", "title": "", "text": "$F(x) = \\int_{-\\infty}^{x}f(t)dt$"}
{"_id": "835011", "title": "", "text": "$(x^4+1)$"}
{"_id": "379623", "title": "", "text": "$   I(\\gamma) = \\int_0^\\gamma \\frac{1}{\\sqrt{\\pi}} \\frac{1}{(\\gamma^2+\\delta)^2} \\mathrm{d} \\gamma = \\frac{\\gamma }{2 \\sqrt{\\pi } \\delta  \\left(\\gamma ^2+\\delta \\right)}+\\frac{1}{2 \\sqrt{\\pi } \\delta ^{3/2}} \\arctan\\left(\\frac{\\gamma }{\\sqrt{\\delta }}\\right) $"}
{"_id": "5039582", "title": "", "text": "$ -\\frac{\\ln[1-(1-e^x)]}{1-e^x} = -\\frac{(-\\sum_{n=1}^{\\infty} \\frac{(1-e^x)^n}{n})}{1-e^x}=\\frac{1}{1-e^x}.\\sum_{n=1}^{\\infty} \\frac{(1-e^x)^n}{n}=\\sum_{n=1}^{\\infty} \\frac{(1-e^x)^{n-1}}{n} $"}
{"_id": "8062113", "title": "", "text": "$m\\in M,m'\\in M^{\\perp}$"}
{"_id": "3169569", "title": "", "text": "$ \\mathbb{E}\\|X\\|_{\\infty}^r \\le C_r d^{r/2}\\mathbb{E}\\|Y\\|_{\\infty}^r, $"}
{"_id": "2020851", "title": "", "text": "$\\varphi=\\varphi(s,t),\\quad \\varphi(s,a)=\\gamma (a),\\quad \\varphi(s,b)=\\gamma (b)\\quad \\text{and}\\quad \\varphi(0,t)=\\gamma (t).$"}
{"_id": "22725", "title": "", "text": "$A_1\\times...\\times A_n$"}
{"_id": "3233078", "title": "", "text": "$\\{\\{a\\}\\} \\subset A$"}
{"_id": "1638865", "title": "", "text": "$ U(\\alpha,\\gamma;z)=\\frac{\\Gamma(1-\\gamma)}{\\Gamma(\\alpha-\\gamma+1)}M(\\alpha,\\gamma;z) + \\frac{\\Gamma(\\gamma-1)}{\\Gamma(\\alpha)}z^{1-\\gamma}M(\\alpha-\\gamma+1,2-\\gamma;z)  \\tag{4} $"}
{"_id": "1600863", "title": "", "text": "$\\int_N^{\\infty}f(x)dx$"}
{"_id": "3029449", "title": "", "text": "$\\mathcal{R}^+$"}
{"_id": "6748283", "title": "", "text": "$Pr(E2|E1)=\\frac{Pr(E_2\\cap E_1)}{Pr(E_1)}$"}
{"_id": "2507167", "title": "", "text": "$E[min(|X|,|Y|)] = \\frac{4}{\\pi}\\int_{x=0}^{\\infty}e^{\\frac{-x^{2}}{2}}(1-e^{\\frac{-x^{2}}{2}})dx$"}
{"_id": "2702449", "title": "", "text": "$x^2-x \\in C$"}
{"_id": "5992018", "title": "", "text": "$f(x) = \\frac{\\mathrm e^x}{1+\\mathrm e^{x^2}}$"}
{"_id": "4585293", "title": "", "text": "$\\lim\\limits_{n}\\mu_{M_0}(B_n)\\geq \\epsilon_2 >0$"}
{"_id": "7303785", "title": "", "text": "$2\\sum_{n\\geq1}\\frac{1}{\\left(1+n^{2}\\right)^{2}}=\\frac{1}{2}\\left(\\pi^{2}\\textrm{csch}^{2}\\left(\\pi\\right)+\\pi\\coth\\left(\\pi\\right)-2\\right).$"}
{"_id": "1024001", "title": "", "text": "$\\eqalign{   J &= s:s \\cr\\cr  dJ &= 2s:ds \\cr     &= 2s:Xda \\cr     &= 2X^Ts:da \\cr\\cr \\frac{\\partial J}{\\partial a} &= 2X^Ts \\cr     &= 2X^T(Xa-y) \\cr\\cr } $"}
{"_id": "3193917", "title": "", "text": "$ \\Gamma(s) = \\frac{1}{\\zeta(s)} \\int_0^{\\infty}\\frac{x^{s-1}}{e^x-1}dx $"}
{"_id": "4725885", "title": "", "text": "$AB=\\begin{pmatrix}A_1B_1 & A_1B_2\\\\A_2B_1 & A_2B_2 \\end{pmatrix}$"}
{"_id": "8670651", "title": "", "text": "$M\\|A\\|_\\infty \\le \\|A\\|_2 \\le \\|A\\|_\\infty$"}
{"_id": "8271825", "title": "", "text": "$y\\in V\\subseteq\\bar{V}\\subseteq X\\setminus\\bar{U}$"}
{"_id": "599142", "title": "", "text": "$\\operatorname{Cov}(U,V)=0$"}
{"_id": "5210785", "title": "", "text": "$\\frac{\\frac{1}{2^{n-2}}}{\\frac{1}{2^n}}=\\frac{1}{4}$"}
{"_id": "5743834", "title": "", "text": "$F(X)=X^2-X+2.$"}
{"_id": "4253754", "title": "", "text": "$t_x\\in [0,1]$"}
{"_id": "1258346", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{\\log\\left(1+\\frac{1}{n}\\right)}{n+1}<\\sum_{n\\geq 1}\\frac{1}{n(n+1)}=1.$"}
{"_id": "442204", "title": "", "text": "$T_n=\\frac{n(n+1)}{2}$"}
{"_id": "2000278", "title": "", "text": "$\\frac 14 k(k + 1)(k + 2)(k + 3) $"}
{"_id": "8122649", "title": "", "text": "$ P F P^T = F $"}
{"_id": "6087666", "title": "", "text": "$\\sum_{\\gamma}a_\\gamma t^{\\gamma}+\\sum_{\\gamma}b_\\gamma t^{\\gamma}=\\sum_\\gamma (a_\\gamma+b_\\gamma)t^\\gamma\\text{,}$"}
{"_id": "437979", "title": "", "text": "$\\sqrt[100]{\\sqrt3 + \\sqrt2} + \\sqrt[100]{\\sqrt3 - \\sqrt2}$"}
{"_id": "9154951", "title": "", "text": "$ S = \\{s\\space |\\space s=\\frac{lcm(a,\\space a+d,\\space a+2d,\\space ...,\\space a+10d)}{a+10d}\\}$"}
{"_id": "3131545", "title": "", "text": "$\\left[\\begin{array}{ccc|c} 1 & -17 & 0 & 3\\\\ 1 & 0 & -6 & 1\\end{array}\\right]$"}
{"_id": "8641458", "title": "", "text": "$\\left\\lfloor \\sum_{k=1}^{100}\\frac{1}{\\sqrt{k}}\\right\\rfloor$"}
{"_id": "1161494", "title": "", "text": "$ax+by = g$"}
{"_id": "9080085", "title": "", "text": "$ F'(x) = \\int_a^x f(t)\\ dt + xf(x) - xf(x) = \\int_a^x f(t)\\ dt $"}
{"_id": "2269150", "title": "", "text": "$|a+b|^r\\leq 2^{r-1}(|a|^r+|b|^r)$"}
{"_id": "8253979", "title": "", "text": "$\\frac{\\hat \\gamma}{a + \\hat \\gamma} A - \\hat \\gamma B$"}
{"_id": "9337037", "title": "", "text": "$f(a+h)=f(a)+L(h)+r(h)$"}
{"_id": "1014165", "title": "", "text": "$\\{a,a+b,a+2b,\\ldots,a+(m-1)b\\}$"}
{"_id": "1218700", "title": "", "text": "$\\{a,a+1,a+2,\\ldots, b\\}$"}
{"_id": "6151083", "title": "", "text": "$\\{\\gamma:\\exists\\beta\\in \\alpha \\cup \\{\\alpha\\} :\\gamma\\in\\beta\\} = \\{\\gamma :\\gamma\\in\\alpha\\}$"}
{"_id": "7295083", "title": "", "text": "$Q:=\\{[a,b)\\times[c,d): a,b,c,d \\in \\mathbb{R}; a < b,c < d\\}$"}
{"_id": "5263840", "title": "", "text": "$n+2, n+3, \\dots, 2n+1$"}
{"_id": "323443", "title": "", "text": "$\\gamma_\\epsilon=\\gamma|_{[\\epsilon, 2\\pi-\\epsilon]}$"}
{"_id": "3605105", "title": "", "text": "$P(m,n) \\implies P(m + 1, n)$"}
{"_id": "2645473", "title": "", "text": "$\\lim_{x \\to c^-} f(x)=\\lim_{x \\to c^+}f(x) = f(c).$"}
{"_id": "8707625", "title": "", "text": "$P(A_2)=1/6$"}
{"_id": "5242002", "title": "", "text": "$ax+b=\\frac{x}{a}-\\frac{b}{a}$"}
{"_id": "7281092", "title": "", "text": "$\\begin{pmatrix} A & B \\\\ -B & A\\end{pmatrix}$"}
{"_id": "9047636", "title": "", "text": "$xy + ax + by = c$"}
{"_id": "8566644", "title": "", "text": "$\\mathbb{R}^n \\in \\mathbb{R}^n$"}
{"_id": "3175598", "title": "", "text": "$ (n+1)!+2, (n+1)!+3, (n+1)!+4, ... (n+1)!+(n+1) $"}
{"_id": "1851931", "title": "", "text": "$f(a+b) = f(a) + f(b),$"}
{"_id": "8535214", "title": "", "text": "$\\int_{0}^{\\infty} f(x) dx \\leq 1,$"}
{"_id": "5446294", "title": "", "text": "$F_k=\\{x\\in \\mathbb{R} | \\ x\\geq0,\\  2\\leq x^2\\leq 2+1/k \\}.$"}
{"_id": "2067159", "title": "", "text": "$\\rho\\left(\\left[\\begin{matrix}0 & A \\\\ -A^{\\dagger} & 0\\end{matrix}\\right]\\right)=\\sqrt{\\rho\\left(A^{\\dagger} A\\right)}$"}
{"_id": "6209088", "title": "", "text": "$\\frac{dy}{dx}=\\frac{\\sqrt{1-y^{2}}}{\\sqrt{1-x^{2}}}$"}
{"_id": "5801841", "title": "", "text": "$|D(u\\circ \\gamma)|\\le L|D\\gamma|$"}
{"_id": "19109", "title": "", "text": "$\\displaystyle\\int\\frac{1}{(x^2+1)^2}dx$"}
{"_id": "850884", "title": "", "text": "$\\forall\\epsilon\\gt 0,\\exists \\delta\\gt 0,|x-y|\\lt \\delta , x,y\\in D\\Rightarrow |f(x)-f(y)|\\lt\\epsilon$"}
{"_id": "837772", "title": "", "text": "$E(Y)=1/2$"}
{"_id": "2944066", "title": "", "text": "$\\cos\\theta = \\frac{a}{x}$"}
{"_id": "5429425", "title": "", "text": "$\\sqrt{1+\\frac{dx}{dy}}dy$"}
{"_id": "5989102", "title": "", "text": "$(N+2)!+2 , (N+2)! + 3,(N+2)!+4,...(N+2)!+(N+2)$"}
{"_id": "8932809", "title": "", "text": "$f''(x) = n(n-1)a_n(x-a)^{n-2}$"}
{"_id": "2666505", "title": "", "text": "$\\mathbf v:\\mathbb R^{n+1}\\to\\mathbb R^n$"}
{"_id": "7578591", "title": "", "text": "$ \\forall x \\in X: f(x)=x \\iff R_f= \\left\\{ \\left(T_\\alpha,T_\\beta\\right)\\subset \\omega_X \\times \\omega_X | T_\\beta \\subset T_\\alpha  \\right\\}$"}
{"_id": "1794187", "title": "", "text": "$ k=\\frac{e^{x+n-1+O(1/(x+n))}}{(x+n-1)^{1-1/(x+n)}}=\\frac{e^{n-1+O(1/(x+n))}}{(x+n-1)^{1-1/(x+n)}}e^x $"}
{"_id": "625822", "title": "", "text": "$\\det{A}=(x-2)^{n-1}(x+2(n-1)).$"}
{"_id": "6694798", "title": "", "text": "$d(x,A)\\leqslant d(x,y)+d(y,A)$"}
{"_id": "2624015", "title": "", "text": "$f(a)=(a+b)^k - a^k -b^k$"}
{"_id": "6294542", "title": "", "text": "$|f'(z)| \\leq |f(z)| \\ ? $"}
{"_id": "7253087", "title": "", "text": "$x=z \\ \\ \\text{or} \\ \\ y=-2\\lambda-1$"}
{"_id": "8380994", "title": "", "text": "$\\mathcal{A}_{+} +\\mathcal{A}_{−} = X$"}
{"_id": "8659882", "title": "", "text": "$\\frac{a+bx}{e^x}$"}
{"_id": "1572041", "title": "", "text": "$S_n = \\left\\{a^{n!}_1, a^{n!}_2, a^{n!}_3, \\dots, a^{n!}_n\\right\\}$"}
{"_id": "3882952", "title": "", "text": "$B = \\{b_1 , b_2, \\ldots, b_n \\} \\subseteq A$"}
{"_id": "8039671", "title": "", "text": "$|f(x+h)-f(x)| \\leq 2|f(x)|$"}
{"_id": "7293985", "title": "", "text": "$\\{E_1,E_2,E_3,\\cdots\\}$"}
{"_id": "2518409", "title": "", "text": "$\\Bbb{P}[X_t \\in B | \\mathcal{F}_s] = \\Bbb{P}[X_t \\in B | X_s]$"}
{"_id": "3385004", "title": "", "text": "$\\color{red}{y=\\dfrac{a+b+c}{2\\sqrt{x}}}$"}
{"_id": "2969859", "title": "", "text": "$ \\frac {a + bx}{1+x^2}$"}
{"_id": "8553697", "title": "", "text": "$(a, a+r,a+2r,a+3r)$"}
{"_id": "4097124", "title": "", "text": "$\\left[{\\partial^2f\\over\\partial X_i\\partial X_k}(A)\\right]_{i,\\,k}=\\left[\\matrix{0&0&0&1\\cr 0&0&-1&0\\cr  0&-1&0&0\\cr  1&0&0&0\\cr}\\right]=:H\\ . $"}
{"_id": "6647749", "title": "", "text": "$X^p=t\\iff X=e^{\\frac{2ik\\pi}{p}}\\sqrt[p]t, k=0,...,p-1$"}
{"_id": "1741281", "title": "", "text": "$\n \\begin{bmatrix} \n 1 & -2 & 3 & 7\\\\\n 0 & 1 & -1 & -2\\\\\n 0 & 0 & 3 & 3\\\\\n \\end{bmatrix}\n $"}
{"_id": "3049144", "title": "", "text": "$xRy\\not =yRx$"}
{"_id": "6133887", "title": "", "text": "$\\displaystyle \\lim_{x \\to c^-} f(x) \\le \\lim_{x \\to c^+} f(x)$"}
{"_id": "3302674", "title": "", "text": "$\\sqrt[n]{Z}=\\sqrt[n]{R}.e^{i\\frac{\\theta+2k\\pi}{n}}$"}
{"_id": "9194291", "title": "", "text": "$f_M(x)=F_M'(x)=3(1-\\exp(-x/200))^2\\frac{\\exp(-x/200)}{200}$"}
{"_id": "6833425", "title": "", "text": "$f_X(x) = \\frac{2x}{10000}$"}
{"_id": "1012534", "title": "", "text": "$\\frac 1{(-2^n)^2} + (\\frac{1}{2^n})^2 = $"}
{"_id": "7370402", "title": "", "text": "$e_N^*(x_n) \\to 0$"}
{"_id": "223456", "title": "", "text": "$F(1/2,1/2,1,x)$"}
{"_id": "1928184", "title": "", "text": "$(a-b,a+b) \\neq a-b$"}
{"_id": "9295748", "title": "", "text": "$F(n+2)^2-2F(n+2)F(n)=F(n-1)F(n+2)$"}
{"_id": "4837325", "title": "", "text": "$\\det(\\alpha_j \\gamma) = \\det(\\alpha_j)$"}
{"_id": "1979466", "title": "", "text": "$F \\subset V \\subset \\overline{V} \\subset U$"}
{"_id": "2933205", "title": "", "text": "$t^a_{\\;b}$"}
{"_id": "3174288", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^2} = \\sum_{n\\geq 1}\\frac{1}{n^2}-2\\sum_{n\\geq 1}\\frac{1}{(2n)^2} = \\frac{1}{2}\\sum_{n\\geq 1}\\frac{1}{n^2}$"}
{"_id": "606707", "title": "", "text": "$\\lambda=\\dfrac{\\bigl\\lvert\\langle x,y\\rangle\\bigr\\rvert}{\\langle x,y\\rangle}$"}
{"_id": "5586600", "title": "", "text": "$[f(f^{-1}(x))]'=f'(f^{-1}(x))(f^{-1})'(x)=1$"}
{"_id": "2384320", "title": "", "text": "$ a^n + a^{n-1} $"}
{"_id": "5198247", "title": "", "text": "$y''y'=y'/y$"}
{"_id": "6797693", "title": "", "text": "$(1/|x_n|)$"}
{"_id": "466355", "title": "", "text": "$|f(x)| \\leq A\\  |f(c)|(x-a)$"}
{"_id": "3504740", "title": "", "text": "$d(x,A) \\leq d(x,y) + d(y,A).$"}
{"_id": "4051764", "title": "", "text": "$\\begin{align} \\int \\frac{1}{(x+1)^2} \\, dx & = \\ln((x+1)^2) \\div (2x+2) + c \\\\                              & = \\frac{2\\ln(x+1)}{2x+2} + c\\\\                              & = \\frac{\\ln(x+1)}{x+1} + c \\end{align}$"}
{"_id": "3366878", "title": "", "text": "$|A\\times B|=ab$"}
{"_id": "1555862", "title": "", "text": "$\\mathcal{A}=\\{A_1,A_2,...\\}$"}
{"_id": "6773573", "title": "", "text": "$\\int_0^\\infty \\frac{1}{(x^2+1)^{3/2}}\\,dx=1$"}
{"_id": "2964904", "title": "", "text": "$\\|A\\|=\\sup_{\\| \\psi \\| = 1} \\|A \\psi \\| =\\sup_{\\psi=1}\\left( \\sup_{\\|\\phi\\|=1}| \\langle \\phi | A | \\psi \\rangle| \\right) =\\sup_{\\|\\phi \\| = \\| \\psi \\| = 1} | \\langle \\phi | A | \\psi \\rangle|$"}
{"_id": "7394038", "title": "", "text": "$\\forall\\varepsilon\\gt0,\\exists \\delta_{1}\\gt0, \\text{ such that } |x| \\lt \\delta_{1} ,|f(x)| \\leq \\varepsilon \\tag{2}$"}
{"_id": "8244893", "title": "", "text": "$f(x) = \\sum_{j=1}^n c_j 1_{A_j}(x)$"}
{"_id": "2934155", "title": "", "text": "$f_{n}\\in D(A_{n})=D(A)$"}
{"_id": "2000148", "title": "", "text": "$a \\leq log(b) \\implies 10^a \\leq 10^{log(b)} \\implies 10^a \\leq b$"}
{"_id": "4160047", "title": "", "text": "$f^{\\otimes p}\\otimes (f^{\\ast-1})^{\\otimes q}: V^{\\otimes p}\\otimes_k (V^\\ast)^{\\otimes q}\\to W^{\\otimes p}\\otimes_k (W^\\ast)^{\\otimes q}$"}
{"_id": "8703732", "title": "", "text": "$x\\to\\infty\\implies x!\\sim \\sqrt{2\\pi x}\\left(\\frac{x}{e}\\right)^x$"}
{"_id": "3898707", "title": "", "text": "$\\int_{0}^{1}\\frac{\\sin(x)\\arctan(x)}{x}dx$"}
{"_id": "2712964", "title": "", "text": "$N= \\left\\lfloor {\\frac{{n\\,m}}{{a\\,b}}} \\right\\rfloor$"}
{"_id": "6645278", "title": "", "text": "$\\alpha^+=0$"}
{"_id": "7899314", "title": "", "text": "$ \\int_0^{x+h}f(t)\\,dt-\\int_0^x f(t)\\,dt= \\int_x^0 f(t)\\,dt+\\int_0^{x+h}f(t)\\,dt-\\int_0^x f(t)\\,dt= \\int_{x}^{x+h}f(t)\\,dt=hf(\\xi) $"}
{"_id": "4373473", "title": "", "text": "$(n+1)^{2}=n^{2}+2n+1\\le n!+2n+1\\le n!+n!+n!=3n!\\le(n+1)n!=(n+1)!$"}
{"_id": "9039224", "title": "", "text": "$(a;b;c;d) = (a;aγ;aγ^{2};aγ^{3}) $"}
{"_id": "8401574", "title": "", "text": "$g(x) = (x-a)^{n+1}$"}
{"_id": "8323897", "title": "", "text": "$d(x,\\ker f) = \\frac{|f(x)|}{\\|f\\|}$"}
{"_id": "3839175", "title": "", "text": "$k[T_1,T_1^{-1}]$"}
{"_id": "3308858", "title": "", "text": "$\\|A\\|_F = \\sqrt{ \\operatorname{Tr}(AA^*)^{\\vphantom{l}}}.$"}
{"_id": "6701340", "title": "", "text": "$\\sum_{n = -\\infty}^\\infty e^{-\\pi n^2 x} = \\sum_{n = -\\infty}^\\infty f(n\\sqrt{x}) = \\sum_{n = -\\infty}^\\infty \\frac{1}{\\sqrt{x}}\\hat{f}\\left(\\frac{n}{\\sqrt{x}}\\right) = \\frac{1}{\\sqrt{x}} \\sum_{n = -\\infty}^\\infty e^{-\\pi n^2/x} = \\frac{1}{\\sqrt{x}}\\vartheta\\left(\\frac{1}{x}\\right).$"}
{"_id": "4279634", "title": "", "text": "$\\color{red}{\\det A} = [1 + n \\beta (\\alpha - \\beta)^{-1}] (\\alpha - \\beta)^n \\color{red}{ = (\\alpha - \\beta)^{n - 1}[\\alpha + (n - 1) \\beta]}.$"}
{"_id": "8330780", "title": "", "text": "$f(K)\\subseteq V\\subseteq\\overline{V}\\subseteq g^{-1}(U)$"}
{"_id": "2948150", "title": "", "text": "$m! = \\sqrt{2\\pi m}(m/e)^m(1 + o(1))$"}
{"_id": "3113196", "title": "", "text": "$p(E|S) = p(E \\& S)/p(S) = p(E).1 / (1 - (1 -p(E).1) + p(E).(1 - 1) = p(E)/p(E) = 1 $"}
{"_id": "7693801", "title": "", "text": "$\\frac{4}{x}+\\frac{6}{2}=x.$"}
{"_id": "1068859", "title": "", "text": "$\\left|\\int_0^\\infty f(x)dx\\right|=0$"}
{"_id": "5102524", "title": "", "text": "$F(\\pi) = F(0) = 0 $"}
{"_id": "1545861", "title": "", "text": "$\\lim_{x→c}⁡〖f(x)〗=L$"}
{"_id": "3285910", "title": "", "text": "$p^*=\\inf_{x\\in X}f_0(x)$"}
{"_id": "3720719", "title": "", "text": "$\\gamma\\tau\\gamma^{-1} = (\\gamma(2), \\gamma(4), \\gamma(5))(\\gamma(3), \\gamma(1), \\gamma(6)).$"}
{"_id": "2052137", "title": "", "text": "$\\color{red}{13}\\mid n^{13}-n$"}
{"_id": "1317771", "title": "", "text": "$M^{-1}=\\left[\\begin{matrix}\\frac{\\beta \\gamma}{(\\alpha-\\beta) (\\alpha-\\gamma)} \\ -\\frac{\\beta+\\gamma}{(\\alpha-\\beta) (\\alpha-\\gamma)} \\ \\frac{1}{(\\alpha-\\beta) (\\alpha-\\gamma)} \\\\ \\frac{\\alpha \\gamma}{(\\beta-\\alpha) (\\beta-\\gamma)} \\ -\\frac{\\alpha+\\gamma}{(\\beta-\\alpha) (\\beta-\\gamma)} \\ \\frac{1}{(\\beta-\\alpha) (\\beta-\\gamma)} \\\\ \\frac{\\alpha \\beta}{(\\gamma-\\alpha) (\\gamma-\\beta)} \\ -\\frac{\\alpha+\\beta}{(\\gamma-\\beta) (\\gamma-\\beta)} \\ \\frac{1}{(\\gamma-\\alpha) (\\gamma-\\beta)}\\end{matrix}\\right]$"}
{"_id": "6812902", "title": "", "text": "$\\frac{1}{2^{2n-2}}$"}
{"_id": "766512", "title": "", "text": "$f(x)=\\frac{e^x f(x)}{e^x}$"}
{"_id": "9287349", "title": "", "text": "$ a_{\\gamma},b_{\\gamma} \\in \\mathbb C$"}
{"_id": "5755825", "title": "", "text": "$\\hat{y} = \\frac{\\hat{x}}{||f||}$"}
{"_id": "709516", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{x+y}$"}
{"_id": "8647915", "title": "", "text": "$ds=\\sqrt{1+\\left(\\frac{dy}{dx}^2 \\right)}\\,dx.$"}
{"_id": "4052535", "title": "", "text": "$f(x;\\theta)=\\frac{\\beta^3}{2}e^{-\\beta(x-\\theta)}(x-\\theta)^2\\mathbf1_{x\\ge\\theta}$"}
{"_id": "4468874", "title": "", "text": "$\\log_a AB = \\log_a A + \\log_a B$"}
{"_id": "3331578", "title": "", "text": "$F_1\\subseteq F_2\\subseteq\\dotso\\subseteq F_n\\subseteq\\dotso$"}
{"_id": "1080869", "title": "", "text": "$x \\to 0^+ \\implies P \\to \\infty, x \\to \\infty  \\implies P \\to \\infty$"}
{"_id": "5437346", "title": "", "text": "$\\int_{\\pi/4}^{x}f(t)\\,dt=\\sin(x)f(0)-\\cos(x)\\int_{0}^{\\pi/4}f(t)\\,dt$"}
{"_id": "2346890", "title": "", "text": "$\\gamma(s+t)=\\gamma(s)\\gamma(t),\\quad\\gamma(0)=I,$"}
{"_id": "338331", "title": "", "text": "$\\log_a(b)$"}
{"_id": "1850136", "title": "", "text": "$K=\\lim\\limits_{x\\to c^-} f(x)<\\lim\\limits_{x\\to c^+} f(x)=L$"}
{"_id": "7925247", "title": "", "text": "$\\gamma_{a,b} + \\gamma_{b,z} = \\gamma_{a,b}$"}
{"_id": "4521314", "title": "", "text": "$\\sum^k_{j=1} I_j = \\sum_{j \\not = i} I_i I_j + \\sum^k_{j=1}I_j I_j$"}
{"_id": "6343039", "title": "", "text": "$\\begin{align} & \\alpha + \\beta + \\gamma = -\\frac ba \\\\ & \\alpha\\beta + \\beta\\gamma + \\alpha\\gamma = \\frac ca \\\\  & \\alpha\\beta\\gamma = -\\frac da \\end{align} $"}
{"_id": "5345642", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}\\frac1{n}\\sum\\limits_{k=1}^{\\infty}\\left\\lfloor\\frac{n}{3^k}\\right\\rfloor=\\frac{1}{2}$"}
{"_id": "5734317", "title": "", "text": "$P(V)\\subseteq V$"}
{"_id": "3720091", "title": "", "text": "$y = (2\\sin(u) - 2\\sin(u)\\cos(u))(1 - v)^2$"}
{"_id": "2116363", "title": "", "text": "$A= \\left[\\begin{array}{ccc|c} 4 & 3 & 2 & 1\\\\ -1 & 2 & -1 & 2 \\\\ 1 & 2 & 3 & -2 \\end{array}\\right]$"}
{"_id": "8903658", "title": "", "text": "$\\int_{-\\pi}^{\\pi}F(x)\\cos nx\\,dx = 2\\int_0^{\\pi}f(x)\\cos nx\\,dx = 0$"}
{"_id": "2836215", "title": "", "text": "$ \\forall V\\in\\tau_{Y}\\colon f^{-1}\\left(V\\right)\\in\\tau_{X}. $"}
{"_id": "3541067", "title": "", "text": "$M=T\\oplus P$"}
{"_id": "7178601", "title": "", "text": "$\\,\\ln(a^b)=b\\ln(a) \\\\a^{log_ax}=x \\\\ log_a(x)=b \\implies a^b=x\\\\$"}
{"_id": "5618313", "title": "", "text": "$\\mathcal O(\\mathbb R^n)$"}
{"_id": "128352", "title": "", "text": "$= \\frac{1}{\\sqrt{2^n}}\\sum_{x}\\bigg[\\bigg[\\sum_{y,z}\\big(\\left|z\\right>\\!\\!\\left<z\\right|\\otimes\\left|y\\oplus f(z)\\right>\\!\\!\\left<y\\right|\\big)\\bigg]\\left|x\\right>\\!\\!\\left|0^n\\right>\\bigg]$"}
{"_id": "4752694", "title": "", "text": "$f_{\\Phi}(\\phi)=\\frac{\\sqrt{1-\\rho^2}}{2\\pi(1-\\rho\\sin2\\phi)},\\quad 0<\\phi<2\\pi$"}
{"_id": "7675649", "title": "", "text": "$\\lim_{x→c} f(x) = L$"}
{"_id": "7512285", "title": "", "text": "$\\begin{bmatrix}\\sigma & -\\omega\\\\\\omega & \\sigma\\end{bmatrix}$"}
{"_id": "2937747", "title": "", "text": "$\\dfrac1{2^{n-2}}$"}
{"_id": "5191248", "title": "", "text": "$xRy \\not\\Rightarrow yRx$"}
{"_id": "5360954", "title": "", "text": "$e^A = I\\gamma^0/0! + (A/\\gamma)\\gamma^1/1! + I\\gamma^2/2! + (A/\\gamma)\\gamma^3/3!+...$"}
{"_id": "2654417", "title": "", "text": "$ f'(x_0) = \\lim_{x \\to x_0^{-}} f'(x) = \\lim_{x \\to x_0^{+}} f'(x). $"}
{"_id": "1223105", "title": "", "text": "$e^x\\ge1+x\\implies e^x\\ge\\left(1+\\frac xk\\right)^k$"}
{"_id": "7765154", "title": "", "text": "$\\Im\\left(\\text{A}\\cdot\\text{a}\\cdot e^{-\\text{a}\\cdot\\text{T}}\\right)=\\Im\\left\\{\\text{A}\\cdot\\text{a}\\cdot\\frac{\\cos\\left(\\text{a}\\zeta\\right)-\\sin\\left(\\text{a}\\zeta\\right)i}{\\exp\\left[\\text{a}\\sigma\\right]}\\right\\}=-\\text{A}\\cdot\\text{a}\\cdot\\frac{\\sin\\left(\\text{a}\\zeta\\right)}{\\exp\\left[\\text{a}\\sigma\\right]}\\tag3$"}
{"_id": "3071381", "title": "", "text": "$HCF(a+b, a-b) \\ge HCF(a,b)$"}
{"_id": "8895264", "title": "", "text": "$P(a<X\\le b, c<Y\\le d)$"}
{"_id": "4647936", "title": "", "text": "$yRx \\land xRy$"}
{"_id": "1072012", "title": "", "text": "$y_n=\\dfrac{x_1+x_2+\\cdots+x_n}{n^a}$"}
{"_id": "394682", "title": "", "text": "$z=\\frac{(1-x)-(1+x)y}{(1+x)+(1-x)y}$"}
{"_id": "143486", "title": "", "text": "$f_X(x)=\\theta x^{\\theta-1},$"}
{"_id": "4384526", "title": "", "text": "$f(n) = n^2-n-1>0$"}
{"_id": "1292710", "title": "", "text": "$A=\\{\\{\\{x\\},\\{x,y\\}\\}\\}$"}
{"_id": "2906032", "title": "", "text": "$\\frac{(k+1)^3}{3} = \\frac{k^3}{3} + k^2 + k + \\frac{1}{3}$"}
{"_id": "7686313", "title": "", "text": "$\\sum_{l=-\\infty}^{\\infty}e^{j2\\pi lNt/T}=\\frac{T}{N}\\sum_{m=-\\infty}^{\\infty}\\delta\\left(t-m\\frac{T}{N}\\right)\\tag{2}$"}
{"_id": "8782971", "title": "", "text": "$\\displaystyle s_n(x) = \\sum_{i=1}^N \\alpha_i \\chi_{Y_{i,n}}(x)$"}
{"_id": "1890189", "title": "", "text": "$ (n+1)! + n! = (n+1)\\cdot n! + n! = ... $"}
{"_id": "4484957", "title": "", "text": "$d(x, z) \\leq d(x, y) + d(y, z) < \\epsilon + \\epsilon = 2\\epsilon$"}
{"_id": "5700262", "title": "", "text": "$\\vec{u} =  \\begin{vmatrix}         \\mathbf i & \\mathbf j & \\mathbf k \\\\         x_1 & y_1 & z_1 \\\\         x_2 & y_2 & z_2 \\\\ \\end{vmatrix}\\\\ $"}
{"_id": "7128721", "title": "", "text": "$(\\gamma\\gamma')f(x)=\\gamma(\\gamma' f)(x)=\\gamma' f(\\gamma x)=f(\\gamma'\\gamma x)$"}
{"_id": "1985710", "title": "", "text": "$\\varepsilon^+$"}
{"_id": "6694804", "title": "", "text": "$d(x,A)-d(x,y)\\leqslant d(y,A)$"}
{"_id": "8304799", "title": "", "text": "$T_p^{0,1}\\mathbb{C}^n$"}
{"_id": "1093410", "title": "", "text": "$ \\sqrt{2\\sqrt[3]{3!\\sqrt[4]{4!\\sqrt[5]{5!..}}}}=2.2902182705436512868 $"}
{"_id": "2958982", "title": "", "text": "$R_{0,1} \\subset \\dots \\subset R_{0,r}$"}
{"_id": "1729923", "title": "", "text": "$\\frac{1}{16} \\left(e^{-i x}-e^{i x}\\right)^4$"}
{"_id": "4894398", "title": "", "text": "$\\lim\\limits_{n\\to \\infty}\\frac{1}{\\ln(n)}\\sum_{r=1}^{n}\\frac{a_r}{r}$"}
{"_id": "7143334", "title": "", "text": "$\\frac{1}{n+1}+\\frac{1}{(n+1)^2}...=\\frac{1}{n}$"}
{"_id": "2466853", "title": "", "text": "$cx + ay = d$"}
{"_id": "3865019", "title": "", "text": "$\\lfloor\\frac{E}{K}\\rfloor =\\lfloor \\frac{E}{K + m}\\rfloor$"}
{"_id": "1253218", "title": "", "text": "$d(x,L)=\\dfrac{\\left| f(x) \\right|}{\\left|\\left|f\\right|\\right|}$"}
{"_id": "2382997", "title": "", "text": "$w:={z-z_1\\over z-z_2}\\ .$"}
{"_id": "1624567", "title": "", "text": "$ds= \\int \\sqrt{1+(\\frac{dy}{dx})^2} dx$"}
{"_id": "2114748", "title": "", "text": "$x^3-x, x^5-x, x^7-x, x^9-x$"}
{"_id": "4403237", "title": "", "text": "$G=\\gamma_1(G) \\ge \\gamma_2(G) \\ge \\cdots \\ge \\gamma_{k+1}(G)=1$"}
{"_id": "4588653", "title": "", "text": "$\\left|\\begin{array}{r}1&x&1\\\\x&1&0\\\\0&1&x\\end{array}\\right|=1$"}
{"_id": "3241844", "title": "", "text": "$\\lim_{x\\to a-} f'(x) = \\lim_{x\\to a+} f'(x) =L$"}
{"_id": "7411124", "title": "", "text": "$Cov(x+y,x-y)$"}
{"_id": "485240", "title": "", "text": "$|ab|=p$"}
{"_id": "1267128", "title": "", "text": "$f(x,y)=\\frac{1}{2\\pi\\sqrt{1-p^2}} \\exp\\left(-\\frac{1}{2(1-p^2)}(x^2-2pxy+y^2)\\right)$"}
{"_id": "4060221", "title": "", "text": "$ \\{ x\\in E \\ \\vert \\ \\forall m\\in M : \\langle x, m\\rangle=0 \\} = \\{ x\\in E \\ \\vert \\ \\forall m\\in span(M) : \\langle x, m\\rangle=0 \\}$"}
{"_id": "8919056", "title": "", "text": "$P[X_n=x_0]=0$"}
{"_id": "4867007", "title": "", "text": "$\\int_{-\\pi}^{\\pi} \\frac{\\cos(nx)}{2\\pi} \\ dx= \\delta_{0,n}, n \\in \\mathbb{N}.$"}
{"_id": "5680541", "title": "", "text": "$f_{r,t}(r,t) = 4 r (1- r \\cos(t))(1- r \\sin(t))$"}
{"_id": "5416176", "title": "", "text": "$\\forall E \\in M, \\lim_{n \\to \\infty }\\mu_n(E)=\\mu(E)< \\infty $"}
{"_id": "5493771", "title": "", "text": "$|f(z)| + |f(−z)| ≤ 2 |z^2| $"}
{"_id": "2194829", "title": "", "text": "$\\mathcal{M} = \\langle D, \\in \\rangle$"}
{"_id": "4470856", "title": "", "text": "$\\int_0^{\\pi} f(t)\\cos(nt)~dt=0$"}
{"_id": "1268514", "title": "", "text": "$\\frac{9}{m}+\\frac{4}{r}+\\frac{10}{h}= 7$"}
{"_id": "3216165", "title": "", "text": "$3^x + 4^x + 5^x = 6^x? $"}
{"_id": "5629577", "title": "", "text": "$\\sum_{n=1}^{n}\\frac{(-1)^n}{n^2}=\\sum_{n \\text{ even}}\\frac{1}{n^2}-\\sum_{n\\text{ odd}}\\frac{1}{n^2}=2\\cdot\\sum_{n \\text{ even}}\\frac{1}{n^2}-\\sum_{n\\geq 1}\\frac{1}{n^2}=\\frac{2}{4}\\sum_{n\\geq 1}\\frac{1}{n^2}-\\sum_{n\\geq 1}\\frac{1}{n^2}$"}
{"_id": "1726322", "title": "", "text": "$\\lim\\limits_{N\\to\\infty}r^{N+1}$"}
{"_id": "3446645", "title": "", "text": "$\\omega^\\alpha=\\alpha$"}
{"_id": "2252448", "title": "", "text": "$E=A_1\\times\\ldots\\times A_k$"}
{"_id": "7247048", "title": "", "text": "$B=\\{\\{a\\}\\}$"}
{"_id": "2219361", "title": "", "text": "$\\frac{6}{x}+\\frac{2}{y}=5$"}
{"_id": "5740606", "title": "", "text": "$g(x)=\\frac{2^x - 1}{2^x}$"}
{"_id": "6880870", "title": "", "text": "$\\gamma(t)=\\frac{\\gamma_2(t)-a}{\\gamma_1(t)-a}+a$"}
{"_id": "3005685", "title": "", "text": "$S^n \\to \\mathbb R^{n+1}$"}
{"_id": "7958943", "title": "", "text": "$z_k=\\sqrt[3]{a}e^{2ik\\pi/3}$"}
{"_id": "6139983", "title": "", "text": "$\\|x_n - x_{n+1}\\| \\to 0$"}
{"_id": "5714372", "title": "", "text": "$d(x,a)>d(x,b)+d(b,a)$"}
{"_id": "3054721", "title": "", "text": "$\\frac{z}{z*}= \\frac{a+bi}{a-bi}$"}
{"_id": "59975", "title": "", "text": "$x^+$"}
{"_id": "6767008", "title": "", "text": "$\\left ( \\begin{array} \\\\F_{k+2} \\\\ F_{k+1} \\\\ \\end{array} \\right ) = \\left ( \\begin{array} \\\\ 1 & 1\\\\1 & 0 \\\\ \\end{array} \\right )^k \\left ( \\begin{array} \\\\F_{2} \\\\ F_1 \\\\ \\end{array} \\right ) = \\left ( \\begin{array} \\\\ 1 & 1\\\\1 & 0 \\\\ \\end{array} \\right )^k \\left ( \\begin{array} \\\\1 \\\\ 1 \\\\ \\end{array} \\right )$"}
{"_id": "5089539", "title": "", "text": "$\\bigg\\lfloor \\lfloor \\frac{b^2}{4} \\rfloor / a \\bigg\\rfloor$"}
{"_id": "9096818", "title": "", "text": "$\\overline{X}(\\mathbb{R})$"}
{"_id": "3047749", "title": "", "text": "$\\operatorname{Re}z^*x_0\\ge\\operatorname{Re}y^*x_0$"}
{"_id": "7250852", "title": "", "text": "$f(x)=\\frac{e^{x/\\sqrt2}}{1+x^2}$"}
{"_id": "5614784", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{1+\\sqrt{2}+...+\\sqrt{n}}{n^{3/2}}$"}
{"_id": "1173749", "title": "", "text": "$(\\gamma(i),\\gamma(j))_{\\gamma(0)} = \\frac{1}{2}(d(\\gamma(i), \\gamma(0)) + d(\\gamma(j), \\gamma(0)) - d(\\gamma(i), \\gamma(j)))$"}
{"_id": "5940604", "title": "", "text": "$\\int_{0}^{\\infty}\\frac{\\sin^{2n+1}(x)}{x} \\ dx=\\frac{\\pi \\binom{2n}{n}}{2^{2n+1}}.$"}
{"_id": "3276535", "title": "", "text": "$(R+1)S^n-RS^n$"}
{"_id": "7290024", "title": "", "text": "$\\zeta(s) = \\frac{1}{\\Gamma(s)} \\sum_{n=0}^\\infty \\frac{B_n}{n!} \\int_0^\\infty x^{s + n - 2} dx$"}
{"_id": "1988966", "title": "", "text": "$V=P\\bigoplus Q$"}
{"_id": "1284908", "title": "", "text": "$(x,y,z)=(1,1,1)\\,.$"}
{"_id": "2152815", "title": "", "text": "$\\lim_{n\\to\\infty}\\left\\lfloor\\sum_{k=1}^n\\frac9{10^k}\\right\\rfloor=0\\\\ \\left\\lfloor\\lim_{n\\to\\infty}\\sum_{k=1}^n\\frac9{10^k}\\right\\rfloor=1\\\\$"}
{"_id": "4598510", "title": "", "text": "$s_n={\\bigl(1+\\sqrt{2}\\bigr)^{n+1} +\\bigl(1-\\sqrt{2}\\bigr)^{n+1}\\over2}\\qquad(n\\geq0)\\ .\\tag{3}$"}
{"_id": "321752", "title": "", "text": "$P(m) \\geq P(n)$"}
{"_id": "1368335", "title": "", "text": "$\\tau^+$"}
{"_id": "4220436", "title": "", "text": "$\\int_{0}^{\\infty} \\dfrac{\\sin(x)}{x}  = \\dfrac{\\pi}{2}$"}
{"_id": "4585224", "title": "", "text": "$\\log_a(A\\cdot B)=\\log_aA+\\log_aB$"}
{"_id": "1785728", "title": "", "text": "$y\\in V\\subseteq \\overline{V}\\subseteq U$"}
{"_id": "4330658", "title": "", "text": "$(a+b,a)=1$"}
{"_id": "320940", "title": "", "text": "$Q \\equiv R \\equiv S$"}
{"_id": "5873624", "title": "", "text": "$\\begin{vmatrix} 1 & x & y \\\\  1 & x_1 & y_1 \\\\  1 & x_2 & y_2 \\end{vmatrix}=0$"}
{"_id": "7615687", "title": "", "text": "$=\\frac{2(x-2)(x^2+1)(x+1)^2+(x+1)^2+4(x^2+1)(x+1)-(x^2+1)}{2(x^2+1)(x+1)^2}$"}
{"_id": "1480088", "title": "", "text": "$f[f(x)^2+f(y)]=xf(x)+y$"}
{"_id": "8164443", "title": "", "text": "$bsa+brp=b$"}
{"_id": "986681", "title": "", "text": "$a_k = \\frac{1}{2^{n-2}}$"}
{"_id": "2521016", "title": "", "text": "$\\displaystyle\\sum_{n \\le x}\\cfrac{1}{n} = \\log x + 1 -\\int_1^\\infty\\cfrac{t-[t]}{t^2}dt + O(\\cfrac{1}{x})$"}
{"_id": "9158885", "title": "", "text": "$\\int_0^{2\\pi} f(x)\\sin nx\\,dx = \\int_0^{2\\pi} f(x)\\cos nx\\,dx = 0$"}
{"_id": "2153328", "title": "", "text": "$x^4+9/x^4$"}
{"_id": "4188048", "title": "", "text": "$by=\\frac{ax+cz}{2}$"}
{"_id": "8829494", "title": "", "text": "$P( \\text{a child is a girl})= P( \\text{a child is a boy})= \\frac12$"}
{"_id": "8481381", "title": "", "text": "$\\ \\color{#0a0}{tm\\equiv 0}\\,\\Rightarrow\\, 1\\,\\equiv\\, sa+\\color{#0a0}{tm}\\,\\equiv\\, sa +\\color{#0a0}0\\equiv\\, sa\\ $"}
{"_id": "7424204", "title": "", "text": "$\\overline{d}(x, z) \\leq \\overline{d}(x, y) + \\overline{d}(y, z)$"}
{"_id": "1883947", "title": "", "text": "$A_{n} \\subset A_{n+1}\\subset A_{n+2}$"}
{"_id": "2705546", "title": "", "text": "$ \\frac{\\|\\alpha(v)\\|^2}{\\|v\\|^2} = \\frac{\\sum_n s_n^2 |\\langle v,e_n\\rangle|^2 }{\\sum_n  |\\langle v,e_n\\rangle|^2 } \\ge \\min_n s_n^2. $"}
{"_id": "7878680", "title": "", "text": "$x[n] = \\delta[n+1] + \\delta[n-1]$"}
{"_id": "5665848", "title": "", "text": "$P[X_{16}=2|X_0=0] $"}
{"_id": "5824572", "title": "", "text": "$x\\to-\\infty, \\cot y=\\infty\\implies y\\to 0$"}
{"_id": "2361235", "title": "", "text": "$P_{x}[X_{0}=x]=1\\quad({}^{\\forall} x\\in E)$"}
{"_id": "1978443", "title": "", "text": "$1-rsr=(rs)^{-1}(rs-r)=-(rs)^{-1}(1-s)r$"}
{"_id": "4232387", "title": "", "text": "$J_n = \\displaystyle \\int_0^{\\pi/2} \\dfrac{\\sin((2n+1)x)}{\\sin(x)}dx$"}
{"_id": "3279659", "title": "", "text": "$\\{A_1(1),A_1(2),A_1(3),\\cdots\\}$"}
{"_id": "5214725", "title": "", "text": "$\\alpha\\cdot\\sup_{\\gamma<\\beta}\\{\\gamma\\}=\\sup_{\\gamma<\\beta}\\{\\alpha\\cdot\\gamma\\}$"}
{"_id": "1127848", "title": "", "text": "$\\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{1}{(2n)^{2^m} -1}$"}
{"_id": "8612459", "title": "", "text": "$s_n(x) = \\sum_j c_j \\chi_{K_j}(x)$"}
{"_id": "7239360", "title": "", "text": "$\\lim_{x \\to c^{+}}f(x) \\neq \\lim_{x \\to c^{-}}f(x),$"}
{"_id": "2851984", "title": "", "text": "$\\dfrac1x + \\dfrac1y= \\dfrac1a$"}
{"_id": "5919321", "title": "", "text": "$P(X_1=0\\vert X_0=2)=0=P(X_1=4\\vert X_0=2)$"}
{"_id": "1106762", "title": "", "text": "$bRx \\lor xRb$"}
{"_id": "1207335", "title": "", "text": "${\\int_{0}^{1}\\frac{(1-3x^2)-\\pi x(1-x^2)\\cot(\\pi x)}{2x^2(1-x^2)}\\,dx}$"}
{"_id": "572145", "title": "", "text": "$\\begin{vmatrix} 1 & 1 & 1\\\\ x & y & z\\\\ x^2 & y^2 & z^2\\\\ \\end{vmatrix}$"}
{"_id": "6565026", "title": "", "text": "$4^{2x}-1= 8^x+3.\\;$"}
{"_id": "1398855", "title": "", "text": "$\\lim_{n\\to\\infty}2\\sin(y_n)$"}
{"_id": "3433491", "title": "", "text": "$\\zeta(s)=\\frac{1}{s-1}\\sum_{n=1}^{\\infty}\\left(\\frac{n}{(n+1)^s}-\\frac{n-s}{n^s}\\right)\\forall \\Re(s)>0$"}
{"_id": "1754592", "title": "", "text": "$\\int_0^\\pi \\sin(t)\\cos(t)dt=0=\\int_0^0 xdx$"}
{"_id": "3663552", "title": "", "text": "$T_n={n(n+1)\\over 2}.$"}
{"_id": "5765080", "title": "", "text": "$(a, a+b)=(a, b)=1$"}
{"_id": "2763308", "title": "", "text": "$V_n=\\frac{n(n+1)}2$"}
{"_id": "1044403", "title": "", "text": "$f(n)= {n+k-1 \\choose k-1}$"}
{"_id": "4321093", "title": "", "text": "$|\\cos \\varphi + \\ii\\sin\\varphi| = \\sqrt{\\cos^2 \\varphi + \\sin^2 \\varphi} = 1$"}
{"_id": "4449272", "title": "", "text": "$D(x,B)\\leqslant d(x,y)+d(y,b)$"}
{"_id": "7384959", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\sum_{m=1}^{\\infty} \\frac{1}{n^2+m^2}=\\infty$"}
{"_id": "5900010", "title": "", "text": "$\\lim_{n \\to \\infty} \\left \\lfloor \\frac{an+b}{cn+d}\\right\\rfloor  = \\left \\lfloor{a\\over c} \\right \\rfloor $"}
{"_id": "6900394", "title": "", "text": "$f^k(B)= A^k B$"}
{"_id": "6167292", "title": "", "text": "$     \\begin{bmatrix}     1 & x & x^2 \\\\     1 & y & y^2 \\\\     1 & z & z^2 \\\\     \\end{bmatrix} $"}
{"_id": "3705413", "title": "", "text": "$\\text{lcm}(m, \\frac{n}{m}) = \\text{lcm}(p, p) = p$"}
{"_id": "2237071", "title": "", "text": "$\\lim_{n\\to \\infty} \\mathbb P(A_n)=0$"}
{"_id": "6155143", "title": "", "text": "$\\ f(n) = n^2-3n+1$"}
{"_id": "4621600", "title": "", "text": "$I_0\\subset I_1\\subset\\dots \\subset I_{n}\\subset I_{n+1}\\subset \\dots,$"}
{"_id": "3345022", "title": "", "text": "$(n!)! \\approx \\left(\\dfrac{\\sqrt{2\\pi n}(n/e)^n}{e}\\right) ^{\\sqrt{2\\pi n}(n/e)^n} $"}
{"_id": "4604956", "title": "", "text": "$f_X(x) = \\theta\\kappa^\\theta/x^{\\theta + 1},$"}
{"_id": "961259", "title": "", "text": "$\\lfloor \\frac{\\lfloor x \\rfloor}{n} \\rfloor = \\lfloor \\frac{x}{n} \\rfloor$"}
{"_id": "6176905", "title": "", "text": "$f\\left(f(x+1) + f\\left( x + f(x)\\right)\\right) = x+2$"}
{"_id": "1549932", "title": "", "text": "$(1+\\frac{1}{x^n})(1+\\frac{1}{y^n}) \\geq (1+2^n)^2$"}
{"_id": "591427", "title": "", "text": "$\\lim_{x\\to a} f(x) =L_1 \\land \\lim_{x\\to b} g(x) =L_2 \\implies \\lim_{x\\to a} f(x) \\pm \\lim_{x\\to b} g(x) =L_1 \\pm L_2$"}
{"_id": "6216163", "title": "", "text": "$ \\begin{aligned} \\cos(e^u)&=(D+a)^{-1}\\bigl[-e^u\\sin(e^u)+a\\cos(e^u)\\bigr]\\qquad(*)\\\\ &=-e^u(D+a+1)^{-1}\\sin(e^u)+a(D+a)^{-1}\\cos(e^u) \\end{aligned} $"}
{"_id": "5209117", "title": "", "text": "$I=\\int_{0}^{\\frac{\\pi}{2}}\\frac{x dx}{\\sin x+\\cos x}=\\int_{0}^{\\frac{\\pi}{2}}\\frac{(\\frac{\\pi}{2}-x) dx}{\\sin(\\frac{\\pi}{2}-x)+\\cos (\\frac{\\pi}{2}-x)} \\implies 2I=\\frac{\\pi}{2}\\int_{0}^{\\frac{\\pi}{2}}\\frac{ dx}{\\sin x+\\cos x}$"}
{"_id": "2700865", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{\\sqrt{1} + \\sqrt{2} + ... + \\sqrt{n}}{n\\sqrt{n}}$"}
{"_id": "296635", "title": "", "text": "$ \\exists \\varepsilon > 0, \\forall \\delta > 0, \\exists r_\\delta\\in(x-\\delta,x+\\delta)\\text{ s.t. } |f(x)-f(r_\\delta)|=f(x) > \\varepsilon$"}
{"_id": "9351341", "title": "", "text": "$\\qquad A_1\\times\\cdots\\times A_n$"}
{"_id": "435550", "title": "", "text": "$d(p,a) \\leq d(p,q)+d(q,a),$"}
{"_id": "8563323", "title": "", "text": "$f(n)=\\binom{n}{a}$"}
{"_id": "311587", "title": "", "text": "$P(0)\\implies P(1)$"}
{"_id": "1346316", "title": "", "text": "$\\{e_i\\}_{1\\leq i\\leq n}$"}
{"_id": "8016539", "title": "", "text": "$f(a+b) = f(a) + f(b).$"}
{"_id": "3352522", "title": "", "text": "$\\mathbb EY_1^k=\\frac12(\\mathbb EX_1^k+\\mathbb EX_2^k)$"}
{"_id": "2554414", "title": "", "text": "$F(x) = \\sum_{i = 1}^{n}a_{i}\\chi_{A_{i}}$"}
{"_id": "1890340", "title": "", "text": "$=\\int \\frac{1+\\tan^2 x-2\\tan x}{1-\\tan^2 x}dx=\\int \\frac{1+\\tan^2 x}{1-\\tan^2 x}dx-\\int\\frac{2\\tan x}{1-\\tan^2 x}dx$"}
{"_id": "1116395", "title": "", "text": "$f_{z_0}^{-1}(re^{i\\theta})=\\sqrt[k]{r}e^{\\frac{i\\theta}k},$"}
{"_id": "6575096", "title": "", "text": "$\\mathbb P[X< x | D_1] = 1$"}
{"_id": "15571", "title": "", "text": "$aX + bY = c$"}
{"_id": "4202723", "title": "", "text": "$d(x,y)=\\frac{|y-x|}{1+|y-x|}$"}
{"_id": "5756297", "title": "", "text": "$\\gamma(A) \\le \\gamma(A+K) \\le \\gamma(A) + \\gamma(K) = \\gamma(A)$"}
{"_id": "4734458", "title": "", "text": "$= P(X_1 = 7 | X_0 = 3) P(X_1 = 3 | X_0 = 1)$"}
{"_id": "3458121", "title": "", "text": "$ (f*g_{\\varepsilon})(x) = \\int\\limits_{-\\infty}^{\\infty}f(t)g_{\\varepsilon}(x-t)dt = \\int\\limits_{x-\\varepsilon}^{x+\\varepsilon}f(t)g_{\\varepsilon}(x-t)dt = g_{\\varepsilon}(x) $"}
{"_id": "7259257", "title": "", "text": "$\\sin\\theta = \\dfrac{x}{1}$"}
{"_id": "5422423", "title": "", "text": "$1+2+3+4+5+6+\\dots=-1/12$"}
{"_id": "6737838", "title": "", "text": "$P(X_n \\leq x_n) \\rightarrow P(X\\leq x)$"}
{"_id": "5473905", "title": "", "text": "$2I=\\int_{-\\pi}^\\pi \\frac{\\sin nx}{\\sin x}dx$"}
{"_id": "9306957", "title": "", "text": "$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.\\tag{$*$}$"}
{"_id": "5943417", "title": "", "text": "$\\int { \\frac{1}{x + x^\\gamma} dx}=\\int {\\left( \\frac{x^{\\gamma-1}+1}{x + x^\\gamma } - \\frac{x^{\\gamma-1}}{x + x^\\gamma } \\right) dx}=f_\\gamma(x)$"}
{"_id": "1813132", "title": "", "text": "$\\exists d_i,  d_s (\\alpha a_{ss}+(1-\\alpha) b_{ss}) \\geq \\sum_{j\\not=s } ( d_s (\\alpha a_{sj}+(1-\\alpha) b_{sj} )  $"}
{"_id": "3843592", "title": "", "text": "$\\sum_{k = 0}^{n - 1} \\cos \\left(\\frac{2\\pi k}{n}\\right) = 0$"}
{"_id": "1590298", "title": "", "text": "$f(a, b) = (a + b, a - b)$"}
{"_id": "5440853", "title": "", "text": "$\\frac{1}{2^{n-1}}<\\epsilon$"}
{"_id": "6996447", "title": "", "text": "$(p_1p_2...p_n)+1$"}
{"_id": "8334190", "title": "", "text": "$=2^{k-2}3^{k+1}=6^k(3/4)<6^k.$"}
{"_id": "6079774", "title": "", "text": "$\\begin{align} \\frac1{e}-\\left (\\frac{x}{1+x}\\right )^x &= \\frac1{e}-\\left ( 1+\\frac1{x}\\right)^{-x}\\\\&=\\frac1{e}-e^{-x \\log{\\left(1+\\frac1{x}\\right)}}\\\\&\\sim \\frac1{e}-e^{-x \\left (\\frac1{x}-\\frac1{2 x^2}\\right )}\\\\ &= \\frac1{e}-\\frac1{e} e^{\\frac1{2 x}}\\\\ &=\\frac1{e} \\left (1-e^{\\frac1{2 x}} \\right ) \\\\ &\\sim -\\frac1{2 e x} \\end{align}$"}
{"_id": "3157859", "title": "", "text": "$B=[e_1~0~0~0]$"}
{"_id": "2949404", "title": "", "text": "$\\lim \\inf a_n+\\lim \\inf b_n=\\lim_{n\\to \\infty}A_n+\\lim_{n\\to \\infty} B_n=\\lim_{n\\to \\infty}(A_n+B_n)\\leq$"}
{"_id": "239334", "title": "", "text": "$\\left\\lfloor \\left\\lfloor \\frac{x}{j} \\right\\rfloor \\frac{1}{k}\\right\\rfloor=\\left\\lfloor \\frac{x}{jk}\\right\\rfloor$"}
{"_id": "2730991", "title": "", "text": "$v = (x,y,z)=(1,1,1)$"}
{"_id": "2773914", "title": "", "text": "$\\bar{f}(a + b) = \\bar{f}(a) + \\bar{f}(b)$"}
{"_id": "887001", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{x}{(x + n^2)^2}-\\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{1}{x + n^2}=\\frac{\\pi ^2 x \\text{csch}^2\\left(\\pi  \\sqrt{x}\\right)-1}{4 x}$"}
{"_id": "7478304", "title": "", "text": "$\\sqrt[4]{c}\\:\\mathrm e^{\\tfrac{i\\pi}4}$"}
{"_id": "1745798", "title": "", "text": "$x(x-1)^{n-1}$"}
{"_id": "6897908", "title": "", "text": "$sec\\ \\theta = \\frac{x}{3}$"}
{"_id": "8952417", "title": "", "text": "$\\sum_{n \\ge 1} \\frac{1}{n^2} = \\sum_{n \\le x} \\frac{1}{n^2}+ \\mathcal{O}(1/x)$"}
{"_id": "4903973", "title": "", "text": "$ax+by = A$"}
{"_id": "7789921", "title": "", "text": "$f(x) = \\frac{2x}{2+x^2}$"}
{"_id": "5680147", "title": "", "text": "$f(x) = \\frac{4^x}{4^x+2},$"}
{"_id": "7682930", "title": "", "text": "$ \\mathbb{E}[X^2 + \\mathbb{E}[X]^2 - 2X\\mathbb{E}[X]] = \\mathbb{E}[X^2] + \\mathbb{E}[X]^2 - 2\\mathbb{E}[X]^2 = \\mathbb{E}[X^2] - \\mathbb{E}[X]^2, $"}
{"_id": "1799253", "title": "", "text": "$A = \\{ \\{ A \\} \\}$"}
{"_id": "5445487", "title": "", "text": "$\\int_0^\\infty\\frac{\\sin(4x)}{x}\\,dx$"}
{"_id": "2239334", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\left(\\frac ne\\right)^n e^{\\frac{\\theta_n}{12n}}$"}
{"_id": "8845069", "title": "", "text": "$d(x,y)=\\frac{x}{y}$"}
{"_id": "1968556", "title": "", "text": "$[a]+[b]=[a+b]=[b+a]=[b]+[a].$"}
{"_id": "4110319", "title": "", "text": "$(1+r+r^2+\\ldots+r^n)(r-1)=r^{n+1}-1$"}
{"_id": "6891323", "title": "", "text": "$\\gcd(a,n)=\\gcd(a,a+b)=\\gcd(a,b)$"}
{"_id": "709136", "title": "", "text": "$1/|y|$"}
{"_id": "870851", "title": "", "text": "$h[n] * (\\delta[n] - A \\delta[n-1])  = \\delta[n]$"}
{"_id": "3411157", "title": "", "text": "$100 + 1000 * (e^{\\frac{-100}{1000}} - e^0)$"}
{"_id": "3212620", "title": "", "text": "$s=\\int \\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2} dx$"}
{"_id": "4032091", "title": "", "text": "$P(X_{12}=2|X_0=0)$"}
{"_id": "4681238", "title": "", "text": "$\\det\\begin{pmatrix} A & B\\\\ B & A \\end{pmatrix}=\\det\\begin{pmatrix} A-B & B-A\\\\ B & A \\end{pmatrix}.$"}
{"_id": "1017211", "title": "", "text": "$\\sqrt 1+\\sqrt 2 +\\sqrt 3 +\\cdots +\\sqrt {2009}$"}
{"_id": "3087364", "title": "", "text": "$S_1\\subseteq S_2\\subseteq S_3\\subseteq S_4\\subseteq S_5\\subseteq\\ldots$"}
{"_id": "2291089", "title": "", "text": "$118!+2, 118!+3, 118!+4, \\ldots, 118!+118$"}
{"_id": "4749445", "title": "", "text": "$x = \\frac{z + ab\\bar{z} -(a+b)}{a-b}$"}
{"_id": "3694732", "title": "", "text": "$\\overline{X}_\\mathbb{R}$"}
{"_id": "4431752", "title": "", "text": "$ \\int_{\\pi}^\\pi f(x)\\,\\cos nx\\,dx=\\int_{\\pi}^\\pi f(x)\\,\\sin nx\\,dx=0, \\quad\\text{for every}\\,\\, n\\in\\mathbb N, $"}
{"_id": "809356", "title": "", "text": "$f3$"}
{"_id": "137492", "title": "", "text": "$1 + 2 + 3 + \\cdots + n = \\frac{n(n+1)}{2}$"}
{"_id": "1210355", "title": "", "text": "$\\sum\\limits_{n=0}^N r^n=\\frac{r^{N+1}-1}{r-1}$"}
{"_id": "5402257", "title": "", "text": "$(x+y+z)^2+(x+y-z)^2+(x-y+z)^2+(x-y-z)^2\\\\+(-x+y+z)^2+(-x+y-z)^2+(-x-y+z)^2+(-x-y-z)^2\\\\=8(x^2+y^2+z^2)$"}
{"_id": "1290858", "title": "", "text": "$\\begin{cases}x=0|-1\\\\y=1|\\ \\ \\ \\ \\ 2\\end{cases}$"}
{"_id": "4834852", "title": "", "text": "$\\sum_{n=-\\infty}^\\infty \\delta(x-n)  = \\sum_{n=-\\infty}^\\infty e^{2i \\pi n x}$"}
{"_id": "7599897", "title": "", "text": "$f(n) = f(2)^{F(n-2)} * f(1)^{F(n-3)}$"}
{"_id": "6401303", "title": "", "text": "$V_1 \\subseteq \\cdots \\subseteq V_n$"}
{"_id": "7259129", "title": "", "text": "$\\tan(\\theta)=\\frac{x}{1}$"}
{"_id": "4660053", "title": "", "text": "$x\\mathrel Ry\\wedge y\\mathrel Rx\\implies x=y.$"}
{"_id": "8409187", "title": "", "text": "$\\gamma = \\min \\{k, \\lfloor \\tfrac{n}{\\mu} \\rfloor \\}$"}
{"_id": "1750897", "title": "", "text": "$\\lim_{a \\to -\\infty} +\\infty = +\\infty$"}
{"_id": "6910911", "title": "", "text": "$ \\mathbb{E} \\left[ \\frac{| \\langle \\gamma, Y \\rangle |}{\\| \\gamma\\| \\| Y \\| } \\right] \\leq \\frac{1}{n}$"}
{"_id": "19110", "title": "", "text": "$\\int \\frac{1}{(x^2+1)^2}\\mathrm dx$"}
{"_id": "1373998", "title": "", "text": "$f(n) = f(n-1)+f(n-2) = f(n-2)+f(n-3) + f(n-2)$"}
{"_id": "4780015", "title": "", "text": "$ A u = b \\iff \\\\ [ A \\mid b ] = \\\\ \\left[ \\begin{array}{rrr|r} 1 & -2 & 5 & 2 \\\\ 1 &  1 & 1 & k \\\\ 2 & -1 & 6 & k^2 \\end{array} \\right] \\to \\left[ \\begin{array}{rrr|r} 1 & -2 & 5 & 2 \\\\ 0 &  3 & -4 & k-2 \\\\ 0 &  3 & -4 & k^2-4 \\end{array} \\right] \\to \\\\ \\left[ \\begin{array}{rrr|r} 1 & -2 &  5 & 2 \\\\ 0 &  3 & -4 & k-2 \\\\ 0 &  0 &  0 & k^2-k-2 \\end{array} \\right] \\to \\left[ \\begin{array}{rrr|r} 1 & -2 &  5 & 2 \\\\ 0 &  1 & -4/3 & (k-2)/3 \\\\ 0 &  0 &  0 & k^2-k-2 \\end{array} \\right] $"}
{"_id": "7581096", "title": "", "text": "$\\int_0^\\infty f_+(x)\\,dx$"}
{"_id": "5454741", "title": "", "text": "$ \\zeta(s)=2^s \\pi^{s-1} sin(\\frac{\\pi s}{2}) \\Gamma(1-s)\\zeta(1-s) $"}
{"_id": "1053518", "title": "", "text": "$ s_n(x)=\\sum\\limits_{j=1}^{m+1}a_j\\chi_{E_j^n}(x). $"}
{"_id": "949816", "title": "", "text": "$1+\\sum_{r=1}^{r=n} r\\cdot r! = (n+1)!$"}
{"_id": "5208319", "title": "", "text": "$\\int \\tan^n x\\,dx = \\frac{1}{n-1}\\tan^{n-1} x - \\int \\tan^{n+2}x\\,dx$"}
{"_id": "5542305", "title": "", "text": "$=61+\\frac1{28+\\dfrac49}=61+\\frac1{28+\\dfrac1{\\dfrac94}}=61+\\frac1{28+\\dfrac1{2+\\dfrac14}}$"}
{"_id": "6216325", "title": "", "text": "$g(t)=\\dfrac{3^t}{3^t+3^{1/2}}$"}
{"_id": "6526700", "title": "", "text": "$|AB| = |CD| = |EF| = r = 1$"}
{"_id": "4535786", "title": "", "text": "$ax+by-d$"}
{"_id": "8728081", "title": "", "text": "$\\mathbb{Q}[x]/(x^4+1) \\cong \\mathbb{Q}(u)$"}
{"_id": "7695057", "title": "", "text": "$p^T\\eta p = F(p)^T \\eta F(p)$"}
{"_id": "6020470", "title": "", "text": "$(n+1)!+2, (n+1)!+3,...(n+1)!+n+1 $"}
{"_id": "1048982", "title": "", "text": "$\\;\\begin{cases} x\\equiv 0\\mod4,\\\\x\\equiv 1\\mod 25.\\end{cases}$"}
{"_id": "6971716", "title": "", "text": "$z=\\sqrt[4]{w}e^{\\frac{2\\pi ki}{4}}\\Longleftrightarrow$"}
{"_id": "5337890", "title": "", "text": "$\\int_0^{x/3} f(t)dt =\\int_0^xf(t)dt$"}
{"_id": "4933510", "title": "", "text": "$n > 1/|x|$"}
{"_id": "4938823", "title": "", "text": "$ \\sum_{k=1}^n \\sqrt{n^4+k}\\sin\\frac{2\\pi k}{n} = n^3 \\frac{1}{n}\\sum_{k=1}^n \\sqrt{1+\\frac{k}{n^4}}\\sin\\frac{2\\pi k}{n} $"}
{"_id": "8400792", "title": "", "text": "$a>2\\implies\\gcd(a+2,a)=\\gcd(a,a+2-a)=\\gcd(a,2)\\leq2$"}
{"_id": "4933499", "title": "", "text": "$\\left| \\int_a^b dX_t \\right| = |X_b-X_a| \\leq c \\left( \\sum_{i=0}^n |X_{t_{i+1}}-X_{t_i}|^{1/\\alpha} \\right)^{\\alpha}$"}
{"_id": "3883819", "title": "", "text": "$d(x,A):=inf $"}
{"_id": "5598029", "title": "", "text": "$\\sum_{n=0}^{\\infty}\\frac{n^2}{n!}=\\sum_{n=0}^{\\infty}\\frac1{(n-1)!}+\\sum_{n=0}^{\\infty}\\frac1{n!}=\\sum_{n=1}^{\\infty}\\frac1{(n-1)!}+\\sum_{n=0}^{\\infty}\\frac1{n!}$"}
{"_id": "7822569", "title": "", "text": "$T_X\\cong T_X^{1,0}$"}
{"_id": "3670778", "title": "", "text": "$D_n=a(a-x)^{n-1}+x\\sum_{k=0}^{n-2}(a-x)^{k}(a-y)^{n-1-k}$"}
{"_id": "1847295", "title": "", "text": "$\\sigma_n= \\frac{a_1+a_2+...+a_n}{n}$"}
{"_id": "8391145", "title": "", "text": "$ \\begin{matrix}\\\\    & 1 &  x& x^2 &   & w & x w &   & w^2 &  & \\\\ 1  & 1 &  .& 1   &   & . & .   &   & .   &  & \\\\ y  & . & -2&     &   & . &     & \\\\ y^2& 1 &   &     &   &   &     & \\\\ \\\\ z  & . & . &     &   & . &     & \\\\ y z& . &   &\\\\ \\\\ z^2& . &   &\\\\ \\end{matrix} $"}
{"_id": "5643655", "title": "", "text": "$\\gcd(a+b,ab)=1.$"}
{"_id": "1454921", "title": "", "text": "$\\sum\\limits_{n=2}^\\infty\\frac{\\cos(n\\pi/3)}{n}$"}
{"_id": "6343653", "title": "", "text": "$\\sum_{k=m}^{k=n}\\frac{1}{(1+r)^k}=\\frac{(r+1)^{1-m}-(r+1)^{-n}}{r}$"}
{"_id": "5935216", "title": "", "text": "$\\|x_n-x_m\\|<1$"}
{"_id": "22613", "title": "", "text": "$ =\\frac{1}{r(\\gamma)}[\\dot{r}{(\\gamma)\\frac {\\partial}{\\partial \\gamma} +r(\\gamma) \\frac{\\partial^2}{\\partial \\gamma^2}+1/r(\\gamma)\\frac {\\partial ^2}{\\partial \\theta^2}}]$"}
{"_id": "11447", "title": "", "text": "$P(m)\\ne P(n)$"}
{"_id": "1208683", "title": "", "text": "$1  / \\vert x \\vert \\le 1/2$"}
{"_id": "8586146", "title": "", "text": "$A = \\n_{\\dot \\gamma} \\dot \\gamma$"}
{"_id": "6653189", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)}{4^{n-1}} =\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)}{2^{2n-2}}= 2\\int_{0}^{\\infty}\\frac{\\sinh(x/2)}{e^x-1}\\,dx =\\int_{0}^{\\infty}e^{-x/2}\\,dx = 2$"}
{"_id": "2922750", "title": "", "text": "$\\lim_{x \\rightarrow c}{\\vert f(x) \\vert} = \\lim_{x \\rightarrow c^-}{\\vert f(x) \\vert} = \\lim_{x \\rightarrow c^+}{\\vert f(x) \\vert} = 1$"}
{"_id": "467146", "title": "", "text": "$ x = \\frac{a}{a+b}$"}
{"_id": "812279", "title": "", "text": "$1+1+1+\\cdots=-\\frac{1}{2}$"}
{"_id": "8224444", "title": "", "text": "$\\left\\{\\frac{(1+x)^{n+1}-(1+x)^r}{x^2}-\\frac{(n-r+1)(1+x)^{r-1}}{x}\\right\\}=\\binom{n+1}{r+1}$"}
{"_id": "7019656", "title": "", "text": "$ \\begin{align} \\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\frac1{m^2+n^2} &\\ge\\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\frac1{m^2+n^2+2mn+m+n}\\\\ &=\\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\left(\\frac1{m+n}-\\frac1{m+n+1}\\right)\\\\ &=\\sum_{m=1}^\\infty\\frac1{m+1}\\\\[6pt] &=\\infty \\end{align} $"}
{"_id": "1588300", "title": "", "text": "$a_n=\\int_{-\\pi}^{\\pi}(1/\\pi)|\\sin(x)|\\cos(nx)dx$"}
{"_id": "6203704", "title": "", "text": "$ \\|A\\|_2 = \\sqrt{\\lambda_\\max(A^t A)} = \\sqrt{\\lambda_\\max(A^2)} = \\sqrt{\\lambda_\\max(A)^2} = \\lambda_\\max(A) $"}
{"_id": "1810804", "title": "", "text": "$ \\begin{align} \\lim_{n\\to\\infty}\\frac n{2^n}\\sum_{k=1}^n\\frac1k &\\le\\lim_{n\\to\\infty}\\frac{n^2}{2^n}\\\\ &=0 \\end{align} $"}
{"_id": "2129320", "title": "", "text": "$\\left(1 + \\frac{x}{n}\\right)^n \\leqslant e^x < \\left(1 - \\frac{x}{n}\\right)^{-n}.$"}
{"_id": "7158975", "title": "", "text": "$Y_a = \\frac{1+aX}{a-X}$"}
{"_id": "5175229", "title": "", "text": "$ \\mathrm{dist}(x_0,M)=\\frac{|F(x_0)|}{\\Vert F\\Vert} $"}
{"_id": "1886664", "title": "", "text": "$(n+1)e^x=(n-1)e^{-x}$"}
{"_id": "3830738", "title": "", "text": "$   \\begin{pmatrix}a&b\\\\-b&a\\end{pmatrix}. $"}
{"_id": "6792489", "title": "", "text": "$f(xf(y))=f(x)y.$"}
{"_id": "1253442", "title": "", "text": "$F=P(H)\\oplus \\text{Coker P}$"}
{"_id": "4181323", "title": "", "text": "$|(x-a)-(y-b)|+|(x-a)+(y-b)|=A$"}
{"_id": "327599", "title": "", "text": "$ |(1-\\lambda)X_i+\\lambda Y_i|^p\\leq (1-\\lambda)|X_i|^p+\\lambda |Y_i|^p. $"}
{"_id": "7170354", "title": "", "text": "$\\{1,x,y,x^2,xy,y^2,x^3,x^2y,\\cdots\\}\\subset\\Lambda(\\mathbb R^\\infty)$"}
{"_id": "7546532", "title": "", "text": "$\\displaystyle\\left(\\frac{n}{2^{k/2}}\\right)^k\\le 1$"}
{"_id": "7324238", "title": "", "text": "$\\left[     \\begin{array}{ccc|c}       1&1&k&3\\\\       1&k&1&-7\\\\k&1&1&4     \\end{array} \\right] $"}
{"_id": "3752253", "title": "", "text": "$y = f(a) + f(b)$"}
{"_id": "9208865", "title": "", "text": "$\\sum_{n=1}^H \\frac{1}{n^2}\\approx\\sum_{n=1}^\\infty \\frac{1}{n^2}$"}
{"_id": "397393", "title": "", "text": "$\\gamma^+$"}
{"_id": "2888421", "title": "", "text": "$\\det\\begin{pmatrix}A & -B \\\\ B & A \\end{pmatrix}=|\\det(A+iB)|^{2}.$"}
{"_id": "2409295", "title": "", "text": "$\\displaystyle\\begin{align*}\\int_0^\\infty \\dfrac{\\sin^3x}{x^3}dx&=\\dfrac32\\int_0^\\infty\\dfrac{\\sin^2x\\cos x}{x^2}dx\\\\&=\\dfrac32\\int_0^\\infty\\dfrac{2\\sin x\\cos^2x-sin^3x}xdx\\\\&=\\dfrac32\\left[2\\int_0^\\infty\\dfrac{\\sin x}xdx-3\\int_0^\\infty\\dfrac{\\sin^3x}xdx\\right]\\\\&=\\dfrac32\\left(\\pi-\\dfrac{3\\pi}4\\right)\\\\&=\\dfrac{3\\pi}8\\end{align*}$"}
{"_id": "180301", "title": "", "text": "$f(x+2xf(y)^2)=yf(x)+f(f(y)+1)$"}
{"_id": "4009066", "title": "", "text": "$m={\\rm ord}(a), n={\\rm ord}(b)$"}
{"_id": "2902417", "title": "", "text": "$\\left|\\frac{f(a)-f(b)}{1-\\overline{f(b)}f(a)}\\right|\\le\\left |\\frac{a-b}{1-\\bar{b}a}\\right|$"}
{"_id": "5144064", "title": "", "text": "$A_1\\times\\ldots\\times A_n \\cong A_1\\oplus\\ldots\\oplus A_n$"}
{"_id": "3908320", "title": "", "text": "$g=\\sum_{j=1}^{n}a_j\\chi_{E_j}$"}
{"_id": "47848", "title": "", "text": "$\\sum_{n\\ge 1} \\frac{1}{n^2}$"}
{"_id": "7691376", "title": "", "text": "$\\sum_{n=0}^\\infty\\frac{1}{(4n)!}$"}
{"_id": "897137", "title": "", "text": "$\\mathbb{E}(X_{t}-X_{s})^{2}\\leq c \\vert t-s\\vert^{2}$"}
{"_id": "7905197", "title": "", "text": "$I_n < \\pi < C_n$"}
{"_id": "1748712", "title": "", "text": "$\\eta(s) = \\frac1{\\Gamma(s)} \\int_0^{\\infty}\\frac{x^{s-1}}{e^x+1}\\mathrm dx$"}
{"_id": "5289664", "title": "", "text": "$\\bigg\\lfloor\\frac{\\lfloor \\frac{x}{m}\\rfloor}{n}\\bigg\\rfloor=\\bigg\\lfloor \\frac{x}{mn}\\bigg\\rfloor?$"}
{"_id": "3922381", "title": "", "text": "$\\frac{(a+b)!(2a)!(2b)!}{a!b!(2a+2b)!}$"}
{"_id": "6710228", "title": "", "text": "$f(y)=1/|y|$"}
{"_id": "210390", "title": "", "text": "$f_X(x) = { 6x(L-x) \\over L^3}, where: 0 \\le x \\le L $"}
{"_id": "1257965", "title": "", "text": "$\\{ e_{1},e_{2},e_{3},\\cdots\\}$"}
{"_id": "8425838", "title": "", "text": "$\\left\\lfloor\\frac{a}{bc}\\right\\rfloor=d$"}
{"_id": "2756840", "title": "", "text": "$p(m)<0<p(m+1)$"}
{"_id": "7067920", "title": "", "text": "$d(x',A)-d(x,A)\\leqslant d(x',x)$"}
{"_id": "19590", "title": "", "text": "$\\cos(\\alpha) = \\frac{|\\langle x,y\\rangle|}{||x|| ||y|| } = 1$"}
{"_id": "3812031", "title": "", "text": "$x\\in V\\subseteq\\cl V\\subseteq U$"}
{"_id": "1006455", "title": "", "text": "$\\int_0^{\\pi}{\\sin x\\over x}$"}
{"_id": "4216149", "title": "", "text": "$Cov(W,V)=0$"}
{"_id": "6482510", "title": "", "text": "$f'(x) = \\frac{e^x(x-n)}{x^{n+1}}>\\frac{e^{n+1}}{n^{n+1}}, \\quad (x > n+1)$"}
{"_id": "8879660", "title": "", "text": "$k,k+B,k+2B,\\dots$"}
{"_id": "8385647", "title": "", "text": "$\\begin{cases}x\\equiv 6 \\pmod{13}\\\\ x\\equiv 2 \\pmod{17} \\end{cases}\\quad \\begin{cases}x\\equiv 6 \\pmod{13}\\\\ x\\equiv -2 \\pmod{17} \\end{cases}\\\\ \\begin{cases}x\\equiv -6 \\pmod{13}\\\\ x\\equiv 2 \\pmod{17} \\end{cases}\\quad \\begin{cases}x\\equiv -6 \\pmod{13}\\\\ x\\equiv -2 \\pmod{17} \\end{cases}$"}
{"_id": "992924", "title": "", "text": "$0\\le \\sum_{n=1}^N \\frac{1}{n^2+3n+2}\\le \\sum_{n=1}^N \\frac{1}{n^2}\\le \\sum_{n=1}^\\infty \\frac{1}{n^2} \\tag 1$"}
{"_id": "846760", "title": "", "text": "$   \\left(  \\begin{array}{cc}   a  &  b   \\\\   -b  &  a      \\end{array}    \\right)  , $"}
{"_id": "4350020", "title": "", "text": "$\\require{cancel} \\frac{r (1-r)\\cancel{(1+r)}}{\\cancel{(1+r)}}$"}
{"_id": "8258811", "title": "", "text": "$U\\oplus T$"}
{"_id": "2192333", "title": "", "text": "$[a+(n-1)\\delta, a+n\\delta]$"}
{"_id": "7800554", "title": "", "text": "$ \\binom{n}{\\gamma n} \\sim \\left( 2 \\pi \\gamma (1 - \\gamma) n \\right)^{- 1/2} \\left( \\gamma^{\\gamma} (1 - \\gamma)^{1 - \\gamma} \\right)^{-n}.$"}
{"_id": "1875665", "title": "", "text": "$ \\sum_{k=1}^n\\frac{2k}{n^2+n}=1 $"}
{"_id": "6513201", "title": "", "text": "$m'(m'(-,-),-)=m'(-,m'(-,-))$"}
{"_id": "1601524", "title": "", "text": "$f(a+b)=f(a)+f(b), f(ab)=f(a)f(b)$"}
{"_id": "1629856", "title": "", "text": "$\\gamma \\wedge \\gamma = (-1)^{n^2} \\gamma \\wedge \\gamma$"}
{"_id": "4490023", "title": "", "text": "$\\lim\\limits_{x\\to a^-}k(x)=\\lim\\limits_{x\\to a^-}g(x)=g(a)=k(a)=f(a)=\\lim\\limits_{x\\to a^+}f(x)=\\lim\\limits_{x\\to a^+}k(x)$"}
{"_id": "1297123", "title": "", "text": "$\\lim_n  \\frac{6}{\\pi^2}\\sum_{k=1}^n\\frac{1}{k^2} =1.$"}
{"_id": "4629596", "title": "", "text": "$P(x) = 1/6$"}
{"_id": "192904", "title": "", "text": "$af+bg=1$"}
{"_id": "877128", "title": "", "text": "$|g(z)| \\leq C(|z|^{3/2}|z-1|^2+|z-1|^{1/2}), z\\in\\mathbb{C}-\\{1\\}$"}
{"_id": "4570996", "title": "", "text": "$p_3(n)=\\binom{n+2}{2}$"}
{"_id": "1377630", "title": "", "text": "$\\sum _{n=1}^{\\infty }n=+\\infty$"}
{"_id": "2926842", "title": "", "text": "$\\sum_{k=1}^R \\cos \\frac{2k \\pi x}{R}$"}
{"_id": "1977344", "title": "", "text": "$ \\int_0^\\pi v(t)\\,dt = \\int_0^\\pi \\cos (2t) \\, dt = 0, $"}
{"_id": "4898373", "title": "", "text": "$G=\\left(\\begin{array}{cc|cc} 1 & 0 &1 &1\\\\ 0 & 1 &1 &-1 \\end{array}\\right)$"}
{"_id": "845218", "title": "", "text": "$\\det(A)\\in\\mathbb{Z}[(a_{i,j})_{i,j}]$"}
{"_id": "3043149", "title": "", "text": "$\\int \\frac{1}{(x^2+bx+c)^n}\\, dx$"}
{"_id": "957206", "title": "", "text": "$A \\times \\ldots \\times A$"}
{"_id": "3108031", "title": "", "text": "$L_1 = \\begin{cases} x= 1+as \\\\ y=-2 +bs \\\\ z=-1+cs \\end{cases}$"}
{"_id": "8992567", "title": "", "text": "$q(n) = \\left(\\frac{n}{10}\\right)^{10}$"}
{"_id": "2483087", "title": "", "text": "$ 1+2r+3r^2+...+nr^{n-1}=\\sum_{k=1}^{n-1}kr^k=\\frac{1-r^{n+1}}{(1-r)^2}+\\frac{-(n+1)r^{n}}{1-r}, \\quad |r|<1. \\tag2 $"}
{"_id": "8120041", "title": "", "text": "$ \\sum_{n=1}^{+\\infty} 1 = +\\infty $"}
{"_id": "7375371", "title": "", "text": "$\\begin{cases} x=2 \\\\ y=0 \\\\ z=-2 \\\\ \\end{cases}$"}
{"_id": "164462", "title": "", "text": "$\\mathbb R[x]/(x^2)$"}
{"_id": "3815", "title": "", "text": "$\\frac{\\sum_{n=0}^{N-1}c_n^N}{c_N^N}=\\frac{1-c_N^N}{c_N^N}=\\frac{1-\\dfrac1{2^N}}{\\dfrac1{2^N}}=2^N-1\\to\\infty.$"}
{"_id": "6337", "title": "", "text": "$(X,\\mathcal{A})$"}
{"_id": "6409796", "title": "", "text": "$6^x+8^x+15^x=9^x+10^x+12^x$"}
{"_id": "7534766", "title": "", "text": "$ 1/|x|^{k}$"}
{"_id": "5065786", "title": "", "text": "$\\int_0^\\infty \\frac{x^{2n - 1}}{(x^2 + 1)^{n + 3}}\\,dx$"}
{"_id": "2476183", "title": "", "text": "$\\|A\\|_2=\\sqrt{\\max_n{\\lambda_n(AA^T)}}$"}
{"_id": "5346559", "title": "", "text": "$f(n)={n\\choose k}$"}
{"_id": "907511", "title": "", "text": "$\\frac{1-a^2}{2}=\\sin(t)\\cos(t)$"}
{"_id": "7253086", "title": "", "text": "$x=y \\ \\ \\text{or} \\ \\ z=-2\\lambda-1$"}
{"_id": "6627769", "title": "", "text": "$\\mathbb{Z}_{2}[X] / (x^4+1)$"}
{"_id": "6625161", "title": "", "text": "$\\begin{bmatrix} 3 & -5\\\\  5 & 3  \\end{bmatrix}x$"}
{"_id": "5997721", "title": "", "text": "$f(x)^2+f(y)^2=f(x+y)\\Big(f(x)+f(y)+a\\Big).\\tag{1}$"}
{"_id": "2728950", "title": "", "text": "$A_1 \\times A_2 \\times \\ldots \\times A_n$"}
{"_id": "6314935", "title": "", "text": "$\\overline{\\sqrt{z}}= \\begin{cases} \\sqrt{|z|}e^{\\frac{-i\\phi}{2}} \\\\ \\sqrt{|z|}e^{\\frac{-i\\phi}{2 + \\pi}} \\end{cases}$"}
{"_id": "6485556", "title": "", "text": "$|f(z^2)|\\geq |f(z)|$"}
{"_id": "9236348", "title": "", "text": "$\\int \\frac1{(x^2+1)^{3/2}} dx $"}
{"_id": "1587950", "title": "", "text": "$\\sum _{k=1}^{n-1} (n-k)\\cos\\frac{2k\\pi}{n} $"}
{"_id": "5923318", "title": "", "text": "$\\frac{a + bc}{b}$"}
{"_id": "3286572", "title": "", "text": "$10/40$"}
{"_id": "1585232", "title": "", "text": "$\\Gamma^+$"}
{"_id": "6081658", "title": "", "text": "$p \\ | \\ n_1n_2$"}
{"_id": "3348271", "title": "", "text": "$\\Rightarrow \\forall x\\in X \\exists y=x\\in X :(xRy \\vee yRx)$"}
{"_id": "7002602", "title": "", "text": "$\\frac{\\mathbb{Z}_4 [x]}{(x^2 + 1)}$"}
{"_id": "3371477", "title": "", "text": "$a_3(p^k) = \\binom{k}{2}$"}
{"_id": "9108923", "title": "", "text": "$\\frac{1-2\\sin\\left(α\\right)\\cos\\left(α\\right)}{1-2\\sin^2\\left(α\\right)}$"}
{"_id": "5473901", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\frac {\\sin nx}{(1+2^x)\\sin x}dx \\:\\:\\:  n \\in  \\mathbb{N}$"}
{"_id": "2182198", "title": "", "text": "$\\frac{8}{a}+\\frac{2}{b} = 1.$"}
{"_id": "6197795", "title": "", "text": "$\\sum_{n \\leq x} \\frac{1}{n} \\lfloor \\frac{x}{n} \\rfloor \\ge \\sum_{n \\leq x} \\frac{1}{n}  (\\frac{x}{n}-1) = x\\sum_{n \\leq x} \\frac{1}{n^2} -\\sum_{n \\leq x} \\frac{1}{n}  = x\\sum_{n \\leq x} \\frac{1}{n^2} -\\ln(x)+O(1) $"}
{"_id": "8633597", "title": "", "text": "$C^{î,*}(D)=C^{î,*}(D_0) \\oplus C^{î-1,*}(D_{1})$"}
{"_id": "1289027", "title": "", "text": "$\\mathbb{R}[X]/(X^4-1)$"}
{"_id": "3299923", "title": "", "text": "$(x-n+1)(x-1)^{n-1}$"}
{"_id": "1025256", "title": "", "text": "$[x,y]=-[y,x]$"}
{"_id": "2520787", "title": "", "text": "$ \\gcd(a,b-a)=\\gcd(-a,b-a)\\leq \\gcd(-a,b)=\\gcd(a,b)$"}
{"_id": "2183100", "title": "", "text": "$f_T(t) = \\frac {1}{m} e^{\\frac{-x}{m}}$"}
{"_id": "5318279", "title": "", "text": "$\\begin{align} a^n \\equiv (a^s)^k \\equiv a^k\\pmod{s} \\tag{2}\\\\ b^n\\equiv b^k\\pmod{s} \\tag{3} \\end{align}$"}
{"_id": "1624449", "title": "", "text": "$S=\\{x_1,x_2,x_3,...,x_n\\}$"}
{"_id": "3749559", "title": "", "text": "$f(t)=\\dfrac{t}{t^{11} + 1}\\tag{1a}$"}
{"_id": "5939467", "title": "", "text": "$a_n=1+\\frac{n(n+1)}{2}$"}
{"_id": "157739", "title": "", "text": "$\\tan \\theta = \\frac{x}{y}$"}
{"_id": "4306619", "title": "", "text": "$f(a+b)=f(a)+f(b)+1$"}
{"_id": "6840151", "title": "", "text": "$ds(t) = \\sqrt{\\left( \\frac{d \\, x(t)}{d \\, t} \\right)^2 + \\left ( \\frac{ d \\, y(t) }{d \\, t} \\right)^2} $"}
{"_id": "534530", "title": "", "text": "$f_2(x) = x^2+x+2$"}
{"_id": "598700", "title": "", "text": "$ar+bs=1$"}
{"_id": "5540820", "title": "", "text": "$\\mathfrak{a}^+$"}
{"_id": "1096520", "title": "", "text": "$\\forall \\epsilon \\gt 0, \\exists \\delta \\gt 0 \\; \\bigl| \\; 0 \\lt | x - a | \\lt \\delta \\implies |\\sum_{k=1}^n f_k(x) - \\sum _{k=1}^n L_k| \\lt \\epsilon$"}
{"_id": "9300366", "title": "", "text": "$f(a+b)=f(a)+_nf(b)$"}
{"_id": "6194883", "title": "", "text": "$ f(x) = \\frac{2x}{\\alpha^2}$"}
{"_id": "4814304", "title": "", "text": "$\\{(-x,-y),(y-x,-x),(-y,x-y),(x-y,x),(y,y-x)\\}$"}
{"_id": "5565312", "title": "", "text": "$ f(a^{-k}) M = f(b) \\\\ M = f(b) f(a^{k}) $"}
{"_id": "5845983", "title": "", "text": "$g(f(x))=\\frac{\\sqrt{1-\\sin{x}}}{1+\\sqrt{1-\\sin{x}}}$"}
{"_id": "8056820", "title": "", "text": "$C_0(X,C_0(Y))$"}
{"_id": "22612", "title": "", "text": "$=\\frac{1}{r(\\gamma)}[\\dot{r}{(\\gamma)\\frac {\\partial}{\\partial \\gamma} +r(\\gamma) \\frac{\\partial^2}{\\partial \\gamma^2}+r(\\gamma)^3 \\frac {\\partial ^2}{\\partial \\theta^2}}] $"}
{"_id": "2047095", "title": "", "text": "$A=\\{\\langle x,y\\rangle\\in\\mathbb R^2\\mid y>x+1\\}$"}
{"_id": "8306424", "title": "", "text": "$\\frac{1}{x}=\\frac{a+b}{ba}$"}
{"_id": "3320405", "title": "", "text": "$(P r)/(1-(1+r)^{-N})$"}
{"_id": "5575722", "title": "", "text": "$C_n = \\frac{1}{2 \\pi} \\int_{-\\pi}^\\pi (\\cos(2x) +\\sin(x)) e^{-inx} \\, dx$"}
{"_id": "2997529", "title": "", "text": "$2|\\alpha|=|\\beta-\\gamma|\\leq|\\beta|+|\\gamma|\\leq 2|\\alpha|$"}
{"_id": "4533235", "title": "", "text": "$x = y = z = 2\\lambda$"}
{"_id": "7974151", "title": "", "text": "$\\lim_{x\\to c}f(x)=l.$"}
{"_id": "1468072", "title": "", "text": "$\\ln(\\lfloor{x}\\rfloor!) = \\sqrt{2\\pi{x}}(\\frac{x}{e})^x + R_k(x)$"}
{"_id": "2214855", "title": "", "text": "$\\lim\\inf \\mu(A_n)$"}
{"_id": "6267680", "title": "", "text": "$ b_{n+1}-b_n=c_n(\\frac{c_n}{n+c_n}-\\frac{1}{n+c_n}). $"}
{"_id": "4346847", "title": "", "text": "$\\color {green} {\\text{Numbers}\\,\\, 20!+2,20!+3,\\cdots ,20!+21 \\,\\,\\text{will do the trick. The following result by Euclid has been known for more than 2000 years.}}$"}
{"_id": "8749054", "title": "", "text": "$\\lbrace\\, a,a+b,a-b,a+2b,a-2b,\\dots\\,\\rbrace$"}
{"_id": "5671589", "title": "", "text": "$V\\subseteq\\ker(P[A])$"}
{"_id": "1357538", "title": "", "text": "$\\int_{\\pi}^{2\\pi}$"}
{"_id": "704454", "title": "", "text": "$\\sum_{n=-\\infty}^{\\infty} e^{-(u+n a)^2} = \\frac{\\sqrt{\\pi}}{a} \\left [1+2 \\sum_{n=1}^{\\infty} e^{-n^2 \\pi^2/a^2} \\cos{\\left (2 \\pi n \\frac{u}{a} \\right )} \\right ]$"}
{"_id": "5040524", "title": "", "text": "$\\left|\\frac{\\sin(x)}{|x|^{3/2}}\\right|^2 \\sim \\frac{1}{|x|}$"}
{"_id": "1010740", "title": "", "text": "$P(B) = 1/6$"}
{"_id": "7600183", "title": "", "text": "$f(x) = \\frac{(-3)^{x-1}}{4^x}$"}
{"_id": "6629487", "title": "", "text": "$F_X(x)=\\frac{2x}{100}$"}
{"_id": "4823592", "title": "", "text": "$e^{-x} \\geqslant \\left(1 - \\frac{x}{n}\\right)^{n}.$"}
{"_id": "2159498", "title": "", "text": "$ y_n = \\frac{x_1 + x_2 + x_n}{n}$"}
{"_id": "7465905", "title": "", "text": "$\\text{Cov}(U,V)=0$"}
{"_id": "3026349", "title": "", "text": "$c_{y}/(b/4)$"}
{"_id": "1377337", "title": "", "text": "$f''(x)=(x-a)^{2n-1}(x-b)^{2m}[2n(x-b)+(2m+1)(x-a)]$"}
{"_id": "3772631", "title": "", "text": "$x^2(z-y)+y^2(x-z)+z^2(y-x)=x^2(z-y)+x(y^2-z^2)+z^2y-y^2z=(z-y)(x^2-x(y+z)+zy)=(z-y)(x-y)(x-z)$"}
{"_id": "7932500", "title": "", "text": "$\\lim \\inf \\mu(A_{n}) >0$"}
{"_id": "6888162", "title": "", "text": "$\\alpha < \\alpha^+$"}
{"_id": "4190544", "title": "", "text": "$\\{(x,y,z,w,m)\\in \\mathbb{R}^5 :x=y,z=w=m\\}=\\{(x,x,z,z,z): x,z\\in \\mathbb{R}\\}$"}
{"_id": "3364827", "title": "", "text": "$     \\begin{vmatrix}     1 & x & x^2 \\\\     1 & y & y^2 \\\\     1 & z & z^2 \\\\     \\end{vmatrix}             +            \\begin{vmatrix}     a & b & c \\\\     d & e & f \\\\     g & h & i \\\\     \\end{vmatrix}. $"}
{"_id": "6109559", "title": "", "text": "$(a, a+k, a+2k...)$"}
{"_id": "2443946", "title": "", "text": "$M_{n}= k(b-a)^{n-1}$"}
{"_id": "6559024", "title": "", "text": "$\\log_a(-b)$"}
{"_id": "87160", "title": "", "text": "$1+2+3+4+...=-\\frac1{12}$"}
{"_id": "8450986", "title": "", "text": "$f(x,y)=\\frac{\\langle x,y \\rangle}{\\|x\\|\\|y\\|}.$"}
{"_id": "791324", "title": "", "text": "$S_{2k}$=$\\ \\lim_{k\\to \\infty}$"}
{"_id": "3822686", "title": "", "text": "$P(F)=1/6$"}
{"_id": "4027565", "title": "", "text": "$bx-ay =d$"}
{"_id": "1384621", "title": "", "text": "$C_R \\cup C_R^+ \\cup C_R^-$"}
{"_id": "2862656", "title": "", "text": "$2\\pi \\mathrm i\\;\\delta'(\\cos(\\varphi))$"}
{"_id": "8367192", "title": "", "text": "$ \\begin{cases} x=2-t\\\\ y=t\\\\ z=3 \\end{cases} $"}
{"_id": "96739", "title": "", "text": "$p|a_1a_2$"}
{"_id": "7155211", "title": "", "text": "$f(a)=f(b)=k$"}
{"_id": "1425465", "title": "", "text": "$P(x) = \\frac{2a}\\pi \\arcsin\\big(\\sin\\big(\\frac{2 \\pi}p x\\big) \\big)$"}
{"_id": "5567082", "title": "", "text": "$\\frac{\\sin \\theta}{4\\theta+\\tan\\theta} = \\frac{\\sin \\theta}{4\\theta+\\frac{\\sin \\theta}{\\cos\\theta}}$"}
{"_id": "3977014", "title": "", "text": "$x_2,x_3 \\in\\mathbb N$"}
{"_id": "1861168", "title": "", "text": "$f(x)=\\frac {2^x}{x^2-x}$"}
{"_id": "1193384", "title": "", "text": "$f(x)=1/|x|^2$"}
{"_id": "3929624", "title": "", "text": "$ f(z^2) = 2 f(z) $"}
{"_id": "3294724", "title": "", "text": "$2*(4^x+2^x)=3^x-6^x+9$"}
{"_id": "7724952", "title": "", "text": "$ \\gcd(a,2b)=\\gcd(a,2)\\gcd(a,b)=\\gcd(a,2). $"}
{"_id": "1913262", "title": "", "text": "$P(E)=1/4$"}
{"_id": "2040379", "title": "", "text": "$\\forall \\epsilon>0, \\exists \\delta>0 / |x-a|<\\epsilon\\implies |f(x)-L|<\\delta$"}
{"_id": "8979049", "title": "", "text": "$ \\quad \\quad \\quad \\quad \\implies s \\equiv ar \\equiv 1 \\pmod p \\implies s^p \\equiv (ar)^p \\equiv 1 \\pmod {p^2}$"}
{"_id": "1964555", "title": "", "text": "$(A_1\\ldots A_n)$"}
{"_id": "3684179", "title": "", "text": "$\\ Cov(Y,Y)+Cov(Y,e^X)$"}
{"_id": "8940293", "title": "", "text": "$\\lim_{x\\to 0}{\\left[ 100\\frac{\\sin^{-1}(x)}{x}\\right]} = 100$"}
{"_id": "8937806", "title": "", "text": "$ \\begin{align} &\\int_0^\\infty\\frac{\\sin^n(x)}{x}\\,\\mathrm{d}x\\\\ &=\\int_0^{2\\pi}\\frac{\\sin^n(x)}{x}\\,\\mathrm{d}x+\\sum_{k=1}^\\infty\\int_0^\\pi\\sin^n(x)\\left(\\frac1{x+2k\\pi}-\\frac1{x+(2k+1)\\pi}\\right)\\,\\mathrm{d}x\\\\ &=\\int_0^{2\\pi}\\frac{\\sin^n(x)}{x}\\,\\mathrm{d}x+\\sum_{k=1}^\\infty\\int_0^\\pi\\sin^n(x)\\frac\\pi{(x+2k\\pi)(x+(2k+1)\\pi)}\\,\\mathrm{d}x\\\\ &\\le\\frac2{\\sqrt{n}}+\\frac1{4\\pi}\\sum_{k=1}^\\infty\\frac1{k^2}\\int_0^\\pi\\sin^n(x)\\,\\mathrm{d}x\\\\ &=\\frac2{\\sqrt{n}}+\\frac\\pi{24}\\int_0^\\pi\\sin^n(x)\\,\\mathrm{d}x\\\\ &\\le\\frac2{\\sqrt{n}}+\\frac{\\pi^2}{24\\sqrt{n}} \\end{align} $"}
{"_id": "1445959", "title": "", "text": "$\\int_0^{\\pi} \\sin(2t) \\phi(t) dt = k \\int_0^{\\pi} \\sin(2t) \\sin(t)dt = 0$"}
{"_id": "9045165", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}\\int_{I} \\chi_{i}=\\lim_{n\\rightarrow\\infty}\\mu(T_{n}).$"}
{"_id": "3108234", "title": "", "text": "$(x/y)^{x-y}=(x^{x-y}/y^{x-y})=(x^x/x^y)*(y^y/y^x)=(x^x*y^y)/(x^y*y^x/)$"}
{"_id": "1110107", "title": "", "text": "$d(x,s)\\leq d(x,y)+d(y,s)$"}
{"_id": "6347235", "title": "", "text": "$i : N \\to \\mathbb{R}^{n+1} \\times \\mathbb{R}^{n+1}$"}
{"_id": "3079707", "title": "", "text": "$y,y',y'',y''',y''''$"}
{"_id": "6409802", "title": "", "text": "$a+b^2-c=0 \\implies  3^x+4^x=5^x$"}
{"_id": "9054453", "title": "", "text": "$\\left[\\begin{array}{ccc|c}2&1&1&5\\\\1&-1&2&1\\\\1&2&-1&4\\end{array}\\right]$"}
{"_id": "1450278", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{\\sqrt{2^{n+1}+1}-\\sqrt{2^{n+1}}}{\\sqrt{2^n+1}-\\sqrt{2^n}}=\\lim_{n \\to \\infty} \\frac{1}{(\\sqrt{2^{n+1}+1}+\\sqrt{2^{n+1}})(\\sqrt{2^n+1}-\\sqrt{2^n})}<1$"}
{"_id": "113253", "title": "", "text": "$\\phi^+$"}
{"_id": "9103242", "title": "", "text": "$ A_n = \\frac{n^2(n+1)^2}4 $"}
{"_id": "362756", "title": "", "text": "$f(x)=\\int_a^xf(t)dt$"}
{"_id": "7292169", "title": "", "text": "$g(x)=x^2-x+2$"}
{"_id": "8389791", "title": "", "text": "$=\\dfrac1{(\\sqrt1+1)(\\sqrt{1+\\sqrt1}+\\sqrt2)}$"}
{"_id": "5346518", "title": "", "text": "$X_0=\\Bbb E(X|\\cal F_0)$"}
{"_id": "6228439", "title": "", "text": "$RS = \\{rs | r \\in R, s \\in S\\}$"}
{"_id": "3756473", "title": "", "text": "$\\begin{align}\\sum_{n=1}^{\\infty} \\frac{1}{n^2+a^2} &= \\frac{1}{n^2} \\frac{1}{1+ \\frac{a^2}{n^2}}\\\\ &= \\sum_{n=1}^{\\infty} \\frac{1}{n^2} \\sum_{k=0}^{\\infty} (-1)^k \\left (\\frac{a^2}{n^2}\\right )^{k} \\\\ &=\\sum_{k=0}^{\\infty} (-1)^k a^{2 k}\\sum_{n=1}^{\\infty} \\frac{1}{n^{2 k+2}} \\\\ &=\\sum_{k=0}^{\\infty} (-1)^k a^{2 k} \\zeta(2 k+2)\\\\ &= \\frac{1}{2 a^2}\\sum_{k=1}^{\\infty} \\frac{B_{2 k} (2 \\pi a)^{2 k}}{(2 k)!} \\\\ &= \\frac{1}{2 a^2} ( \\pi a \\coth{\\pi a} - 1)\\\\  \\end{align}$"}
{"_id": "586609", "title": "", "text": "$\\gcd(a+b,ab)=1$"}
{"_id": "4680197", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}\\sum\\limits^n_{i=0}F_{2i}-\\phi F_{2i-1}$"}
{"_id": "5170398", "title": "", "text": "$\\dots=[-2]_{\\sim}=[-1]_{\\sim}=[0]_{\\sim}=[1]_{\\sim}=[2]_{\\sim}=\\,\\dots$"}
{"_id": "5726946", "title": "", "text": "$\\mathbb R^{n+1} - S\\times \\mathbb{R}^1$"}
{"_id": "8224608", "title": "", "text": "$I=\\int \\frac{\\sin ^{-1}(x)}{x}\\,dx$"}
{"_id": "4314303", "title": "", "text": "$ d(x,z) \\le d(x,y) + d(y,z) \\quad \\Longrightarrow \\quad f(x) \\le d(x,y) + f(y) $"}
{"_id": "1777282", "title": "", "text": "$f(z^{2})=f(z)^{2}$"}
{"_id": "6390441", "title": "", "text": "$F(x)= \\int_a^x f(t)\\, dt,\\,\\,  x\\in [a,b].$"}
{"_id": "4610641", "title": "", "text": "$\\{\\omega\\in\\Omega| \\lim_{n\\rightarrow\\infty} X_n(\\omega)$"}
{"_id": "5051575", "title": "", "text": "$\\sqrt{1!\\sqrt{2!\\sqrt{3!\\sqrt{...}}}}$"}
{"_id": "637827", "title": "", "text": "$\\omega(t_n)=\\omega^*(t_n)$"}
{"_id": "5245548", "title": "", "text": "$|AB|=0$"}
{"_id": "6146515", "title": "", "text": "$9^x + 6^x = 2× 4^x$"}
{"_id": "1512076", "title": "", "text": "$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{\\cos(nx)\\cos(my)}{n^2+m^2}$"}
{"_id": "5934252", "title": "", "text": "$\\int  \\frac { \\tan { x }  }{ \\left( \\sin { x }  \\right) ^{ 2 }+2\\left( \\cos { x }  \\right) ^{ 2 } } dx=\\int  \\frac { \\tan { x }  }{ \\cos ^{ 2 }{ x } \\left( 2+\\tan ^{ 2 }{ x }  \\right)  } dx=\\int  \\frac { \\tan { x }  }{ \\left( 2+\\tan ^{ 2 }{ x }  \\right)  } d\\tan { x } \\\\ $"}
{"_id": "9240443", "title": "", "text": "$f_X(x)=\\frac{1}{\\theta}1_{[-\\theta, \\theta]}(x)$"}
{"_id": "4771615", "title": "", "text": "$A(V) \\subseteq V$"}
{"_id": "3425517", "title": "", "text": "$\\displaystyle \\sum\\limits_{n=0}^{\\infty} (-1)^{n}e^{-\\alpha n^2} = \\frac{1}{2} + \\sqrt{\\frac{\\pi}{\\alpha}}\\sum\\limits_{n=0}^{\\infty} e^{-\\frac{(2n+1)^2\\pi^2}{4\\alpha}}$"}
{"_id": "3748025", "title": "", "text": "$= \\frac{(-1)^{n+1}n(n+1)}{2} + \\frac{2((-1)^{(n+2)}(n+1)^2)}2 =(-1)^{(n+2)}\\left(\\frac{-n(n+1)}{2} +\\frac{2(n+1)^2)}2\\right) =(-1)^{(n+2)}\\left(\\frac{(n+1)(n+2)}{2}\\right)$"}
{"_id": "9156836", "title": "", "text": "$ \\left(\\frac{1}{2}\\right) > \\left(\\frac{1}{3}\\right)^{3/2} $"}
{"_id": "4708093", "title": "", "text": "$\\gamma,\\gamma'<\\beta$"}
{"_id": "2223019", "title": "", "text": "$ \\Sigma = diag(\\sigma_1, \\sigma_2, \\sigma_3,....,\\sigma_i)$"}
{"_id": "7465256", "title": "", "text": "$\\{a,a+1,a+2,a+3\\}$"}
{"_id": "5074663", "title": "", "text": "$\\int_0^\\pi f(x) \\sin x dx = \\int_0^\\pi f(x) \\cos x dx =0.$"}
{"_id": "6927164", "title": "", "text": "$ \\sum_{d|N} \\frac1{d^2}\\leq\\sum_{n=1}^N\\frac1{n^2}\\leq 2 $"}
{"_id": "908523", "title": "", "text": "$P(S\\cap E) = P(S| E)P(E)$"}
{"_id": "281328", "title": "", "text": "$ U_n=\\frac{(1+\\rho)^n-(1+\\rho)^N}{1-(1+\\rho)^N}$"}
{"_id": "2645401", "title": "", "text": "$ds=\\sqrt{1+\\bigg(\\frac{dx}{dy}\\bigg)^2}dy.$"}
{"_id": "3201206", "title": "", "text": "$\\mathcal F=\\{\\{\\{1\\}\\}\\}$"}
{"_id": "8637856", "title": "", "text": "$H\\ge \\mathbb Z+(a/2^\\gamma)\\mathbb Z=\\frac{(2^\\gamma+a)\\mathbb Z}{2^\\gamma}=\\frac{gcd(2^\\gamma,a)\\mathbb Z}{2^\\gamma}$"}
{"_id": "3218475", "title": "", "text": "$g(x)=\\sum_{J\\in{P}}c_J\\chi_J(x).$"}
{"_id": "1514212", "title": "", "text": "$|ab| = |ba|$"}
{"_id": "234019", "title": "", "text": "$\\int_0^\\infty f(x)\\,dx = \\infty.$"}
{"_id": "8787485", "title": "", "text": "$d(x,z)\\le d(x,y)+d(y,z)<2/m_x=1/n_x,$"}
{"_id": "5775793", "title": "", "text": "$\\ds{{\\rm g}\\pars{t}      =\\int_{\\gamma - \\infty\\ic}^{\\gamma + \\infty\\ic}       {\\Gamma\\pars{a + 1} \\over s^{a + 1}}\\,\\expo{st}\\,{\\dd s \\over 2\\pi\\ic}\\,,       \\qquad\\gamma > 0}$"}
{"_id": "7539512", "title": "", "text": "$\\rm\\ \\lfloor \\lfloor a/b \\rfloor / c\\rfloor =  \\lfloor a/(bc)\\rfloor. $"}
{"_id": "1889870", "title": "", "text": "$\\left(1+\\frac xn\\right)^n \\geq 2^{x}$"}
{"_id": "2047024", "title": "", "text": "$a^2-db^2=1$"}
{"_id": "7877515", "title": "", "text": "$\\langle \\alpha ^-,\\alpha^+\\rangle$"}
{"_id": "734504", "title": "", "text": "$|\\triangle ABC| = rs$"}
{"_id": "974513", "title": "", "text": "$d(x,A)=\\inf _{y\\in A}d(x,y)$"}
{"_id": "5566256", "title": "", "text": "$E(X\\mid\\mathcal F_0)$"}
{"_id": "4398852", "title": "", "text": "$\\ f(x) = x^2 + x - 2\\ $"}
{"_id": "6102947", "title": "", "text": "$S/\\equiv$"}
{"_id": "2666936", "title": "", "text": "$t\\gamma - \\lfloor t\\gamma \\rfloor$"}
{"_id": "9246160", "title": "", "text": "$\\lim_{N \\to \\infty}\\sum\\limits_{n=1}^{N} \\frac{n^2}{2^n}$"}
{"_id": "1297489", "title": "", "text": "$p_1 p_2 + 1$"}
{"_id": "4175953", "title": "", "text": "$                      p_j(M)=p_j(M)^{2}. $"}
{"_id": "3647755", "title": "", "text": "$\\begin{pmatrix} \\det{(A)} & \\det{(B)} \\\\ \\det{(C)} & \\det{(D)} \\end{pmatrix} $"}
{"_id": "1307321", "title": "", "text": "$\\Bbb P[X_{n+1}=j\\,|\\,\\mathcal F_n] = p(n,i,j)$"}
{"_id": "5887992", "title": "", "text": "$w={\\normalsize e}^{\\Large{\\frac{2\\pi i}{N}}}$"}
{"_id": "1904435", "title": "", "text": "$(\\frac{1}{3})^k * (\\frac{2}{3})^{n-k}$"}
{"_id": "3201886", "title": "", "text": "$\\lim_{n\\to\\infty} \\frac{\\sqrt{n}-1}{\\sqrt{n}+1}$"}
{"_id": "1397784", "title": "", "text": "$f(x)=2+x-x^2$"}
{"_id": "9197989", "title": "", "text": "${\\bf \\hat{x}} = (\\hat{x}, \\hat{y})$"}
{"_id": "5718684", "title": "", "text": "$\\lim_{n\\to\\infty}f_n(0)=\\lim_{n\\to\\infty} n^{1/p}= \\infty.$"}
{"_id": "8384127", "title": "", "text": "$ \\mathcal F(x) = \\int_a^x f(t)dt $"}
{"_id": "6433411", "title": "", "text": "$k^{\\log_k(a)}=a$"}
{"_id": "2467953", "title": "", "text": "$100!+2,100!+3,\\dots$"}
{"_id": "2092271", "title": "", "text": "$d_1(x,z)\\leqslant d_1(x,y)+d_1(y,z)$"}
{"_id": "3016914", "title": "", "text": "$\\sum_{m=-\\infty}^\\infty e^{-a \\pi (x-m)^2} = f(x) = g(x) = \\sum_{n=-\\infty}^\\infty a^{-1/2}e^{-\\pi n^2 /a} e^{2 i \\pi n x}$"}
{"_id": "8575647", "title": "", "text": "$s(x)=\\sum_{i=1}^{n}c_iK_{E_i}(x)$"}
{"_id": "4242331", "title": "", "text": "$F(x)=\\int_a^xf(t),dt$"}
{"_id": "8532301", "title": "", "text": "$\\sup_{n\\geq n_0}|E|X_n|^{\\alpha}-E|X|^{\\alpha}|\\leq 3\\varepsilon$"}
{"_id": "1475556", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{1}{2^{n-1}}$"}
{"_id": "779016", "title": "", "text": "$f\\left( f(x)^2+f(y) \\right)=xf(x)+y$"}
{"_id": "6769557", "title": "", "text": "$S=\\{s_1,s_2,s_3,...,s_k\\}$"}
{"_id": "885721", "title": "", "text": "$f(n)=(n+1)$"}
{"_id": "6723894", "title": "", "text": "$\\sum_{t=1}^T (1+r)^{T-t+1} = (1+r)^3 + (1+r)^2 + (1+r)^1.$"}
{"_id": "684800", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{n^{2}}=\\sum_{n ~\\text{odd}}\\frac{1}{n^{2}} - \\sum_{n ~\\text{even}}\\frac{1}{n^{2}}.$"}
{"_id": "4796350", "title": "", "text": "$\\det(\\psi_\\gamma(H))=\\det(\\psi_\\beta(H))\\det(Q)^2.$"}
{"_id": "54843", "title": "", "text": "$P[X_{0}=1|X_{3}=0]*P[X_{3}=0] $"}
{"_id": "7012738", "title": "", "text": "$(1+i)=\\sqrt{2}e^{\\frac{i\\pi}{4}}$"}
{"_id": "8502699", "title": "", "text": "$\\ln (n+\\gamma) + \\gamma\n = \\ln \\ n + ln(1+\\gamma/n)+\\gamma\n = \\ln \\ n + \\gamma + \\gamma/n + O(1/n^2)\n $"}
{"_id": "7991523", "title": "", "text": "$  \\|A\\|  = \\sqrt{\\lambda_{\\mathrm{max}}(A^T A)}, $"}
{"_id": "4854299", "title": "", "text": "$\\frac{\\pi^2}6=\\sum_{n=1}^\\infty\\frac1{n^2}=\\sum_{n=1}^\\infty\\frac1{(2n)^2}+\\sum_{n=1}^\\infty\\frac1{(2n-1)^2}=\\frac14\\sum_{n=1}^\\infty\\frac1{n^2}+\\sum_{n=1}^\\infty\\frac1{(2n-1)^2}\\implies$"}
{"_id": "7241073", "title": "", "text": "$E=\\{e_1, e_2=ie_1, e_3=ie_1, e_4=i e_2 \\}$"}
{"_id": "1795822", "title": "", "text": "$|AB| = a$"}
{"_id": "1298942", "title": "", "text": "$p_X(x) = \\dfrac {2^{x}}{3^{x+1}}$"}
{"_id": "6022789", "title": "", "text": "$\\sum_{k=1}^N\\frac{1}{2^k} = 1 - \\frac{1}{2^N}$"}
{"_id": "663636", "title": "", "text": "$ \\left( \\frac{1}{2} \\right)^{1/2e} $"}
{"_id": "4245947", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\frac1{\\left (n+\\frac{3}{2} \\right )^2-\\frac{5}{4}} = \\frac12 \\sum_{n=-\\infty}^{\\infty} \\frac1{\\left (n+\\frac{3}{2} \\right )^2-\\frac{5}{4}} - 1 +1$"}
{"_id": "4023283", "title": "", "text": "$\\epsilon \\Delta x = \\Delta y - f'(a)\\Delta x$"}
{"_id": "758602", "title": "", "text": "$f(x)=\\int_a^xg(t)dt$"}
{"_id": "8449139", "title": "", "text": "$\\lim_{x \\rightarrow k} f(x) = \\lim_{x \\rightarrow k^+} f(x) = \\lim_{x \\rightarrow k^-} f(x)$"}
{"_id": "1119194", "title": "", "text": "$\\int_{-\\infty}^{\\infty} f(t)u(t)dt = \\int_{-\\infty}^{0} f(t)u(t)dt+\\int_{0}^{\\infty} f(t)u(t)dt= \\int_{0}^{\\infty} f(t)dt $"}
{"_id": "6701872", "title": "", "text": "$\\left(\\frac12\\right)^{n-2}.$"}
{"_id": "1117476", "title": "", "text": "$\\lim_{N\\to \\infty}\\sum_{n=1}^{N}\\frac{1}{n^2} \\to 0$"}
{"_id": "391472", "title": "", "text": "$\\theta(x)\\ :=\\ \\sum_{n=-\\infty}^\\infty e^{-n^2\\pi x}\\ .$"}
{"_id": "5321270", "title": "", "text": "$|y| \\le e^{-1/|x|}$"}
{"_id": "2081778", "title": "", "text": "$\\sec\\theta= \\frac x2$"}
{"_id": "5368194", "title": "", "text": "$\\frac{(n-1)^2}2<\\frac{n(n-1)}2<\\frac{n^2}2<\\frac{n(n+1)}2<\\frac{(n+1)^2}2$"}
{"_id": "4581182", "title": "", "text": "$E|X| \\le \\mathbb{E} |X|^3  $"}
{"_id": "4013622", "title": "", "text": "$e_i \\mapsto \\lambda_i e_i, 1 \\le i \\le n.$"}
{"_id": "8654986", "title": "", "text": "$\\int_{-\\pi}^{\\pi}v(x)\\cos nx\\mathbb dx=\\int_{-\\pi}^{\\pi}v(x)\\sin nx\\mathbb dx=0$"}
{"_id": "2838465", "title": "", "text": "$U,V\\subseteq A$"}
{"_id": "8230522", "title": "", "text": "$s\\cdot s \\equiv 1 \\Rightarrow s'\\cdot s + s\\cdot s'\\equiv 0 \\Rightarrow s\\cdot s'\\equiv 0.$"}
{"_id": "7004541", "title": "", "text": "$\\frac{a+b x}{x^5+x+1}$"}
{"_id": "7915283", "title": "", "text": "$x+\\left(\\frac{dy}{dx}\\right)= \\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2 }$"}
{"_id": "5692231", "title": "", "text": "$P(i) \\implies P(i+1).$"}
{"_id": "1106463", "title": "", "text": "$x_n ={n(n+1) \\above 1.5pt 2}$"}
{"_id": "3951230", "title": "", "text": "$\\Delta\\subseteq V\\subseteq U$"}
{"_id": "818649", "title": "", "text": "$\\sum_{n=1}^\\infty e^{-\\pi n^2x}\\le e^{-\\pi x}\\sum_{n=0}^\\infty e^{-\\pi nx} $"}
{"_id": "3348259", "title": "", "text": "$\\vdash (\\forall x,\\forall y :  (xRy \\lor yRx)) \\Rightarrow (\\forall x : xRx)$"}
{"_id": "4060881", "title": "", "text": "$\\forall n>0 : C (n) \\implies C (n+1)$"}
{"_id": "1661804", "title": "", "text": "$\\int _{0}^{1}{\\tan^{-1}\\left(x\\right) \\over {1+x}}dx$"}
{"_id": "2664893", "title": "", "text": "$ \\bbox[lightyellow] {   P_{\\,M}(r,p,n) = \\sum\\limits_{0\\, \\le \\,\\,s\\,\\, \\le \\,n} {p^{\\,s} \\left( {1 - p} \\right)^{\\,n - s} \\sum\\limits_{\\left( {0\\, \\le } \\right)\\,\\,k\\,\\,\\left( { \\le \\,{s \\over r}\\, \\le \\,n - s + 1} \\right)} {\\left( { - 1} \\right)^k \\left( \\matrix{   n - s + 1 \\cr    k \\cr}  \\right)\\left( \\matrix{   n - k\\left( {r + 1} \\right) \\cr    s - k\\left( {r + 1} \\right) \\cr}  \\right)} }  }$"}
{"_id": "5400206", "title": "", "text": "$\\sum_{n\\ge 1}\\frac1n=\\sum_{n\\ge 1}\\frac1{2n}+\\sum_{n\\ge 1}\\frac1{2n+1}\\;.$"}
{"_id": "4642200", "title": "", "text": "$J(\\gamma) = (\\gamma - 1, \\gamma)$"}
{"_id": "813288", "title": "", "text": "$1 + 2 + 3 + 4 + 5 + \\cdots = -{\\frac{1}{12}}$"}
{"_id": "8548932", "title": "", "text": "$\\left\\{ \\begin{array}{c}n+1\\\\k-1\\end{array} \\right\\}\\left\\{ \\begin{array}{c}n+1\\\\k+1\\end{array} \\right\\}<\\left\\{ \\begin{array}{c}n+1\\\\k\\end{array} \\right\\}^2$"}
{"_id": "5602196", "title": "", "text": "$\\#B^{\\#A}=4^6$"}
{"_id": "1649717", "title": "", "text": "$C=\\{\\langle x,y\\rangle\\in X\\times y:xy\\ge 1\\}\\;;$"}
{"_id": "4680527", "title": "", "text": "$log_{b^{n}} x = \\frac {1}{n} log_{b} x$"}
{"_id": "8623529", "title": "", "text": "$S \\subseteq \\cup_{\\gamma \\in \\Gamma}\\ U_\\gamma$"}
{"_id": "7049504", "title": "", "text": "$E_2^{2,0}=E_2^{0,2}\\rightarrow E_3^{0,2} \\rightarrow \\ldots\\rightarrow E_{\\infty}^{0,2}$"}
{"_id": "5074461", "title": "", "text": "$aX_0 + bY_0 = d$"}
{"_id": "7684513", "title": "", "text": "$n!=\\sqrt{2\\pi n}(n/e)^n\\left(1+O\\left(\\frac{1}{n}\\right)\\right)$"}
{"_id": "230734", "title": "", "text": "$           \\int_a^b f(t)dt  = \\int_a^b t f(t)dt = 0 $"}
{"_id": "8909805", "title": "", "text": "$(\\frac{\\gamma u}{(1+\\gamma)c})^2 = \\frac{\\gamma^2-1}{(\\gamma+1)^2} = \\frac{\\gamma-1}{\\gamma+1}$"}
{"_id": "1158877", "title": "", "text": "$ \\left\\{ \\matrix{   P_{\\,0} (x) = x,\\quad P_{\\,1} (x) = x^{\\,2}  \\hfill \\cr    P_{\\,n + 2} (x) = x^{\\,2} P\\,'_{\\,n + 1} (x) + \\left( {n + 1} \\right)x\\,P_{\\,n + 1} (x) - \\left( {n + 1} \\right)P\\,'_{\\,n} (x) \\hfill \\cr}  \\right. $"}
{"_id": "1396316", "title": "", "text": "$\\mathrm{d}S=\\sqrt{1+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}x}\\right)^2}\\mathrm{d}x$"}
{"_id": "6082565", "title": "", "text": "$\\displaystyle f(1-k)=\\frac{9^{1-k}}{9^{1-k}+3}=\\frac9{9+3\\cdot9^k}=\\frac3{9^k+3}$"}
{"_id": "644237", "title": "", "text": "$|f(x)| \\le \\max_{z \\in A} |f(z)|$"}
{"_id": "8416763", "title": "", "text": "$\\begin{pmatrix}a & -\\overline{b} \\\\ b & \\overline{a} \\end{pmatrix}.$"}
{"_id": "40356", "title": "", "text": "$f(a+b)\\le f(a)+f(b)$"}
{"_id": "2107710", "title": "", "text": "$P[X_{n+1} \\in S' | X_n = s] = 1$"}
{"_id": "4122003", "title": "", "text": "$\\quote(+A+\\quote\\to+B+\\quote)$"}
{"_id": "2398479", "title": "", "text": "$f(f(x)^2 y) = x^3 f(xy)$"}
{"_id": "794961", "title": "", "text": "$\\frac{I}{kq} = \\int_{0}^{1}\\frac{1}{(x^2+1)\\sqrt{x^2+2}}\\,dx$"}
{"_id": "7384961", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\sum_{m=1}^{\\infty}$"}
{"_id": "693075", "title": "", "text": "$A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\dots$"}
{"_id": "4271788", "title": "", "text": "$\\lim_{x\\to x_0^-}f'(x)=\\lim_{x\\to x_0^+}f'(x)=l$"}
{"_id": "3298692", "title": "", "text": "$\\mathbb{R}^n=\\mathbb{R}\\times\\mathbb{R}^{n-1}$"}
{"_id": "5351549", "title": "", "text": "$x=\\frac{1}{4}-\\frac{y}{4}$"}
{"_id": "607112", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\mu (A_n) = \\mu (\\bigcup A_n)$"}
{"_id": "2074885", "title": "", "text": "$(a+b, a-b) \\ge (a, b)$"}
{"_id": "400841", "title": "", "text": "$\\lim_ {n\\to \\infty}\\ \\frac{1}{2^n}$"}
{"_id": "7562616", "title": "", "text": "$\\sum_{\\gamma<\\alpha+1} a_\\gamma=\\sum_{\\gamma<\\alpha} a_\\gamma + a_\\alpha \\le \\prod_{\\gamma<\\alpha} a_\\gamma + a_\\alpha \\overset{(1)}\\le \\prod_{\\gamma<\\alpha+1} a_{\\gamma}.$"}
{"_id": "7635855", "title": "", "text": "$P(X_t=3|X_0=1)=1-e^{-\\mu t}$"}
{"_id": "3117500", "title": "", "text": "$F_3(n)$"}
{"_id": "6133888", "title": "", "text": "$\\displaystyle \\lim_{x \\to c^-} f(x) < \\lim_{x \\to c^+} f(x)$"}
{"_id": "2364979", "title": "", "text": "$ \\begin{align} \\int_0^{\\pi/2}\\frac{x\\log(\\sin(x))}{\\sin(x)}\\,\\mathrm{d}x &=\\int_0^1\\frac{\\sin^{-1}(u)}{\\sqrt{1-u^2}}\\frac{\\log(u)}u\\,\\mathrm{d}u\\\\ &=\\int_0^1\\sum_{k=0}^\\infty\\frac1{(2k+1)}\\frac{4^k}{\\binom{2k}{k}}u^{2k}\\log(u)\\,\\mathrm{d}u\\\\ &=-\\sum_{k=0}^\\infty\\frac1{(2k+1)^3}\\frac{4^k}{\\binom{2k}{k}}\\tag{3} \\end{align} $"}
{"_id": "7770651", "title": "", "text": "$z=( \\alpha_{1},..., \\alpha_{n},  \\beta_{1},....,\\beta_{n})$"}
{"_id": "2539242", "title": "", "text": "$(f+g)(x+ab)=f(x+ab)+g(x+ab)=f(x)+g(x)=(f+g)(x)$"}
{"_id": "369203", "title": "", "text": "$\\dfrac{p}{1-(\\varepsilon\\cos\\,\\varphi)^2}$"}
{"_id": "6817061", "title": "", "text": "$  \\alpha_2^{-} = \\frac{ (1-f)^3 }{ 1 + ( 1 - f)^2 } $"}
{"_id": "5897609", "title": "", "text": "$ax+by=t$"}
{"_id": "1008142", "title": "", "text": "$\\begin{pmatrix}     4    & -1\\\\     2    & 1  \\\\   \\end{pmatrix}$"}
{"_id": "8940090", "title": "", "text": "$A_1....A_M$"}
{"_id": "1353763", "title": "", "text": "$\\sum_{n=m+1}^{\\infty}\\left\\|e_{n}-f_{n}\\right\\|^{2}<1. \\tag{III}$"}
{"_id": "8342604", "title": "", "text": "$\\lim\\limits_{n\\to\\infty}\\sqrt[3]{n+\\sqrt{n}}-\\sqrt[3]{n}$"}
{"_id": "4681236", "title": "", "text": "$\\det\\begin{pmatrix} A & B\\\\ B & A \\end{pmatrix}=\\det(A-B)\\det(A+B).$"}
{"_id": "7458104", "title": "", "text": "$F_\\nu (\\xi) = \\frac{1}{\\Gamma(\\nu)}\\int_0^\\infty{\\frac{x^{\\nu-1}}{e^x/\\xi -1}dx}$"}
{"_id": "4219394", "title": "", "text": "$U\\cap L=\\{\\langle x,y\\rangle\\in L:x_0\\le x<b\\}\\;,$"}
{"_id": "3268618", "title": "", "text": "$\\|x^i_n-w_i\\|\\le \\epsilon$"}
{"_id": "953286", "title": "", "text": "$p_1p_2...p_n +1$"}
{"_id": "1184551", "title": "", "text": "$\\alpha, \\omega \\alpha, \\omega^{2} \\alpha$"}
{"_id": "4361797", "title": "", "text": "$\\frac{|\\langle x,y\\rangle|}{||x||\\cdot||y||}$"}
{"_id": "421442", "title": "", "text": "$\\delta_x(x)=1$"}
{"_id": "8177068", "title": "", "text": "$\\mathbf x = (x,y),$"}
{"_id": "3643043", "title": "", "text": "$X_1 \\subseteq X_2 \\subseteq \\cdots$"}
{"_id": "3117831", "title": "", "text": "$f_{Z}(z)=f_{U}(u)=\\frac{1}{2}(1-e^{\\frac{-1}{a}})e^{\\frac{-1}{a}(u)}$"}
{"_id": "5525523", "title": "", "text": "$P[X_i \\geq 0] = 1$"}
{"_id": "3901117", "title": "", "text": "$ \\lim_{x \\to c^-} f(x) = \\lim_{x \\to c^+} f(x).$"}
{"_id": "4976581", "title": "", "text": "$\\left\\lbrace \\begin{array}{ll}  y=\\alpha(2x) \\\\  x=\\alpha(2y) \\\\  x^2+y^2-2=0 \\end{array} \\right.$"}
{"_id": "2238676", "title": "", "text": "$R = \\{\\begin{bmatrix}x \\\\y\\end{bmatrix}\\in \\mathbb R^2:a \\le x\\le b \\, \\, , c\\le y \\le d\\,  \\}$"}
{"_id": "213694", "title": "", "text": "$|I_j| = (b_j - a_j) = |\\bar{I_j}|$"}
{"_id": "8844484", "title": "", "text": "$\\hat{\\mu} = \\frac{x_{1} + x_{2} + ... + x_{n}}{n}$"}
{"_id": "7958335", "title": "", "text": "$ \\left[   \\begin{array}{ccc|c}     3 & 1 & 1 & 14\\\\     -1 & -2 & -3 & -9\\\\     5 & -1 & 5 & 30   \\end{array} \\right] $"}
{"_id": "1688440", "title": "", "text": "$sin(2πt/p(t))$"}
{"_id": "7266983", "title": "", "text": "$\\log_a(bc) = \\log_a(b) \\log_a(c)$"}
{"_id": "2182904", "title": "", "text": "$ ds = \\sqrt{1+\\bigg(\\frac{dy}{dx}\\bigg)^2}\\, dx. $"}
{"_id": "3654703", "title": "", "text": "$ d(x,y)= \\frac{||x-y||}{1+||x-y||}$"}
{"_id": "649606", "title": "", "text": "$P(3) \\implies P(4),$"}
{"_id": "1358084", "title": "", "text": "$ N_{\\,1} (n + 1) = N_{\\,1} (n) + N_{\\,1} (n - 1)\\quad \\left| \\matrix{   \\;N_{\\,1} (0) = 0 \\hfill \\cr    \\;N_{\\,1} (1) = 1\\; \\hfill \\cr}  \\right. $"}
{"_id": "5192884", "title": "", "text": "$m \\equiv 1 \\mod p$"}
{"_id": "7749955", "title": "", "text": "$z=\\sqrt[4]{2}e^{\\frac{7\\pi i}{6}}$"}
{"_id": "223820", "title": "", "text": "$\\Pi^{+}$"}
{"_id": "4997065", "title": "", "text": "$\\begin{align}\\mathsf E(XY^j\\mid XY^j<w) ~=~& \\dfrac{\\mathsf E (XY^j~\\mathbf 1_{XY^j<w}) }{ \\mathsf P(XY^j<w)} \\\\[1ex]~=~&\\dfrac{\\mathsf E (X~\\mathsf E(Y^j~\\mathbf 1_{Y<(w/X)^{1/j}}\\mid X)) }{ \\mathsf P(Y<(w/X)^{1/j})} &\\text{only if}~&j > 0 \\\\[1ex] ~=~& \\bbox[gainsboro]{\\dfrac{\\int\\limits_0^\\infty~x~\\mathsf E( Y^j~\\mathbf 1_{Y<(w/x)^{1/j}}\\mid X=x)~f_X(x)~\\mathrm d x }{\\int\\limits_0^\\infty~\\mathsf P(Y<(w/x)^{1/j}\\mid X=x)~f_X(x)~\\mathrm d x }} \\\\[1ex] ~=~& \\dfrac{\\int\\limits_0^\\infty~x~f_X(x)\\int\\limits_0^{(w/x)^{1/j}} y^j~f_{Y\\mid X}(y\\mid x)~\\mathrm d~y~\\mathrm d x }{\\int\\limits_0^\\infty~f_X(x)\\int\\limits_0^{(w/x)^{1/j}} f_{Y\\mid X}(y\\mid x)~\\mathrm d~y~\\mathrm d x } \\end{align}$"}
{"_id": "4408021", "title": "", "text": "$ x'+y'=(x'+y)'=(y+x')'=(y+x)''=(x+y)''=(x+y')'=(y'+x)'=y'+x'; $"}
{"_id": "6023560", "title": "", "text": "$f(A+B)=f(A)+f(B).$"}
{"_id": "3332959", "title": "", "text": "$\\frac{x_1+x_2+...+x_n}{n}\\geq \\sqrt[n]{x_1x_2...x_n}$"}
{"_id": "7620065", "title": "", "text": "$f(r) = 1-r -\\dfrac{1-r^m}{(1+r)^m}=(1-r)\\left(1-\\dfrac{1}{(1+r)^m}-\\dfrac{r^2}{(1+r)^m}-\\cdots-\\dfrac{r^{m-1}}{(1+r)^m}\\right)>0 \\iff (1+r)^m > 1+r+r^2+\\cdots +r^{m-1}$"}
{"_id": "2122282", "title": "", "text": "$ = \\left[     \\begin{array}{ccc|c}       1&-2&3&2\\\\       0&3&-2&k-2\\\\       0&0&0&k^2-k-6     \\end{array} \\right] $"}
{"_id": "4960824", "title": "", "text": "$a_n=\\frac{n(n+1)}{2},$"}
{"_id": "409725", "title": "", "text": "$ \\pi_* ( F_* m ) = ( \\pi \\circ F)_* m = (f \\circ \\pi)_* m = f_* ( \\pi_* m) =f_* \\mu = \\mu $"}
{"_id": "3837186", "title": "", "text": "$S=\\{A_1,A_2,A_3,\\dots,A_n\\}$"}
{"_id": "7542083", "title": "", "text": "$a=\\frac{x_1+x_2+\\cdots+x_n}{n}.$"}
{"_id": "5736690", "title": "", "text": "$t^1=(t_x^1,t_y^1)$"}
{"_id": "541581", "title": "", "text": "$f_3'$"}
{"_id": "4781937", "title": "", "text": "$ \\sum x_n >0$"}
{"_id": "6492478", "title": "", "text": "$f;g: A_1\\to A_1$"}
{"_id": "3908752", "title": "", "text": "$\\displaystyle{\\frac{b \\gamma_2}{\\gamma_1}=\\lambda=\\frac{a \\gamma_1}{\\gamma_2}}$"}
{"_id": "9339919", "title": "", "text": "$\\frac{1-r^N}{1-r}=1+r+r^2+r^3+\\dots+r^{N-1}$"}
{"_id": "2229706", "title": "", "text": "$B = \\{ \\{ a\\}, a  \\}$"}
{"_id": "7672726", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\sum_{m=1}^{\\infty} \\frac{1}{m^4(m^2+n^2)}$"}
{"_id": "8118680", "title": "", "text": "$M_1=\\left(\\begin{array}{} A & B\\\\ B &A \\end{array}\\right)$"}
{"_id": "3518612", "title": "", "text": "$\\lVert T_x\\rVert=\\lVert x\\rVert$"}
{"_id": "8883832", "title": "", "text": "$y_n=\\frac{x_1+x_2+⋯+x_n}{n}\\to l$"}
{"_id": "3632347", "title": "", "text": "$I_n:=\\int_{0}^{\\pi}(\\sin x)^ndx$"}
{"_id": "7401451", "title": "", "text": "$d=\\{(x,y)\\in\\mathbb{R}^2\\ |\\ 0 \\le x \\le 3, 0 \\le y \\le 2\\}$"}
{"_id": "4885719", "title": "", "text": "$\\left\\vert a-b \\right\\vert = r.$"}
{"_id": "7608903", "title": "", "text": "$a'x+b'y=d$"}
{"_id": "4535075", "title": "", "text": "$\\det (A) = \\left[a+(n-1)b\\right](a-b)^{n-1}$"}
{"_id": "6503184", "title": "", "text": "$\\int_{-\\pi}^\\pi\\frac{\\sin(nx)}{2^n}dx=0,$"}
{"_id": "9157656", "title": "", "text": "$\\forall\\varepsilon\\gt0,\\quad \\exists\\delta\\gt0,\\quad \\underline{\\forall x\\in D},\\quad |x-x_0|\\lt\\delta\\implies|f(x)-f(x_0)|\\lt\\varepsilon.$"}
{"_id": "7435409", "title": "", "text": "$\\psi(a)=\\big\\{\\{a\\}\\big\\}$"}
{"_id": "6833990", "title": "", "text": "$\\left[\\begin{array}{rrr|r} 2 & 1 & 1 & 5\\\\ 1 & -1 & 1 & 3 \\\\ -2 & p & 2 & q\\end{array}\\right]$"}
{"_id": "1391347", "title": "", "text": "$(\\mathbb{F}_p[X]/(X^2))^*$"}
{"_id": "2078365", "title": "", "text": "$\\overline{\\mathbb{R}}\\times\\overline{\\mathbb{R}}$"}
{"_id": "3518912", "title": "", "text": "$ Y =   \\begin{pmatrix}   A & -B \\\\   B & A  \\\\   \\end{pmatrix}. $"}
{"_id": "7327839", "title": "", "text": "$\\mid f(z^2)\\mid\\geq \\mid f(z)\\mid$"}
{"_id": "2808324", "title": "", "text": "$||x^{k+1} - y^{k+1}|| \\leq \\int_{t=0}^{1}||F'(x^{k})^{-1}\\left( F'(x^{k}) - F'(y^{k} + t(x^{k} - y^{k})) \\right) (x^{k} - y^{k}) \\text{d}t|| + \\\\ \\quad \\quad \\quad ||F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\\Delta y^{k}||$"}
{"_id": "3969611", "title": "", "text": "$p'(m')=p'(m)$"}
{"_id": "4283565", "title": "", "text": "$ \\int_{-\\pi}^\\pi \\frac {\\sin(n\\theta)} {\\sin(\\theta)}\\,d\\theta. $"}
{"_id": "3218711", "title": "", "text": "$\\Bbb R^n \\wedge \\Bbb R^n$"}
{"_id": "4083323", "title": "", "text": "$A\\subset B\\subset \\bar B\\subset U$"}
{"_id": "4708083", "title": "", "text": "$F_{\\gamma,\\gamma+1}$"}
{"_id": "8236965", "title": "", "text": "$ \\sin(\\pi s) \\Gamma(s) = \\frac{i}{2} \\sum_{n=1}^{\\infty} \\frac{\\mu(n)}{n^s}  \\oint_C \\frac{(-x)^{s-1}}{e^x-1} dx . \\qquad (*)  $"}
{"_id": "655634", "title": "", "text": "$ N(n)=n^2-n+2 $"}
{"_id": "6307602", "title": "", "text": "$\\mathbb ET = |a|b$"}
{"_id": "718703", "title": "", "text": "$c_r(E) = e(E_{\\mathbb R})$"}
{"_id": "3740105", "title": "", "text": "$(y)^{y'}=(y')^{y}$"}
{"_id": "7093894", "title": "", "text": "$\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{a} + \\frac{1}{b}$"}
{"_id": "2143737", "title": "", "text": "$\\chi_{A_I}:= \\prod_{j=1}^n \\chi_{A_j}^{I_j}.$"}
{"_id": "9098446", "title": "", "text": "$N = \\lfloor \\frac{x-p}{\\delta}\\rfloor \\leq \\lfloor \\frac{b-a}{\\delta}\\rfloor$"}
{"_id": "8881686", "title": "", "text": "$=\\dfrac1{2x}\\int\\dfrac{2x}{(x^2+1)^n}dx-\\int\\left(\\dfrac{d(1/2x)}{dx}\\int\\dfrac{2x}{(x^2+1)^n}dx\\right)dx=?$"}
{"_id": "5641557", "title": "", "text": "$ \\delta A (x+\\delta x)+A\\delta x=0, $"}
{"_id": "6388309", "title": "", "text": "$f'(x) = \\frac{2}{3}e^{\\frac{x}{3}} + \\frac{1}{2}e^{-\\frac{x}{2}} > 0 $"}
{"_id": "2141991", "title": "", "text": "$\\|A\\|_2 = \\sqrt{\\lambda _{\\max}(A^TA)}$"}
{"_id": "8125185", "title": "", "text": "$r (f(x)+g(x))=r f(x) + r g(x)$"}
{"_id": "3783717", "title": "", "text": "$P[X_n\\geq 1/2]=1$"}
{"_id": "3500930", "title": "", "text": "$ \\mu\\left(\\bigcup_{n\\geq 1}A_n\\right)=\\lim_{n\\to\\infty}\\mu(A_n). $"}
{"_id": "8583974", "title": "", "text": "$Cov(X,Y)=-0.7$"}
{"_id": "603544", "title": "", "text": "$n,n+Q,n+2Q,\\ldots$"}
{"_id": "5261779", "title": "", "text": "$ F(x)=\\int_a^xf(t)dt-\\frac{a+x}{2}\\int_a^xf(t)dt. $"}
{"_id": "8565431", "title": "", "text": "$p_1, p_2 \\mid b$"}
{"_id": "9118335", "title": "", "text": "$c =|[a,b]| \\leq |\\mathbb{R}  - \\{r\\}| \\leq | \\mathbb{R}| = c$"}
{"_id": "5245635", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} \\frac{a_{1}+a_{2}+...+a_{n}}{n} = A$"}
{"_id": "2605859", "title": "", "text": "$\\left(\\frac{\\partial^2}{\\partial x^2} +2 \\frac{\\partial}{\\partial x}\\right) \\{i\\sin(x)+\\cos(x)\\} = (-1+2i)(i\\sin(x)+\\cos(x))$"}
{"_id": "21879", "title": "", "text": "$\\tan\\mu = \\frac{\\mu}{2}$"}
{"_id": "2126312", "title": "", "text": "$1 + \\frac{1}{2^n - 2} = \\frac{2^n - 1}{2^n - 2} = \\frac{1}{2} \\frac{2^n - 1}{2^{n-1} - 1}$"}
{"_id": "5877917", "title": "", "text": "$f_X(x) = \\theta e^{- \\theta x},\\ x >0$"}
{"_id": "1169823", "title": "", "text": "$A_1 \\subset A_2 \\subset \\dots \\subset A_k \\subset A_{k+1} \\subset ...$"}
{"_id": "5467105", "title": "", "text": "$ |f(y_1)|\\le\\frac{1}{2}|f(x_1)|$"}
{"_id": "488682", "title": "", "text": "$ J_n= \\int_0^\\frac{\\pi}{2} \\frac{\\sin(2nx)}{\\sin x}\\:\\mathrm{d}x,\\quad I_n = \\int_0^\\frac{\\pi}{2} \\frac{\\sin(2n+1)x}{\\sin x}\\:\\mathrm{d}x $"}
{"_id": "8245788", "title": "", "text": "$F = A \\frac{(1+i)^{2n}-(1+i)^n}{i(1+i)^n}$"}
{"_id": "795411", "title": "", "text": "$2*5/3=2*2=4;$"}
{"_id": "1513264", "title": "", "text": "$\\dfrac3a + \\dfrac4b = 1$"}
{"_id": "7985090", "title": "", "text": "$1/|t|^n$"}
{"_id": "2244117", "title": "", "text": "$\\frac{\\pi^2}{6} = \\sum_{n=1}^\\infty \\frac{1}{n^2} = \\sum_{n=1}^\\infty \\frac{1}{(2n)^2} +\\sum_{n=0}^\\infty \\frac{1}{(2n+1)^2} $"}
{"_id": "7369582", "title": "", "text": "$I_n = \\int_{-\\infty}^\\infty \\frac{|\\sin \\left( \\frac{x}{n} \\right) \\sin(x)|}{\\frac{x}{n} x} \\, dx.$"}
{"_id": "6309238", "title": "", "text": "$n!+n \\geq n!$"}
{"_id": "5306036", "title": "", "text": "$ \\langle a \\wedge b, c \\wedge d \\rangle := \\begin{vmatrix} \\langle a,c \\rangle & \\langle a,d \\rangle \\\\ \\langle b,c \\rangle & \\langle b,d \\rangle \\end{vmatrix} $"}
{"_id": "2893812", "title": "", "text": "$ 2\\pi \\int x ds = 2\\pi \\int x \\sqrt{dx^2 + dy^2} $"}
{"_id": "1598458", "title": "", "text": "$F(A+B)\\cong F(A)+F(B)$"}
{"_id": "3191533", "title": "", "text": "$ \\begin{cases} x=-2s\\\\ y=0\\\\ z=s\\\\ \\end{cases} $"}
{"_id": "2930540", "title": "", "text": "$ \\sqrt{1+  \\left(\\frac{dy }{ dx} \\right)^2}= \\frac{d^2 y }{dx^2} $"}
{"_id": "6587931", "title": "", "text": "$\\displaystyle S(n) = \\sum_{k = 1}^n k = \\dfrac{n(n+1)}{2}$"}
{"_id": "8918858", "title": "", "text": "$f=\\{\\{\\{a\\},\\{a,f\\}\\}\\}$"}
{"_id": "1930549", "title": "", "text": "$I_n=\\int \\frac{\\sin(2nx)}{\\sin x}dx$"}
{"_id": "5692405", "title": "", "text": "$\\frac{1}{x}+\\frac{1}{y}=-\\frac{1}{6}$"}
{"_id": "6212199", "title": "", "text": "$\\int\\frac{e^x \\sin(2x)}{\\cos^2(2x)}\\mathrm{d}x = \\int\\frac{e^x 2\\sin(x)\\cos(x)}{(cos^2(x)-sin^2(x))^2}\\mathrm{d}x=\\int\\frac{e^x 2\\sin(x)\\cos(x)}{(\\cos(x)-\\sin(x))^2(\\cos(x)-\\sin(x))^2}\\mathrm{d}x=\\int\\frac{e^x 2\\sin(x)\\cos(x)}{(1 +2\\cos(x)\\sin(x))(1-2\\cos(x)\\sin(x))}\\mathrm{d}x$"}
{"_id": "8435754", "title": "", "text": "$A_1 \\subseteq A_2 \\subseteq \\cdots \\subseteq A_m \\subseteq \\cdots$"}
{"_id": "8310102", "title": "", "text": "$\\int \\frac{\\gamma^{2 x} dx}{\\gamma^x + 1} = \\frac{1}{\\log \\gamma}(\\gamma^x - \\log(\\gamma^x + 1)) + C .$"}
{"_id": "6071099", "title": "", "text": "$n\\le f_j(x) < n+1$"}
{"_id": "1019139", "title": "", "text": "$\\lim_{n\\to\\infty}\\frac{1+(\\sqrt{n}+1)^{3}+2\\sqrt{n}}{n+\\sin(n)}$"}
{"_id": "9321939", "title": "", "text": "$x*(y*z) = x*(y+z+2) = x + (y+z+2) + 2 = (x+y+2) + z + 2 = (x*y)*z$"}
{"_id": "1606279", "title": "", "text": "$t_n=\\dfrac{s_1+s_2+\\ldots+s_n}{n}$"}
{"_id": "2486479", "title": "", "text": "$\\|A\\|_2=\\sqrt{\\max_n\\lambda_n(AA^t)} $"}
{"_id": "3066130", "title": "", "text": "$(\\mathbb{R}^n\\times\\mathbb{R}^n,\\mathcal{L}(\\mathbb{R}^n)\\times\\mathcal{L}(\\mathbb{R}^n))$"}
{"_id": "6740020", "title": "", "text": "$a_n= \\frac{n (n + 1) (2n + 1)} {6}$"}
{"_id": "16597", "title": "", "text": "$\\lim_{n \\rightarrow \\infty}m(A_n) =m(A)$"}
{"_id": "2888428", "title": "", "text": "$\\begin{pmatrix}A & -B \\\\ B & A \\end{pmatrix}$"}
{"_id": "817847", "title": "", "text": "$\\int \\frac{x}{({x^2+1})^6}dx$"}
{"_id": "6984895", "title": "", "text": "$k=2 \\rightarrow \\left[\\begin{array}{ccc}1&0&1\\end{array}\\right], k=3 \\rightarrow \\left[\\begin{array}{ccc}1&1&1\\end{array}\\right]$"}
{"_id": "4241138", "title": "", "text": "$Y_n = \\frac{X_1 + X_2 + \\ldots + X_n}{n} - n$"}
{"_id": "6692757", "title": "", "text": "$f(x;\\,\\theta)=\\frac{2x}{\\theta^2}\\qquad \\text{for }0<x<\\theta.$"}
{"_id": "5632864", "title": "", "text": "$ax+by+cy=d$"}
{"_id": "7648004", "title": "", "text": "$\\int_0^1 \\frac{\\sin^4(x)}{x^2}$"}
{"_id": "4979600", "title": "", "text": "$ (x,(y,z)) = (x,\\{\\{y\\},\\{y,z\\}\\})=\\{\\{x\\},\\{x,\\{\\{y\\},\\{y,z\\}\\}\\}\\}$"}
{"_id": "4079307", "title": "", "text": "$\\tag{2} x^5+1=(x+1)(x- e^{\\frac{\\pi i}{5}})(x- e^{\\frac{-\\pi i}{5}})(x- e^{\\frac{3\\pi i}{5}})(x- e^{\\frac{-3\\pi i}{5}}).$"}
{"_id": "2222467", "title": "", "text": "$P(E)=\\frac{1}{105}$"}
{"_id": "5200925", "title": "", "text": "$ x * y = y*(y*(x*y)) = y * ((x * (x*y))*(x*y)) = y *x. $"}
{"_id": "3586873", "title": "", "text": "$f(\\mathbf{x})=\\sum^N_{k=1} a_k\\chi_{E_k}(\\mathbf{x})$"}
{"_id": "5106144", "title": "", "text": "$d'(a) = b'(b'(b'(a))) = b'(b'(b)) = b'(c) = d\\\\ d'(b) = b'(b'(b'(b))) = b'(b'(c)) = b'(d) = a\\\\ d'(c) = b'(b'(b'(c))) = b'(b'(d)) = b'(a) = b\\\\ d'(d) = b'(b'(b'(d))) = b'(b'(a)) = b'(b) = c.$"}
{"_id": "8308512", "title": "", "text": "$S=(\\gamma e)^{\\frac{1}{\\gamma}}\\Gamma_L(1+\\frac{1}{\\gamma},\\frac{1}{\\gamma})$"}
{"_id": "3571838", "title": "", "text": "$\\left(\\begin{array}{rr} a& -b\\\\b&a\\end{array}\\right),$"}
{"_id": "7342", "title": "", "text": "$x_4=10$"}
{"_id": "1107619", "title": "", "text": "$F_{1/3}(x)=\\int\\frac{dx}{\\left(1+x^{1/3}\\right)^{1/3}}=\\frac9{40}\\left(9-6x^{1/3}+5x^{2/3}\\right)\\left(1+x^{1/3}\\right)^{2/3}$"}
{"_id": "8448614", "title": "", "text": "$X\\mapsto\\mathbb{E}[X|\\mathcal{F}_0]$"}
{"_id": "5459614", "title": "", "text": "$|X_n-X|^{2r} \\leq 2^{2r} (|X_n|^{2r}+|X|^{2r}),$"}
{"_id": "3566184", "title": "", "text": "$x*y*z+x'*y*z+x*y'*z+x*y*z' = y*z+x*z+x*y$"}
{"_id": "1228874", "title": "", "text": "$A_1 \\times \\ldots \\times A_k$"}
{"_id": "6155199", "title": "", "text": "$a|b_1b_2$"}
{"_id": "1275244", "title": "", "text": "$\\lim_{n\\to \\infty}\\sum_{i=1}^n  \\frac{1}{n+i}$"}
{"_id": "6169013", "title": "", "text": "$ \\int\\frac{1}{(1+x^2)^k} dx\\qquad(1)$"}
{"_id": "3450289", "title": "", "text": "$P(X_2 \\leq x \\mid X_1 = x)\\to0$"}
{"_id": "8336132", "title": "", "text": "$dy/dx=f(x,y), R=\\{(x, y):a\\le x\\le b, c \\le y \\le d\\}$"}
{"_id": "6525522", "title": "", "text": "$\\{\\gamma:\\gamma=\\aleph_\\gamma\\}$"}
{"_id": "8141655", "title": "", "text": "$\\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ s.t. } |x - a| < \\delta \\implies |f(x) - f(a)| < \\epsilon$"}
{"_id": "8260774", "title": "", "text": "$\\gamma_a \\gamma_b = \\gamma_{ab} = \\gamma_{ba} = \\gamma_b \\gamma_a$"}
{"_id": "3946086", "title": "", "text": "$ \\frac{d}{dx}( \\frac{1}{n}(1-e^{-x})^{n})=(1-e^{-x})^{n-1}e^{-x}$"}
{"_id": "5425094", "title": "", "text": "$f(n+2)=2(f(n+2))^2 -f(n+2)f(n)-2012$"}
{"_id": "1268649", "title": "", "text": "$\\frac1x+\\frac1y=\\frac1{n!}$"}
{"_id": "902765", "title": "", "text": "$ \\begin{array} {|c|c|} \\hline M^1                  & \\begin{bmatrix} 3 & 1 \\\\ 5 & 3 \\end{bmatrix} \\\\ \\hline M^2 = (M^1)^2        & \\begin{bmatrix} 14 & 6 \\\\ 30 & 14 \\end{bmatrix} \\\\ \\hline M^4 = (M^2)^2        & \\begin{bmatrix} 76 & 68 \\\\ 40 & 76 \\end{bmatrix} \\\\ \\hline M^8 = (M^4)^2        & \\begin{bmatrix} 96 & 36 \\\\ 80 & 96 \\end{bmatrix} \\\\ \\hline M^{16} = (M^8)^2       & \\begin{bmatrix} 96 & 12 \\\\ 60 & 96 \\end{bmatrix} \\\\ \\hline M^{32} = (M^{16})^2    & \\begin{bmatrix} 36 & 4 \\\\ 30 & 36 \\end{bmatrix} \\\\ \\hline M^{34} = M^{32}M^2     & \\begin{bmatrix} 24 & 70 \\\\ 60 & 24 \\end{bmatrix} \\\\ \\hline \\end{array}$"}
{"_id": "2526941", "title": "", "text": "$f'(x) = \\sqrt{(\\frac {dL}{dx})^2 - 1}$"}
{"_id": "279549", "title": "", "text": "${\\mathbb R}^{n+1}-\\{0\\}$"}
{"_id": "965889", "title": "", "text": "$ \\mathbb{E} [X^2] \\geq \\mathbb{E} [|X|]^2 $"}
{"_id": "8993922", "title": "", "text": "$M M M M M$"}
{"_id": "7912205", "title": "", "text": "$y = 4*[x^9*x^6]$"}
{"_id": "46745", "title": "", "text": "$D^+$"}
{"_id": "5612115", "title": "", "text": "$\\int_0^\\infty F(x) ~ dx$"}
{"_id": "3760817", "title": "", "text": "$\\overline H:=\\bigl\\{\\langle x,y\\rangle\\in\\Bbb R^2:x\\ge0\\bigr\\},$"}
{"_id": "4083334", "title": "", "text": "$A\\subseteq B\\subseteq\\overline B\\subseteq U\\cap B(0,r)\\subseteq U$"}
{"_id": "5026629", "title": "", "text": "$f(t)=\\frac1\\pi \\int_0^\\pi \\frac{\\cos(xt)}{\\sqrt{1+\\sin^2(x/2)}}\\,dx \\tag 1$"}
{"_id": "7584479", "title": "", "text": "$P[X\\in A|\\mathcal{F}_t] = P[X\\in A|\\mathcal{F}_s]$"}
{"_id": "2728854", "title": "", "text": "$4^n/(4^n+2 )$"}
{"_id": "4408848", "title": "", "text": "$\\forall \\mathbf{x}\\in\\mathbb{R}^{n+1}$"}
{"_id": "315747", "title": "", "text": "$[x,y]=1$"}
{"_id": "3459089", "title": "", "text": "$S=\\{a,a+d,a+2d,\\ldots\\}$"}
{"_id": "921171", "title": "", "text": "$(X,Y)\\mapsto [X,Y]=XY-YX$"}
{"_id": "8322050", "title": "", "text": "$\\left \\lfloor{\\frac{1}{n}\\sum_{k=1}^n\\sqrt{k}}\\right \\rfloor=\\left \\lfloor{\\left(\\frac{2}{3}+\\frac{1}{6n}\\right)\\sqrt{n+1}}\\right \\rfloor$"}
{"_id": "3763537", "title": "", "text": "$\\frac{\\left( 1+\\tan x\\right) ( 1+\\tan ^{2}x) }{(1-\\tan x) \\left( 1+\\tan ^{2}x\\right) }=\\frac{(1+\\tan x)^{2}\\left( 1-\\tan x\\right) }{\\left( 1-\\tan x\\right) \\left( 1+\\tan ^{2}x\\right) },\\qquad |\\tan x|< 1,$"}
{"_id": "8940290", "title": "", "text": "$\\frac{\\sin^{-1}(x)}{x}\\geq1$"}
{"_id": "3879033", "title": "", "text": "$Cov_1(x,y)>0$"}
{"_id": "1659869", "title": "", "text": "$ a\\equiv b^s, $"}
{"_id": "541501", "title": "", "text": "$1+2+3+4+\\cdots=$"}
{"_id": "5397660", "title": "", "text": "$\\|A\\|_2^2=\\rho(A^HA)$"}
{"_id": "6231038", "title": "", "text": "$(3)(9)^{\\frac{1}{3}}(27)^{\\frac{1}{9}}...=x^3$"}
{"_id": "5613219", "title": "", "text": "$f(x) = \\frac{5}{x^2}$"}
{"_id": "3378593", "title": "", "text": "$d_1(x,y)=\\left|f(x)-f(y)\\right|$"}
{"_id": "9223561", "title": "", "text": "$\\mathbb R^n- \\mathbb R$"}
{"_id": "4427242", "title": "", "text": "$G(x) = \\int_a^x F(t)dt$"}
{"_id": "7811867", "title": "", "text": "$\\int_0^1 \\frac{\\log \\left(1-x^2\\right) \\sin ^{-1}(x)^2}{x^2} \\, dx$"}
{"_id": "1776127", "title": "", "text": "$r = a^{\\log_a(r)}$"}
{"_id": "208784", "title": "", "text": "$\\Phi^+$"}
{"_id": "7862733", "title": "", "text": "$\\left\\lfloor \\frac{ \\left\\lfloor x/2 \\right\\rfloor }{ 2 }\\right\\rfloor=\\left\\lfloor m+\\frac{1}{2}\\right\\rfloor=m$"}
{"_id": "9155039", "title": "", "text": "$a^{log_b n}$"}
{"_id": "5113759", "title": "", "text": "$M = \\begin{pmatrix} a^2 & a & 1 \\\\ b^2 & b & 1 \\\\ c^2 & c & 1 \\end{pmatrix}$"}
{"_id": "5851549", "title": "", "text": "$a_n = an^{log_b(n)}$"}
{"_id": "1601779", "title": "", "text": "$ S = \\frac{1}{2} \\sum_{n=1}^{\\infty}\\sum_{m=1}^{\\infty} \\left(\\frac{m}{n} + \\frac{n}{m} \\right) \\frac{(-1)^{m+n}}{m^{2} + n^{2}} = \\frac{1}{2} \\left( \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1}}{n} \\right)^{2} = \\frac{1}{2}\\log^{2} 2. $"}
{"_id": "2409290", "title": "", "text": "$\\begin{align*}\\int_0^\\infty\\dfrac{\\sin^5x}xdx&=\\dfrac1{16}\\left[\\int_0^\\infty \\dfrac{\\sin5x}xdx+20\\int_0^\\infty\\dfrac{\\sin^3x}xdx - 5\\int_0^\\infty\\dfrac{\\sin x}xdx\\right]\\\\&=\\dfrac1{16}\\left[\\dfrac \\pi 2 + 5\\pi - \\dfrac{5\\pi}2\\right]\\\\&=\\dfrac{3\\pi}{16}\\end{align*}$"}
{"_id": "6383829", "title": "", "text": "$\\dot{x}_{ag}=\\left[\\matrix{\\dot{x}\\\\ \\dot{f}}\\right]=\\left[\\matrix{Ax+Bu\\\\ 0}\\right]=\\left[\\matrix{A & 0\\\\ 0 & 0 }\\right]x_{ag}+\\left[\\matrix{B\\\\ 0}\\right]u$"}
{"_id": "872335", "title": "", "text": "$\\lim_{x \\to c} |f(x)| = L$"}
{"_id": "8074152", "title": "", "text": "$\\int_{0}^\\pi \\frac{\\sin(nx)}{\\sin x} dx$"}
{"_id": "7749953", "title": "", "text": "$z=\\sqrt[4]{2}e^{\\frac{\\pi i}{6}}$"}
{"_id": "8302380", "title": "", "text": "$C=\\mathbb{R}^{n+1} -\\mathbb{R}^{n-1}$"}
{"_id": "5053075", "title": "", "text": "$\\prod_{k=1}^{n-1}\\cos\\frac{2k\\pi}n$"}
{"_id": "7506345", "title": "", "text": "$1+2+3+4\\dots = A-\\frac{1}{12}$"}
{"_id": "5738333", "title": "", "text": "$a\\in V_a\\subseteq \\overline{V_a}\\subseteq U_a$"}
{"_id": "6026655", "title": "", "text": "$ n! = \\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n e^{\\Theta(1/n)} $"}
{"_id": "446656", "title": "", "text": "$\\lim_{x\\rightarrow 2^-}f(x)=\\lim_{x\\rightarrow 2^+}f(x)=\\lim_{x\\rightarrow 2}f(x)=3,$"}
{"_id": "1753480", "title": "", "text": "$\\mathbb R^{n+1} = \\mathbb R^n\\times\\mathbb R$"}
{"_id": "3513108", "title": "", "text": "$f(n)=\\Big(\\frac{n(n+1)}2\\Big)^2$"}
{"_id": "5267384", "title": "", "text": "$X=\\frac{c(1+r)^{T+1}+(1-c)r^2}{r(1+r)((c-r)(1+r)^{T}+(1-c)r)}$"}
{"_id": "437020", "title": "", "text": "$F(x) = \\int_a^x f(t) dt, a \\le x \\le b$"}
{"_id": "5234550", "title": "", "text": "$Y Y = Y (Y Y ).$"}
{"_id": "8472807", "title": "", "text": "$||A||_2 <= \\sqrt{n}||A||_\\infty$"}
{"_id": "4008039", "title": "", "text": "$ \\sum_{n\\geq 1}\\frac{1}{n^{2+\\cos n}}, \\qquad \\sum_{n\\geq 1}\\frac{1}{n^{2-\\cos n}} $"}
{"_id": "28546", "title": "", "text": "$d(p^{1-\\gamma} T^\\gamma)=(1-\\gamma) p^{-\\gamma} T^\\gamma dp + \\gamma p^{1-\\gamma} T^{\\gamma-1} dT$"}
{"_id": "8848411", "title": "", "text": "$f'(-\\pi)=0=f'(\\pi)$"}
{"_id": "94429", "title": "", "text": "$\\sin^2\\varphi + \\cos^2\\varphi = 1$"}
{"_id": "5810922", "title": "", "text": "$ \\frac{Y(y)-Y''(y)}{Y(y)}$"}
{"_id": "5239461", "title": "", "text": "$\\frac{\\sqrt{1+y}}{1+\\sqrt{1+y}}=\\frac{\\sqrt{1-y}}{1-\\sqrt{1-y}}$"}
{"_id": "1423077", "title": "", "text": "$f_n=\\left(nf_1+(n-1)f_0\\right)(-1)^{n-1}$"}
{"_id": "7999415", "title": "", "text": "$x^3,x^5,x^9$"}
{"_id": "791489", "title": "", "text": "$f_3(n)$"}
{"_id": "4082021", "title": "", "text": "$\\phi(x)=\\frac{1}{\\sqrt{2\\pi}} e^{\\frac{-x^2}{2}}$"}
{"_id": "5863325", "title": "", "text": "$x = (p_1,p_2,p_3,\\dots,p_n)$"}
{"_id": "1686715", "title": "", "text": "$g(x)=x^2+x+2$"}
{"_id": "2611657", "title": "", "text": "$\\|A\\|_2^2 = Tr(A^TA)$"}
{"_id": "2022338", "title": "", "text": "$\\sigma_i(A) = \\sqrt{\\lambda_i (A^T A)}$"}
{"_id": "3422871", "title": "", "text": "$\\sum\\limits_{r=1}^{n}\\frac{1}{r(r+1)(r+2)}=?$"}
{"_id": "2661673", "title": "", "text": "$\\left\\lfloor \\frac{x}{n}\\right\\rfloor = \\left\\lfloor \\frac{\\lfloor x\\rfloor}{n}\\right\\rfloor$"}
{"_id": "1613354", "title": "", "text": "$x+1=\\frac {a+b}b$"}
{"_id": "8281075", "title": "", "text": "$\\int_{0}^{1} \\frac{1}{(1+x^2)^n} dx$"}
{"_id": "9135892", "title": "", "text": "$[x,y]=[h,k]g$"}
{"_id": "6629442", "title": "", "text": "$ \\int \\text{d} \\Omega_{1} \\text{d} \\Omega_{2} f(\\gamma) = 8 \\pi^{2} \\int \\sin \\gamma \\, \\text{d}\\gamma f(\\gamma).  $"}
{"_id": "8006331", "title": "", "text": "$\\alpha\\omega=\\alpha$"}
{"_id": "5043804", "title": "", "text": "$\\chi(n) = \\left(\\frac{n}{d}\\right)$"}
{"_id": "8937807", "title": "", "text": "$ \\begin{align} \\int_0^\\pi\\frac{\\sin^n(x)}{x}\\,\\mathrm{d}x &\\le\\int_0^\\pi\\left(\\frac{4x(\\pi-x)}{\\pi^2}\\right)^n\\frac{\\mathrm{d}x}x\\\\ &=4^n\\int_0^1x^{n-1}(1-x)^n\\,\\mathrm{d}x\\\\ &=4^n\\frac{\\Gamma(n)\\Gamma(n+1)}{\\Gamma(2n+1)}\\\\ &=\\frac{4^n}{n\\binom{2n}{n}}\\\\ &=2\\prod_{k=1}^{n-1}\\frac{k}{k+\\frac12}\\\\ &\\le2\\prod_{k=1}^{n-1}\\sqrt{\\frac{k}{k+1}}\\\\ &=\\frac2{\\sqrt{n}} \\end{align} $"}
{"_id": "7563092", "title": "", "text": "$d(\\gamma(0), \\gamma(s)) + d(\\gamma(s), \\gamma(t))+d(\\gamma(t), \\gamma(1))\\le \\left(s + (t-s) + (1-t)\\right)\\, d(\\gamma(0), \\gamma(1)) = d(\\gamma(0), \\gamma(1))$"}
{"_id": "6280941", "title": "", "text": "$S = \\{w^0, w^1, w^2, ....., w^n\\}$"}
{"_id": "9336480", "title": "", "text": "$\\sum_{i=1}^n \\|e_i-v_i\\|^2 < 1$"}
{"_id": "2481030", "title": "", "text": "$\\left\\{ \\matrix{   \\sin \\phi  = {{{1 \\over {\\sqrt 3 }}} \\over {\\sqrt {{1 \\over 3} + 1} }} = {1 \\over 2} \\hfill \\cr    \\cos \\phi  = {1 \\over {\\sqrt {{1 \\over 3} + 1} }} = {{\\sqrt 3 } \\over 2} \\hfill \\cr}  \\right.\\,\\,\\,\\,\\, \\to \\,\\,\\,\\,\\,\\phi  = {\\pi  \\over 6}$"}
{"_id": "5848702", "title": "", "text": "$\\sum_{n=1}^\\infty\\sum_{\\ell=n}^\\infty$"}
{"_id": "304107", "title": "", "text": "$ 1/|x|^{n-\\alpha} $"}
{"_id": "3134424", "title": "", "text": "$ \\operatorname{id}^{*} \\circ f = \\left( f^{*} \\circ \\operatorname{id} \\right)^{*} = \\left( f^{*} \\right)^{*} = f, \\\\ f \\circ \\operatorname{id}^{*} = \\left( \\operatorname{id} \\circ f^{*} \\right)^{*} = \\left (f^{*} \\right)^{*} = f $"}
{"_id": "5391772", "title": "", "text": "$\\sum_{n=-\\infty}^{-1} + \\sum_{n=1}^{\\infty}$"}
{"_id": "705229", "title": "", "text": "$\\beta=\\{v_1,v_2,\\dots,v_n\\}$"}
{"_id": "3215737", "title": "", "text": "$\\sum_{N=2}^\\infty Np_N=\\sum_{N=2}^\\infty\\frac{N(N-1)}{N!} =\\sum_{N=2}^\\infty\\frac1{(N-2)!}=\\cdots.$"}
{"_id": "963512", "title": "", "text": "$\\lim_{n \\to \\infty}\\frac{1}{n} \\sum_{r=1}^n f\\left ({r\\over n} \\right) $"}
{"_id": "4223130", "title": "", "text": "$\\sum \\chi_{I_j}\\le 2$"}
{"_id": "1534594", "title": "", "text": "$A_0\\subseteq A_1\\subseteq \\cdot \\cdot \\cdot \\subseteq A_n \\subseteq \\cdot \\cdot \\cdot $"}
{"_id": "3574721", "title": "", "text": "$\\lim_{n \\rightarrow \\infty} f(x_n)= \\sup_{x<m} f(x).$"}
{"_id": "1102078", "title": "", "text": "$V=S\\oplus T$"}
{"_id": "478978", "title": "", "text": "$\\int\\frac{1}{(x^2+x+1)^{1/2}} dx$"}
{"_id": "913289", "title": "", "text": "$\\omega+\\alpha=\\alpha$"}
{"_id": "3629736", "title": "", "text": "$ \\frac{\\;\\;\\frac{a^{3/2}}{b^3}\\;\\;}{\\frac{a^{-1}}{b^2}} $"}
{"_id": "334196", "title": "", "text": "$(x+a_n)(n-1)(x+a)^{n-2} =(n-1)(x+a)^{n-1} $"}
{"_id": "6507393", "title": "", "text": "$S_n=\\frac{n(n+1)}{2}.$"}
{"_id": "2874255", "title": "", "text": "$\\lim_{x\\rightarrow2^+}f'(x) = \\lim_{x\\rightarrow2^-}f'(x)$"}
{"_id": "61508", "title": "", "text": "$\\frac{2\\sin(a)+\\sec(a)}{1+\\tan(a)}=\\frac{1+\\tan(a)}{\\sec(a)}$"}
{"_id": "6722284", "title": "", "text": "$K \\subset D\\subset \\bar{D} \\subset A$"}
{"_id": "7635946", "title": "", "text": "$f_X (x) = e^{-x/\\theta}/\\theta$"}
{"_id": "2608411", "title": "", "text": "$b_k = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)\\sin(nx) dx.$"}
{"_id": "3419927", "title": "", "text": "$Cor(X,Y)=0$"}
{"_id": "6995424", "title": "", "text": "$(t^∗)^{\\mathcal A}(a) = f_n^{\\mathcal A}(s_1(x/ \\tau)^{\\mathcal A}(a), \\ldots,  s_n(x/ \\tau)^{\\mathcal A}(a))$"}
{"_id": "9062430", "title": "", "text": "$=\\left[\\begin{array}{ccc|c}1&-2&2&0\\\\2&k&-1&3\\\\1&-1&3&-5\\end{array}\\right]$"}
{"_id": "1759571", "title": "", "text": "$ \\sum_{k=1}^n \\cos \\frac{2 \\pi k}{n} = 0 $"}
{"_id": "6301038", "title": "", "text": "$\\displaystyle y=\\frac{bx}{x-a}$"}
{"_id": "199554", "title": "", "text": "$ \\begin{pmatrix} 2 & 2 & -5 & 2 \\\\ 1 & 1 & -2 & 1 \\\\ 1 & 0 & -3 & 4  \\end{pmatrix}  $"}
{"_id": "3329515", "title": "", "text": "$[f] \\in \\mathbb{R}[x] / (x^2 +1)$"}
{"_id": "6456636", "title": "", "text": "$p^r-p^{r-1}.$"}
{"_id": "1946427", "title": "", "text": "$\\overline{\\mathbb R},~$"}
{"_id": "5504050", "title": "", "text": "$10^{40}=2^{40}\\times5^{40}$"}
{"_id": "8184414", "title": "", "text": "$\\sum\\limits_{n=0}^{\\infty}\\sum\\limits_{r=0}^{n}\\left(\\frac{1}{(n-r)!}a^{n-r}\\right)\\left(\\frac{1}{r!}b^{r}\\right)$"}
{"_id": "6308985", "title": "", "text": "$t f(p) + (1-t) f(p')$"}
{"_id": "4995340", "title": "", "text": "$\\sum_{n=1}^{\\infty} (-1)^n=\\sum_{n=1}^{\\infty} (-1)^{2n}-\\sum_{n=1}^{\\infty} (-1)^{2n+1} ``=\" \\infty-\\infty.$"}
{"_id": "1021574", "title": "", "text": "$\\text{re } \\gamma_k = \\text{im } \\gamma_k = 0$"}
{"_id": "1644490", "title": "", "text": "$n!=\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^{n}$"}
{"_id": "1814933", "title": "", "text": "$^+e$"}
{"_id": "3744283", "title": "", "text": "$\\lim_{n\\to\\infty}{\\dfrac{1}{n}\\sum_{k=1}^{n}{\\left(\\dfrac{n}{k}-\\left\\lfloor\\dfrac{n}{k}\\right\\rfloor\\right)}}$"}
{"_id": "2062739", "title": "", "text": "$Cov(X,v)=0$"}
{"_id": "6625478", "title": "", "text": "$\\lim_{n \\to \\infty} \\inf t_n = t_*$"}
{"_id": "4217041", "title": "", "text": "$1-(P(A^c)+P(B^c))=1-((1-P(A))+(1-P(B)))$"}
{"_id": "4611049", "title": "", "text": "$M = F \\oplus T$"}
{"_id": "1837427", "title": "", "text": "$\\lim_{n \\to \\infty} \\frac{1}{\\sqrt{n}} \\sum_{k=1}^n \\frac{1}{\\sqrt{k}}$"}
{"_id": "5021930", "title": "", "text": "$\\lim\\limits_{n\\to \\infty} \\sqrt{x_n} = \\sqrt{\\lim\\limits_{n\\to \\infty} x_n}$"}
{"_id": "8008821", "title": "", "text": "$ \\det M=(k-\\lambda)^{n-1}(\\lambda(n-1)+k). $"}
{"_id": "6587942", "title": "", "text": "$a_n=\\frac{n(n+1)}{2}=T_n$"}
{"_id": "2666748", "title": "", "text": "$n! \\sim \\sqrt{2\\pi n}\\cdot (\\frac{n}{e})^n$"}
{"_id": "2222592", "title": "", "text": "$\\{a,b,a+b,a-b\\}$"}
{"_id": "2222465", "title": "", "text": "$P(E)=\\frac{1}{3}$"}
{"_id": "6930993", "title": "", "text": "$E[(X+Y)^p] \\le 2^p (E[X^p] + E[Y^p])$"}
{"_id": "2525188", "title": "", "text": "$f_n(x)=\\frac{\\sin x\\sin nx}{x^2}$"}
{"_id": "3790174", "title": "", "text": "$\\textbf{Cov}(X,Y)=0$"}
{"_id": "4388298", "title": "", "text": "$\\frac{1}x+\\frac{2}y=\\frac{3}n,$"}
{"_id": "5532005", "title": "", "text": "$\\sum_{n=1}^{\\infty} \\frac{1}{n^2+2^n}=\\sum_{n=1}^{k} \\frac{1}{n^2+2^n}+\\sum_{n=k+1}^{\\infty} \\frac{1}{n^2+2^n}$"}
{"_id": "1677294", "title": "", "text": "$x_n < \\frac{n(n+1)}{2} $"}
{"_id": "6329144", "title": "", "text": "$ M=\\left (\\begin{array}{cc} A &-B\\\\ B & A \\end{array} \\right ) $"}
{"_id": "7349655", "title": "", "text": "$\\det{A^T}=\\sum_{\\tau\\in\\Sigma_n}\\epsilon(\\tau)a_{\\tau(1),1}\\cdots  a_{\\tau(n),n}=\\det{A}$"}
{"_id": "7078725", "title": "", "text": "$e_i,\\; m+1\\le i\\le n$"}
{"_id": "9017664", "title": "", "text": "$[x, y] = \\mathscr{C}(a, b)$"}
{"_id": "4684120", "title": "", "text": "$C_1\\subseteq V\\subseteq \\overline{V}\\subseteq C_2^c$"}
{"_id": "1251302", "title": "", "text": "$F=S+T$"}
{"_id": "9314708", "title": "", "text": "$\\frac{ \\alpha\\beta}{\\gamma} = \\frac{\\alpha\\beta\\gamma}{\\gamma^2} = - \\frac{q}{\\gamma^2} $"}
{"_id": "452389", "title": "", "text": "$\\frac{d}{dx}(x-r)^mq(x)=(x-r)^{m-1}(mq(x)+(x-r)q'(x))$"}
{"_id": "8060081", "title": "", "text": "$\\|u\\|_p = \\frac{|\\langle u,v\\rangle |}{\\|v\\|_q}$"}
{"_id": "3495635", "title": "", "text": "$d(x,u)=d(y,v)$"}
{"_id": "3857728", "title": "", "text": "$\\frac{f}{||f||+||g||}$"}
{"_id": "2551289", "title": "", "text": "$(1,1,-1,1,f(1,1,-1,1)) \\in \\mathbb{R}^5$"}
{"_id": "4393114", "title": "", "text": "$\\det \\begin{pmatrix} x^2 & y^2 & z^2 \\\\ x & y & z \\\\ 1 & 1 & 1 \\end{pmatrix} = (x-y)(x-z)(y-z)$"}
{"_id": "8225489", "title": "", "text": "$F(x) = \\frac{1}{2}\\Big(\\frac{1}{(1-x)^3} - \\frac{1}{(1-x)(1-x^2)}\\Big) = \\frac{(1-x^2) - (1-x)^2}{2(1-x^2)(1-x)^3} = \\frac{x}{(1-x^2)(1-x)^2}$"}
{"_id": "898326", "title": "", "text": "$\\left\\{ {\\matrix{    {{a_n},\\mathbb{N}_{even}}  \\cr     {0,\\mathbb{N}_{odd}}  \\cr   } } \\right.$"}
{"_id": "6710470", "title": "", "text": "$w=\\frac{a+bi+1}{a-bi+1}$"}
{"_id": "859512", "title": "", "text": "$b_j(x) = (f(x) - \\frac{1}{|I_j|} \\int_{I_j} f(y) dy) \\chi_{I_j}(x)$"}
{"_id": "9281088", "title": "", "text": "$P(n-3)\\implies P(n)$"}
{"_id": "4567431", "title": "", "text": "$\\{a,a+c,a+2c,a+3c,a+4c,a+5c,\\ldots\\}.$"}
{"_id": "6504020", "title": "", "text": "$B = \\{ (x,y) \\in \\mathbb{R}^2 |   y \\leq x \\leq y^2+1,  0 \\leq y \\leq 1 \\}$"}
{"_id": "7115769", "title": "", "text": "$\\phi: \\mathbb{Z}/N\\mathbb{Z}\\rightarrow E[N](S)$"}
{"_id": "7066741", "title": "", "text": "$\\forall~\\epsilon >0,~\\exists~\\delta > 0 :~ |x - a|<\\delta \\implies |fx - L|<\\epsilon$"}
{"_id": "6021103", "title": "", "text": "$Cov(y_i,y_j)=0$"}
{"_id": "5319505", "title": "", "text": "$I=A_1\\times...\\times A_m$"}
{"_id": "9336125", "title": "", "text": "$\\begin{align}&\\quad 2^{k+1} \\\\[1ex]&= 2(2^k) \\\\[1ex]&\\geq 2(k+1) \\tag 1 \\\\[1ex]&= (k+1)+(k+1)\\\\[1ex]&\\geq (k+1)+1 \\tag 2\\end{align}$"}
{"_id": "6506104", "title": "", "text": "$\\sum_{i=1}^n(-1)^{i+1}i(i+1)$"}
{"_id": "8469975", "title": "", "text": "$T_x X = T_x U$"}
{"_id": "6644993", "title": "", "text": "$S = 1 + 2+ 4 + 8 + 16 + \\cdots$"}
{"_id": "3684200", "title": "", "text": "$\\lim_{n\\to\\infty} ((1+\\frac{1}{n^2})(1+\\frac{2}{n^2})(1+\\frac{3}{n^2})\\cdots(1+\\frac {n}{n^2}))$"}
{"_id": "6060829", "title": "", "text": "$x= \\frac{1}{4}, y= \\frac{1}{4}$"}
{"_id": "3569867", "title": "", "text": "$\\frac{a+b}{b}=1+\\frac{a}{b}=1+\\frac{1}{\\frac{b}{a}}=1+\\frac{1}{\\frac{a+b}{b}}$"}
{"_id": "7735210", "title": "", "text": "$ \\mathcal G_1\\supset\\mathcal G_2\\supset\\cdots\\supset\\mathcal G_n\\supset\\mathcal G_{n+1}\\supset\\cdots $"}
{"_id": "2353130", "title": "", "text": "$ \\frac{(1-p)^x\\cdot p}{1-(1-p)}=(1-p)^x.$"}
{"_id": "6816011", "title": "", "text": "$\\lim_{n\\to \\infty}\\mu(A_n)=1$"}
{"_id": "7767097", "title": "", "text": "$\\mathcal{O}\\subset \\mathbb{R}^n\\times \\mathbb{R}\\rightarrow \\mathbb{R}^n$"}
{"_id": "318870", "title": "", "text": "$\\langle X_i : i < \\alpha^{+} \\rangle$"}
{"_id": "6210264", "title": "", "text": "$ M_\\gamma = \\begin{pmatrix}\\cos(\\gamma) & -\\sin(\\gamma)\\\\ \\sin(\\gamma) & \\cos(\\gamma)\\end{pmatrix}, \\qquad \\qquad \\gamma  = m \\pi/2 $"}
{"_id": "8429538", "title": "", "text": "$\\sqrt{z} = \\sqrt{|z|} \\exp(i\\frac{\\varphi}{2})$"}
{"_id": "5106118", "title": "", "text": "$ \\lim_{n\\to\\infty} e^{-n}\\sum_{k=1}^n \\frac{n^k}{k!} $"}
{"_id": "8325145", "title": "", "text": "$\\sum\\limits_{N\\geq 1} \\frac{N}{N^2} = \\sum\\limits_{N\\geq 1} \\frac{1}{N}=\\infty$"}
{"_id": "4831588", "title": "", "text": "$\\forall \\epsilon \\gt 0, \\exists \\delta \\gt 0 ~~s.t.~~ |x-2| \\lt \\delta ~~implies~~ |\\alpha(x) - \\alpha(2)| = |x-2| < \\epsilon  $"}
{"_id": "8801254", "title": "", "text": "$k\\cong (k[X]/(X^n))/(X)$"}
{"_id": "8841899", "title": "", "text": "$ \\zeta(1-s) = \\frac{1}{\\pi} (2 \\pi)^{1-s} \\cos(\\frac{\\pi s}{2}) \\Gamma(s) \\zeta(s) $"}
{"_id": "1366401", "title": "", "text": "$c_1(\\mathcal T_X)=c_1(TX) = c_1(T^{1,0}X \\oplus T^{0,1}X) = c_1(T^{1,0}X)+c_1(T^{0,1}X)$"}
{"_id": "4099038", "title": "", "text": "$h(n)=\\sum_{j=1}^nA_{\\psi(j)}$"}
{"_id": "8768600", "title": "", "text": "$\\exists \\varepsilon > 0, \\exists \\delta > 0 : |x-a| < \\delta \\Rightarrow |f(x)-f(a)| \\geq \\varepsilon$"}
{"_id": "2377169", "title": "", "text": "$\\int_0^\\infty f(x)\\ dx=\\infty$"}
{"_id": "5053986", "title": "", "text": "$\\mathfrak{A} = \\sum_{n=-\\infty}^{\\infty} 2^{-n^{2}}$"}
{"_id": "51", "title": "", "text": "$>$"}
{"_id": "3928064", "title": "", "text": "$ D_3~=~cov(x,y) $"}
{"_id": "6289286", "title": "", "text": "$\\lim_{x\\to \\alpha^+} f(x) \\not\\exists$"}
{"_id": "6557954", "title": "", "text": "$20!+18, 20!+19, \\cdots, 20!+23$"}
{"_id": "2149468", "title": "", "text": "$|y - x_{n+1}| < \\epsilon/2$"}
{"_id": "3080380", "title": "", "text": "$1-2\\left ( \\frac{1}{2}^{n-1} \\right )$"}
{"_id": "2661672", "title": "", "text": "$\\left \\lfloor \\dfrac{\\left \\lfloor en! \\right \\rfloor}{n} \\right  \\rfloor=\\left \\lfloor e(n-1)! \\right \\rfloor$"}
{"_id": "4971934", "title": "", "text": "$gcd(a+b,a-b)=1$"}
{"_id": "7265464", "title": "", "text": "$\\sum_{n\\geq1}\\frac{1}{2n^{2}}+\\sum_{n\\geq1}\\left(\\frac{1}{2\\left(4+n^{2}\\right)}-\\frac{4}{\\left(4+n^{2}\\right)^{2}}\\right)$"}
{"_id": "1447551", "title": "", "text": "$ c_n = 3+\\frac{n(n+1)}{2} = \\frac{n(n+1) + 6}{2} = \\frac{n^2 +n + 6}{2} $"}
{"_id": "3669507", "title": "", "text": "$\\inf\\{F_\\gamma(\\gamma)\\mid\\gamma<\\alpha\\}\\le \\inf\\{F_\\gamma(\\beta)\\mid\\beta\\le\\gamma<\\alpha\\}=0$"}
{"_id": "6291335", "title": "", "text": "$\\gamma,\\dot\\gamma,\\ddot\\gamma$"}
{"_id": "7956447", "title": "", "text": "$\\mathrm{Spec}(\\mathbf{C}[X]/(X^2+1))=\\mathrm{Spec}(\\mathbf{C})\\coprod\\mathrm{Spec}(\\mathbf{C})$"}
{"_id": "2998613", "title": "", "text": "$n{n-1 \\choose 2}={n \\choose 2}{(n-2)}$"}
{"_id": "7436945", "title": "", "text": "$x-3\\mid x^3-3$"}
{"_id": "8497829", "title": "", "text": "$p_1p_2...p_r+1$"}
{"_id": "2529656", "title": "", "text": "$ \\begin{align} \\int_0^\\infty\\frac{\\sin^2(x)}{x}\\,\\mathrm{d}x &=\\sum_{k=1}^\\infty\\int_{(k-1)\\pi}^{k\\pi}\\frac{\\sin^2(x)}{x}\\,\\mathrm{d}x\\\\ &\\ge\\sum_{k=1}^\\infty\\frac1{k\\pi}\\int_{(k-1)\\pi}^{k\\pi}\\sin^2(x)\\,\\mathrm{d}x\\\\ &=\\sum_{k=1}^\\infty\\frac1{2k} \\end{align} $"}
{"_id": "4722741", "title": "", "text": "$\\lim \\limits_{n \\to \\infty} (\\sum \\limits_{r=1}^{n^2} \\frac{1}{(n+r)a_{n,r}}-\\ln n)= - \\infty$"}
{"_id": "3001525", "title": "", "text": "$\\frac{(1+r)^n - 1}{(1+r)^n \\times r}=y=8.491875$"}
{"_id": "91039", "title": "", "text": "$n!+2, n!+3, \\ldots, n!+n$"}
{"_id": "7532452", "title": "", "text": "$ \\color{blue}{\\sqrt{1+\\frac{d^2y}{dx^2}}=x+\\frac{dy}{dx}}\\implies 1+\\frac{d^2y}{dx^2}=\\bigg[x+\\frac{dy}{dx}\\bigg]^2 $"}
{"_id": "7671219", "title": "", "text": "$\\omega+1 = \\sqrt{3}e^{\\frac{i\\pi}{6}}$"}
{"_id": "2127522", "title": "", "text": "$xRy\\implies yRx,\\quad$"}
{"_id": "3157250", "title": "", "text": "$v_1^{1,1}, v_1^{1,2}, v_2^{2,1}$"}
{"_id": "1878075", "title": "", "text": "$x = t^{1/2n}$"}
{"_id": "5956074", "title": "", "text": "$P(n) \\implies P(f(n))$"}
{"_id": "8741790", "title": "", "text": "$n! = 1+\\sum\\limits_{r=1}^{n-1}( r\\cdot r!)$"}
{"_id": "2498308", "title": "", "text": "$e^x = \\lim \\left({1 + \\frac x n}\\right)^n$"}
{"_id": "7317886", "title": "", "text": "$\\left| {\\begin{array}{*{20}{c}} x&y& \\cdots &y\\\\ y&x& \\cdots &y\\\\  \\vdots &y& \\ddots &y\\\\ y&y&y&x \\end{array}} \\right| = {(x - y)^{n - 1}}\\left[ {x + (n - 1)y} \\right]$"}
{"_id": "268710", "title": "", "text": "$A_1\\subseteq A_2\\subseteq A_3\\subseteq A_4 ...\\subseteq A_n$"}
{"_id": "1348270", "title": "", "text": "$ \\left[     \\begin{array}{ccc|c}       1&1&k&6\\\\       1&k&1&3\\\\       k&1&1&7     \\end{array} \\right] $"}
{"_id": "3688242", "title": "", "text": "$x^9 + x^6 + x^4 - x^2 + 1=0$"}
{"_id": "1586855", "title": "", "text": "$Cov(X,Y) > 0$"}
{"_id": "4602062", "title": "", "text": "$E|X|^r < \\infty$"}
{"_id": "225616", "title": "", "text": "$\\phi(x) = \\sum_{k=1}^N a_k \\chi_{A_k}(x)$"}
{"_id": "8636355", "title": "", "text": "$\\gamma(s)=\\gamma(0) + \\int_0^s C\\,ds = \\gamma(0) + Cs$"}
{"_id": "8251939", "title": "", "text": "$P(X_{n+1}=y | X_n=x)=P(X_1=y | X_0=x)$"}
{"_id": "506603", "title": "", "text": "$x^6+3 x^5+6 x^4+3 x^3+9 x+9.$"}
{"_id": "5080123", "title": "", "text": "$F=P\\oplus P'$"}
{"_id": "8402865", "title": "", "text": "$ p_1p_2\\mid n\\iff p_1\\mid n\\text{ and }p_2\\mid n\\tag{3} $"}
{"_id": "1618172", "title": "", "text": "$\\varphi^+$"}
{"_id": "5023361", "title": "", "text": "$\\tan(a)=\\frac{\\sin(a)}{\\cos(a)}=\\frac{\\sin(a)}{\\sqrt{1-\\sin^2(a)}}$"}
{"_id": "1968531", "title": "", "text": "$20 + 10 + 10 = 40$"}
{"_id": "8594255", "title": "", "text": "$|u_n (x)|\\leq \\frac {2|x|}{n^2+x^2}\\leq \\frac{2|x|}{n^2} .$"}
{"_id": "251865", "title": "", "text": "$A_0, A_1, ..., A_n, A_{n+1}$"}
{"_id": "3517348", "title": "", "text": "$R e^{\\frac{i\\pi}{n}}$"}
{"_id": "1301732", "title": "", "text": "$\\frac{1}{x}+\\frac{2}{y}+\\frac{2}{z}=4$"}
{"_id": "3565218", "title": "", "text": "$f_{x_{max},x_{max-1}}(x,y)=\\frac{N!}{(N-2)!}e^{-x-y}(1-e^{-y})^{N-2}.$"}
{"_id": "5775481", "title": "", "text": "$ p(M) \\leq \\lambda(M) \\leq \\pi(M) $"}
{"_id": "3218963", "title": "", "text": "$\\implies (1+\\frac {x}{n})^n\\le e^x $"}
{"_id": "6800367", "title": "", "text": "$S = \\{S_1, S_2, S_3, \\dots, S_n\\}$"}
{"_id": "2761133", "title": "", "text": "$\\,\\frac{\\pi}{\\sin(\\pi s)}+\\Gamma(1-s)\\,\\zeta(1-s)\\,$"}
{"_id": "4708751", "title": "", "text": "$\\sum_{n=1}^\\infty\\frac1{n^2}=\\sum_{n=1}^{a-1}\\frac1{n^2}+\\underbrace{\\int_a^\\infty\\frac1{x^2}~\\mathrm dx}_{=1/a}+\\frac1{2a^2}+\\sum_{k=1}^p\\frac{B_{2k}}{a^{2k+1}}+R_p$"}
{"_id": "4892365", "title": "", "text": "$(a-b,a^2-ab+b^2)=1$"}
{"_id": "5643791", "title": "", "text": "$\\text{P}(A) = 1/6$"}
{"_id": "4457932", "title": "", "text": "$(e_1, e_2, e_3, \\ldots)$"}
{"_id": "2893566", "title": "", "text": "$(a+b,a-b)=(15,1)\\; \\text {or} \\;(5,3)$"}
{"_id": "7105371", "title": "", "text": "$\\frac{(1+x^2)^{n+1} - 1}{x^2(1+x^2)^{n+1}}$"}
{"_id": "7709220", "title": "", "text": "$2^kp_1p_2\\ldots p_s$"}
{"_id": "9277775", "title": "", "text": "$[x,y]=rx+sy$"}
{"_id": "2084733", "title": "", "text": "$gx-x,x\\in M,g\\in G$"}
{"_id": "3629537", "title": "", "text": "$\\sum_{n\\leq x}\\frac{\\varphi(n)}{n^{2}}=\\sum_{n\\leq x}\\frac{1}{n^{2}}\\sum_{d\\mid n}\\mu(d)\\frac{n}{d}=\\sum_{{q,d}\\atop{qd\\leq x}}\\frac{\\mu(d)}{d^{2}q}=\\sum_{d\\leq x}\\frac{\\mu(d)}{d^{2}}\\sum_{q\\leq\\frac{x}{d}}\\frac{1}{q} \\tag{4}$"}
{"_id": "4262278", "title": "", "text": "$(a+b)(a-b)(a^{-2})=1$"}
{"_id": "5448858", "title": "", "text": "$x^3-x\\ge x^3 -x$"}
{"_id": "8925919", "title": "", "text": "$1+2+3+4+\\dots = -1/12.$"}
{"_id": "5409909", "title": "", "text": "$f_i(x)=\\frac{\\langle x, v_i \\rangle}{\\|v_i\\|^2}$"}
{"_id": "8061849", "title": "", "text": "$(f'(f^{-1}(x)))'=f''(f^{-1}(x))·(f^{-1}(x))'$"}
{"_id": "5561050", "title": "", "text": "$=\\lim_{h\\to0^+}\\frac{(1+2h)^2-(1+6h-h^2)}{h(1+2h+\\sqrt{1+6h-h^2})}$"}
{"_id": "119847", "title": "", "text": "$p^r - p^{r-1}$"}
{"_id": "160177", "title": "", "text": "$f(z)=f(z/2+z/2)=f(z/2)^2\\ne0$"}
{"_id": "254759", "title": "", "text": "$D=\\{ (x,y) \\in \\mathbb{R}^2: a\\leq x\\leq b, c\\leq y \\leq d   \\}$"}
{"_id": "9267761", "title": "", "text": "$\\forall m \\leq n \\ P(m) \\implies P(n')$"}
{"_id": "1738067", "title": "", "text": "$\\frac{1}{2^n}\\rightarrow \\frac{1}{2^{n-1}}$"}
{"_id": "8594695", "title": "", "text": "$ (\\gamma'')^\\perp = \\langle \\gamma'' , \\gamma\\rangle \\gamma + \\langle \\gamma'', \\nu\\rangle \\nu.   $"}
{"_id": "5538111", "title": "", "text": "$f(xy)=f(f(x)+f(y))$"}
{"_id": "1155781", "title": "", "text": "$S=\\{\\alpha f+ \\beta g + \\gamma h: |\\alpha|\\le a, |\\beta|\\le b, |\\gamma|\\le c\\}$"}
{"_id": "2119575", "title": "", "text": "$ax + my = d$"}
{"_id": "2494054", "title": "", "text": "$\\int_k^{k+1}f(\\frac{u}{n})g(u)du=\\int_0^1f(\\frac{t+k}{n})g(t+k)dt=\\int_0^1f(\\frac{t+k}{n})g(t)dt$"}
{"_id": "4391622", "title": "", "text": "$|ab|=12$"}
{"_id": "2115709", "title": "", "text": "$|\\nabla(u^2)|^2=2u^2|\\nabla u|^2$"}
{"_id": "4892238", "title": "", "text": "$\\Big\\lfloor\\sum_{r=1}^N\\log_br\\Big\\rfloor+1.$"}
{"_id": "2185634", "title": "", "text": "$\\sqrt[n]{x_1 x_2 \\cdots x_n} \\leq \\frac{x_1 +x_2 + \\cdots + x_n}{n} $"}
{"_id": "3181319", "title": "", "text": "$cov(X/Y,Y)=0$"}
{"_id": "1087840", "title": "", "text": "$H = \\{(x,y)| a \\le x \\le b; c \\le y \\le d\\}$"}
{"_id": "3404466", "title": "", "text": "$\\dfrac {\\sqrt3 - \\sqrt2}{\\sqrt3 - \\sqrt2}$"}
{"_id": "8680989", "title": "", "text": "$L_2 = \\{ r \\in \\mathbb{Q} : 0 < r < 2 \\} $"}
{"_id": "1451692", "title": "", "text": "$Cov(X,Y)\\neq 0.$"}
{"_id": "965900", "title": "", "text": "$\\mathbb E[X^2]\\geqslant \\mathbb E[|X|]^2. $"}
{"_id": "5480292", "title": "", "text": "$ \\left\\lfloor\\frac{\\left\\lfloor\\frac ac\\right\\rfloor}b\\right\\rfloor=  \\left\\lfloor\\frac{qb+q''}c\\right\\rfloor=q.$"}
{"_id": "6656043", "title": "", "text": "$G:\\mathbb R^{n+1}\\to\\mathbb R^n$"}
{"_id": "308660", "title": "", "text": "$\\mathbb{C}^{+}$"}
{"_id": "6784797", "title": "", "text": "$f(x)=\\frac{x^5}{1+x^6}$"}
{"_id": "9177513", "title": "", "text": "$\\therefore \\int_0^\\pi f^2(x) \\, dx = \\int_0^\\pi f(x) \\left(\\sum_{n=0}^\\infty \\cos(nx)\\right) \\, dx = \\sum_{n=0}^\\infty \\int_0^\\pi f(x) \\cos(nx) \\, dx = 0$"}
{"_id": "4079175", "title": "", "text": "$\\lim_{k \\to \\infty}|\\frac{(k+1)^3}{k^3}| = |\\frac{{(k(1+\\frac{1}{k}))}^3}{k^3}| = |\\frac{k^3(1+\\frac{1}{k})^3}{k^3}| = 1$"}
{"_id": "2950726", "title": "", "text": "$ \\int_{0}^{1}\\frac{\\log^2 (x+1)}{x}$"}
{"_id": "3187331", "title": "", "text": "$f(n)=n^2+n+1$"}
{"_id": "232370", "title": "", "text": "$\\begin{align} \\int_0^\\infty f_X(x)\\,dx&=\\frac{4c}{3}\\Gamma(4)=8c \\end{align}$"}
{"_id": "4144493", "title": "", "text": "$ w(x) = 2 \\sqrt{\\frac{x}{c}}$"}
{"_id": "4407038", "title": "", "text": "$\\frac{14}{x}+\\frac{4}{y}=\\frac{11}{6}$"}
{"_id": "7649784", "title": "", "text": "$\\lim_{n\\to\\infty} \\mu(A_n)=\\mu(\\bigcup_{n} A_n)$"}
{"_id": "2274548", "title": "", "text": "$P(p) \\implies P(p+1)$"}
{"_id": "5146105", "title": "", "text": "$m_r \\equiv \\mathbb{E}|X|^r$"}
{"_id": "4537749", "title": "", "text": "$|\\langle x,y\\rangle | \\leq \\epsilon ||x||^2 + C_{\\epsilon}||y||^2$"}
{"_id": "603030", "title": "", "text": "$ f (xf (y)) f (y)=f (x+y) $"}
{"_id": "6706535", "title": "", "text": "$A=\\int_{0}^{\\pi }\\frac{xsinx}{1+sin^2x}dx =\\int_{0}^{\\pi }\\frac{(\\pi-x)sin(\\pi-x)}{1+sin^2(\\pi-x)}dx\\\\\\int_{0}^{\\pi }\\frac{(\\pi-x)sin(\\pi-x)}{1+sin^2(\\pi-x)}dx\\\\=\\int_{0}^{\\pi }\\frac{(\\pi-x)sin(x)}{1+sin^2(x)}dx\\\\=\\int_{0}^{\\pi }\\frac{(\\pi)sin(x)}{1+sin^2(x)}dx+\\int_{0}^{\\pi }\\frac{(-x)sin(x)}{1+sin^2(x)}dx\\\\A=\\int_{0}^{\\pi }\\frac{(\\pi)sin(x)}{1+sin^2(x)}dx-A\\\\$"}
{"_id": "8427931", "title": "", "text": "$f(xf(y))=f(x)y$"}
{"_id": "7923372", "title": "", "text": "$ \\begin{align} \\frac{(1+x)^{n+1}-(1+y)^{n+1}}{x-y} &=\\sum_{k=0}^{n+1}\\binom{n+1}{k}\\frac{x^k-y^k}{x-y}\\\\ &=\\sum_{k=0}^{n+1}\\binom{n+1}{k}\\sum_{j=1}^kx^{j-1}y^{k-j}\\tag{7} \\end{align} $"}
{"_id": "2386742", "title": "", "text": "$f(xf(y)+f(x))=yf(x)+x$"}
{"_id": "4676908", "title": "", "text": "$P(AB) = 1/6,$"}
{"_id": "205120", "title": "", "text": "$\\left\\{e_1,e_2,e_3,...\\right\\}$"}
{"_id": "2776505", "title": "", "text": "$a_n=\\frac{x_1+x_2+...+x_n}{n}.$"}
{"_id": "7529261", "title": "", "text": "$ \\frac{1-\\dfrac{b^{n}}{a^{n}}}{1-\\dfrac{b}{a}} =\\frac{\\dfrac{a^n-b^n}{a^n}}{\\dfrac{a-b}{a}}= \\frac{a^n-b^n}{a^n}\\frac{a}{a-b}= \\frac{1}{a^{n-1}}\\frac{a^n-b^n}{a-b} $"}
{"_id": "20695", "title": "", "text": "$ \\Bigl|\\int_0^x\\sin (t^2)\\,dt\\Bigr|\\leq\\int_0^x|\\sin(t^2)|\\,dt $"}
{"_id": "5194328", "title": "", "text": "$xy \\vee xz \\vee yz$"}
{"_id": "1892381", "title": "", "text": "$a_n =\\int_{0}^{1}  \\frac{x^n \\sin(\\pi x)}{1-x} \\, \\mathrm{d}x$"}
{"_id": "427533", "title": "", "text": "$d(x,z) < d(x,y) + d(y,z)$"}
{"_id": "552024", "title": "", "text": "$f(a+b)=f(a) \\circ f(b)$"}
{"_id": "136066", "title": "", "text": "$f \\colon K[T_1^{\\pm 1}, \\dots, T_r^{\\pm 1}] \\to \\mathcal O_X(U)$"}
{"_id": "8126725", "title": "", "text": "$\\lambda<\\alpha^+$"}
{"_id": "6643083", "title": "", "text": "$\\int_{-\\pi}^{\\pi}\\dfrac{\\sin nx}{(1+\\pi^{x})\\sin x}dx$"}
{"_id": "4407504", "title": "", "text": "$\\Pr[X\\ge x]=0$"}
{"_id": "148236", "title": "", "text": "$x\\in V\\subseteq U$"}
{"_id": "6213827", "title": "", "text": "$ 0 \\leq \\varepsilon\\cdot \\mathbb{P}\\{ \\lvert X - X_n\\rvert \\geq \\varepsilon\\} \\leq \\mathbb{E} \\lvert X - X_n\\rvert. $"}
{"_id": "1441707", "title": "", "text": "$f(x)=\\frac { x }{ { e }^{ x }+1 } $"}
{"_id": "9116278", "title": "", "text": "$x_{1},x_{2},x_{3}\\in \\Bbb{N}\\setminus\\{0\\}$"}
{"_id": "1858954", "title": "", "text": "$c=\\{\\{\\emptyset\\}\\}$"}
{"_id": "1559142", "title": "", "text": "$F3$"}
{"_id": "6103059", "title": "", "text": "$n! \\ge \\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n$"}
{"_id": "4606790", "title": "", "text": "$(e_i,f_j)^{-1}(e_k,1)(e_i,f_j)$"}
{"_id": "5704607", "title": "", "text": "$\\gcd(nn!, n!+1)=1$"}
{"_id": "4168407", "title": "", "text": "$\\left|\\int_\\gamma f(z)\\right|\\le\\int_\\gamma |f(z)|\\le M\\int_\\gamma |z^n|\\le M'|z|^{n+1}$"}
{"_id": "3640475", "title": "", "text": "$xRy \\implies yRy$"}
{"_id": "1360425", "title": "", "text": "$X=\\left\\{x_{1},x_{2},x_{3},.....,x_{N}\\right\\}$"}
{"_id": "1565861", "title": "", "text": "$m = p_1 p_2 \\cdots p_k$"}
{"_id": "6817197", "title": "", "text": "$M = F \\oplus T(M)$"}
{"_id": "7932766", "title": "", "text": "$\\int 2\\pi r ds$"}
{"_id": "1494912", "title": "", "text": "$\\mathbb{E}|X - m|^3 \\leq \\mathbb{E}|X|^3 (1 + \\frac{m}{\\sigma})^3$"}
{"_id": "7559458", "title": "", "text": "$x/4 = \\tan \\theta$"}
{"_id": "7051092", "title": "", "text": "$s_{n}(x) = \\sum \\limits_{i = 1}^{m} \\alpha_{i}*\\chi_{A_{i}}(x)$"}
{"_id": "1575008", "title": "", "text": "$\\sqrt{1+\\left(\\frac{df}{dt}\\right)^2} = \\operatorname{const.}$"}
{"_id": "2123228", "title": "", "text": "$\\{x\\}=f^{-1}(f(\\{x\\}))=f^{-1}(\\{f(x)\\})=f^{-1}(\\{f(x')\\})=f^{-1}(f(\\{x'\\}))=\\{x'\\}$"}
{"_id": "3047071", "title": "", "text": "$n! + 2, n! + 3, n! + 4... n! + n$"}
{"_id": "6644256", "title": "", "text": "$Var=|t-s|$"}
{"_id": "2932131", "title": "", "text": "$P[X_t=x|T=t]= \\pi(x)$"}
{"_id": "8391233", "title": "", "text": "$P = \\frac {k^3 - (k-1)^3}{216}$"}
{"_id": "4883397", "title": "", "text": "$x=\\frac{ab-t^2}{2t+a+b}$"}
{"_id": "740296", "title": "", "text": "$f_n(x)= \\dfrac{\\sin 2 \\pi x \\,\\sin 2\\pi n x}{x^2}$"}
{"_id": "6469277", "title": "", "text": "$ A+iB\\mapsto \\begin{pmatrix} A & -B \\cr B & A \\end{pmatrix}. $"}
{"_id": "3608657", "title": "", "text": "$A = \\{a_1,a_2,a_3,\\ldots,a_n\\}$"}
{"_id": "6645384", "title": "", "text": "$P:=(e_2,e_3,....,e_n,e_1)$"}
{"_id": "1831016", "title": "", "text": "$\\sum_{n=0}^\\infty \\frac{1}{2^{n+2}}$"}
{"_id": "1946800", "title": "", "text": "$A_1\\subseteq A_2\\subseteq A_3\\subseteq\\dots $"}
{"_id": "7546266", "title": "", "text": "$\\lim_{n\\to\\infty} \\mu(A_n)=\\mu(X).$"}
{"_id": "7977654", "title": "", "text": "$x! \\sim \\left(\\frac{x}{e}\\right)^x\\sqrt{2\\pi{x}}\\left(1 + \\frac{1}{12x}\\right)$"}
{"_id": "4202431", "title": "", "text": "$P(X\\mid A_1 \\cap A_2 \\cap \\cdots \\cap A_n \\cap A_{n+1})$"}
{"_id": "1910776", "title": "", "text": "$(P ∨ ¬P) = (F ∨ T) = T$"}
{"_id": "6119858", "title": "", "text": "$f\\big(xy + f(y)\\big) = y\\,f(x)$"}
{"_id": "7665770", "title": "", "text": "$\\begin{align}&(3)\\;\\;4k^2-1=(2k-1)(2k+1)\\implies\\frac1{4k^2-1}=\\frac12\\left(\\frac1{2k-1}-\\frac1{2k+1}\\right)\\\\{}\\\\ &(4)\\;\\;\\sum_{l=0}^k\\binom kl\\frac1{2^{k+l}}=\\frac1{2^k}\\sum_{l=0}^k\\binom kl\\frac1{2^l}=\\frac1{2^k}\\left(1+\\frac12\\right)^k\\end{align}$"}
{"_id": "4864095", "title": "", "text": "$\\frac1x+\\frac1y=a$"}
{"_id": "2098299", "title": "", "text": "$\\left(1+\\frac {x^2}n\\right)^n\\ge \\left(1+\\frac{x^2}{2}\\right)^2$"}
{"_id": "6378346", "title": "", "text": "$x_{\\gamma}  = \\frac{1}{1+\\gamma}x_{\\gamma=0}.$"}
{"_id": "4650635", "title": "", "text": "$f(x)=\\frac{10x}{x^2+1}$"}
{"_id": "5785392", "title": "", "text": "$ z \\mapsto \\vartheta(z + \\frac{1+\\tau}{2};\\tau)$"}
{"_id": "1092474", "title": "", "text": "$|1/x| = 1/|x|$"}
{"_id": "4588755", "title": "", "text": "$\\gamma=\\gamma(t)=(\\gamma_1(t),...,\\gamma_n(t))$"}
{"_id": "3626051", "title": "", "text": "$\\binom{n}{2}2^{n-2}$"}
{"_id": "4949000", "title": "", "text": "$f_X(x) = \\frac{2(\\theta -x)}{\\theta^2}, \\ \\ \\ \\ 0<x<\\theta $"}
{"_id": "2234034", "title": "", "text": "$\\sum_{\\substack{{k_1,k_2,\\dots, k_n\\geq0}\\\\{k_1+2k_2+3k_3+\\cdots+nk_n=n} }}\\frac{(k_1+k_2+\\dots+k_n)!}{k_1!k_2!\\dots k_n!} = 2^{n-1}.$"}
{"_id": "6578536", "title": "", "text": "$a^{({}^2a)^b*({}^2b)^a}*b^{({}^2b)^a*({}^2a)^b}$"}
{"_id": "9365185", "title": "", "text": "$\\omega^a_{\\;b}$"}
{"_id": "3105443", "title": "", "text": "$f(x)=\\frac{2^x}{2^x+1}$"}
{"_id": "2478958", "title": "", "text": "$f(ab)=f(a)+f(b), f(1)=0$"}
{"_id": "2527311", "title": "", "text": "$d(x,y)\\leqslant d(x,x)+d(y,x)$"}
{"_id": "7886379", "title": "", "text": "$a^{log_ax} = x$"}
{"_id": "185570", "title": "", "text": "$\\sum_{n=1}^\\infty\\frac1{2^n}$"}
{"_id": "7133880", "title": "", "text": "$\\int\\sum_{n=-\\infty}^\\infty e^{-\\pi{n^2}/ x}  dx $"}
{"_id": "2140692", "title": "", "text": "$\\Delta = \\begin{vmatrix} x & y & z \\\\ x_1 & y_1 & z_1 \\\\ x_2 & y_2 & z_2 \\end{vmatrix}$"}
{"_id": "7301625", "title": "", "text": "$A=Q\\sqrt{A^TA}$"}
{"_id": "5552981", "title": "", "text": "$\\sin(x)=\\frac{\\sqrt{(1-p^2)}}{1}$"}
{"_id": "7896864", "title": "", "text": "$xRy\\wedge yRx\\wedge x\\neq y$"}
{"_id": "5240641", "title": "", "text": "$0 = d^2 = (\\partial + \\bar\\partial)(\\partial +\\bar \\partial) = \\partial ^2 + (\\partial \\bar\\partial + \\bar\\partial \\partial) + \\bar\\partial^2,$"}
{"_id": "936100", "title": "", "text": "$ \\zeta(2) = \\sum_{n\\geq 1}\\frac{1}{n^2} = \\color{red}{\\sum_{n\\geq 1}\\frac{3}{n^2\\binom{2n}{n}}}\\tag{A}$"}
{"_id": "193674", "title": "", "text": "$F(x) = \\int_a^x f(t)\\,dt$"}
{"_id": "6847146", "title": "", "text": "$\\sum_{k=1}^{n-1}  {\\cos {\\frac{2k\\pi}{n}}} = -1$"}
{"_id": "1158607", "title": "", "text": "$[x,y]^{\\alpha}=[x,y]$"}
{"_id": "9244813", "title": "", "text": "$P+H=F$"}
{"_id": "4992320", "title": "", "text": "$T_{n}=\\frac{n(n+1)(n+2)}{6}$"}
{"_id": "5887798", "title": "", "text": "$P(M) = 1/6$"}
{"_id": "5403652", "title": "", "text": "$p(M)=p(\\overline{M})=\\frac{1}{2}$"}
{"_id": "6978388", "title": "", "text": "$\\lfloor \\frac{a}{c} \\rfloor+\\lfloor \\frac{b}{c} \\rfloor \\le \\lfloor \\frac{a+b}{c} \\rfloor $"}
{"_id": "1502524", "title": "", "text": "$\\lim_{x \\to c} f(x) = L,$"}
{"_id": "4716804", "title": "", "text": "$0=\\inf_{n \\geq 1} \\nu(E_n) = \\lim_{n \\to \\infty} \\nu(E_n).$"}
{"_id": "9151297", "title": "", "text": "$\\int_0^{+\\infty} f(t) \\sin(t) dt$"}
{"_id": "2298084", "title": "", "text": "$\\begin{cases} x = 0\\\\ x+y = 0\\\\ y = 0 \\end{cases} \\Rightarrow \\begin{cases} x = 0\\\\ y = 0 \\end{cases}$"}
{"_id": "2520607", "title": "", "text": "$x^3+x^2,x$"}
{"_id": "1862043", "title": "", "text": "$\\int_0^\\infty f(x)\\,dx.$"}
{"_id": "5984346", "title": "", "text": "$\\dfrac{\\tan^2\\theta}{\\tan\\theta\\sin\\theta}-\\dfrac{\\sin^2\\theta}{\\tan\\theta\\sin\\theta}\\Rightarrow \\dfrac{\\tan^2\\theta-\\sin^2\\theta}{\\tan\\theta\\sin\\theta}$"}
{"_id": "7879154", "title": "", "text": "$R_{1}(x) = o((x - a)^{n - 1})$"}
{"_id": "718036", "title": "", "text": "$F(x)=\\int_a^xf(t)~dt\\;,$"}
{"_id": "2766195", "title": "", "text": "$\\frac{T_4}{T_2}=\\frac {a+3d}{a+d}=\\frac {\\frac{a}{d}+3}{\\frac{a}{d}+1}=\\frac {r+3}{r+1}=\\frac {\\frac{1}{3}+3}{\\frac{1}{3}+1}=\\frac{5}{2}$"}
{"_id": "6261528", "title": "", "text": "$ \\tag{1} [T_a,T_b] = T_a $"}
{"_id": "2808323", "title": "", "text": "$x^{k+1} - y^{k+1} = x^{k} + \\Delta x^{k} - y^{k} - \\Delta y^{k} \\\\ = F'(x^{k})^{-1}\\left( F'(x^{k})(x^{k} - y^{k}) - \\int_{0}^{1}F'(y^{k} + t(x^{k}-y^{k}))(x^{k}-y^{k})\\text{d}t \\right) + \\\\ \\quad \\quad F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\\Delta{y}^{k}.$"}
{"_id": "1937469", "title": "", "text": "$\\limsup_{n\\to\\infty} \\mu_n(K) \\leq \\mu(K)$"}
{"_id": "1677712", "title": "", "text": "$3^x + 4^x = 5^x$"}
{"_id": "6406858", "title": "", "text": "$\\begin{vmatrix} 1 & a & a^2 \\\\ 1 & b & b^2 \\\\ 1 & c & c^2 \\end{vmatrix}\\neq0$"}
{"_id": "6873096", "title": "", "text": "$f^2(x) = \\dfrac{7x + 18}{3x + 10}$"}
{"_id": "8085735", "title": "", "text": "$(RS+R)^* = (RS)^*R(RS)^*R(RS)^*....R(RS)^* = R^*(RS)R^*(RS)...R^*$"}
{"_id": "9254823", "title": "", "text": "$P_2 = P \\oplus P$"}
{"_id": "4920230", "title": "", "text": "$a^\\sqrt{log_ab}=b^\\sqrt{log_ba}$"}
{"_id": "3402886", "title": "", "text": "$\\begin{align}   \\int\\limits_0^{\\infty} f(x)\\,\\mathrm{d}x \\;\\;&=\\;\\;      \\int\\limits_0^1 f(x)\\,\\mathrm{d}x +      \\int\\limits_1^{\\infty} f(u)\\,\\mathrm{d}u     \\\\\\;\\;&=\\;\\;\\int\\limits_0^1 f(x)\\,\\mathrm{d}x +      \\int\\limits_0^1 \\frac{f\\!\\left(\\frac{1}{x}\\right)}{x^2}\\,\\mathrm{d}x \\;\\;=\\;\\;      \\int\\limits_0^1 \\frac{x^2f(x)+f\\!\\left(\\frac{1}{x}\\right)}{x^2}\\,\\mathrm{d}x \\end{align}$"}
{"_id": "8609132", "title": "", "text": "$ \\sum_{n=1}^{+\\infty}\\frac1{1+n}=\\sum_{n=2}^{+\\infty}\\frac1{n}=+\\infty $"}
{"_id": "8511281", "title": "", "text": "$R = \\mathbb{R}[x]/(x^n + 1)$"}
{"_id": "8869789", "title": "", "text": "$ \\begin{cases} x=39t\\\\ y=-36 t\\\\ z=20t \\end{cases} $"}
{"_id": "1582039", "title": "", "text": "$X,C_0,C_1$"}
{"_id": "2883164", "title": "", "text": "$dL^2 = \\left(1 + \\left(\\frac{dy}{dx}\\right)^2\\right) dx^2$"}
{"_id": "6734128", "title": "", "text": "$\\int_{-\\pi/2}^{\\pi/2}=\\int_{\\pi}^0$"}
{"_id": "6965606", "title": "", "text": "$=\\log\\left(\\frac{\\left(1+\\tan\\frac{x}{2}\\right)^2}{\\left(1-\\tan\\frac{x}{2}\\right)\\left(1+\\tan\\frac{x}{2}\\right)}\\right)$"}
{"_id": "624752", "title": "", "text": "$\\left(1+\\frac1n\\right)^n\\ge 2$"}
{"_id": "8515030", "title": "", "text": "$g'(a_1)=f'(a_1)$"}
{"_id": "4077306", "title": "", "text": "$C = \\begin{bmatrix}cov(x,x)&cov(x,y)&cov(x,z)\\\\cov(y,x)&cov(y,y)&cov(y,z)\\\\cov(z,x)&cov(z,y)&cov(z,z)\\end{bmatrix}$"}
{"_id": "1136855", "title": "", "text": "$\\mathbb{Cov}(X,Y) = 0$"}
{"_id": "4661519", "title": "", "text": "$\\int_{-\\infty}^\\infty \\frac{x^{2n}}{(x^2 + 1)^{n + 1}}\\ dx.$"}
{"_id": "3515355", "title": "", "text": "$\\lim\\limits_{n\\to \\infty}\\ \\frac{1+\\sqrt[2]{2}+\\sqrt[3]{3}+\\cdots+\\sqrt[n]{n}}{n}=1$"}
{"_id": "2845051", "title": "", "text": "$x^3 - x^2 \\in \\mathbb{Z}$"}
{"_id": "419388", "title": "", "text": "$\\overline{x}:=\\frac{x_1+…+x_n}{n}$"}
{"_id": "928964", "title": "", "text": "$d(x,M)=\\inf_{y\\in M}{\\|x-y\\|}=\\frac{|\\langle f,x\\rangle|}{\\|f\\|}.$"}
{"_id": "7743403", "title": "", "text": "$A=B=C=D=\\mathbb{R}$"}
{"_id": "5103333", "title": "", "text": "$\\dots\\subseteq\\overline{V_{i+1}}\\subseteq\\overline{V_i}\\subseteq\\dots$"}
{"_id": "7848775", "title": "", "text": "$d_{f}(x,y)=|f(x)-f(y)|$"}
{"_id": "524969", "title": "", "text": "$P(k-1)\\implies P(k)$"}
{"_id": "2728855", "title": "", "text": "$\\sum_{i=1}^{1000}f\\left(\\frac{i}{1000}\\right),\\qquad f(x) = \\frac{4^x}{4^x+2}$"}
{"_id": "3010142", "title": "", "text": "$F(x) - F(a) = \\int_a^x f(t) dt$"}
{"_id": "6478368", "title": "", "text": "$F([x,y]) = x$"}
{"_id": "217400", "title": "", "text": "$a_n=n^2-n+2$"}
{"_id": "228432", "title": "", "text": "$({\\color{red}2},3)=1$"}
{"_id": "922758", "title": "", "text": "$ E(D^2L^2)=E(D^2)E(L^2) = [Var(D)+E(D)^2][Var(L)+E(L)^2] = Var(D)Var(L)+Var(D)E(L^2)+Var(L)E(D)^2+E(D)^2E(L)^2$"}
{"_id": "3002074", "title": "", "text": "$q^n-q^{l+1}$"}
{"_id": "5515520", "title": "", "text": "$1(1!)+\\dots+n(n!) = (n+1)!-1$"}
{"_id": "4163711", "title": "", "text": "$\\frac{1}{2^{x - 2}} = 2^{2 - x}$"}
{"_id": "5908737", "title": "", "text": "$Z_n=\\frac{X_1+X_2+...+X_n}{n^{1/p}}$"}
{"_id": "2762392", "title": "", "text": "$\\frac{1}{X} + \\frac{1}{Y} = 7$"}
{"_id": "9110968", "title": "", "text": "$\\lim_\\gamma \\delta_\\gamma = \\bigcup_\\gamma \\delta_\\gamma = \\sup_\\gamma \\delta_\\gamma$"}
{"_id": "2765357", "title": "", "text": "$40-10$"}
{"_id": "1868250", "title": "", "text": "$ M=\\left( \\begin{array}{cc} A&-\\vec d^T \\\\ \\vec c& b \\\\ \\end{array} \\right) $"}
{"_id": "5621561", "title": "", "text": "$I_{i_0} \\subseteq J_{j_0}$"}
{"_id": "3425516", "title": "", "text": "$\\sum\\limits_{n=-\\infty}^{\\infty} (-1)^{n}e^{-\\alpha n^2} = \\sqrt{\\frac{\\pi}{\\alpha}}\\sum\\limits_{n=-\\infty}^{\\infty} e^{-\\frac{(2n+1)^2\\pi^2}{4\\alpha}}$"}
{"_id": "3428076", "title": "", "text": "$f(a)+f(b)\\not=f(a+b)$"}
{"_id": "3303375", "title": "", "text": "$e^{x/n} > (1+\\frac{x}{n})$"}
{"_id": "7787623", "title": "", "text": "$2a, 2b,  a+b, a-b $"}
{"_id": "6481267", "title": "", "text": "$T(\\sum e_n) = \\sum x_n$"}
{"_id": "1061378", "title": "", "text": "$ G=\\frac{X_1+X_2+...+X_n}{n}, $"}
{"_id": "4190543", "title": "", "text": "$\\{(x,y,z,w,l)\\in \\mathbb{R}^5 :x=y,z=w=l\\}$"}
{"_id": "3204394", "title": "", "text": "$\\mathbf x = [x, y]$"}
{"_id": "8878739", "title": "", "text": "$\\lim_{n\\to \\infty}f(x_n) =\\inf f(x)\\geq -\\infty$"}
{"_id": "2050893", "title": "", "text": "$n! = (1 + o(1)) \\sqrt{ 2 \\pi n} \\left( \\frac{n}{e} \\right)^n$"}
{"_id": "5091654", "title": "", "text": "$\\large{a^{log_a(c)} = c}$"}
{"_id": "5010636", "title": "", "text": "$\\{\\dots,a-p,a,a+p,a+2p,\\dots\\}$"}
{"_id": "5636809", "title": "", "text": "$2a(2a(2b+a)-b(2b+a))=2a(2b+a)(2a-b)$"}
{"_id": "8742088", "title": "", "text": "$\\gamma = \\gamma(x) = \\frac{1}{x^2}\\tilde{\\gamma}(x)$"}
{"_id": "4083626", "title": "", "text": "$f(a+b) \\leqslant f(a) + f(b)$"}
{"_id": "94428", "title": "", "text": "$ = \\frac{(1 - \\sin \\varphi)^2}{ 1 - \\sin^2 \\varphi} = \\frac{(1 - \\sin \\varphi)^2}{(1 - \\sin \\varphi)(1+\\sin \\varphi)} = \\frac{1-\\sin \\varphi}{1+\\sin\\varphi}$"}
{"_id": "7254390", "title": "", "text": "$f(x)={e^x\\over e^x+1}, x\\in \\mathbb R$"}
{"_id": "7962186", "title": "", "text": "$\\dim A + \\dim B = \\kappa + 1 = \\kappa = \\kappa + 2 = \\dim C + \\dim D$"}
{"_id": "5511766", "title": "", "text": "$F(x) = \\int_a^x f(t)\\,\\mathrm{d}t.$"}
{"_id": "5199770", "title": "", "text": "$z^n=w=se^{i\\phi}$"}
{"_id": "3752446", "title": "", "text": "$ \\left.(\\gamma\\circ s^{-1})'\\right|_{s(t)} = \\left.(\\gamma'\\circ s^{-1})(s^{-1})'\\right|_{s(t)} = \\left.\\frac{\\gamma'\\circ s^{-1}}{s'\\circ s^{-1}}\\right|_{s(t)} = \\frac{\\gamma'}{s'} = \\frac{\\text{d}\\gamma/\\text{d}t}{\\text{d}s/\\text{d}t} = \\frac{\\text{d}\\gamma}{\\text{d}s} $"}
{"_id": "3515238", "title": "", "text": "$\\frac{10/9}3 + 1 = \\frac{1+3+9+27}{27}$"}
{"_id": "3365696", "title": "", "text": "$x = \\frac{- y^2 + 2ay - az}{a - z}.$"}
{"_id": "2249723", "title": "", "text": "$(S^{n}-S^{n-1} )$"}
{"_id": "8118681", "title": "", "text": "$M_2=\\left(\\begin{array}{} A & B\\\\ B^{H} &C \\end{array}\\right)$"}
{"_id": "4853031", "title": "", "text": "$\\langle\\beta,m\\rangle\\in D$"}
{"_id": "2782812", "title": "", "text": "$\\text{Cov}(Y,Z)=0$"}
{"_id": "3424456", "title": "", "text": "$a_n= {n(n+3)(n-1) \\above 1.5pt 2}$"}
{"_id": "6984894", "title": "", "text": "$k=0 \\rightarrow \\left[\\begin{array}{ccc}1&2&1\\end{array}\\right], k=1 \\rightarrow \\left[\\begin{array}{ccc}1&-1&1\\end{array}\\right]$"}
{"_id": "8726966", "title": "", "text": "$ P\\left( \\sup_{t \\geq 0 } \\exp\\left\\{a W_t - \\frac12 a^2 t \\right\\} > x \\mid \\mathcal{F}_0 \\right) = 1 \\wedge \\frac{1}{x} \\quad\\text{a.s.,}$"}
{"_id": "3125749", "title": "", "text": "$\\left  \\lfloor \\frac{a}{b} \\right \\rfloor + \\left  \\lfloor \\frac{c}{d}  \\right \\rfloor =\\left  \\lfloor \\frac{c+a}{b*d} \\right \\rfloor$"}
{"_id": "6836880", "title": "", "text": "$\\big[{T^{-1}\\big]}^{\\alpha}_{\\beta} = (\\big[{I\\big]}^{\\beta}_{\\alpha})^{-1}\\big[{T^{-1}\\big]}^{\\beta}_{\\alpha}\\big[{I\\big]}^{\\beta}_{\\alpha}  =\\big[{I\\big]}^{\\alpha}_{\\beta}\\big[{T^{-1}\\big]}^{\\beta}_{\\alpha}\\big[{I\\big]}^{\\beta}_{\\alpha}$"}
{"_id": "5602561", "title": "", "text": "$(1)\\ \\tan^2(\\frac{1}{2}\\theta) = \\frac{\\tan(\\theta) - \\sin(\\theta)}{\\tan(\\theta)+\\sin(\\theta)}$"}
{"_id": "4351772", "title": "", "text": "$\\sum_{n=1}^{N}\\frac{1}{2^{n-1}}\\to 2$"}
{"_id": "6492766", "title": "", "text": "$ \\gamma(A) = \\begin{cases} 0 & \\text{if }A\\text{ is countable}\\\\ 1 & \\text{otherwise.} \\end{cases} $"}
{"_id": "9131124", "title": "", "text": "$\\mathbb R^{N+1}\\times \\mathbb R^N$"}
{"_id": "1783943", "title": "", "text": "$f(x)=\\frac{a+x}{b+x}=1+\\frac{a-b}{b+x}$"}
{"_id": "2461005", "title": "", "text": "$\\Rightarrow a=\\lambda a \\gamma\\Rightarrow \\lambda=\\frac {1}{\\gamma}$"}
{"_id": "4582084", "title": "", "text": "$\\cos(x,y)=\\frac{\\langle x,y\\rangle}{||x||\\cdot||y||}$"}
{"_id": "1074492", "title": "", "text": "$P_n(x)=\\int_{-x}^{1-x}f(x+t)Q_{n}(t)dt=\\int_{0}^{1}f(t)Q_{n}(t-x)dt$"}
{"_id": "6107692", "title": "", "text": "$[T_1, T_2]:=T_1\\circ T_2-T_2\\circ T_1$"}
{"_id": "6153548", "title": "", "text": "$X=S\\oplus T$"}
{"_id": "8290131", "title": "", "text": "$\\{ \\dot \\gamma, \\ddot \\gamma, \\dots, \\gamma^{(m)} \\}$"}
{"_id": "7480718", "title": "", "text": "$\\sqrt{x}=\\sqrt{r}e^{ \\frac{i\\theta}{2}}$"}
{"_id": "2731860", "title": "", "text": "$\\int_{-\\frac{\\pi}{2}}^{0} {\\frac{1}{x\\sin(x)}} dx+\\int_{0}^{\\frac{\\pi}{2}} {\\frac{1}{x\\sin(x)}} dx$"}
{"_id": "7561925", "title": "", "text": "$\\gamma^* = \\gamma([a,b])$"}
{"_id": "5585176", "title": "", "text": "$L=\\lim_{n\\to\\infty}\\frac1{\\ln n}\\sum_{r=1}^{n^4}\\frac1r,\\qquad M=\\lim_{n\\to\\infty}\\left\\lfloor\\frac1{\\ln n}\\sum_{r=1}^{n^4}\\frac1r\\right\\rfloor$"}
{"_id": "5911178", "title": "", "text": "$\\frac{(1+\\sin(\\theta)+i\\cos(\\theta))^2}{(1+\\sin(\\theta)-i\\cos(\\theta))(1+\\sin(\\theta)+i\\sin(\\theta))}$"}
{"_id": "6941936", "title": "", "text": "$\\pi \\colon \\mathbb{R}^{n+1} \\rightarrow \\mathbb{R}^n$"}
{"_id": "4527144", "title": "", "text": "$\\forall\\epsilon \\gt0 ,\\, \\exists\\delta\\gt 0 :|x-0|\\lt\\delta\\Rightarrow|\\frac{2+\\sin x}{3-\\cos x}x-0|\\lt\\epsilon$"}
{"_id": "693113", "title": "", "text": "$ \\frac{\\tan \\theta \\ + \\ \\cot \\theta}{\\sec \\theta \\ \\cdot \\ \\sin \\theta} \\ = \\ \\frac{\\tan \\theta \\ + \\ \\cot \\theta}{\\tan \\theta } \\ = \\ 1 \\ + \\ \\frac{\\cos^2 \\theta }{\\sin^2 \\theta } \\ = \\ \\frac{\\sin^2 \\theta \\ + \\ \\cos^2 \\theta }{\\sin^2 \\theta } \\ = \\ \\frac{1}{\\sin^2 \\theta } $"}
{"_id": "1741995", "title": "", "text": "$|f '(z)| < |f(z)|$"}
{"_id": "2387873", "title": "", "text": "$n! > \\sqrt{2{\\pi}n}(\\frac{n}{e})^n$"}
{"_id": "799312", "title": "", "text": "$A_1\\subset A_2\\subset\\cdots$"}
{"_id": "1105374", "title": "", "text": "$ax+by=z$"}
{"_id": "5214498", "title": "", "text": "$|f(z)+f(-z)|=|f(z)|+|f(-z)|=2|z|^2$"}
{"_id": "834418", "title": "", "text": "$\\lim_{x \\to \\ c-}f(x)=l$"}
{"_id": "3152020", "title": "", "text": "$|ab|=3$"}
{"_id": "7889324", "title": "", "text": "$P(t) \\implies P(t+1)$"}
{"_id": "7132930", "title": "", "text": "$\\alpha a + (\\beta + \\gamma + \\delta) a' = (\\alpha + \\beta + \\gamma + \\delta)e\\\\ a'=\\frac{(\\alpha + \\beta + \\gamma + \\delta)e - \\alpha a}{\\beta + \\gamma + \\delta}$"}
{"_id": "9098321", "title": "", "text": "$ \\left | F(\\gamma(t)) \\cdot \\gamma'(t) \\right | \\leq \\left | F(\\gamma(t)) \\right | \\left | \\gamma'(t) \\right | $"}
{"_id": "718000", "title": "", "text": "$|f(z)| \\leq |f'(z)|$"}
{"_id": "2021663", "title": "", "text": "$\\neg B\\lor \\neg\\neg B$"}
{"_id": "3876817", "title": "", "text": "$[1,0]x=0$"}
{"_id": "5600389", "title": "", "text": "$f(e^{-\\gamma})=e^{e^{-\\gamma}-\\gamma}$"}
{"_id": "1882580", "title": "", "text": "$\\alpha = \\alpha^+- \\alpha^-$"}
{"_id": "7861671", "title": "", "text": "$y = (y,y',y'')$"}
{"_id": "226568", "title": "", "text": "$u^+$"}
{"_id": "758963", "title": "", "text": "$\\frac{4}{x}+\\frac{7}{x-3}+\\frac{1}{2}=1$"}
{"_id": "4570311", "title": "", "text": "$\\text{Im}(\\gamma_1) = \\text{Im}(\\gamma_2)$"}
{"_id": "2309585", "title": "", "text": "$(\\tilde{x},\\tilde{y},\\tilde{z})=(1,1,1)$"}
{"_id": "5333066", "title": "", "text": "$d(x,A) \\le d(x,a) \\le d(x,y) - d(y,a),$"}
{"_id": "2333664", "title": "", "text": "$\\left[\\matrix{\\bar x\\cr \\bar y\\cr}\\right]=\\left[\\matrix{x_0\\cr y_0\\cr}\\right]+d\\ \\left[\\matrix{\\cos\\phi&-\\sin\\phi \\cr \\sin\\phi&\\cos\\phi\\cr}\\right]\\>\\left[\\matrix{x\\cr y\\cr}\\right]\\ .$"}
{"_id": "1266898", "title": "", "text": "$\\neg \\neg p \\lor \\neg \\neg q$"}
{"_id": "6206591", "title": "", "text": "$||x||=\\sqrt{|\\langle x,x\\rangle|}$"}
{"_id": "6382678", "title": "", "text": "$x[n] = \\delta[n+1]+\\delta[n]+\\delta[n-4]$"}
{"_id": "1097403", "title": "", "text": "$\\gamma, \\ddot \\gamma \\in \\dot \\gamma ^\\perp$"}
{"_id": "1242566", "title": "", "text": "$\\gcd(a) = a$"}
{"_id": "4409552", "title": "", "text": "$\\frac{1}{\\pi}\\int_0^\\pi |\\cos(x)|dx=\\frac{1}{\\pi}\\int_0^{\\pi/2} \\cos(x)dx + \\frac{1}{\\pi}\\int_{\\pi/2}^{\\pi} -\\cos(x)dx=\\frac{1}{\\pi}[\\sin(\\pi/2)-\\sin(0)-\\sin(\\pi)+\\sin(\\pi/2)]=\\frac{1}{\\pi}(1-0-0+1)=\\frac{2}{\\pi}$"}
{"_id": "7011256", "title": "", "text": "$ n! \\approx \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n $"}
{"_id": "7052708", "title": "", "text": "$\\varepsilon = \\frac{|\\langle v,f \\rangle|}{2 \\| f \\|_{L^2}}$"}
{"_id": "3178852", "title": "", "text": "$\\left(\\text{a}-\\text{b}\\right)^2=\\left(\\text{a}-\\text{b}\\right)\\left(\\text{a}-\\text{b}\\right)=\\text{a}\\text{a}-\\text{a}\\text{b}-\\text{a}\\text{b}-\\text{b}\\cdot\\left(-\\text{b}\\right)=\\text{a}^2+\\text{b}^2-2\\text{a}\\text{b}$"}
{"_id": "2631177", "title": "", "text": "$a(n) = n^2 - n/2 + O(1)$"}
{"_id": "4110415", "title": "", "text": "$\\lim\\limits_{n \\rightarrow \\infty} \\frac{\\sqrt{n+1} + 3}{\\sqrt{n+2} - 4}$"}
{"_id": "8428004", "title": "", "text": "$\\lim\\limits_{n\\to \\infty }\\frac{1}{n}\\sum\\limits_{j=1}^na_j\\:=\\:\\lim\\limits_{n\\to \\infty }\\left(a_n\\right)$"}
{"_id": "3250573", "title": "", "text": "$(x' + y') + z' = (x + y)' + z' = ((x+y) + z) = (x + (y+z))' = x' + (y+z)' = x' + (y' + z')$"}
{"_id": "5936072", "title": "", "text": "$\\nabla f=O(1/|x|)$"}
{"_id": "7487719", "title": "", "text": "$A = \\frac{R(1+r)^{-1}(1-(1+r)^{-n})}{1-(1+r)^{-1}} \\; .$"}
{"_id": "6701875", "title": "", "text": "$P(X=0) = \\left(\\frac12\\right)^{n-2}$"}
{"_id": "2821281", "title": "", "text": "$\\gamma_\\epsilon\\subseteq\\gamma$"}
{"_id": "3979023", "title": "", "text": "$\\psi = \\frac{-xa\\:+\\:by}{a\\:-x} \\tag{11}$"}
{"_id": "7943848", "title": "", "text": "$f_X(x) = \\frac{2}{3}e^{-2x/3}$"}
{"_id": "4828109", "title": "", "text": "$A_1\\cap A_{n+1},\\ldots,A_n\\cap A_{n+1}$"}
{"_id": "2193208", "title": "", "text": "$\\int_0^{\\infty} x f(x) $"}
{"_id": "3838972", "title": "", "text": "$1-f,f\\in M$"}
{"_id": "3262912", "title": "", "text": "$\\frac{dl}{d \\beta} = \\sqrt{r^2 + \\frac{dr}{d \\beta}^2}$"}
{"_id": "4117693", "title": "", "text": "$[c_0, c_1]$"}
{"_id": "6797933", "title": "", "text": "$ \\lim_{n\\to\\infty} \\frac{1}{\\log n}\\sum_{m=1}^n \\frac{1}{m} = 1. $"}
{"_id": "3654367", "title": "", "text": "$(x^4-x^2+1)$"}
{"_id": "3222061", "title": "", "text": "$y=\\frac{a+bx} {c+dx}+d$"}
{"_id": "4609660", "title": "", "text": "$I'(a)=\\int_0^{\\pi/2}\\frac{\\ln(\\tan x)}{\\left(1+\\tan^ax\\right)^2}dx$"}
{"_id": "2549240", "title": "", "text": "$\\begin{cases}x=2t+1\\\\y=t\\\\z=3t-1\\end{cases}$"}
{"_id": "1500077", "title": "", "text": "$\\sum^\\infty_{-\\infty} |\\hat{f}(n)|^2 < \\infty$"}
{"_id": "7302484", "title": "", "text": "$\\sqrt{1+\\frac{\\partial y}{\\partial x}^2}$"}
{"_id": "3033457", "title": "", "text": "$(\\gamma', \\gamma \\times \\gamma', \\gamma)$"}
{"_id": "5455922", "title": "", "text": "$\\sum_{n=1}^\\infty=\\infty $"}
{"_id": "9077979", "title": "", "text": "$ \\begin{align} &\\small n\\frac{\\frac1{n-1}-\\frac1n}{(n-1)^{\\frac{1-k}k}+(n-1)^{\\frac{2-k}k}n^{-\\frac1k}+(n-1)^{\\frac{3-k}k}n^{-\\frac2k}+\\cdots+(n-1)^{-\\frac2k}n^{\\frac{3-k}k}+(n-1)^{-\\frac1k}n^{\\frac{2-k}k}+n^{\\frac{1-k}k}}\\\\[9pt] &\\le\\frac{\\frac1{n-1}}{kn^{\\frac{1-k}k}}\\\\[6pt] &=\\frac{n^{-\\frac1k}}k\\frac n{n-1}\\\\[12pt] &\\to0 \\end{align} $"}
{"_id": "6796434", "title": "", "text": "$f(x) = \\displaystyle\\frac{2e^{2x}}{x^2}$"}
{"_id": "2662493", "title": "", "text": "$ \\left(\\left(\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{    x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}\\right)_{\\left(\\left(x_x^x\\right)_{x_x^x}^    {x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x    }}}^{\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^    x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}}\\right)_{\\left(\\left(\\left(x_x^x\\right)_{x_x^    x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_    x^x}}\\right)_{\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x    ^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}}^{\\left(\\left(x_x^x\\right)_{x_x^x}^{    x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}    }}}^{\\left(\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}    ^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}\\right)_{\\left(\\left(x_x^x\\right)_{x_x^x    }^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x    ^x}}}^{\\left(\\left(x_x^x\\right)_{x_x^x}^{x_x^x}\\right)_{\\left(x_x^x\\right)_{x_x^x}^{x_    x^x}}^{\\left(x_x^x\\right)_{x_x^x}^{x_x^x}}}} $"}
{"_id": "2540676", "title": "", "text": "$3 \\mid a^3-a$"}
{"_id": "5890227", "title": "", "text": "$\\lim_{x\\rightarrow0}f(x+c)=L$"}
{"_id": "8257662", "title": "", "text": "$1/|x|^{n-a}$"}
{"_id": "3848654", "title": "", "text": "$A_1^C,A_2^C,\\dots,A_{n-1}^C,A_n$"}
{"_id": "739188", "title": "", "text": "$\\lim_{n\\rightarrow\\infty}\\sum_{k=1}^{n}a_k=\\lim_{n\\rightarrow\\infty}\\sum_{k=1}^{n}k=\\lim_{n\\rightarrow\\infty}\\frac{n(n+1)}{2}\\rightarrow\\infty$"}
{"_id": "5669745", "title": "", "text": "$\\pi:\\mathbb{R}^{n+1}\\to \\mathbb{P}^n$"}
{"_id": "8454251", "title": "", "text": "$[x,y]=x\\overline{y}$"}
{"_id": "4408671", "title": "", "text": "$\\lim_{n\\to\\infty}n\\sum_{r=1}^n\\frac1{(n+r)^2}=\\lim_{n\\to\\infty}\\frac1n\\sum_{r=1}^n\\frac1{\\left(1+\\dfrac rn\\right)^2} $"}
{"_id": "3810538", "title": "", "text": "$x^2,x^3\\in A$"}
{"_id": "24296", "title": "", "text": "$\\infty > \\mu(A) \\geq \\mu(A \\cap B) = \\mu(A)\\mu(B) = \\infty$"}
{"_id": "2409288", "title": "", "text": "$\\displaystyle \\int_0^\\infty \\dfrac{\\sin^5x}xdx=\\dfrac{3\\pi}{16}$"}
{"_id": "2630013", "title": "", "text": "$<  \\gamma+\\beta\\cup \\lbrace  \\gamma+\\beta\\rbrace$"}
{"_id": "4329780", "title": "", "text": "$\\log_a(1) = 0, \\log_a(b) = 1 + \\log_a(b/a)$"}
{"_id": "2654567", "title": "", "text": "$z=-1+2\\lambda$"}
{"_id": "8048714", "title": "", "text": "$0 = z = 1 + 2λ$"}
{"_id": "4530245", "title": "", "text": "$\\tan \\theta = \\frac{\\sqrt 5}{2};$"}
{"_id": "8594698", "title": "", "text": "$\\langle\\gamma'',  \\nu\\rangle = h(\\gamma',\\gamma')$"}
{"_id": "2092223", "title": "", "text": "$d(x,z) \\leq d(x,y) + d(y,z);$"}
{"_id": "5521125", "title": "", "text": "$5^x+7^x=12^x.$"}
{"_id": "7198595", "title": "", "text": "$c_0 + c_1x$"}
{"_id": "450734", "title": "", "text": "$x^3,x^2,x,1$"}
{"_id": "8095639", "title": "", "text": "$\\bar{X}_n = {X_1 + X_2 + \\cdots + X_n \\over n}$"}
{"_id": "5027482", "title": "", "text": "$y_n=\\frac{x_1-x_2-...-x_n}{n}$"}
{"_id": "1711212", "title": "", "text": "$\\lim\\limits_{n\\to \\infty} \\frac{1+2\\sqrt2+3\\sqrt3+\\ldots+n\\sqrt n}{n^2 \\sqrt{n}}$"}
{"_id": "1317096", "title": "", "text": "$\\frac1n\\sum_{i=1}^nx_i=\\frac{x_1+x_2+\\ldots+x_n}n\\;.$"}
{"_id": "3850306", "title": "", "text": "$|f(z)| \\le 2 |z|^2$"}
{"_id": "4808211", "title": "", "text": "$a\\in \\Bbb F$"}
{"_id": "8336290", "title": "", "text": "$\\frac{\\partial l(\\theta, X)}{\\partial \\theta}\\frac{\\partial l(\\theta, X)}{\\partial \\theta}^T$"}
{"_id": "3343667", "title": "", "text": "$5^x+7^x+11^x=6^x+8^x+9^x$"}
{"_id": "2460964", "title": "", "text": "$\\left| \\sum_{k=1}^n c_k - \\sum_{n=1} ^\\infty a_n \\sum_{m=1}^\\infty b_m\\right| \\le 2\\left( \\sum_{n=1}^\\infty |a_n| + \\sum_{m=1}^\\infty |b_m| \\right)\\epsilon$"}
{"_id": "543656", "title": "", "text": "$\\int_{0}^{\\infty} \\frac{\\sin^{4}(x) }{x^{2}} \\, dx $"}
{"_id": "5143537", "title": "", "text": "$ \\sum_{n=1}^{\\infty} \\sum_{k=1}^{\\infty}\\frac{1}{(n^{2}+k^{2})^{2}}=\\frac{{\\pi}^{2}}{4}\\sum_{n=1}^{\\infty}\\frac{\\coth^2(\\pi n)}{n^{2}}+\\frac{\\pi}{4}\\sum_{n=1}^{\\infty}\\frac{\\coth(\\pi n)}{n^{3}}-\\frac{{\\pi}^{2}}{4}\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}-\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{1}{n^{4}}$"}
{"_id": "6760376", "title": "", "text": "$f(f(x)+f(y))=af(x)+f(a)+f(x)-f(y)x$"}
{"_id": "2230513", "title": "", "text": "$\\det \\begin{pmatrix} A & B \\\\ B & A \\end{pmatrix} = \\det{(A+B)}\\det{(A-B)}$"}
{"_id": "7328907", "title": "", "text": "$\\sum\\limits_{k=1}^{n^2}\\frac{n^{r-1}}{n^r+k^r}=\\sum\\limits_{k=1}^{n}\\frac{n^{r-1}}{n^r+k^r}+\\sum\\limits_{k=n}^{n^2}\\frac{n^{r-1}}{n^r+k^r} = \\frac1n\\sum\\limits_{k=1}^{n}\\frac{1}{1+(\\frac kn)^r}+\\sum\\limits_{k=n}^{n^2}\\frac{n^{r-1}}{n^r+k^r}$"}
{"_id": "2393366", "title": "", "text": "$\\beta A \\alpha^{1-\\gamma}=((\\beta A \\alpha^{1-\\gamma})^{1/\\gamma})^\\gamma=(\\beta^{1/\\gamma}A^{1/\\gamma}\\alpha^{(1/\\gamma)-1})^\\gamma=c^\\gamma$"}
{"_id": "5789791", "title": "", "text": "$S:=A_1\\times \\ldots\\times A_n$"}
{"_id": "35736", "title": "", "text": "$\\mathbb{Q}(\\sqrt{-d})$"}
{"_id": "779009", "title": "", "text": "$f(x^{2}+f(y))=f(x^{2})+f(f(y))$"}
{"_id": "902622", "title": "", "text": "$A \\subseteq B \\subseteq C \\subseteq A$"}
{"_id": "8596021", "title": "", "text": "$\\forall \\epsilon >0,\\quad\\exists\\delta:|x-a|<\\delta\\implies|f(x)-f(a)|<\\epsilon$"}
{"_id": "7984920", "title": "", "text": "$\\lim\\limits_{k \\to \\infty} \\inf J(u_k)\\geq J(v)$"}
{"_id": "6509899", "title": "", "text": "$\\overline{\\{x\\}}=\\mathbb R$"}
{"_id": "9253276", "title": "", "text": "$P(x)=x^2+x+2$"}
{"_id": "573783", "title": "", "text": "$[t^n_i, t^n_{i + 1}]$"}
{"_id": "198190", "title": "", "text": "$(\\overline{\\mathbb{R}},\\tau(\\mathbb{\\overline{R}})$"}
{"_id": "1508304", "title": "", "text": "$F(x) = \\int_{x-T}^x f(t)\\,\\mathrm{d}t$"}
{"_id": "8113524", "title": "", "text": "$\\frac{1-\\gamma}{\\gamma'-\\gamma}\\big[A-\\frac{1-\\gamma'}{1-\\gamma}B\\big]\\in\\mathscr{W}$"}
{"_id": "7955851", "title": "", "text": "$w=\\{\\{a\\},\\{a,b\\}\\}$"}
{"_id": "5819956", "title": "", "text": "$ q(T) = q((T^{*})^{*}) = [q(T^{*})]^{*} = \\mathbf{0}^{*} = \\mathbf{0}. $"}
{"_id": "880660", "title": "", "text": "$\\int \\frac{1}{x^2+x+1}\\mathrm{d}x$"}
{"_id": "4263058", "title": "", "text": "$\\int_{-\\infty}^\\infty \\frac{1}{(x^2+1)^3} dx$"}
{"_id": "404956", "title": "", "text": "$(x(y+n-2)-(n-1))(y-1)^{n-2}$"}
{"_id": "498744", "title": "", "text": "$a-x,\\:\\:x\\in \\left(-\\infty ,a\\right)$"}
{"_id": "6220559", "title": "", "text": "$E=F (a_{1},a_{2},a_{3},....,a_{n})$"}
{"_id": "6575086", "title": "", "text": "$P[X< x |D_1] = 1$"}
{"_id": "6885787", "title": "", "text": "$A_1\\subset A_2, A_2\\subset A_3, \\ldots , A_{n-1}\\subset A_n$"}
{"_id": "4686709", "title": "", "text": "$f_1(x)=\\frac{dF(x)}{dx}$"}
{"_id": "4331749", "title": "", "text": "$\\mathbb Z[i]\\cong \\mathbb Z[x]/(x^2+1).$"}
{"_id": "307262", "title": "", "text": "$ax+\\sin x=L$"}
{"_id": "3615953", "title": "", "text": "$[x,y] = {\\bar x}^T H y.$"}
{"_id": "462975", "title": "", "text": "$P(m+1)\\implies P(m)$"}
{"_id": "4669783", "title": "", "text": "$n(\\gamma,a)=-n(-\\gamma,a).$"}
{"_id": "3534662", "title": "", "text": "$A_0\\times\\dots\\times A_n\\times \\Omega\\times\\Omega\\times\\dots$"}
{"_id": "1320418", "title": "", "text": "$9^x - 6^x = 4^x.4^{1/2}$"}
{"_id": "3974385", "title": "", "text": "$\\begin{cases} x \\equiv 1 \\pmod4\\\\ x\\equiv -1 \\pmod{25} \\end{cases}$"}
{"_id": "859382", "title": "", "text": "$ L = \\lim_{x \\to 0^-} f(x) = \\lim_{x \\to 0^+} f(-x) = \\lim_{x \\to 0^+} -f(x) = -L. $"}
{"_id": "8654359", "title": "", "text": "$(x-y, x+y)=(x,x)-(y,x)+(x,y)-(y,y)=(x,y)-(y,x)$"}
{"_id": "1559722", "title": "", "text": "$ \\gamma_1 \\geq 0 \\\\ \\gamma_2 \\geq 0$"}
{"_id": "5156443", "title": "", "text": "$\\implies s_N=\\frac{N(N+1)}{2}$"}
{"_id": "8402410", "title": "", "text": "$(\\mathbb{Z}[x] / (x^2-x))_U \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2$"}
{"_id": "6569532", "title": "", "text": "$P\\{\\sup_{t\\geq 0} M_t > x \\,| \\,F_0\\} = 1\\wedge \\frac{M_0}{x}$"}
{"_id": "1141488", "title": "", "text": "$ \\sigma_n:=\\frac{x_1+\\cdots+x_n}{n} $"}
{"_id": "51028", "title": "", "text": "$(1 + {x \\over n})^n \\geq 1 + x$"}
{"_id": "7683878", "title": "", "text": "$G=\\gamma_0(G) \\geq \\gamma_1(G) \\geq \\cdots \\geq \\gamma_{c}(G)=1,$"}
{"_id": "6613170", "title": "", "text": "$-\\dfrac{1}{2\\pi}\\int_{0}^{2\\pi}\\psi(s)\\ln{\\left(4\\sin^2{\\dfrac{t-s}{2}}\\right)}ds\\approx \\displaystyle\\sum_{i=0}^{2n-1}\\psi(t_{j})R_{j}(t)$"}
{"_id": "6314010", "title": "", "text": "$=\\pi\\int_0^\\pi f(\\sin t) dt- \\int_0^\\pi tf(\\sin t) dt$"}
{"_id": "7178879", "title": "", "text": "$\\|y_n-x_n\\|\\leq r$"}
{"_id": "7604976", "title": "", "text": "$|P(m) - P(m+t)| < \\epsilon$"}
{"_id": "8527245", "title": "", "text": "$\\sqrt{\\rho(A^t A)}$"}
{"_id": "24678", "title": "", "text": "$\\gcd(a+b, a-b) = 1$"}
{"_id": "8848398", "title": "", "text": "$f(\\pi)=f'(\\pi)=0$"}
{"_id": "2017849", "title": "", "text": "$\\gamma _s(a)=\\gamma (a),\\quad \\gamma _s(b)=\\gamma (b)\\quad \\text{and}\\quad \\gamma _0=\\gamma ,$"}
{"_id": "6859514", "title": "", "text": "$\\left|\\int_{\\gamma} f dz\\right|\\leqslant L(\\gamma)|f|_\\gamma.$"}
{"_id": "6139593", "title": "", "text": "$dm=|ab|$"}
{"_id": "5259538", "title": "", "text": "$v(r e^{i \\theta}) = \\frac{4r (1-r^2) \\sin (\\theta)}{(1-2r\\cos (\\theta)+r^2)^2}.$"}
{"_id": "3927529", "title": "", "text": "$ |S(\\gamma)(x,y)|^{2}\\le S(\\gamma)(x,x)S(\\gamma)(y,y),\\forall x,y\\in \\mathbb{R} $"}
{"_id": "7084686", "title": "", "text": "$A(\\sum x_n e_n) = \\sum \\lambda_n x_n e_n$"}
{"_id": "190003", "title": "", "text": "$ds=\\sqrt {1+(\\frac {dy}{dx})^2}dx$"}
{"_id": "4556866", "title": "", "text": "$\\sum_n\\frac{1}{n^p}>\\sum_n\\frac{1}{n}$"}
{"_id": "47441", "title": "", "text": "$M = \\frac{a+b+c+{\\dots}n}{n}$"}
{"_id": "8074160", "title": "", "text": "$ I_n = \\int_0^\\pi \\frac{\\sin(nt)}{\\sin t} dt \\\\ I_{n+2}-I_n =\\int_0^\\pi \\frac{\\sin((n+2)t)-\\sin(nt)}{\\sin t} dt $"}
{"_id": "1182366", "title": "", "text": "$\\frac{1}{||f||}$"}
{"_id": "7616866", "title": "", "text": "$\\eta^+$"}
{"_id": "2155205", "title": "", "text": "$\\lim_{x\\to c} g(x)=L=\\lim_{x\\to c} h(x)$"}
{"_id": "7815823", "title": "", "text": "$\\int_0^\\pi \\frac{x}{\\sin{x}}dx=\\int_0^{\\frac{\\pi}{2}} \\frac{x}{\\sin{x}}dx+\\int_{\\frac{\\pi}{2}}^\\pi \\frac{x}{\\sin{x}}dx$"}
{"_id": "7474881", "title": "", "text": "$‎‎‎\\mathbb{R^+}‎$"}
{"_id": "7810117", "title": "", "text": "$A=\\frac {1}{8} + \\frac {1}{64} + ..... = \\frac {1}{8} + \\frac {1}{8} \\cdot  (\\frac {1}{8} + \\frac {1}{64} + .....) = \\frac {1}{8} + \\frac {A}{8} $"}
{"_id": "130661", "title": "", "text": "$g(x) = \\int_a^x f(t)\\, dt$"}
{"_id": "9172905", "title": "", "text": "$\\mathbb{F}^*_{8} = (\\mathbb{F}_2[x]/(x^3+x^2+1))^* $"}
{"_id": "7695698", "title": "", "text": "$ \\left[ \\matrix{1 & 3/2 & -1/2 & 2 & | & a/2\\cr                   0 & 1 & -1/7 & 12/7 & | & 3a/7 -2b/7\\cr                   0 &  0   & 1 & -17/9 & | & -2a/3 + 4b/9 +7c/9 \\cr                   0 &  0   &  0 &    0 & | & -a+b+c+d\\cr}\\right]$"}
{"_id": "7583024", "title": "", "text": "$\\Pr[X \\ge x] \\ge 1-k/q<)$"}
{"_id": "2268667", "title": "", "text": "$ P(\\{\\omega \\in \\Omega: \\lim_{n \\to \\infty}X_n(\\omega) = X(\\omega)\\})=1. $"}
{"_id": "2689330", "title": "", "text": "$ D : \\mathbb{R}^{n+1} \\to \\mathbb{R}^n $"}
{"_id": "2254087", "title": "", "text": "$ \\left( {e^{\\,x} } \\right)^{\\,{n \\over m}}  = e^{\\,\\,{n \\over m}\\,x} \\quad \\left| \\matrix{   \\,n,m \\in Z \\hfill \\cr    \\;0 \\ne m \\hfill \\cr}  \\right. $"}
{"_id": "4076896", "title": "", "text": "$\\lim\\limits_{n\\rightarrow\\infty}\\frac{\\sqrt{1+\\frac{1}{n}}+\\sqrt{1+\\frac{2}{n}}+...+\\sqrt{1+\\frac{n}{n}}}{n}$"}
{"_id": "2894800", "title": "", "text": "$ \\forall s:\\text{Re}(s)>0,\\qquad \\zeta(s)=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\sum_{n\\geq 1}\\frac{(-1)^{n+1}}{n^s}=\\left(1-\\frac{2}{2^s}\\right)^{-1}\\frac{1}{\\Gamma(s)}\\int_{0}^{+\\infty}\\frac{t^{s-1}}{e^t+1}\\,dt $"}
{"_id": "4948188", "title": "", "text": "$x=\\frac{a+b}{a-b}$"}
{"_id": "7925251", "title": "", "text": "$\\int_{\\gamma_{a,b} + \\gamma_{b,z}} f = \\int_{\\gamma_{a,z}} f = F_a(z).$"}
{"_id": "1360173", "title": "", "text": "$X=\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}$"}
{"_id": "8306678", "title": "", "text": "$A_n = \\frac{n(n+1)(n+2)}{6}$"}
{"_id": "7366588", "title": "", "text": "$\\frac{dx}{ds} = \\frac{1}{\\frac{ds}{dx}} = \\frac{1}{\\sqrt{1+ \\big(\\frac{dy}{dx}\\big)^2}}$"}
{"_id": "2681134", "title": "", "text": "$cov(X+Y,X-Y) = cov(X,X) + cov (X,-Y)+cov(Y,X)+cov(Y,-Y)$"}
{"_id": "727879", "title": "", "text": "$(n-1)C_n, (n-2)C_n, \\ldots, \\frac{n+1}{2}C_n$"}
{"_id": "2003408", "title": "", "text": "$ e^x > \\left(1 + \\frac{x}{n}\\right)^{n}. $"}
{"_id": "4468656", "title": "", "text": "$\\mathcal A= \\mathcal A+x + (-x)$"}
{"_id": "6965150", "title": "", "text": "$ S:= \\{T_{1,1}, T_{2,1}, \\ldots, T_{n,1}\\} $"}
{"_id": "2514750", "title": "", "text": "$\\ell : ax + by = c$"}
{"_id": "8395718", "title": "", "text": "$\\int_{-\\pi}^{\\pi} \\frac{\\sin nx}{(1+2^x)\\sin x}dx$"}
{"_id": "181807", "title": "", "text": "$f(x)=\\frac{x}{\\theta^2}\\exp(\\frac{-x}{\\theta})$"}
{"_id": "7475480", "title": "", "text": "$  T^\\prime_{mix}(\\varepsilon):=  \\inf \\Big  \\{t \\ge 0:   \\max_{x,y \\in \\mathbb{Z}^d_n}\\Big|\\frac{1}{2}\\Big(      \\mathbb{P}(X_t = y | X_0=x)+\\mathbb{P}(X_{t+1}= y | X_0=x)\\Big)-       \\pi(y)\\Big|     \\le \\varepsilon \\Big\\} $"}
{"_id": "1977923", "title": "", "text": "$a^b =alog(alog(\\log(b)+\\log(\\log(a)))) $"}
{"_id": "6244754", "title": "", "text": "$ |x^n - a^n| < |(a + \\delta)^n - a^n| < |(a + a\\delta)^n - a^n| = |a^n((1 + \\delta)^n - 1)| $"}
{"_id": "9119778", "title": "", "text": "$H \\subset B \\subset{\\bar{B}} \\subset G$"}
{"_id": "6246826", "title": "", "text": "$ \\begin{cases} x=-2 \\\\ x=-2 \\\\ y = -5 \\end{cases}\\Longleftrightarrow $"}
{"_id": "8473227", "title": "", "text": "$ F_U(u) = \\mathbb P(U\\leq u) = \\mathbb P \\left[\\left( 1+\\frac{\\gamma X}{\\sigma} \\right)^{-\\frac{1}{\\gamma}}\\leq u\\right] = F_{\\gamma, \\sigma}\\left[ \\frac{\\sigma}{\\gamma}\\left( u^{-\\gamma}-1 \\right) \\right] = 1-u. $"}
{"_id": "5032534", "title": "", "text": "$R=A_1\\times\\cdots\\times A_n$"}
{"_id": "339530", "title": "", "text": "$|f(z)| \\le C ( 1+|z|^n)$"}
{"_id": "3001521", "title": "", "text": "$y=\\frac{(1+r)^n - 1}{(1+r)^n\\times r}$"}
{"_id": "5898386", "title": "", "text": "$ \\begin{cases} x=t\\\\ y=t\\\\ z=0 \\end{cases} $"}
{"_id": "8421703", "title": "", "text": "$m,m',m''$"}
{"_id": "722852", "title": "", "text": "$\\int_0^\\infty f(x)=\\infty$"}
{"_id": "4519815", "title": "", "text": "$x \\in V_0 \\subseteq \\overline{ V_0 } \\subseteq U_0$"}
{"_id": "7521204", "title": "", "text": "$ = \\int_{-1}^{1}f(t)\\delta(t-t_0)dt+\\int_{-1}^{1}f(t)\\delta(t+t_o)dt = f(t_0)+f(-t_0) = 0$"}
{"_id": "2386782", "title": "", "text": "$f(xf(1)+f(x))=f(x)+x$"}
{"_id": "8919280", "title": "", "text": "$\\iff b-\\dfrac{a+b}2>t-\\dfrac{a+b}2\\ge a-\\dfrac{a+b}2\\iff\\dfrac{b-a}2>t-\\dfrac{a+b}2\\ge-\\dfrac{b-a}2$"}
{"_id": "7914458", "title": "", "text": "$\\|A\\|_2\\le\\sqrt{\\|A^TA\\|_{\\infty}}$"}
{"_id": "6968318", "title": "", "text": "$Cov(I,X) = 0$"}
{"_id": "2331646", "title": "", "text": "$i_n:\\pi(X_n)\\rightarrow\\pi(X)$"}
{"_id": "7373110", "title": "", "text": "$6^{x} + 8^{x} + 9^{x} = 12^{x}$"}
{"_id": "6841528", "title": "", "text": "$y =  log_a(b)$"}
{"_id": "4268486", "title": "", "text": "$ \\frac{\\sqrt3 + 2}{2\\sqrt3}(1+\\sqrt3)^n + \\frac{\\sqrt3-2}{2\\sqrt3}*(1-\\sqrt3)^n $"}
{"_id": "7574241", "title": "", "text": "$\\infty > \\sum_{k \\in \\mathbb{N}} \\lVert  a_k-b_k \\lVert \\geq 1$"}
{"_id": "4729514", "title": "", "text": "$f(a + b) = f(a) + f(b) + ab$"}
{"_id": "2356975", "title": "", "text": "$f(x) =\\frac{x}{9+x^2}$"}
{"_id": "6204572", "title": "", "text": "$e(z,x,y) = (1,1,1)$"}
{"_id": "5956067", "title": "", "text": "$P(k) \\not \\implies P(k+1)$"}
{"_id": "5989468", "title": "", "text": "$x\\sim y\\iff xRy\\land yRx$"}
{"_id": "3761069", "title": "", "text": "$a x + b y + c x = d$"}
{"_id": "5214722", "title": "", "text": "$\\alpha+\\sup_{\\gamma<\\beta}\\{\\gamma\\}=\\sup_{\\gamma<\\beta}\\{\\alpha+\\gamma\\}$"}
{"_id": "7125860", "title": "", "text": "$2\\int_{0}^{\\infty}f(x)dx=1$"}
{"_id": "8662880", "title": "", "text": "$\\frac{x}{x-a} + x = \\frac{b}{b-a}+ b$"}
{"_id": "306973", "title": "", "text": "$\\sum\\limits_{k=0}^{n-1}\\dfrac{1}{\\cos^2\\frac{\\pi k}{n}}=n^2$"}
{"_id": "1664609", "title": "", "text": "$ \\frac{\\pi}{\\sin(\\pi/p)} $"}
{"_id": "7556806", "title": "", "text": "$ \\frac{(n-p+1)!}{p!(n-2p+1)!} $"}
{"_id": "3187068", "title": "", "text": "$\\sum_{n=1}^\\infty\\sum_{k=1}^\\infty\\frac{1}{n^k k^n}$"}
{"_id": "7379514", "title": "", "text": "$\\|x\\| \\leq \\lim_{n\\to \\infty} \\inf \\|x_n\\|$"}
{"_id": "5999172", "title": "", "text": "$p_2 \\mid p_1^d -1$"}
{"_id": "832237", "title": "", "text": "$as+bt=d$"}
{"_id": "2086901", "title": "", "text": "$Cov(X,-Y)=-Cov(X,Y)$"}
{"_id": "3120594", "title": "", "text": "$ f(n) = \\binom{n}{k} => f(n) $"}
{"_id": "8820991", "title": "", "text": "$\\operatorname{Tr}(AABB)-\\operatorname{Tr}(ABAB)\\leq \\operatorname{Tr}(A^{2})\\left[ \\operatorname{Tr}(ABB)-\\operatorname{Tr}(AB)\\operatorname{Tr}(AB)% \\right]? $"}
{"_id": "4260441", "title": "", "text": "$\\lim_{x\\downarrow \\pi} f(x)=|\\sin(\\pi)+\\cos(\\pi)|=\\lim_{x\\uparrow \\pi}f(x)=\\lim_{x\\to \\pi}f(x)=1$"}
{"_id": "2214528", "title": "", "text": "$(ax+by=c)$"}
{"_id": "8931354", "title": "", "text": "$   A_1 \\oplus \\underbrace{\\bigoplus_i (D_i' \\oplus D_i'') \\oplus \\bigoplus_j D_j}_{C_1}   \\simeq A_2 \\oplus \\underbrace{\\bigoplus_j (D_j' \\oplus D_j'') \\oplus \\bigoplus_i D_i}_{C_2} $"}
{"_id": "7569933", "title": "", "text": "$\\gamma(a,t)=\\gamma(a+2\\pi,t)$"}
{"_id": "7317826", "title": "", "text": "$|\\sum x_n|\\le\\sum|x_n|$"}
{"_id": "1647357", "title": "", "text": "$||A||_2 = \\sqrt{r(A^TA)}$"}
{"_id": "194004", "title": "", "text": "$[x,y]=0$"}
{"_id": "4560042", "title": "", "text": "$\\sum_{n=1}^\\infty na_n = \\sum_{n=1}^\\infty n(s_n-s_{n-1}) = \\sum_{n=1}^\\infty ns_n - \\sum_{n=1}^\\infty ns_{n-1} = \\sum_{n=1}^\\infty ns_n - \\sum_{n=1}^\\infty (n+1)s_n = -\\sum_{n=1}^\\infty s_n$"}
{"_id": "7750168", "title": "", "text": "$ 1=1\\cdot1=\\left(a^{-1/2}a^{1/2}\\right)\\cdot\\left(a^{1/2}a^{-1/2}\\right) =a^{-1/2}\\cdot\\left(a^{1/2}a^{1/2}\\right)\\cdot a^{-1/2}=a^{-1/2}\\cdot a\\cdot a^{-1/2} $"}
{"_id": "2699731", "title": "", "text": "$P(X_i = 1 | X_j = 1) \\geq p^*$"}
{"_id": "3726149", "title": "", "text": "$ \\zeta(2) = \\sum_{n\\geq 1}\\frac{1}{n^2} = \\sum_{n\\geq 1}\\frac{H_n}{n 2^{n-1}}.$"}
{"_id": "2052072", "title": "", "text": "$p_1p_2=0$"}
{"_id": "1884049", "title": "", "text": "$\\zeta(s)=\\sum_{n=1}^{\\infty} \\frac{1}{n^s}=\\frac{1}{1^s}+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+...\\Longleftrightarrow$"}
{"_id": "4410541", "title": "", "text": "$\\frac{2}{a} + \\frac{4}{b} = 1$"}
{"_id": "5891114", "title": "", "text": "$ P \\left( X \\geq n \\right) = \\begin{cases} \\frac{1}{ {2}^{n - 2} } & n \\geq 2 \\\\ 1 & n \\leq 1 \\end{cases} $"}
{"_id": "3739867", "title": "", "text": "$\\left\\lfloor\\frac n3\\left\\lfloor\\frac{n-1}2\\right\\rfloor\\right\\rfloor-\\varepsilon$"}
{"_id": "1188392", "title": "", "text": "$cov(X,E(Y|X))=cov(X,Y)$"}
{"_id": "8827667", "title": "", "text": "$(J_{(0,0)},J_{(1,0)},J_{(0,1)},\\dots )$"}
{"_id": "3522288", "title": "", "text": "$2|x+y-z|+2|x-y+z|+2|-x+y+z|$"}
{"_id": "2391438", "title": "", "text": "$|X|^p \\leq |X-X_n|^p + |X_n|^p$"}
{"_id": "1328977", "title": "", "text": "$\\frac{\\zeta'(s)}{\\zeta(s)}=log(2)+log(\\pi)+\\frac{\\pi}{2}cot(\\frac{\\pi s}{2})-\\frac{\\Gamma'(1-s)}{\\Gamma(1-s)}-\\frac{\\zeta'(1-s)}{\\zeta(1-s)}$"}
{"_id": "2651644", "title": "", "text": "$a = 2/4 = 3/6 = ... = (1*k)/(2*k)$"}
{"_id": "811825", "title": "", "text": "$f(x) = 1/27$"}
{"_id": "4053012", "title": "", "text": "$ADD(x,y) = x + y$"}
{"_id": "8852932", "title": "", "text": "$s[\\mathbf{1}_{\\{\\tau=0\\}\\cup\\{s\\geq z\\}}]=s\\left[1-\\mathbf{1}_{\\{\\tau=1\\}\\cap\\{s<z\\}}\\right]=s\\left[1-\\tau\\mathbf{1}_{\\left(-\\infty,z\\right)}\\left(s\\right)\\right]=s-\\tau s\\mathbf{1}_{\\left(-\\infty,z\\right)}\\left(s\\right)$"}
{"_id": "8868114", "title": "", "text": "$3^x+a^x\\ge 6^x+9^x$"}
{"_id": "8990946", "title": "", "text": "$\\bigcup_rC_r =G\\supset\\ldots\\supset C_r\\supset C_{r+1}\\supset \\ldots C_0\\supset C_1\\supset C_2\\supset\\ldots \\supset 0=\\bigcap_rC_r\\,.$"}
{"_id": "1697542", "title": "", "text": "$ \\forall \\epsilon > 0, \\exists \\delta > 0, \\forall x \\in X : |x - y| < \\delta \\implies |f(x) - f(y)| < \\epsilon $"}
{"_id": "6868978", "title": "", "text": "$D = (1+x)^{n+1} - (1+x)^n - nx^2 $"}
{"_id": "4135361", "title": "", "text": "$\\binom{N}{2} = \\Omega(N^2)$"}
{"_id": "8033324", "title": "", "text": "$d(x,y)=\\sqrt{\\langle x,y \\rangle}$"}
{"_id": "3563671", "title": "", "text": "$a <x <b;c <y <d$"}
{"_id": "5259438", "title": "", "text": "$F([x,y,z])=([x^2,y^2,z^2,xy,yz,zx])$"}
{"_id": "3177787", "title": "", "text": "$\\ell_p(t) = f(p) + f'(p)(t-p)$"}