{"_id": "Acoustic_theory:28", "title": "", "text": "$\\begin{aligned}\\displaystyle\\nabla\\cdotsymbol{\\tau}\\equiv\\cfrac{\\partial s_{ij}}{\\partial x_{i}}&\\displaystyle=\\mu\\left[\\cfrac{\\partial}{\\partial x_{i}}\\left(\\cfrac{\\partial u_{i}}{\\partial x_{j}}+\\cfrac{\\partial u_{j}}{\\partial x_{i}}\\right)\\right]+\\lambda~{}\\left[\\cfrac{\\partial}{\\partial x_{i}}\\left(\\cfrac{\\partial u_{k}}{\\partial x_{k}}\\right)\\right]\\delta_{ij}\\\\ &\\displaystyle=\\mu~{}\\cfrac{\\partial^{2}u_{i}}{\\partial x_{i}\\partial x_{j}}+\\mu~{}\\cfrac{\\partial^{2}u_{j}}{\\partial x_{i}\\partial x_{i}}+\\lambda~{}\\cfrac{\\partial^{2}u_{k}}{\\partial x_{k}\\partial x_{j}}\\\\ &\\displaystyle=(\\mu+\\lambda)~{}\\cfrac{\\partial^{2}u_{i}}{\\partial x_{i}\\partial x_{j}}+\\mu~{}\\cfrac{\\partial^{2}u_{j}}{\\partial x_{i}^{2}}\\\\ &\\displaystyle\\equiv(\\mu+\\lambda)~{}\\nabla(\\nabla\\cdot\\mathbf{u})+\\mu~{}\\nabla^{2}\\mathbf{u}~{}.\\end{aligned}$"}
{"_id": "Generalized_beta_distribution:31", "title": "", "text": "$\\begin{aligned}\\displaystyle G=\\left({\\frac{1}{2\\mu}}\\right)\\operatorname{E}(|Y-X|)=\\left(P{\\frac{1}{2\\mu}}\\right)\\int_{0}^{\\infty}\\int_{0}^{\\infty}|x-y|f(x)f(y)\\,dxdy\\\\ \\displaystyle=1-\\frac{\\int_{0}^{\\infty}(1-F(y))^{2}\\,dy}{\\int_{0}^{\\infty}(1-F(y))\\,dy}\\\\ \\displaystyle P=\\left(\\frac{1}{2\\mu}\\right)\\operatorname{E}(|Y-\\mu|)=\\left(\\frac{1}{2\\mu}\\right)\\int_{0}^{\\infty}|y-\\mu|f(y)\\,dy\\\\ \\displaystyle T=\\operatorname{E}(\\ln(Y/\\mu)^{Y/\\mu})=\\int_{0}^{\\infty}(y/\\mu)\\ln(y/\\mu)f(y)\\,dy\\end{aligned}$"}
{"_id": "Fundamental_theorem_of_calculus:68", "title": "", "text": "$\\int_{2}^{5}x^{2}\\,dx=F(5)-F(2)=\\frac{5^{3}}{3}-\\frac{2^{3}}{3}=\\frac{125}{3}-\\frac{8}{3}=\\frac{117}{3}=39.$"}
{"_id": "Wetting:2", "title": "", "text": "$\\gamma_{\\alpha\\theta}\\cos{\\alpha}+\\gamma_{\\theta\\beta}\\cos{\\beta}+\\gamma_{\\alpha\\beta}\\ =0$"}
{"_id": "Reciprocal_rule:1", "title": "", "text": "$\\displaystyle\\frac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\frac{1}{g(x)}\\right)=\\frac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\frac{f(x)}{g(x)}\\right)$"}
{"_id": "Cramer's_rule:96", "title": "", "text": "$\\begin{vmatrix}b_{1}&a_{12}\\\\ b_{2}&a_{22}\\end{vmatrix}=\\begin{vmatrix}a_{11}x_{1}&a_{12}\\\\ a_{21}x_{1}&a_{22}\\end{vmatrix}=x_{1}\\begin{vmatrix}a_{11}&a_{12}\\\\ a_{21}&a_{22}\\end{vmatrix}$"}
{"_id": "TITAN2D:0", "title": "", "text": "${\\underbrace{\\partial h\\over\\partial t}}_{\\begin{smallmatrix}\\,\\text{Change}\\\\ \\,\\text{in mass}\\\\ \\,\\text{over time}\\end{smallmatrix}}+\\underbrace{{\\partial\\overline{hu}\\over\\partial x}+{\\partial\\overline{hv}\\over\\partial y}}_{\\begin{smallmatrix}\\,\\text{Total spatial}\\\\ \\,\\text{variation of}\\\\ \\,\\text{x,y mass fluxes}\\end{smallmatrix}}=0$"}
{"_id": "Field_trace:19", "title": "", "text": "$ax^{2}+bx+c=0,\\,\\text{ with }a\\neq 0,$"}
{"_id": "William_Brouncker,_2nd_Viscount_Brouncker:2", "title": "", "text": "$\\frac{1}{1+\\frac{1^{2}}{2+\\frac{3^{2}}{2}}}=\\frac{13}{15}=1-\\frac{1}{3}+\\frac{1}{5}.$"}
{"_id": "Föppl–von_Kármán_equations:16", "title": "", "text": "$E_{ij}:=\\frac{1}{2}\\left[\\frac{\\partial u_{i}}{\\partial x_{j}}+\\frac{\\partial u_{j}}{\\partial x_{i}}+\\frac{\\partial u_{k}}{\\partial x_{i}}\\,\\frac{\\partial u_{k}}{\\partial x_{j}}\\right]\\,.$"}
{"_id": "Graph_algebra:6", "title": "", "text": "$x,y\\in V,(x,y)\\in E$"}
{"_id": "Metallicity:10", "title": "", "text": "$[\\mathrm{Fe}/\\mathrm{H}]=\\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_{\\mathrm{star}}}-\\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_{\\mathrm{sun}}}$"}
{"_id": "Divided_differences:41", "title": "", "text": "$f[x_{0},\\dots,x_{n}]=\\sum_{j=0}^{n}\\frac{f(x_{j})}{q^{\\prime}(x_{j})}.$"}
{"_id": "Henry's_law:7", "title": "", "text": "$\\rm atm\\,$"}
{"_id": "Posterior_probability:22", "title": "", "text": "$f_{X\\mid Y=y}(x)={f_{X}(x)L_{X\\mid Y=y}(x)\\over{\\int_{-\\infty}^{\\infty}f_{X}(x)L_{X\\mid Y=y}(x)\\,dx}}$"}
{"_id": "Hypercube:7", "title": "", "text": "${n\\choose m}=\\frac{n!}{m!\\,(n-m)!}$"}
{"_id": "Classical_central-force_problem:22", "title": "", "text": "$\\hat{symbol\\varphi}\\cdot\\mathbf{\\hat{r}}=-\\sin\\varphi\\cos\\varphi+\\cos\\varphi\\sin\\varphi=0$"}
{"_id": "Duality_(electrical_circuits):2", "title": "", "text": "$v_{C}(t)=V_{0}+{1\\over C}\\int_{0}^{t}i_{C}(\\tau)\\,d\\tau\\iff i_{L}(t)=I_{0}+{1\\over L}\\int_{0}^{t}v_{L}(\\tau)\\,d\\tau$"}
{"_id": "Expander_code:157", "title": "", "text": "$O(mr)\\,$"}
{"_id": "Taylor's_law:80", "title": "", "text": "$s^{2}=ap^{b}(1-p)^{c}$"}
{"_id": "Pell_number:53", "title": "", "text": "$41-29\\sqrt{2}=-0.01219\\ldots$"}
{"_id": "Acoustic_wave_equation:29", "title": "", "text": "$\\frac{\\partial}{\\partial t}(\\rho_{0}+\\rho_{0}s)+\\frac{\\partial}{\\partial x}(\\rho_{0}u+\\rho_{0}su)=0$"}
{"_id": "Petersen_matrix:4", "title": "", "text": "$A+B\\rightarrow S$"}
{"_id": "Eigendecomposition_of_a_matrix:18", "title": "", "text": "$\\begin{bmatrix}1&0\\\\ 1&3\\\\ \\end{bmatrix}\\begin{bmatrix}a&b\\\\ c&d\\\\ \\end{bmatrix}=\\begin{bmatrix}a&b\\\\ c&d\\\\ \\end{bmatrix}\\begin{bmatrix}x&0\\\\ 0&y\\\\ \\end{bmatrix}$"}
{"_id": "Peano_axioms:30", "title": "", "text": "$\\forall x,y\\in Nx<y\\Rightarrow\\exists z\\in Nx+z=y$"}
{"_id": "Carl_Johan_Malmsten:20", "title": "", "text": "$L(s)\\equiv\\sum_{n=0}^{\\infty}\\frac{(-1)^{n}}{(2n+1)^{s}}\\qquad\\qquad L(1-s)=L(s)\\Gamma(s)2^{s}\\pi^{-s}\\sin\\frac{\\pi s}{2},$"}
{"_id": "Bloch_group:10", "title": "", "text": "$\\operatorname{B}^{\\bullet}:A(K)\\stackrel{d}{\\longrightarrow}\\wedge^{2}K^{*}$"}
{"_id": "Approximations_of_π:185", "title": "", "text": "$\\pi=\\cfrac{4}{1+\\cfrac{1^{2}}{3+\\cfrac{2^{2}}{5+\\cfrac{3^{2}}{7+\\ddots}}}}\\!$"}
{"_id": "Bessel's_correction:36", "title": "", "text": "$\\begin{aligned}\\displaystyle\\operatorname{E}(s^{2})&\\displaystyle=\\operatorname{E}\\left(\\sum_{i=1}^{n}\\frac{(x_{i}-\\overline{x})^{2}}{n-1}\\right)\\\\ &\\displaystyle=\\frac{1}{n-1}\\operatorname{E}\\left(\\sum_{i=1}^{n}\\left[x_{i}-\\mu-\\left(\\overline{x}-\\mu\\right)\\right]^{2}\\right)\\\\ &\\displaystyle=\\frac{1}{n-1}\\left[\\sum_{i=1}^{n}\\operatorname{Var}\\left(x_{i}\\right)-n\\operatorname{Var}\\left(\\overline{x}\\right)\\right]\\end{aligned}$"}
{"_id": "Vizing's_theorem:61", "title": "", "text": "$O(m\\sqrt{n}\\log n)$"}
{"_id": "Nernst_equation:56", "title": "", "text": "$\\mathrm{Ox}+ne^{-}\\rightarrow\\mathrm{Red},$"}
{"_id": "Image_derivatives:23", "title": "", "text": "$\\mathbf{k}=\\mathbf{u}^{T}M=(M^{T}\\mathbf{v})^{T}$"}
{"_id": "Satake_diagram:1", "title": "", "text": "$\\mathfrak{g}=\\mathfrak{k}\\oplus\\mathfrak{p}$"}
{"_id": "Dirichlet_beta_function:0", "title": "", "text": "$\\beta(s)=\\sum_{n=0}^{\\infty}\\frac{(-1)^{n}}{(2n+1)^{s}},$"}
{"_id": "Proof_that_π_is_irrational:57", "title": "", "text": "$\\int_{-x}^{x}(x^{2}-y^{2})^{n}\\cos(y)\\,dy.$"}
{"_id": "Dynamization:22", "title": "", "text": "$Q_{D}\\left(n\\right)=O(Q_{S}\\left(n\\right)\\log\\left(n\\right))\\,\\!$"}
{"_id": "List_of_New_Testament_papyri:50", "title": "", "text": "$\\mathfrak{P}^{27}$"}
{"_id": "Surface_plasmon_polariton:5", "title": "", "text": "$\\omega_{P}=\\sqrt{\\frac{ne^{2}}{{\\varepsilon_{0}}m^{*}}}$"}
{"_id": "Weight_function:40", "title": "", "text": "$\\frac{\\int_{\\Omega}f(x)\\ w(x)dx}{\\int_{\\Omega}w(x)\\ dx}$"}
{"_id": "Pell_number:14", "title": "", "text": "$\\frac{577}{408}=1+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2}}}}}}}.$"}
{"_id": "Universal_joint:17", "title": "", "text": "$\\hat{\\mathbf{x}}_{2}=[-\\cos\\beta\\sin\\gamma_{2}\\,,\\,\\cos\\gamma_{2}\\,,\\,\\sin\\beta\\sin\\gamma_{2}]$"}
{"_id": "Misiurewicz_point:25", "title": "", "text": "$\\{0,i,i-1,-i,i-1,-i...\\}\\,$"}
{"_id": "Probability:3", "title": "", "text": "$=1-\\tfrac{1}{6}=\\tfrac{5}{6}$"}
{"_id": "Subtractor:12", "title": "", "text": "$D=y\\oplus x$"}
{"_id": "Dirichlet_L-function:0", "title": "", "text": "$L(s,\\chi)=\\sum_{n=1}^{\\infty}\\frac{\\chi(n)}{n^{s}}.$"}
{"_id": "Localizing_subcategory:2", "title": "", "text": "$0\\rightarrow A^{\\prime}\\rightarrow A\\rightarrow A^{\\prime\\prime}\\rightarrow 0$"}
{"_id": "Crystal_system:64", "title": "", "text": "$a=d\\neq b=c,\\alpha=\\zeta=120^{\\circ},\\beta=\\gamma=\\delta=\\epsilon=90^{\\circ},$"}
{"_id": "Monic_polynomial:10", "title": "", "text": "$C:=\\{b\\in B:\\exists\\,p(x)\\in A[x]\\,,\\hbox{ which is monic and such that }p(b)=0\\}\\,.$"}
{"_id": "Concordance_correlation_coefficient:11", "title": "", "text": "$s_{xy}=\\frac{1}{N}\\sum_{n=1}^{N}(x_{n}-\\bar{x})(y_{n}-\\bar{y}).$"}
{"_id": "Conditional_convergence:3", "title": "", "text": "$1-{1\\over 2}+{1\\over 3}-{1\\over 4}+{1\\over 5}-\\cdots=\\sum\\limits_{n=1}^{\\infty}{(-1)^{n+1}\\over n}$"}
{"_id": "Edge-graceful_labeling:2", "title": "", "text": "$\\{0,1\\dots p-1\\}$"}
{"_id": "Young's_lattice:2", "title": "", "text": "$\\displaystyle\\underbrace{1+\\cdots\\cdots\\cdots+1}_{n\\,\\text{ terms}}$"}
{"_id": "Neutral_particle_oscillation:1", "title": "", "text": "${{\\nu}_{e}}+n\\to p+e^{-}$"}
{"_id": "Quantum_gate:17", "title": "", "text": "$X=\\begin{bmatrix}0&1\\\\ 1&0\\end{bmatrix}$"}
{"_id": "Casus_irreducibilis:22", "title": "", "text": "$4x^{3}-3x-\\cos(\\theta)=0.$"}
{"_id": "Iterated_logarithm:0", "title": "", "text": "$\\log^{*}n:=\\begin{cases}0&\\mbox{if }~{}n\\leq 1;\\\\ 1+\\log^{*}(\\log n)&\\mbox{if }~{}n>1\\end{cases}$"}
{"_id": "Inverse_Galois_problem:58", "title": "", "text": "$αβ+βγ+γα=−2$"}
{"_id": "Logan_plot:17", "title": "", "text": "${{\\int_{0}^{t}\\mathrm{ROI}(\\tau)d\\tau}\\over\\mathrm{ROI}(t)}$"}
{"_id": "Bram_van_Leer:4", "title": "", "text": "$j+\\frac{1}{2}$"}
{"_id": "Gravitational_lensing_formalism:27", "title": "", "text": "$A_{ij}=\\frac{\\partial\\beta_{i}}{\\partial\\theta_{j}}=\\delta_{ij}-\\frac{\\partial\\alpha_{i}}{\\partial\\theta_{j}}=\\delta_{ij}-\\frac{\\partial^{2}\\psi}{\\partial\\theta_{i}\\partial\\theta_{j}}$"}
{"_id": "Duckworth–Lewis_method:2", "title": "", "text": "$\\begin{matrix}\\,\\text{Total}\\\\ \\,\\text{resources}\\\\ \\,\\text{available}\\end{matrix}\\ \\ \\ =\\ \\ \\ \\begin{matrix}\\,\\text{Resources}\\\\ \\,\\text{at start}\\\\ \\,\\text{of innings}\\end{matrix}\\ \\ \\ \\ -\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\begin{matrix}\\,\\text{Resources lost by}\\\\ \\,\\text{first interruption}\\end{matrix}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\begin{matrix}\\,\\text{Resources lost by}\\\\ \\,\\text{second interruption}\\end{matrix}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\begin{matrix}\\,\\text{Resources lost by}\\\\ \\,\\text{third interruption}\\end{matrix}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -\\ \\ \\ \\ \\ \\,\\text{etc}...$"}
{"_id": "Matrix_calculus:180", "title": "", "text": "$\\left(\\frac{\\partial\\mathbf{u}}{\\partial x}\\right)^{\\top}$"}
{"_id": "Algorithms_for_calculating_variance:17", "title": "", "text": "$\\mathrm{variance}=s^{2}=\\displaystyle\\frac{\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}}{n-1}\\!$"}
{"_id": "Group_cohomology:3", "title": "", "text": "$0\\to L\\to M\\to N\\to 0$"}
{"_id": "Hinge_loss:22", "title": "", "text": "$\\ell(y)=\\begin{cases}\\frac{1}{2}-ty&\\,\\text{if}~{}~{}ty\\leq 0,\\\\ \\frac{1}{2}(1-ty)^{2}&\\,\\text{if}~{}~{}0<ty\\leq 1,\\\\ 0&\\,\\text{if}~{}~{}1\\leq ty\\end{cases}$"}
{"_id": "Curvelet:31", "title": "", "text": "$V(\\frac{N\\omega}{2\\pi})$"}
{"_id": "List_of_formulae_involving_π:48", "title": "", "text": "$\\pi=<mtpl>{{1}}+\\frac{{1}}{{2}}+\\frac{{1}}{{3}}+\\frac{{1}}{{4}}-\\frac{{1}}{{5}}+\\frac{{1}}{{6}}+\\frac{{1}}{{7}}+\\frac{{1}}{{8}}+\\frac{{1}}{{9}}-\\frac{{1}}{{10}}+\\frac{{1}}{{11}}+\\frac{{1}}{{12}}-\\frac{{1}}{{13}}+\\cdots\\!$"}
{"_id": "Beer–Lambert_law:1", "title": "", "text": "$T=e^{-\\sum_{i=1}^{N}\\sigma_{i}\\int_{0}^{\\ell}n_{i}(z)\\mathrm{d}z}=10^{-\\sum_{i=1}^{N}\\varepsilon_{i}\\int_{0}^{\\ell}c_{i}(z)\\mathrm{d}z},$"}
{"_id": "Lawson_criterion:2", "title": "", "text": "${}^{2}_{1}\\mathrm{D}+\\,^{3}_{1}\\mathrm{T}\\rightarrow\\,^{4}_{2}\\mathrm{He}\\left(3.5\\,\\mathrm{MeV}\\right)+\\,^{1}_{0}\\mathrm{n}\\left(14.1\\,\\mathrm{MeV}\\right)$"}
{"_id": "Langmuir_probe:14", "title": "", "text": "$\\langle v_{e}\\rangle=\\frac{\\int_{v_{e0}}^{\\infty}f(v_{x})\\,v_{x}\\,dv_{x}}{\\int_{-\\infty}^{\\infty}f(v_{x})\\,dv_{x}}$"}
{"_id": "Edmonds'_algorithm:71", "title": "", "text": "$O(E\\log V)$"}
{"_id": "Quadratic_Gauss_sum:20", "title": "", "text": "$an^{2}+bn+q=0$"}
{"_id": "Cesàro_summation:27", "title": "", "text": "$\\lim_{\\lambda\\to\\infty}\\int_{0}^{\\lambda}\\left(1-\\frac{x}{\\lambda}\\right)^{\\alpha}f(x)\\,dx$"}
{"_id": "XOR_swap_algorithm:16", "title": "", "text": "$(A\\oplus B)\\oplus A=A\\oplus(A\\oplus B)$"}
{"_id": "Fat_object:2", "title": "", "text": "$\\left(\\frac{\\,\\text{volume of}\\ o}{\\,\\text{volume of largest ball enclosed in}\\ o}\\right)^{1/d}$"}
{"_id": "Group_extension:2", "title": "", "text": "$1\\rightarrow K\\stackrel{i}{\\rightarrow}G\\stackrel{\\pi}{\\rightarrow}H\\rightarrow 1$"}
{"_id": "Thermodynamic_potential:89", "title": "", "text": "$x_{j}=\\left(\\frac{\\partial\\Phi}{\\partial y_{j}}\\right)_{\\{y_{i\\neq j}\\}}$"}
{"_id": "Representation_theory_of_finite_groups:1", "title": "", "text": "$n=\\begin{bmatrix}0&1\\\\ 1&0\\end{bmatrix}$"}
{"_id": "Method_of_steepest_descent:42", "title": "", "text": "$\\left.\\frac{\\partial^{2}S(z)}{\\partial z_{i}\\partial z_{j}}\\right|_{z=0}=2h_{ij}(0);$"}
{"_id": "Loewner_differential_equation:43", "title": "", "text": "$\\displaystyle{|a_{3}|=2\\int_{0}^{\\infty}|\\Re\\alpha^{2}|\\,dt+4\\left(\\int_{0}^{\\infty}\\Re\\alpha\\,dt\\right)^{2}}\\leq 2\\int_{0}^{\\infty}|\\Re\\alpha^{2}|\\,dt+4\\left(\\int_{0}^{\\infty}e^{-t}\\,dt\\right)\\left(\\int_{0}^{\\infty}e^{t}(\\Re\\alpha)^{2}\\,dt\\right)=1+4\\int_{0}^{\\infty}(e^{-t}-e^{-2t})(\\Re\\kappa)^{2}\\,dt\\leq 3,$"}
{"_id": "Cayley's_ruled_cubic_surface:0", "title": "", "text": "$3z-3xy+x^{3}=0.$"}
{"_id": "Net_present_value:9", "title": "", "text": "$\\mathrm{NPV}(i,N)=\\sum_{t=0}^{N}\\frac{R_{t}}{(1+i)^{t}}$"}
{"_id": "Prime_zeta_function:28", "title": "", "text": "$-0.026838601\\ldots$"}
{"_id": "Sliding_mode_control:254", "title": "", "text": "$0=\\dot{\\sigma}=a_{11}\\mathord{\\overbrace{e_{1}}^{{}=0}}+A_{12}\\mathbf{e}_{2}-\\mathord{\\overbrace{v_{\\,\\text{eq}}}^{v(\\sigma)}}=A_{12}\\mathbf{e}_{2}-v_{\\,\\text{eq}}.$"}
{"_id": "Pi:122", "title": "", "text": "$\\pi=\\frac{4}{1}-\\frac{4}{3}+\\frac{4}{5}-\\frac{4}{7}+\\frac{4}{9}-\\frac{4}{11}+\\frac{4}{13}\\cdots.$"}
{"_id": "Conjugate_prior:157", "title": "", "text": "$\\mathbf{C}=\\sum_{i=1}^{n}(\\mathbf{x_{i}}-\\mathbf{\\bar{x}})(\\mathbf{x_{i}}-\\mathbf{\\bar{x}})^{T}$"}
{"_id": "Pinch_(plasma_physics):0", "title": "", "text": "$\\nabla\\times\\vec{B}=\\mu_{0}\\vec{J}$"}
{"_id": "Normalization_property_(abstract_rewriting):8", "title": "", "text": "$\\rightarrow\\ \\ldots\\,$"}
{"_id": "Random_permutation_statistics:6", "title": "", "text": "$\\scriptstyle\\mathfrak{P}_{2}$"}
{"_id": "Divergence:2", "title": "", "text": "$\\underline{\\underline{\\epsilon}}$"}
{"_id": "Egyptian_fraction:57", "title": "", "text": "$\\frac{1}{k}+\\frac{1}{k}=\\frac{1}{k}+\\frac{1}{k+1}+\\frac{1}{k(k+1)}.$"}
{"_id": "Qualitative_variation:110", "title": "", "text": "$SBD=\\sqrt{\\frac{b+c}{a+b+c+d}}$"}
{"_id": "Golden_ratio:14", "title": "", "text": "$\\varphi=\\frac{1-\\sqrt{5}}{2}=-0.6180\\,339887\\dots$"}
{"_id": "Conditional_mutual_information:22", "title": "", "text": "$X_{*}\\mathfrak{P}=\\mathfrak{P}\\big(X^{-1}(\\cdot)\\big).$"}
{"_id": "Electromagnetic_reverberation_chamber:18", "title": "", "text": "$\\langle X\\rangle_{\\infty}$"}
{"_id": "Josephus_problem:57", "title": "", "text": "$f(n)=2(n-2^{\\lfloor\\log_{2}(n)\\rfloor})+1$"}
{"_id": "Tobit_model:28", "title": "", "text": "$y_{2i}=\\begin{cases}y_{2i}^{*}&\\textrm{if}\\;y_{1i}^{*}>0\\\\ 0&\\textrm{if}\\;y_{1i}^{*}\\leq 0.\\end{cases}$"}
{"_id": "Frequency_modulation:4", "title": "", "text": "$=A_{c}\\cos\\left(2\\pi f_{c}t+2\\pi f_{\\Delta}\\int_{0}^{t}x_{m}(\\tau)d\\tau\\right)$"}
{"_id": "Well-quasi-ordering:98", "title": "", "text": "$\\forall x,y\\in X,x\\leq y\\wedge x\\in S\\Rightarrow y\\in S$"}
{"_id": "Antimetric_electrical_network:10", "title": "", "text": "$\\left[\\mathbf{S}\\right]=\\begin{bmatrix}S_{11}&S_{12}\\\\ S_{12}&S_{11}\\end{bmatrix}$"}
{"_id": "Hilbert_transform:38", "title": "", "text": "$H\\left(\\frac{du}{dt}\\right)=\\frac{d}{dt}H(u)$"}
{"_id": "Ideal_gas:5", "title": "", "text": "$V=\\left(\\frac{kba}{3}\\right)\\left(\\frac{Tn}{P}\\right)$"}
{"_id": "Ion_source:7", "title": "", "text": "$A^{+}+B\\to A+B^{+}$"}
{"_id": "Student's_t-test:19", "title": "", "text": "$t\\text{score}=\\frac{(\\widehat{\\beta}-\\beta_{0})\\sqrt{n-2}}{\\sqrt{\\,\\text{SSR}/\\sum_{i=1}^{n}\\left(x_{i}-\\overline{x}\\right)^{2}}}.$"}
{"_id": "Freshman's_dream:6", "title": "", "text": "${\\left({{p}\\atop{n}}\\right)}=\\frac{p!}{n!(p-n)!}.$"}
{"_id": "Large_eddy_simulation:119", "title": "", "text": "$\\frac{\\partial\\bar{u_{i}}}{\\partial t}+\\bar{u_{j}}\\frac{\\partial\\bar{u_{i}}}{\\partial x_{j}}=-\\frac{1}{\\rho}\\frac{\\partial\\bar{p}}{\\partial x_{i}}+\\nu\\frac{\\partial^{2}\\bar{u_{i}}}{\\partial x_{j}\\partial x_{j}}-\\frac{\\partial\\tau_{ij}}{\\partial x_{j}}.$"}
{"_id": "Diversification_(finance):35", "title": "", "text": "$\\underbrace{\\,\\text{Var}(R_{P})}_{\\equiv\\sigma^{2}_{P}}=\\mathbb{E}[R_{P}-\\mathbb{E}[R_{P}]]^{2}$"}
{"_id": "Conformastatic_spacetimes:25", "title": "", "text": "$(14.a)\\quad\\nabla^{2}\\psi=\\,(\\nabla\\psi)^{2}$"}
{"_id": "Ludwig_Boltzmann:2", "title": "", "text": "$W=N!\\prod_{i}\\frac{1}{N_{i}!}$"}
{"_id": "Logistic_regression:73", "title": "", "text": "$Y_{i}=\\begin{cases}1&\\,\\text{if }Y_{i}^{1\\ast}>Y_{i}^{0\\ast},\\\\ 0&\\,\\text{otherwise.}\\end{cases}$"}
{"_id": "Fibonacci_number:51", "title": "", "text": "$\\varphi=1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\;\\;\\ddots\\,}}}$"}
{"_id": "Argumentation_framework:23", "title": "", "text": "$\\forall a,b\\in E,(a,b)\\not\\in R$"}
{"_id": "1::2_−_1::4_+_1::8_−_1::16_+_⋯:0", "title": "", "text": "$\\frac{1}{2}-\\frac{1}{4}+\\frac{1}{8}-\\frac{1}{16}+\\cdots=\\frac{1/2}{1-(-1/2)}=\\frac{1}{3}.$"}
{"_id": "Cache-oblivious_matrix_multiplication:20", "title": "", "text": "$\\Theta(mnp)$"}
{"_id": "*-autonomous_category:10", "title": "", "text": "$A^{*}\\otimes B^{*}\\to(B\\otimes A)^{*}$"}
{"_id": "Convergence_of_random_variables:74", "title": "", "text": "$X_{n}\\ \\xrightarrow{p}\\ X$"}
{"_id": "Indefinite_inner_product_space:50", "title": "", "text": "$x,\\,y\\in K$"}
{"_id": "Sundial:58", "title": "", "text": "$\\sin G=\\cos L\\cos D\\cos R-\\sin L\\sin R=-\\cos L\\cos D\\sin I+\\sin L\\cos I$"}
{"_id": "XOR_swap_algorithm:2", "title": "", "text": "$A\\oplus B=B\\oplus A$"}
{"_id": "Rounding:6", "title": "", "text": "$q=\\left\\lceil y-0.5\\right\\rceil=-\\left\\lfloor-y+0.5\\right\\rfloor\\,$"}
{"_id": "Hermite's_problem:1", "title": "", "text": "$x=\\sum_{n=0}^{\\infty}\\frac{a_{n}}{10^{n}}.$"}
{"_id": "Continued_fraction:313", "title": "", "text": "$\\pi=3+\\cfrac{1}{7+\\cfrac{1}{15+\\cfrac{1}{1+\\cfrac{1}{292+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\cfrac{1}{3+\\cfrac{1}{1+\\ddots}}}}}}}}}}}$"}
{"_id": "Alternatization:16", "title": "", "text": "$\\forall x,y\\in S,$"}
{"_id": "Quadratic_equation:76", "title": "", "text": "$ax^{2}+bx+c=0.$"}
{"_id": "Generalized_hypergeometric_function:12", "title": "", "text": "$\\,{}_{p}F_{q}(a_{1},\\ldots,a_{p};b_{1},\\ldots,b_{q};z)=\\sum_{n=0}^{\\infty}\\frac{(a_{1})_{n}\\dots(a_{p})_{n}}{(b_{1})_{n}\\dots(b_{q})_{n}}\\,\\frac{z^{n}}{n!}$"}
{"_id": "Degrees_of_freedom_(statistics):15", "title": "", "text": "$\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}=\\begin{Vmatrix}X_{1}-\\bar{X}\\\\ \\vdots\\\\ X_{n}-\\bar{X}\\end{Vmatrix}^{2}.$"}
{"_id": "Elastic_energy:12", "title": "", "text": "$\\sigma_{ij}=\\left(\\frac{\\partial f}{\\partial\\varepsilon_{ij}}\\right)_{T},$"}
{"_id": "Spence's_function:11", "title": "", "text": "$\\operatorname{Li}_{2}\\left(\\frac{1}{3}\\right)-\\frac{1}{6}\\operatorname{Li}_{2}\\left(\\frac{1}{9}\\right)=\\frac{{\\pi}^{2}}{18}-\\frac{\\ln^{2}3}{6}$"}
{"_id": "Elliptic_partial_differential_equation:1", "title": "", "text": "$B^{2}-AC<0.$"}
{"_id": "Mortgage_yield:0", "title": "", "text": "$\\mbox{Mortgage Yield: ri such that P}~{}=\\sum_{n=1}^{N}\\frac{C(t)}{(1+ri/1200)^{t-1}}$"}
{"_id": "Gauss's_continued_fraction:75", "title": "", "text": "$\\frac{\\pi}{4}=\\cfrac{1}{1+\\cfrac{1^{2}}{2+\\cfrac{3^{2}}{2+\\cfrac{5^{2}}{2+\\ddots}}}}=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+-\\dots$"}
{"_id": "Tonelli–Shanks_algorithm:59", "title": "", "text": "$2^{S^{\\prime}}$"}
{"_id": "Vector_Laplacian:16", "title": "", "text": "$\\nabla^{2}\\mathbf{E}-\\mu_{0}\\epsilon_{0}\\frac{\\partial^{2}\\mathbf{E}}{\\partial t^{2}}=0.$"}
{"_id": "MQV:3", "title": "", "text": "$L=\\left\\lceil\\frac{\\lfloor\\log_{2}n\\rfloor+1}{2}\\right\\rceil$"}
{"_id": "Higher-dimensional_gamma_matrices:110", "title": "", "text": "$\\gamma\\text{chir}=\\gamma_{0}\\gamma_{1}=\\sigma_{3}=\\gamma\\text{chir}^{\\dagger}~{}.$"}
{"_id": "Talagrand's_concentration_inequality:9", "title": "", "text": "$x,y\\in\\Omega$"}
{"_id": "Isotopes_of_rutherfordium:0", "title": "", "text": "$\\,{}^{238}_{92}\\mathrm{U}\\ +\\,^{24}_{12}\\mathrm{Mg}\\to\\,^{259}_{104}\\mathrm{Rf}\\ +3\\,^{1}_{0}\\mathrm{n}$"}
{"_id": "Continuous_foam_separation:10", "title": "", "text": "$\\text{ Enrichment ratio}=\\left(\\frac{\\text{Protein concentration in the foam}}{\\text{Protein concentration in the initial feed}}\\right)$"}
{"_id": "Quotition_and_partition:3", "title": "", "text": "$6=\\underbrace{3+3}_{\\,\\text{2 parts}}.$"}
{"_id": "Generalized_normal_distribution:25", "title": "", "text": "$g^{\\prime}(\\beta)=-\\frac{\\psi(1/\\beta)}{\\beta^{2}}-\\frac{\\psi^{\\prime}(1/\\beta)}{\\beta^{3}}+\\frac{1}{\\beta^{2}}-\\frac{\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta}(\\log|x_{i}-\\mu|)^{2}}{\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta}}+\\frac{(\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta}\\log|x_{i}-\\mu|)^{2}}{(\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta})^{2}}+\\frac{\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta}\\log|x_{i}-\\mu|}{\\beta\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta}}-\\frac{\\log(\\frac{\\beta}{N}\\sum_{i=1}^{N}|x_{i}-\\mu|^{\\beta})}{\\beta^{2}},$"}
{"_id": "Jarque–Bera_test:2", "title": "", "text": "$C=\\frac{\\hat{\\mu}_{4}}{\\hat{\\sigma}^{4}}=\\frac{\\frac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{4}}{\\left(\\frac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}\\right)^{2}},$"}
{"_id": "Convolution:1", "title": "", "text": "$\\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty}f(\\tau)\\,g(t-\\tau)\\,d\\tau$"}
{"_id": "Double_auction:1", "title": "", "text": "$1-1/k$"}
{"_id": "Weierstrass's_elliptic_functions:48", "title": "", "text": "$[\\wp^{\\prime}(z)]^{2}|_{z=0}\\sim\\frac{4}{z^{6}}-\\frac{24}{z^{2}}\\sum\\frac{1}{(m\\omega_{1}+n\\omega_{2})^{4}}-80\\sum\\frac{1}{(m\\omega_{1}+n\\omega_{2})^{6}}$"}
{"_id": "Cyclotomic_fast_Fourier_transform:10", "title": "", "text": "$L(x)=\\sum_{i}l_{i}x^{p^{i}},l_{i}\\in\\mathrm{GF}(p^{m}).$"}
{"_id": "D'Agostino's_K-squared_test:1", "title": "", "text": "$\\displaystyle g_{1}=\\frac{m_{3}}{m_{2}^{3/2}}=\\frac{\\frac{1}{n}\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^{3}}{\\left(\\frac{1}{n}\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^{2}\\right)^{3/2}}\\ ,$"}
{"_id": "Thermal_oxidation:1", "title": "", "text": "$\\rm Si+O_{2}\\rightarrow SiO_{2}\\,$"}
{"_id": "Lyapunov_stability:44", "title": "", "text": "$\\forall\\epsilon>0\\ \\exists\\delta>0\\ \\forall y\\in X\\ \\left[d(x,y)<\\delta\\Rightarrow\\forall n\\in\\mathbf{N}\\ d\\left(f^{n}(x),f^{n}(y)\\right)<\\epsilon\\right].$"}
{"_id": "Nuclear_forensics:0", "title": "", "text": "$A+n\\to B^{*}+\\gamma$"}
{"_id": "Plus-minus_sign:6", "title": "", "text": "$\\cos(A+B)=\\cos(A)\\cos(B)+\\sin(A)\\sin(B)\\,$"}
{"_id": "Ext_functor:7", "title": "", "text": "$0\\rightarrow B\\rightarrow E\\rightarrow A\\rightarrow 0$"}
{"_id": "Prime_zeta_function:36", "title": "", "text": "$0.00304936208\\ldots$"}
{"_id": "Dilution_of_precision_(GPS):15", "title": "", "text": "$Q=\\left(A^{T}A\\right)^{-1}$"}
{"_id": "Bernoulli_polynomials:61", "title": "", "text": "$S_{\\nu}(x)=\\sum_{k=0}^{\\infty}\\frac{\\sin((2k+1)\\pi x)}{(2k+1)^{\\nu}}$"}
{"_id": "Poisson_limit_theorem:8", "title": "", "text": "$p_{n}(k)=\\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}.$"}
{"_id": "Symmetric_difference:5", "title": "", "text": "$A\\oplus B.$"}
{"_id": "Zermelo–Fraenkel_set_theory:19", "title": "", "text": "$\\forall x\\forall y\\exists z(x\\in z\\land y\\in z).$"}
{"_id": "1_−_2_+_4_−_8_+_⋯:2", "title": "", "text": "$\\frac{a_{0}}{2}-\\frac{\\Delta a_{0}}{4}+\\frac{\\Delta^{2}a_{0}}{8}-\\frac{\\Delta^{3}a_{0}}{16}+\\cdots=\\frac{1}{2}-\\frac{1}{4}+\\frac{1}{8}-\\frac{1}{16}+\\cdots.$"}
{"_id": "Bicentric_quadrilateral:48", "title": "", "text": "$\\frac{1}{IA^{2}}+\\frac{1}{IC^{2}}=\\frac{1}{IB^{2}}+\\frac{1}{ID^{2}}=\\frac{1}{r^{2}}$"}
{"_id": "Hadamard_code:11", "title": "", "text": "$\\frac{1}{2}-\\epsilon$"}
{"_id": "History_of_trigonometry:6", "title": "", "text": "$\\sin\\left(a+b\\right)=\\sin a\\cos b+\\cos a\\sin b$"}
{"_id": "Rotation_group_SO(3):169", "title": "", "text": "$Z=\\alpha X+\\beta Y+\\gamma[X,Y],$"}
{"_id": "Contraction_(operator_theory):43", "title": "", "text": "$\\displaystyle{T={1\\over 2}I-{1\\over 2}\\int_{0}^{\\infty}e^{-t}T(t)\\,dt.}$"}
{"_id": "Backhouse's_constant:0", "title": "", "text": "$1+\\cfrac{1}{2+\\cfrac{1}{5+\\cfrac{1}{5+\\cfrac{1}{4+\\ddots}}}}$"}
{"_id": "Beer–Lambert_law:11", "title": "", "text": "$T=e^{-\\int_{0}^{\\ell}\\mu(z)\\mathrm{d}z}=10^{-\\int_{0}^{\\ell}\\mu_{10}(z)\\mathrm{d}z},$"}
{"_id": "Microcanonical_ensemble:23", "title": "", "text": "$f(x)=\\begin{cases}1,&\\mathrm{if}~{}|x|<\\tfrac{1}{2},\\\\ 0,&\\mathrm{otherwise.}\\end{cases}$"}
{"_id": "Trigonometry:11", "title": "", "text": "$\\cos(A\\pm B)=\\cos A\\ \\cos B\\mp\\sin A\\ \\sin B$"}
{"_id": "Vertical_axis_wind_turbine:9", "title": "", "text": "$\\alpha=\\tan^{-1}\\left(\\frac{\\sin\\theta}{\\cos\\theta+\\lambda}\\right)$"}
{"_id": "Fat_object:33", "title": "", "text": "$\\frac{\\,\\text{radius of circle}}{\\,\\text{length of chord}}=\\frac{R}{2R\\sin\\frac{\\theta}{2}}=\\frac{1}{2\\sin\\frac{\\theta}{2}}\\approx\\frac{1}{2\\theta/2}=\\frac{1}{\\theta}$"}
{"_id": "Serial_module:7", "title": "", "text": "$U\\rightarrow V$"}
{"_id": "Helmholtz_theorem_(classical_mechanics):4", "title": "", "text": "$\\left\\langle\\cdot\\right\\rangle_{t}$"}
{"_id": "Contrast_transfer_function:30", "title": "", "text": "$\\alpha_{s}=\\arctan\\left(\\frac{b}{R}\\right)-\\arctan\\left(\\frac{b}{R+r_{s}}\\right)$"}
{"_id": "Rational_motion:20", "title": "", "text": "$H^{2n}(t)]=\\sum\\limits_{k=0}^{2n}{B_{k}^{2n}(t)[H_{k}]},$"}
{"_id": "LCP_array:43", "title": "", "text": "$O(m+\\log n)$"}
{"_id": "Mereology:3", "title": "", "text": "$Pxy\\leftrightarrow\\forall z[Ozx\\rightarrow Ozy].$"}
{"_id": "Exclusive_or:48", "title": "", "text": "$(A\\oplus B)$"}
{"_id": "Kabsch_algorithm:3", "title": "", "text": "$U=(A^{T}A)^{1/2}A^{-1}$"}
{"_id": "Preconditioner:44", "title": "", "text": "$Ax-b=0.$"}
{"_id": "Euclidean_division:22", "title": "", "text": "$d=-\\left\\lfloor\\frac{m}{2}\\right\\rfloor$"}
{"_id": "Lambda_calculus_definition:69", "title": "", "text": "$(\\forall z:z\\not\\in FV(y)z\\not\\in BV(b))\\to\\operatorname{beta-redex}[\\lambda x.b\\ y]=b[x:=y]$"}
{"_id": "Common_integrals_in_quantum_field_theory:1", "title": "", "text": "$G^{2}=\\left(\\int_{-\\infty}^{\\infty}e^{-{1\\over 2}x^{2}}\\,dx\\right)\\cdot\\left(\\int_{-\\infty}^{\\infty}e^{-{1\\over 2}y^{2}}\\,dy\\right)=2\\pi\\int_{0}^{\\infty}re^{-{1\\over 2}r^{2}}\\,dr=2\\pi\\int_{0}^{\\infty}e^{-w}\\,dw=2\\pi.$"}
{"_id": "Polygamma_function:37", "title": "", "text": "$\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(z+k)^{m+1}}=\\frac{1}{(-2)^{m+1}m!}\\left[\\psi^{(m)}\\left(\\frac{z}{2}\\right)-\\psi^{(m)}\\left(\\frac{z+1}{2}\\right)\\right]$"}
{"_id": "Free_electron_model:61", "title": "", "text": "$\\Delta V\\Delta{k}=(2\\pi)^{3}$"}
{"_id": "Karplus–Strong_string_synthesis:0", "title": "", "text": "$-2\\pi$"}
{"_id": "Jacobian_matrix_and_determinant:83", "title": "", "text": "$\\mathbf{J}_{\\mathbf{F}}(x_{1},x_{2},x_{3})=\\begin{bmatrix}\\dfrac{\\partial y_{1}}{\\partial x_{1}}&\\dfrac{\\partial y_{1}}{\\partial x_{2}}&\\dfrac{\\partial y_{1}}{\\partial x_{3}}\\\\ \\dfrac{\\partial y_{2}}{\\partial x_{1}}&\\dfrac{\\partial y_{2}}{\\partial x_{2}}&\\dfrac{\\partial y_{2}}{\\partial x_{3}}\\\\ \\dfrac{\\partial y_{3}}{\\partial x_{1}}&\\dfrac{\\partial y_{3}}{\\partial x_{2}}&\\dfrac{\\partial y_{3}}{\\partial x_{3}}\\\\ \\dfrac{\\partial y_{4}}{\\partial x_{1}}&\\dfrac{\\partial y_{4}}{\\partial x_{2}}&\\dfrac{\\partial y_{4}}{\\partial x_{3}}\\end{bmatrix}=\\begin{bmatrix}1&0&0\\\\ 0&0&5\\\\ 0&8x_{2}&-2\\\\ x_{3}\\cos x_{1}&0&\\sin x_{1}\\end{bmatrix}.$"}
{"_id": "Range_query_(data_structures):55", "title": "", "text": "$\\Omega\\left(\\frac{\\log n}{\\log(Sw/n)}\\right)$"}
{"_id": "Conway's_LUX_method_for_magic_squares:0", "title": "", "text": "$\\mathrm{L}:\\quad\\begin{smallmatrix}4&&1\\\\ &\\swarrow&\\\\ 2&\\rightarrow&3\\end{smallmatrix}\\qquad\\mathrm{U}:\\quad\\begin{smallmatrix}1&&4\\\\ \\downarrow&&\\uparrow\\\\ 2&\\rightarrow&3\\end{smallmatrix}\\qquad\\mathrm{X}:\\quad\\begin{smallmatrix}1&&4\\\\ &\\searrow\\!\\!\\!\\!\\!\\!\\nearrow&\\\\ 3&&2\\end{smallmatrix}$"}
{"_id": "Hodge_theory:3", "title": "", "text": "$\\langle\\ ,\\ \\rangle_{k}$"}
{"_id": "Big_O_notation:38", "title": "", "text": "$\\mathbb{R}^{n}$"}
{"_id": "Budan's_theorem:144", "title": "", "text": "$a+\\cfrac{1}{b+\\cfrac{1}{c+\\cfrac{1}{\\ddots}}}$"}
{"_id": "Greisen–Zatsepin–Kuzmin_limit:3", "title": "", "text": "$\\gamma_{\\rm CMB}+p\\rightarrow\\Delta^{+}\\rightarrow n+\\pi^{+}.$"}
{"_id": "Functional_predicate:4", "title": "", "text": "$\\forall A,\\exists B,\\forall C,\\forall D,P(C,D)\\rightarrow(C\\in A\\rightarrow D\\in B).$"}
{"_id": "Matrix_calculus:14", "title": "", "text": "$\\frac{\\partial\\mathbf{y}}{\\partial x}=\\begin{bmatrix}\\frac{\\partial y_{1}}{\\partial x}\\\\ \\frac{\\partial y_{2}}{\\partial x}\\\\ \\vdots\\\\ \\frac{\\partial y_{m}}{\\partial x}\\\\ \\end{bmatrix}.$"}
{"_id": "Centimetre–gram–second_system_of_units:68", "title": "", "text": "$\\frac{1}{\\frac{1}{r}-\\frac{1}{R}}$"}
{"_id": "Routing_and_wavelength_assignment:22", "title": "", "text": "$O(pn(m+n\\log n))$"}
{"_id": "Hyperfine_structure:25", "title": "", "text": "$\\scriptstyle{\\underline{\\underline{Q}}}$"}
{"_id": "True_arithmetic:12", "title": "", "text": "$\\underline{\\#(\\theta)}$"}
{"_id": "Partition_function_(statistical_mechanics):24", "title": "", "text": "$Z=\\frac{1}{N!h^{3N}}\\int\\,\\exp[-\\beta H(p_{1}\\cdots p_{N},x_{1}\\cdots x_{N})]\\;d^{3}p_{1}\\cdots d^{3}p_{N}\\,d^{3}x_{1}\\cdots d^{3}x_{N}$"}
{"_id": "Dynamic_connectivity:7", "title": "", "text": "$O(lgn)$"}
{"_id": "Analyticity_of_holomorphic_functions:3", "title": "", "text": "$f(z)=\\sum_{n=0}^{\\infty}(z-a)^{n}{1\\over 2\\pi i}\\int_{C}{f(w)\\over(w-a)^{n+1}}\\,\\mathrm{d}w.$"}
{"_id": "Smith_chart:22", "title": "", "text": "$\\gamma=\\alpha+j\\beta\\,$"}
{"_id": "Positive-definite_matrix:50", "title": "", "text": "$M=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}$"}
{"_id": "John's_equation:7", "title": "", "text": "$\\frac{\\partial^{2}u}{\\partial x_{i}\\partial y_{j}}-\\frac{\\partial^{2}u}{\\partial y_{i}\\partial x_{j}}=0$"}
{"_id": "Craig's_theorem:1", "title": "", "text": "$\\underbrace{A_{i}\\land\\dots\\land A_{i}}_{i}$"}
{"_id": "Rewriting:20", "title": "", "text": "$x\\stackrel{*}{\\rightarrow}y$"}
{"_id": "Gravitational_lensing_formalism:108", "title": "", "text": "$q_{xy}=\\frac{\\sum(x-\\bar{x})(y-\\bar{y})w(x-\\bar{x},y-\\bar{y})I(x,y)}{\\sum w(x-\\bar{x},y-\\bar{y})I(x,y)}$"}
{"_id": "Frequency_modulation:7", "title": "", "text": "$\\int_{0}^{t}x_{m}(\\tau)d\\tau=\\frac{A_{m}\\cos(2\\pi f_{m}t)}{2\\pi f_{m}}\\,$"}
{"_id": "Completing_the_square:37", "title": "", "text": "$\\begin{array}[]{c}2x^{2}+7x+6\\,=\\,0\\\\ x^{2}+\\tfrac{7}{2}x+3\\,=\\,0\\\\ \\left(x+\\tfrac{7}{4}\\right)^{2}-\\tfrac{1}{16}\\,=\\,0\\\\ \\left(x+\\tfrac{7}{4}\\right)^{2}\\,=\\,\\tfrac{1}{16}\\\\ x+\\tfrac{7}{4}=\\tfrac{1}{4}\\quad\\,\\text{or}\\quad x+\\tfrac{7}{4}=-\\tfrac{1}{4}\\\\ x=-\\tfrac{3}{2}\\quad\\,\\text{or}\\quad x=-2.\\end{array}$"}
{"_id": "Convex_conjugate:14", "title": "", "text": "$f^{\\star}\\left(x^{*}\\right)=\\begin{cases}0,&\\left|x^{*}\\right|\\leq 1\\\\ \\infty,&\\left|x^{*}\\right|>1.\\end{cases}$"}
{"_id": "De_Morgan's_laws:26", "title": "", "text": "$A^{c}\\cup B^{c}\\subseteq(A\\cap B)^{c}$"}
{"_id": "Square_matrix:10", "title": "", "text": "$\\det(\\mathsf{A}-\\lambda\\mathsf{I})=0.$"}
{"_id": "Tukey_lambda_distribution:10", "title": "", "text": "$L_{6}\\lambda=-\\frac{2}{1+\\lambda}+\\frac{60}{2+\\lambda}-\\frac{420}{3+\\lambda}+\\frac{1120}{4+\\lambda}-\\frac{1260}{5+\\lambda}+\\frac{504}{6+\\lambda}.$"}
{"_id": "List_of_relativistic_equations:90", "title": "", "text": "$\\frac{h\\nu^{\\prime}}{c}=\\gamma\\frac{h\\nu}{c}-\\gamma\\beta\\left\\|p\\right\\|\\cos\\theta=\\gamma\\frac{h\\nu}{c}-\\gamma\\beta\\frac{h\\nu}{c}\\cos\\theta$"}
{"_id": "Path_integrals_in_polymer_science:104", "title": "", "text": "$G(\\vec{R},\\vec{R}^{\\prime};N)\\equiv\\frac{\\displaystyle\\int_{\\vec{R}_{0}=\\vec{R}^{\\prime}}^{\\vec{R}_{N}=\\vec{R}}\\mathcal{D}\\vec{R}(n)\\exp\\left[-\\frac{3}{2l^{2}}\\displaystyle\\int_{0}^{N}dn\\left(\\frac{\\partial\\vec{R}_{n}}{\\partial n}\\right)^{2}-\\beta\\displaystyle\\int_{0}^{N}duU_{e}[\\vec{R}(n)]\\right]}{\\displaystyle\\int d\\vec{R}^{\\prime}\\displaystyle\\int d\\vec{R}\\displaystyle\\int_{\\vec{R}_{0}=\\vec{R}^{\\prime}}^{\\vec{R}_{N}=\\vec{R}}\\mathcal{D}\\vec{R}_{n}\\exp\\left[-\\frac{3}{2l^{2}}\\displaystyle\\int_{0}^{N}dn\\left(\\frac{\\partial\\vec{R}_{n}}{\\partial n}\\right)^{2}\\right]}$"}
{"_id": "Fractional_calculus:10", "title": "", "text": "$\\begin{aligned}\\displaystyle(J^{\\alpha})(J^{\\beta}f)(x)&\\displaystyle=\\frac{1}{\\Gamma(\\alpha)}\\int_{0}^{x}(x-t)^{\\alpha-1}(J^{\\beta}f)(t)\\;dt\\\\ &\\displaystyle=\\frac{1}{\\Gamma(\\alpha)\\Gamma(\\beta)}\\int_{0}^{x}\\int_{0}^{t}(x-t)^{\\alpha-1}(t-s)^{\\beta-1}f(s)\\;ds\\;dt\\\\ &\\displaystyle=\\frac{1}{\\Gamma(\\alpha)\\Gamma(\\beta)}\\int_{0}^{x}f(s)\\left(\\int_{s}^{x}(x-t)^{\\alpha-1}(t-s)^{\\beta-1}\\;dt\\right)ds\\end{aligned}$"}
{"_id": "List_of_United_States_presidential_elections_by_Electoral_College_margin:0", "title": "", "text": "$\\mbox{absolute margin of victory}~{}=\\begin{cases}0;&w\\leq\\frac{c}{2}\\\\ w-\\max\\{r,\\frac{c}{2}\\};&w>\\frac{c}{2}\\end{cases}$"}
{"_id": "Rectangular_function:28", "title": "", "text": "$\\therefore\\mathrm{rect}(t)=\\Pi(t)=\\lim_{n\\rightarrow\\infty,n\\in\\mathbb{(}Z)}\\frac{1}{(2t)^{2n}+1}=\\begin{cases}0&\\mbox{if }~{}|t|>\\frac{1}{2}\\\\ \\frac{1}{2}&\\mbox{if }~{}|t|=\\frac{1}{2}\\\\ 1&\\mbox{if }~{}|t|<\\frac{1}{2}.\\\\ \\end{cases}$"}
{"_id": "Kronecker_delta:0", "title": "", "text": "$\\delta_{ij}=\\begin{cases}0&\\,\\text{if }i\\neq j,\\\\ 1&\\,\\text{if }i=j.\\end{cases}$"}
{"_id": "Jaccard_index:3", "title": "", "text": "$A\\triangle B=(A\\cup B)-(A\\cap B)$"}
{"_id": "Free_electron_model:22", "title": "", "text": "$\\omega_{p}=\\sqrt{\\frac{ne^{2}}{\\epsilon_{0}m}}$"}
{"_id": "Multiplication:57", "title": "", "text": "$a^{n}=\\underbrace{a\\times a\\times\\cdots\\times a}_{n}$"}
{"_id": "Anisotropic_Network_Model:4", "title": "", "text": "$H_{ij}=\\begin{bmatrix}{\\partial^{2}V_{ij}\\over\\partial x_{i}\\partial x_{j}}&{\\partial^{2}V_{ij}\\over\\partial x_{i}\\partial y_{j}}&{\\partial^{2}V_{ij}\\over\\partial x_{i}\\partial z_{j}}\\\\ {\\partial^{2}V_{ij}\\over\\partial y_{i}\\partial x_{j}}&{\\partial^{2}V_{ij}\\over\\partial y_{i}\\partial y_{j}}&{\\partial^{2}V_{ij}\\over\\partial y_{i}\\partial z_{j}}\\\\ {\\partial^{2}V_{ij}\\over\\partial z_{i}\\partial x_{j}}&{\\partial^{2}V_{ij}\\over\\partial z_{i}\\partial y_{j}}&{\\partial^{2}V_{ij}\\over\\partial z_{i}\\partial z_{j}}\\end{bmatrix}$"}
{"_id": "Sylvester's_sequence:9", "title": "", "text": "$1=\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{43}+\\frac{1}{1807}+\\cdots.$"}
{"_id": "Hemorheology:49", "title": "", "text": "$\\mu_{a}={{\\mu_{0}}{{(1-0.5kH)}^{-2}}}$"}
{"_id": "S-matrix:16", "title": "", "text": "$\\begin{pmatrix}B\\\\ C\\end{pmatrix}=\\begin{pmatrix}S_{11}&S_{12}\\\\ S_{21}&S_{22}\\end{pmatrix}\\begin{pmatrix}A\\\\ D\\end{pmatrix}.$"}
{"_id": "Order_statistic:46", "title": "", "text": "$f_{X_{(k)}}(x)=\\frac{n!}{(k-1)!(n-k)!}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)$"}
{"_id": "Tobit_model:29", "title": "", "text": "$y_{1i}=\\begin{cases}y_{1i}^{*}&\\textrm{if}\\;y_{1i}^{*}>0\\\\ 0&\\textrm{if}\\;y_{1i}^{*}\\leq 0.\\end{cases}$"}
{"_id": "Mason's_invariant:1", "title": "", "text": "$\\begin{bmatrix}Z^{\\prime}_{11}&Z^{\\prime}_{12}\\\\ Z^{\\prime}_{21}&Z^{\\prime}_{22}\\end{bmatrix}=\\begin{bmatrix}n_{11}&n_{12}\\\\ n_{21}&n_{22}\\end{bmatrix}\\begin{bmatrix}Z_{11}&Z_{12}\\\\ Z_{21}&Z_{22}\\end{bmatrix}\\begin{bmatrix}n_{11}&n_{12}\\\\ n_{21}&n_{22}\\end{bmatrix}$"}
{"_id": "Upper-convected_time_derivative:1", "title": "", "text": "${\\stackrel{\\triangledown}{\\mathbf{A}}}$"}
{"_id": "Heegner_number:48", "title": "", "text": "$\\begin{aligned}\\displaystyle e^{\\pi\\sqrt{19}}&\\displaystyle\\approx(5x)^{3}-6.000\\,010\\dots\\\\ \\displaystyle e^{\\pi\\sqrt{43}}&\\displaystyle\\approx(5x)^{3}-6.000\\,000\\,010\\dots\\\\ \\displaystyle e^{\\pi\\sqrt{67}}&\\displaystyle\\approx(5x)^{3}-6.000\\,000\\,000\\,061\\dots\\\\ \\displaystyle e^{\\pi\\sqrt{163}}&\\displaystyle\\approx(5x)^{3}-6.000\\,000\\,000\\,000\\,000\\,034\\dots\\end{aligned}$"}
{"_id": "Apéry's_constant:0", "title": "", "text": "$1+\\frac{1}{4+\\cfrac{1}{1+\\cfrac{1}{18+\\cfrac{1}{\\ddots\\qquad{}}}}}$"}
{"_id": "Legendre–Clebsch_condition:3", "title": "", "text": "$\\frac{\\partial^{2}H}{\\partial u^{2}}>0$"}
{"_id": "Silhouette_edge:0", "title": "", "text": "$ax+by+cz+d=\\begin{cases}>0&\\,\\text{front facing}\\\\ =0&\\,\\text{parallel}\\\\ <0&\\,\\text{back facing}\\end{cases}$"}
{"_id": "Bring_radical:2", "title": "", "text": "$x^{5}+c_{2}x^{2}+c_{1}x+c_{0}=0\\,$"}
{"_id": "Lieb-Robinson_bounds:109", "title": "", "text": "$<A>_{\\Omega}$"}
{"_id": "Petri_net:47", "title": "", "text": "$M_{0}=\\begin{bmatrix}1&0&2&1\\end{bmatrix}$"}
{"_id": "Rate_of_return:42", "title": "", "text": "$\\mbox{NPV}~{}=\\sum_{t=0}^{n}\\frac{C_{t}}{(1+r)^{t}}=0$"}
{"_id": "Active_contour_model:1", "title": "", "text": "$i=0\\ldots n-1$"}
{"_id": "List_of_formulae_involving_π:37", "title": "", "text": "$\\frac{1}{2^{6}}\\sum_{n=0}^{\\infty}\\frac{{(-1)}^{n}}{2^{10n}}\\left(-\\frac{2^{5}}{4n+1}-\\frac{1}{4n+3}+\\frac{2^{8}}{10n+1}-\\frac{2^{6}}{10n+3}-\\frac{2^{2}}{10n+5}-\\frac{2^{2}}{10n+7}+\\frac{1}{10n+9}\\right)=\\pi\\!$"}
{"_id": "Particle_in_a_box:58", "title": "", "text": "$\\mathrm{Var}(x)=\\int_{-\\infty}^{\\infty}(x-\\langle x\\rangle)^{2}P_{n}(x)\\,dx=\\frac{L^{2}}{12}\\left(1-\\frac{6}{n^{2}\\pi^{2}}\\right)$"}
{"_id": "Jiles-Atherton_model:34", "title": "", "text": "$M\\text{an}\\text{aniso}=M\\text{s}\\frac{\\int_{0}^{\\pi}\\!e^{0.5(E(1)+E(2))}\\sin(\\theta)\\cos(\\theta)\\,d\\theta}{\\int_{0}^{\\pi}\\!e^{0.5(E(1)+E(2))}\\sin(\\theta)\\,d\\theta}$"}
{"_id": "Karp–Flatt_metric:4", "title": "", "text": "$e=\\frac{\\frac{1}{\\psi}-\\frac{1}{p}}{1-\\frac{1}{p}}$"}
{"_id": "Black–Scholes_model:65", "title": "", "text": "$\\frac{\\partial^{2}C}{\\partial S^{2}}$"}
{"_id": "Rank_of_an_abelian_group:1", "title": "", "text": "$0\\to A\\to B\\to C\\to 0\\;$"}
{"_id": "Morse–Kelley_set_theory:18", "title": "", "text": "$\\forall a\\,\\forall s\\,[(Ma\\and\\forall x\\,[x\\in s\\leftrightarrow\\exists y\\,(x\\in y\\and y\\in a)])\\rightarrow Ms].$"}
{"_id": "Gauss–Legendre_algorithm:3", "title": "", "text": "$\\begin{aligned}\\displaystyle a_{n+1}&\\displaystyle=\\frac{a_{n}+b_{n}}{2},\\\\ \\displaystyle\\par <code> b_{n+1}&\\displaystyle = \\sqrt{a_{n} b_{n}},\\\\ \\displaystyle</code><br/><code> t_{n+1}&\\displaystyle = t_{n} - p_{n}(a_{n}-a_{n+1})^{2},\\\\ \\displaystyle</code><br/><code> p_{n+1}&\\displaystyle = 2p_{n}.</code><br/><code>\\end{aligned}</code>$"}
{"_id": "Conversion_between_quaternions_and_Euler_angles:10", "title": "", "text": "$\\begin{bmatrix}\\cos\\theta\\cos\\psi&-\\cos\\phi\\sin\\psi+\\sin\\phi\\sin\\theta\\cos\\psi&\\sin\\phi\\sin\\psi+\\cos\\phi\\sin\\theta\\cos\\psi\\\\ \\cos\\theta\\sin\\psi&\\cos\\phi\\cos\\psi+\\sin\\phi\\sin\\theta\\sin\\psi&-\\sin\\phi\\cos\\psi+\\cos\\phi\\sin\\theta\\sin\\psi\\\\ -\\sin\\theta&\\sin\\phi\\cos\\theta&\\cos\\phi\\cos\\theta\\\\ \\end{bmatrix}$"}
{"_id": "Ewald_summation:66", "title": "", "text": "$O(N\\,\\log N)$"}
{"_id": "Fürer's_algorithm:0", "title": "", "text": "$O(n\\log n\\log\\log n)$"}
{"_id": "Infinitesimal_strain_theory:111", "title": "", "text": "$\\frac{\\partial^{2}\\epsilon_{y}}{\\partial z^{2}}+\\frac{\\partial^{2}\\epsilon_{z}}{\\partial y^{2}}=2\\frac{\\partial^{2}\\epsilon_{yz}}{\\partial y\\partial z}\\,\\!$"}
{"_id": "Super-resolution_optical_fluctuation_imaging:8", "title": "", "text": "$\\langle\\cdots\\rangle_{t}$"}
{"_id": "Centroid:27", "title": "", "text": "$C_{k}=\\frac{\\int zS_{k}(z)\\;dz}{\\int S_{k}(z)\\;dz}$"}
{"_id": "Beck's_monadicity_theorem:0", "title": "", "text": "$U:C\\to D$"}
{"_id": "Simple_linear_regression:108", "title": "", "text": "$\\displaystyle\\sum_{i=1}^{n}\\left(y_{i}-\\bar{y}\\right)\\left(x_{i}-\\bar{x}\\right)-\\hat{\\beta}\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^{2}=0$"}
{"_id": "Dinic's_algorithm:48", "title": "", "text": "$O(VE\\log V)$"}
{"_id": "Graphoid:13", "title": "", "text": "$\\langle X,Z,Y\\rangle_{D}$"}
{"_id": "Spectral_concentration_problem:3", "title": "", "text": "$\\lambda(T,W)=\\frac{\\int_{-W}^{W}{\\|U(f)\\|}^{2}\\,df}{\\int_{-1/2}^{1/2}{\\|U(f)\\|}^{2}\\,df}.$"}
{"_id": "Zemor's_decoding_algorithm:214", "title": "", "text": "$O(N\\log{N})$"}
{"_id": "Whistle:38", "title": "", "text": "${}^{\\lambda}\\!\\!\\diagup\\!\\!{}_{4}\\;$"}
{"_id": "Accumulation_function:6", "title": "", "text": "$a(t)=e^{\\int_{0}^{t}\\delta_{u}\\,du}$"}
{"_id": "Singly_and_doubly_even:7", "title": "", "text": "$\\tanh\\frac{1}{2}=\\frac{e-1}{e+1}=0+\\cfrac{1}{2+\\cfrac{1}{6+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{\\ddots}}}}}$"}
{"_id": "List_of_rules_of_inference:22", "title": "", "text": "$\\underline{\\lnot\\psi\\quad\\quad\\quad}\\,\\!$"}
{"_id": "Zinc_smelting:2", "title": "", "text": "$\\mathrm{ZnO+SO_{3}\\rightarrow ZnSO_{4}}$"}
{"_id": "Characteristic_admittance:0", "title": "", "text": "$Y_{0}=\\sqrt{\\frac{G+j\\omega C}{R+j\\omega L}}$"}
{"_id": "Gamma_function:75", "title": "", "text": "${n\\choose k}=\\frac{n!}{k!(n-k)!}.$"}
{"_id": "Mean_of_a_function:25", "title": "", "text": "$\\exp\\left(\\frac{1}{\\hbox{Vol}(U)}\\int_{U}\\log f\\right).$"}
{"_id": "1::4_+_1::16_+_1::64_+_1::256_+_⋯:7", "title": "", "text": "$1+\\frac{1}{4}+\\frac{1}{4^{2}}+\\cdots+\\frac{1}{4^{n}}=\\frac{1-\\left(\\frac{1}{4}\\right)^{n+1}}{1-\\frac{1}{4}}.$"}
{"_id": "Solving_quadratic_equations_with_continued_fractions:0", "title": "", "text": "$ax^{2}+bx+c=0,\\,\\!$"}
{"_id": "E_(mathematical_constant):157", "title": "", "text": "$e=1+\\cfrac{2}{1+\\cfrac{1}{6+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\cfrac{1}{22+\\cfrac{1}{26+\\ddots\\,}}}}}}}.$"}
{"_id": "Euler–Mascheroni_constant:42", "title": "", "text": "${\\gamma=\\sum_{k=2}^{\\infty}(-1)^{k}\\frac{\\left\\lfloor\\log_{2}k\\right\\rfloor}{k}=\\frac{1}{2}-\\frac{1}{3}+2\\left(\\frac{1}{4}-\\frac{1}{5}+\\frac{1}{6}-\\frac{1}{7}\\right)+3\\left(\\frac{1}{8}-\\frac{1}{9}+\\frac{1}{10}-\\frac{1}{11}+\\dots-\\frac{1}{15}\\right)+\\dots}$"}
{"_id": "Network_analysis_(electrical_circuits):48", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ V_{0}\\end{bmatrix}=\\begin{bmatrix}z(j\\omega)_{11}&z(j\\omega)_{12}\\\\ z(j\\omega)_{21}&z(j\\omega)_{22}\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ I_{0}\\end{bmatrix}$"}
{"_id": "Chain_rule:42", "title": "", "text": "$\\displaystyle\\frac{d}{dx}\\left(\\frac{f(x)}{g(x)}\\right)$"}
{"_id": "Gravitoelectromagnetism:7", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\frac{1}{\\epsilon_{0}c^{2}}\\mathbf{J}+\\frac{1}{c^{2}}\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Symmetry_in_quantum_mechanics:13", "title": "", "text": "$X_{j}=\\left.\\frac{\\partial g}{\\partial\\xi_{j}}\\right|_{\\xi_{j}=0}$"}
{"_id": "Residence_time_distribution:28", "title": "", "text": "$E(t)=\\frac{C(t)}{\\int_{0}^{\\infty}C(t)\\ dt}$"}
{"_id": "Curvilinear_coordinates:9", "title": "", "text": "$h_{i}=\\left|\\frac{\\partial\\mathbf{r}}{\\partial q_{i}}\\right|$"}
{"_id": "Displacement_current:28", "title": "", "text": "$symbol{\\nabla\\times}\\left(symbol{\\nabla\\times B}\\right)=\\mu_{0}\\epsilon_{0}\\frac{\\partial}{\\partial t}symbol{\\nabla\\times E}\\ .$"}
{"_id": "Faithful_representation:3", "title": "", "text": "$V^{\\otimes n}=\\underbrace{V\\otimes V\\otimes\\cdots\\otimes V}_{n\\,\\text{ times}}$"}
{"_id": "Interior_reconstruction:36", "title": "", "text": "$\\begin{bmatrix}f\\\\ g\\end{bmatrix}=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\end{bmatrix}$"}
{"_id": "Preference_ranking_organization_method_for_enrichment_evaluation:40", "title": "", "text": "$P_{j}(d_{j})=\\begin{cases}0&\\,\\text{if }d_{j}\\leq 0\\\\ 1&\\,\\text{if }d_{j}>0\\end{cases}$"}
{"_id": "Domatic_number:31", "title": "", "text": "$n^{O(\\log\\log n)}$"}
{"_id": "Van_Wijngaarden_transformation:6", "title": "", "text": "$\\scriptstyle 1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\cdots=\\frac{\\pi}{4}=0.7853981\\ldots$"}
{"_id": "Metric_tensor:13", "title": "", "text": "$\\begin{aligned}\\displaystyle ds^{2}&\\displaystyle=\\begin{bmatrix}du&dv\\end{bmatrix}\\begin{bmatrix}E&F\\\\ F&G\\end{bmatrix}\\begin{bmatrix}du\\\\ dv\\end{bmatrix}\\\\ &\\displaystyle=\\begin{bmatrix}du^{\\prime}&dv^{\\prime}\\end{bmatrix}\\begin{bmatrix}\\frac{\\partial u}{\\partial u^{\\prime}}&\\frac{\\partial u}{\\partial v^{\\prime}}\\\\ \\frac{\\partial v}{\\partial u^{\\prime}}&\\frac{\\partial v}{\\partial v^{\\prime}}\\end{bmatrix}^{\\mathrm{T}}\\begin{bmatrix}E&F\\\\ F&G\\end{bmatrix}\\begin{bmatrix}\\frac{\\partial u}{\\partial u^{\\prime}}&\\frac{\\partial u}{\\partial v^{\\prime}}\\\\ \\frac{\\partial v}{\\partial u^{\\prime}}&\\frac{\\partial v}{\\partial v^{\\prime}}\\end{bmatrix}\\begin{bmatrix}du^{\\prime}\\\\ dv^{\\prime}\\end{bmatrix}\\\\ &\\displaystyle=\\begin{bmatrix}du^{\\prime}&dv^{\\prime}\\end{bmatrix}\\begin{bmatrix}E^{\\prime}&F^{\\prime}\\\\ F^{\\prime}&G^{\\prime}\\end{bmatrix}\\begin{bmatrix}du^{\\prime}\\\\ dv^{\\prime}\\end{bmatrix}\\\\ &\\displaystyle=(ds^{\\prime})^{2}.\\end{aligned}$"}
{"_id": "Differential_geometry_of_surfaces:52", "title": "", "text": "$\\int_{\\Delta}K\\,dA=\\alpha+\\beta+\\gamma-\\pi.$"}
{"_id": "Polynomial:8", "title": "", "text": "$-5x^{2}y\\,$"}
{"_id": "Partition_of_sums_of_squares:9", "title": "", "text": "$\\begin{aligned}\\displaystyle\\sum_{i=1}^{n}(y_{i}-\\overline{y})^{2}&\\displaystyle=\\sum_{i=1}^{n}(y_{i}-\\overline{y}+\\hat{y}_{i}-\\hat{y}_{i})^{2}=\\sum_{i=1}^{n}((\\hat{y}_{i}-\\bar{y})+\\underbrace{(y_{i}-\\hat{y}_{i})}_{\\hat{\\varepsilon}_{i}})^{2}\\\\ &\\displaystyle=\\sum_{i=1}^{n}((\\hat{y}_{i}-\\bar{y})^{2}+2\\hat{\\varepsilon}_{i}(\\hat{y}_{i}-\\bar{y})+\\hat{\\varepsilon}_{i}^{2})\\\\ &\\displaystyle=\\sum_{i=1}^{n}(\\hat{y}_{i}-\\bar{y})^{2}+\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}^{2}+2\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}(\\hat{y}_{i}-\\bar{y})\\\\ &\\displaystyle=\\sum_{i=1}^{n}(\\hat{y}_{i}-\\bar{y})^{2}+\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}^{2}+2\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}(\\hat{\\beta}_{0}+\\hat{\\beta}_{1}x_{i1}+\\cdots+\\hat{\\beta}_{p}x_{ip}-\\overline{y})\\\\ &\\displaystyle=\\sum_{i=1}^{n}(\\hat{y}_{i}-\\bar{y})^{2}+\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}^{2}+2(\\hat{\\beta}_{0}-\\overline{y})\\underbrace{\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}}_{0}+2\\hat{\\beta}_{1}\\underbrace{\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}x_{i1}}_{0}+\\cdots+2\\hat{\\beta}_{p}\\underbrace{\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}x_{ip}}_{0}\\\\ &\\displaystyle=\\sum_{i=1}^{n}(\\hat{y}_{i}-\\bar{y})^{2}+\\sum_{i=1}^{n}\\hat{\\varepsilon}_{i}^{2}=\\mathrm{ESS}+\\mathrm{RSS}\\\\ \\end{aligned}$"}
{"_id": "Cayley–Dickson_construction:18", "title": "", "text": "$pq=(qp)^{\\prime}$"}
{"_id": "Variance:21", "title": "", "text": "$p(k)={n\\choose k}p^{k}(1-p)^{n-k},$"}
{"_id": "Experimental_uncertainty_analysis:37", "title": "", "text": "$\\hat{\\sigma}_{i}\\,\\,\\,=\\,\\,\\,\\sqrt{{{\\,\\,\\sum\\limits_{k=1}^{n}{\\left({x_{k}-\\bar{x}_{i}}\\right)^{2}}}\\over{n-1}}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hat{\\sigma}_{i,j}\\,\\,\\,=\\,\\,\\,\\sqrt{{{\\,\\,\\sum\\limits_{k=1}^{n}{\\left({x_{k}-\\bar{x}_{i}}\\right)\\left({x_{k}-\\bar{x}_{j}}\\right)}}\\over{n-1}}}$"}
{"_id": "Huzita–Hatori_axioms:15", "title": "", "text": "$as^{2}+bs+c=0\\,$"}
{"_id": "Rushbrooke_inequality:5", "title": "", "text": "$\\alpha,\\alpha^{\\prime},\\beta,\\gamma,\\gamma^{\\prime}$"}
{"_id": "M::M::c_queue:3", "title": "", "text": "$\\pi_{k}=\\begin{cases}\\pi_{0}\\dfrac{(c\\rho)^{k}}{k!},&\\mbox{if }~{}0<k<c\\\\ \\pi_{0}\\dfrac{\\rho^{k}c^{c}}{c!},&\\mbox{if }~{}c\\leq k\\end{cases}$"}
{"_id": "Elementary_matrix:5", "title": "", "text": "$T_{i,j}(m)=\\begin{bmatrix}1&&&&&&&\\\\ &\\ddots&&&&&&\\\\ &&1&&&&&\\\\ &&&\\ddots&&&&\\\\ &&m&&1&&\\\\ &&&&&&\\ddots&\\\\ &&&&&&&1\\end{bmatrix}$"}
{"_id": "Surface_of_revolution:16", "title": "", "text": "$\\begin{aligned}\\displaystyle A&\\displaystyle{}=2\\pi\\int_{0}^{\\pi}\\sin(t)\\sqrt{\\left(\\cos(t)\\right)^{2}+\\left(\\sin(t)\\right)^{2}}\\,dt\\\\ &\\displaystyle{}=2\\pi\\int_{0}^{\\pi}\\sin(t)\\,dt\\\\ &\\displaystyle{}=4\\pi.\\end{aligned}$"}
{"_id": "Scattering_parameters:115", "title": "", "text": "$\\begin{pmatrix}b_{1}\\\\ a_{1}\\end{pmatrix}=\\begin{pmatrix}T_{11}&T_{12}\\\\ T_{21}&T_{22}\\end{pmatrix}\\begin{pmatrix}a_{2}\\\\ b_{2}\\end{pmatrix}\\,$"}
{"_id": "Lindhard_theory:14", "title": "", "text": "$\\omega+i\\delta\\to 0$"}
{"_id": "Einstein_solid:23", "title": "", "text": "$\\Omega={\\left(q+N^{\\prime}-1\\right)!\\over q!(N^{\\prime}-1)!}$"}
{"_id": "Universal_joint:44", "title": "", "text": "$\\tan\\gamma_{4}=\\cos\\beta/\\tan\\gamma_{2}=1/\\tan\\gamma_{1}=\\tan(\\gamma_{1}+\\pi/2)\\,$"}
{"_id": "Binary_logarithm:80", "title": "", "text": "$\\lfloor\\log_{2}(0)\\rfloor=-1$"}
{"_id": "Hadamard_matrix:5", "title": "", "text": "$H_{1}=\\begin{bmatrix}1\\end{bmatrix},$"}
{"_id": "Axiom_of_regularity:8", "title": "", "text": "$\\forall x\\,(x\\neq\\emptyset\\rightarrow\\exists y\\in x\\,(y\\cap x=\\emptyset))$"}
{"_id": "Random_structure_function:4", "title": "", "text": "$\\mathfrak{P}(\\{d\\})$"}
{"_id": "Plasma_fusion_preface:154", "title": "", "text": "$v=\\sqrt{\\frac{ke^{2}}{mr}}$"}
{"_id": "Bayesian_programming:107", "title": "", "text": "$Pr\\begin{cases}Ds\\begin{cases}Sp(\\pi)\\begin{cases}Va:\\\\ S^{0},\\cdots,S^{T},O^{0},\\cdots,O^{T}\\\\ Dc:\\\\ \\begin{cases}&P\\left(S^{0}\\wedge\\cdots\\wedge O^{T}|\\pi\\right)\\\\ =&\\left[\\begin{array}[]{c}P\\left(S^{0}\\wedge O^{0}|\\pi\\right)\\\\ \\prod_{t=1}^{T}\\left[P\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)\\times P\\left(O^{t}|S^{t}\\wedge\\pi\\right)\\right]\\end{array}\\right]\\end{cases}\\\\ Fo:\\\\ \\begin{cases}P\\left(S^{t}\\mid S^{t-1}\\wedge\\pi\\right)\\equiv G\\left(S^{t},A\\bullet S^{t-1},Q\\right)\\\\ P\\left(O^{t}\\mid S^{t}\\wedge\\pi\\right)\\equiv G\\left(O^{t},H\\bullet S^{t},R\\right)\\end{cases}\\end{cases}\\\\ Id\\end{cases}\\\\ Qu:\\\\ P\\left(S^{T}\\mid O^{0}\\wedge\\cdots\\wedge O^{T}\\wedge\\pi\\right)\\end{cases}$"}
{"_id": "Maximum_coverage_problem:20", "title": "", "text": "$1-\\frac{1}{e}$"}
{"_id": "Generalized_Gauss–Bonnet_theorem:4", "title": "", "text": "$\\wedge^{\\hbox{even}}\\,T^{*}M$"}
{"_id": "Bloom_filter:29", "title": "", "text": "$1-\\frac{1}{m}.$"}
{"_id": "Pell_number:63", "title": "", "text": "$3363-2378\\sqrt{2}=0.00014\\ldots$"}
{"_id": "Battle_of_the_sexes_(game_theory):9", "title": "", "text": "$M_{O}=1-\\tfrac{3}{5}=\\tfrac{2}{5}$"}
{"_id": "Equivariant_sheaf:20", "title": "", "text": "$V_{x}\\overset{\\simeq}{\\to}V_{gx}$"}
{"_id": "Euler_spiral:16", "title": "", "text": "$\\begin{aligned}\\displaystyle x&\\displaystyle=\\int_{0}^{L}\\cos\\theta\\,ds\\\\ &\\displaystyle=\\int_{0}^{L}\\cos\\left[(as)^{2}\\right]ds\\end{aligned}$"}
{"_id": "Random_projection:2", "title": "", "text": "$O(ckN)$"}
{"_id": "Solution_of_triangles:91", "title": "", "text": "$a=\\arccos\\left(\\frac{\\cos\\alpha+\\cos\\beta\\cos\\gamma}{\\sin\\beta\\sin\\gamma}\\right)$"}
{"_id": "Beta_distribution:284", "title": "", "text": "$\\alpha=2:\\quad\\operatorname{E}\\left[-\\frac{1}{N}\\frac{\\partial^{2}\\ln\\mathcal{L}(\\alpha,\\beta,a,c|Y)}{\\partial\\alpha\\partial a}\\right]={\\mathcal{I}}_{\\alpha,a}$"}
{"_id": "Internal_set_theory:4", "title": "", "text": "$\\forall^{\\mathrm{st}}x\\,\\exists^{\\mathrm{st}}y\\,\\forall^{\\mathrm{st}}t\\,(t\\in y\\leftrightarrow(t\\in x\\land\\phi(t,u_{1},\\dots,u_{n})))$"}
{"_id": "Nyquist_stability_criterion:98", "title": "", "text": "$-1/k$"}
{"_id": "Divide-and-conquer_eigenvalue_algorithm:34", "title": "", "text": "$T=\\begin{bmatrix}Q_{1}&\\\\ &Q_{2}\\end{bmatrix}\\left(\\begin{bmatrix}D_{1}&\\\\ &D_{2}\\end{bmatrix}+\\beta zz^{T}\\right)\\begin{bmatrix}Q_{1}^{T}&\\\\ &Q_{2}^{T}\\end{bmatrix}$"}
{"_id": "Exponentiation_by_squaring:1", "title": "", "text": "$\\lfloor\\log_{2}n\\rfloor$"}
{"_id": "Ptolemy's_theorem:27", "title": "", "text": "$\\sin\\alpha\\sin\\beta\\cos\\beta\\cos\\gamma+\\sin\\alpha\\cos^{2}\\beta\\sin\\gamma+\\cos\\alpha\\sin^{2}\\beta\\cos\\gamma+\\cos\\alpha\\sin\\beta\\cos\\beta\\sin\\gamma$"}
{"_id": "Hurwitz's_theorem_(composition_algebras):16", "title": "", "text": "$Re(a b)=Re(b a)$"}
{"_id": "AN_codes:92", "title": "", "text": "$A=(r^{n}-1)/B=3$"}
{"_id": "Draft:Leaving_Certificate_Mathematics_Ordinary_Level:45", "title": "", "text": "$-1-2i$"}
{"_id": "Riemannian_manifold:27", "title": "", "text": "$g^{M}_{p}(u,v)=g^{N}_{f(p)}(df(u),df(v))\\qquad\\forall p\\in M,\\forall u,v\\in T_{p}M.$"}
{"_id": "Autocatalytic_reaction:21", "title": "", "text": "$B+X\\rightarrow Y+D$"}
{"_id": "Stieltjes_constants:34", "title": "", "text": "$\\gamma_{1}\\!\\left(\\!\\frac{1}{\\,2\\,}\\!\\right)=-2\\gamma\\ln 2-\\ln^{2}\\!2+\\gamma_{1}\\,=\\,-1.353459680\\ldots$"}
{"_id": "Largest_remainder_method:0", "title": "", "text": "$\\frac{\\mbox{total}~{}\\;\\mbox{votes}~{}}{\\mbox{total}~{}\\;\\mbox{seats}~{}}$"}
{"_id": "Plücker_embedding:0", "title": "", "text": "$\\begin{aligned}\\displaystyle\\iota\\colon\\mathbf{Gr}(r,K^{n})&\\displaystyle{}\\rightarrow\\mathbf{P}(\\wedge^{r}K^{n})\\\\ \\displaystyle\\operatorname{span}(v_{1},\\ldots,v_{r})&\\displaystyle{}\\mapsto K(v_{1}\\wedge\\cdots\\wedge v_{r})\\end{aligned}$"}
{"_id": "Functional_(mathematics):17", "title": "", "text": "$F(y)=\\frac{\\int_{x_{0}}^{x_{1}}y(x)\\;\\mathrm{d}x}{\\int_{x_{0}}^{x_{1}}(1+[y(x)]^{2})\\;\\mathrm{d}x}$"}
{"_id": "Linear_belief_function:24", "title": "", "text": "$M^{\\downarrow Y}(X,Y)=\\left[{\\begin{array}[]{*{20}c}{\\mu_{2}}\\\\ {\\Sigma_{22}}\\\\ \\end{array}}\\right]$"}
{"_id": "Chain_rule:81", "title": "", "text": "$\\frac{\\partial^{2}y}{\\partial x_{i}\\partial x_{j}}=\\sum_{k}\\left(\\frac{\\partial y}{\\partial u_{k}}\\frac{\\partial^{2}u_{k}}{\\partial x_{i}\\partial x_{j}}\\right)+\\sum_{k,\\ell}\\left(\\frac{\\partial^{2}y}{\\partial u_{k}\\partial u_{\\ell}}\\frac{\\partial u_{k}}{\\partial x_{i}}\\frac{\\partial u_{\\ell}}{\\partial x_{j}}\\right).$"}
{"_id": "Greisen–Zatsepin–Kuzmin_limit:4", "title": "", "text": "$\\nu+\\bar{\\nu}\\rightarrow Z\\rightarrow\\,\\text{hadrons}$"}
{"_id": "Discriminant_of_an_algebraic_number_field:2", "title": "", "text": "$\\Delta_{K}=\\left\\{\\begin{array}[]{ll}d&\\,\\text{if }d\\equiv 1\\;\\;(\\mathop{{\\rm mod}}4)\\\\ 4d&\\,\\text{if }d\\equiv 2,3\\;\\;(\\mathop{{\\rm mod}}4).\\\\ \\end{array}\\right.$"}
{"_id": "Pell_number:61", "title": "", "text": "$1393-985\\sqrt{2}=-0.00035\\ldots$"}
{"_id": "History_of_algebra:3", "title": "", "text": "$x^{2}+px+q=0$"}
{"_id": "Frobenius_endomorphism:11", "title": "", "text": "$\\frac{p!}{k!(p-k)!},$"}
{"_id": "Lambda_calculus_definition:21", "title": "", "text": "$(X\\ Y)[x:=y]=X[x:=y]\\ Y[x:=y]$"}
{"_id": "Blood_sugar:0", "title": "", "text": "$\\mathrm{Glucose}+\\mathrm{Alkaline\\ copper\\ tartarate}\\xrightarrow{\\mathrm{Reduction}}\\mathrm{Cuprous\\ oxide}$"}
{"_id": "Quadric_(projective_geometry):100", "title": "", "text": "$\\mathfrak{P}_{n}(K)$"}
{"_id": "Periodic_points_of_complex_quadratic_mappings:46", "title": "", "text": "$\\alpha_{1}=\\frac{1}{2}-m$"}
{"_id": "Score_(statistics):29", "title": "", "text": "$L(\\theta;A,B)=\\frac{(A+B)!}{A!B!}\\theta^{A}(1-\\theta)^{B},$"}
{"_id": "Rate–distortion_theory:10", "title": "", "text": "$R(D)=\\left\\{\\begin{matrix}\\frac{1}{2}\\log_{2}(\\sigma_{x}^{2}/D),&\\mbox{if }~{}0\\leq D\\leq\\sigma_{x}^{2}\\\\ \\\\ 0,&\\mbox{if }~{}D>\\sigma_{x}^{2}.\\end{matrix}\\right.$"}
{"_id": "Complex_network_zeta_function:77", "title": "", "text": "$\\textstyle(1-p^{2})^{N-2}$"}
{"_id": "Multiple_zeta_function:3", "title": "", "text": "$\\zeta(s,t)=\\sum_{n=1}^{\\infty}\\frac{H_{n,t}}{(n+1)^{s}}$"}
{"_id": "Ideal_gas:48", "title": "", "text": "$\\frac{S}{kN}=\\ln\\left(\\frac{VT^{\\hat{c}_{V}}}{N\\Phi}\\right)$"}
{"_id": "Commutation_matrix:5", "title": "", "text": "$\\mathbf{M}=\\begin{bmatrix}a&b\\\\ c&d\\\\ \\end{bmatrix}$"}
{"_id": "Compound_interest:31", "title": "", "text": "$L=P\\sum_{j=1}^{n}\\frac{1}{(1+i)^{j}}$"}
{"_id": "Pell_number:51", "title": "", "text": "$17-12\\sqrt{2}=0.02943\\ldots$"}
{"_id": "Binary_moment_diagram:0", "title": "", "text": "$\\begin{cases}\\,\\text{if }x\\begin{cases}\\,\\text{if }y,3\\\\ \\,\\text{if }\\neg y,2\\end{cases}\\\\ \\,\\text{if }\\neg x\\begin{cases}\\,\\text{if }y\\,\\text{ , }1\\\\ \\,\\text{if }\\neg y\\,\\text{ , }0\\end{cases}\\end{cases}$"}
{"_id": "Rational_number:16", "title": "", "text": "$a_{0}+\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{\\ddots+\\cfrac{1}{a_{n}}}}},$"}
{"_id": "Knuth's_up-arrow_notation:15", "title": "", "text": "$\\begin{matrix}a\\ \\underbrace{\\uparrow\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n}\\ b=\\underbrace{a\\ \\underbrace{\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\\ (a\\ \\underbrace{\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\\ (\\dots\\ \\underbrace{\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\\ a))}_{b\\,\\text{ copies of }a}\\end{matrix}$"}
{"_id": "Divergent_series:0", "title": "", "text": "$1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\cdots=\\sum_{n=1}^{\\infty}\\frac{1}{n}.$"}
{"_id": "Sound_power:7", "title": "", "text": "$L_{W}=L_{p}+10\\log_{10}\\!\\left(\\frac{4\\pi r^{2}}{A_{0}}\\right)\\!~{}\\mathrm{dB},$"}
{"_id": "Lorentz_covariance:12", "title": "", "text": "$\\delta^{a}_{b}=\\begin{cases}1&\\mbox{if }~{}a=b,\\\\ 0&\\mbox{if }~{}a\\neq b.\\end{cases}$"}
{"_id": "Pearson_product-moment_correlation_coefficient:17", "title": "", "text": "$r=r_{xy}=\\frac{\\sum^{n}_{i=1}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{\\sqrt{\\sum^{n}_{i=1}(x_{i}-\\bar{x})^{2}}\\sqrt{\\sum^{n}_{i=1}(y_{i}-\\bar{y})^{2}}}$"}
{"_id": "Heyting_algebra:42", "title": "", "text": "$\\forall x,y\\in H:\\qquad\\lnot(x\\vee y)=\\lnot x\\wedge\\lnot y.$"}
{"_id": "Jacobian_conjecture:0", "title": "", "text": "$J_{F}=\\left|\\begin{matrix}\\frac{\\partial f_{1}}{\\partial X_{1}}&\\cdots&\\frac{\\partial f_{1}}{\\partial X_{N}}\\\\ \\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial f_{N}}{\\partial X_{1}}&\\cdots&\\frac{\\partial f_{N}}{\\partial X_{N}}\\end{matrix}\\right|,$"}
{"_id": "Polar_curve:4", "title": "", "text": "$\\frac{1}{p!}\\Delta_{Q}^{p}f(p,q,r)=\\frac{1}{(n-p)!}\\Delta_{P}^{n-p}f(a,b,c).$"}
{"_id": "Light-front_quantization_applications:47", "title": "", "text": "$A+B\\to C+D$"}
{"_id": "Geodesics_on_an_ellipsoid:19", "title": "", "text": "${\\color{white}.}\\qquad\\begin{aligned}\\displaystyle\\frac{s}{b}&\\displaystyle=\\int_{0}^{\\sigma}\\frac{\\sqrt{1-e^{2}\\cos^{2}\\beta(\\sigma^{\\prime};\\alpha_{0})}}{1-f}\\,d\\sigma^{\\prime}\\\\ &\\displaystyle=\\int_{0}^{\\sigma}\\sqrt{1+k^{2}\\sin^{2}\\sigma^{\\prime}}\\,d\\sigma^{\\prime},\\end{aligned}$"}
{"_id": "Airy_function:2", "title": "", "text": "$y^{\\prime\\prime}-xy=0.$"}
{"_id": "Random_permutation_statistics:96", "title": "", "text": "$\\mathcal{Q}=\\mathfrak{P}\\left(\\sum_{q\\geq 1}\\mathfrak{C}_{=q}(\\mathcal{Z})\\times\\sum_{p=1}^{q}{q\\choose p}\\mathcal{U}^{p}\\right).$"}
{"_id": "Hurwitz's_theorem_(composition_algebras):66", "title": "", "text": "$(j c)b=j(b c)$"}
{"_id": "Normal_number:13", "title": "", "text": "$0.a_{1}a_{2}a_{3}\\ldots$"}
{"_id": "Newman–Penrose_formalism:55", "title": "", "text": "$\\delta\\Delta-\\Delta\\delta=-\\bar{\\nu}D+(\\tau-\\bar{\\alpha}-\\beta)\\Delta+(\\mu-\\gamma+\\bar{\\gamma})\\delta+\\bar{\\lambda}\\bar{\\delta}\\,,$"}
{"_id": "Castigliano's_method:5", "title": "", "text": "$=\\int_{0}^{L}{\\frac{Px^{2}}{EI}dx}$"}
{"_id": "5_(number):9", "title": "", "text": "$0.\\overline{384615}$"}
{"_id": "Complement_(set_theory):10", "title": "", "text": "$\\left(A\\cup B\\right)^{c}=A^{c}\\cap B^{c}.$"}
{"_id": "Rodrigues'_formula:2", "title": "", "text": "$L_{n}(x)=\\frac{e^{x}}{n!}\\frac{d^{n}}{dx^{n}}\\left(e^{-x}x^{n}\\right)=\\frac{1}{n!}\\left(\\frac{d}{dx}-1\\right)^{n}x^{n},$"}
{"_id": "Hyperbolic_triangle:25", "title": "", "text": "$\\cos C=-\\cos A\\cos B+\\sin A\\sin B\\cosh c,$"}
{"_id": "Flesch–Kincaid_readability_tests:0", "title": "", "text": "$206.835-1.015\\left(\\frac{\\,\\text{total words}}{\\,\\text{total sentences}}\\right)-84.6\\left(\\frac{\\,\\text{total syllables}}{\\,\\text{total words}}\\right).$"}
{"_id": "Capital_structure_substitution_theory:0", "title": "", "text": "$\\left[\\frac{\\partial D}{\\partial n}\\right]_{\\,\\text{x,t}}=-P_{\\,\\text{x,t}}$"}
{"_id": "Hom_functor:5", "title": "", "text": "$U:C\\to\\,\\textbf{Set}$"}
{"_id": "Principalization_(algebra):79", "title": "", "text": "$D_{\\tau(\\mathfrak{P})}=\\tau D_{\\mathfrak{P}}\\tau^{-1}$"}
{"_id": "Lab_color_space:1", "title": "", "text": "$f(t)=\\begin{cases}t^{1/3}&\\,\\text{if }t>(\\frac{6}{29})^{3}\\\\ \\frac{1}{3}\\left(\\frac{29}{6}\\right)^{2}t+\\frac{4}{29}&\\,\\text{otherwise}\\end{cases}$"}
{"_id": "Maxwell–Boltzmann_distribution:17", "title": "", "text": "$\\sqrt{\\langle v^{2}\\rangle}=\\left(\\int_{0}^{\\infty}v^{2}\\,f(v)\\,dv\\right)^{1/2}=\\sqrt{\\frac{3kT}{m}}=\\sqrt{\\frac{3RT}{M}}=\\sqrt{\\frac{3}{2}}v_{p}$"}
{"_id": "Self-focusing:4", "title": "", "text": "$\\omega_{p}=\\sqrt{\\frac{ne^{2}}{\\gamma m\\epsilon_{0}}}$"}
{"_id": "Neural_cryptography:12", "title": "", "text": "$\\theta_{N}(x)=\\begin{cases}0&\\,\\text{if }x\\leq N/2,\\\\ 1&\\,\\text{if }x>N/2.\\end{cases}$"}
{"_id": "Chevalley_basis:15", "title": "", "text": "$\\beta_{0}+\\gamma_{0}=\\beta+\\gamma$"}
{"_id": "Jacobi_symbol:40", "title": "", "text": "$O(\\log a\\log b)$"}
{"_id": "Mathematical_morphology:23", "title": "", "text": "$(A\\ominus B)\\ominus C=A\\ominus(B\\oplus C)$"}
{"_id": "Alignments_of_random_points:0", "title": "", "text": "$\\frac{n!}{(n-k)!k!}\\left({\\frac{w}{d}}\\right)^{k-2}$"}
{"_id": "Lagrange_inversion_theorem:27", "title": "", "text": "$W(e^{1+z})=1+\\frac{z}{2}+\\frac{z^{2}}{16}-\\frac{z^{3}}{192}-\\frac{z^{4}}{3072}+\\frac{13z^{5}}{61440}-\\frac{47z^{6}}{1474560}-\\frac{73z^{7}}{41287680}+\\frac{2447z^{8}}{1321205760}+O(z^{9}).$"}
{"_id": "Skew_gradient:3", "title": "", "text": "$\\nabla u(x,y)\\cdot\\nabla^{\\perp}u(x,y)=0,\\rVert\\nabla u\\rVert=\\rVert\\nabla^{\\perp}u\\rVert$"}
{"_id": "Tanc_function:1", "title": "", "text": "$\\operatorname{Im}\\left(\\frac{\\tan(x+iy)}{x+iy}\\right)$"}
{"_id": "Solution_of_triangles:52", "title": "", "text": "$\\alpha=180^{\\circ}-\\beta-\\gamma$"}
{"_id": "Solar_neutrino:2", "title": "", "text": "$d+p\\to{{}^{3}}He+\\gamma$"}
{"_id": "Elementary_algebra:196", "title": "", "text": "$\\displaystyle 2x-2x-y$"}
{"_id": "Egyptian_fraction:62", "title": "", "text": "$O(\\log\\log y)$"}
{"_id": "Delay_spread:2", "title": "", "text": "$\\tau_{\\,\\text{rms}}=\\sqrt{\\frac{\\int_{0}^{\\infty}(\\tau-\\overline{\\tau})^{2}A_{c}(\\tau)d\\tau}{\\int_{0}^{\\infty}A_{c}(\\tau)d\\tau}}$"}
{"_id": "Loop_representation_in_gauge_theories_and_quantum_gravity:0", "title": "", "text": "$\\nabla\\cdot\\vec{E}={\\rho\\over\\epsilon_{0}}\\qquad\\nabla\\times\\vec{B}-\\epsilon_{0}\\mu_{0}{\\partial\\vec{E}\\over\\partial t}=\\mu_{0}\\vec{J}\\qquad\\nabla\\times\\vec{E}+{\\partial\\vec{B}\\over\\partial t}=0\\qquad\\nabla\\cdot\\vec{B}=0$"}
{"_id": "Universal_Century_technology:0", "title": "", "text": "${}^{3}_{2}\\mathrm{He}+{}^{2}_{1}\\mathrm{H}\\to{}^{4}_{2}\\mathrm{He}+\\mathrm{p}$"}
{"_id": "Thorium_fuel_cycle:3", "title": "", "text": "$\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow{}_{\\ 90}^{231}\\mathrm{Th}+2\\mathrm{n}\\xrightarrow{\\beta^{-}}{}_{\\ 91}^{231}\\mathrm{Pa}+\\mathrm{n}\\rightarrow{}_{\\ 91}^{232}\\mathrm{Pa}\\xrightarrow{\\beta^{-}}{}_{\\ 92}^{232}\\mathrm{U}$"}
{"_id": "De_Rham_curve:60", "title": "", "text": "$i=0\\ldots n-2.$"}
{"_id": "Duhamel's_principle:23", "title": "", "text": "$=\\int_{0}^{\\infty}G(\\tau)F(t-\\tau)\\,d\\tau$"}
{"_id": "Power_series:24", "title": "", "text": "$=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}a_{i}b_{j}(x-c)^{i+j}$"}
{"_id": "Bézier_curve:20", "title": "", "text": "${n\\choose i}=\\frac{n!}{i!(n-i)!}$"}
{"_id": "Hodge_dual:59", "title": "", "text": "$\\lambda\\wedge\\theta\\in\\bigwedge^{n}V.$"}
{"_id": "Graver_basis:39", "title": "", "text": "$O\\left(m^{g(k)}n^{k}\\right)$"}
{"_id": "Singular_integral_operators_of_convolution_type:110", "title": "", "text": "$\\frac{1}{p}+\\frac{1}{q}.$"}
{"_id": "Compound_Poisson_process:15", "title": "", "text": "$\\nu^{*n}=\\underbrace{\\nu*\\cdots*\\nu}_{n\\,\\text{ factors}}$"}
{"_id": "Gauss–Newton_algorithm:73", "title": "", "text": "$\\frac{\\partial^{2}S}{\\partial\\beta_{j}\\partial\\beta_{k}}$"}
{"_id": "Omnibus_test:0", "title": "", "text": "$F=\\tfrac{{\\displaystyle\\sum_{j=1}^{k}n_{j}\\left(\\bar{y}_{j}-\\bar{y}\\right)^{2}}/{(k-1)}}{{\\displaystyle{\\sum_{j=1}^{k}}{\\sum_{i=1}^{n_{j}}}\\left(y_{ij}-\\bar{y}_{j}\\right)^{2}}/{(n-k)}}$"}
{"_id": "Riemann_series_theorem:24", "title": "", "text": "$1-\\frac{1}{2}-\\frac{1}{4}+\\frac{1}{3}-\\frac{1}{6}-\\frac{1}{8}+\\frac{1}{5}-\\frac{1}{10}-\\frac{1}{12}+\\cdots$"}
{"_id": "Local_inverse:14", "title": "", "text": "$\\begin{bmatrix}x_{1}\\\\ y_{1}\\end{bmatrix}=\\begin{bmatrix}E&F\\\\ G&H\\end{bmatrix}\\begin{bmatrix}f-By_{0}\\\\ g-Dy_{0}\\end{bmatrix}$"}
{"_id": "Numeric_precision_in_Microsoft_Excel:0", "title": "", "text": "$\\sqrt{\\frac{\\Sigma(x-\\bar{x})^{2}}{n}}=\\sqrt{\\frac{\\Sigma\\left[x-\\left(\\Sigma x\\right)/n\\right]^{2}}{n}}\\ ,$"}
{"_id": "Pi:63", "title": "", "text": "$\\pi=3+\\textstyle\\frac{1}{7+\\textstyle\\frac{1}{15+\\textstyle\\frac{1}{1+\\textstyle\\frac{1}{292+\\textstyle\\frac{1}{1+\\textstyle\\frac{1}{1+\\textstyle\\frac{1}{1+\\ddots}}}}}}}$"}
{"_id": "Mathematical_proof:14", "title": "", "text": "$2n+ 1 = 2(1) + 1 = 3$"}
{"_id": "List_of_formulae_involving_π:99", "title": "", "text": "$\\pi=\\cfrac{4}{1+\\cfrac{1^{2}}{2+\\cfrac{3^{2}}{2+\\cfrac{5^{2}}{2+\\cfrac{7^{2}}{2+\\ddots}}}}}\\,$"}
{"_id": "Acoustic_wave_equation:48", "title": "", "text": "$p=-\\rho{\\partial\\over\\partial t}\\Phi$"}
{"_id": "Conjectural_variation:15", "title": "", "text": "$P^{*}=\\frac{1}{2}$"}
{"_id": "Scalability:4", "title": "", "text": "$\\frac{1}{\\alpha+\\frac{1-\\alpha}{P}}$"}
{"_id": "Axiom_schema_of_predicative_separation:0", "title": "", "text": "$\\forall x\\;\\exists y\\;\\forall z\\;(z\\in y\\leftrightarrow z\\in x\\wedge\\phi(z))$"}
{"_id": "Deal–Grove_model:21", "title": "", "text": "$(B/A)_{0}\\ \\left(\\frac{\\mu m}{hr}\\right)$"}
{"_id": "Charge_number:1", "title": "", "text": "$-1\\cdot e$"}
{"_id": "Approximations_of_π:212", "title": "", "text": "$\\pi=\\frac{1}{2^{6}}\\sum_{n=0}^{\\infty}\\frac{(-1)^{n}}{2^{10n}}\\left(-\\frac{2^{5}}{4n+1}-\\frac{1}{4n+3}+\\frac{2^{8}}{10n+1}-\\frac{2^{6}}{10n+3}-\\frac{2^{2}}{10n+5}-\\frac{2^{2}}{10n+7}+\\frac{1}{10n+9}\\right)\\!$"}
{"_id": "Kurtosis:30", "title": "", "text": "$=\\frac{(n+1)\\,n}{(n-1)\\,(n-2)\\,(n-3)}\\;\\frac{\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{4}}{k_{2}^{2}}-3\\,\\frac{(n-1)^{2}}{(n-2)(n-3)}$"}
{"_id": "Primitive_equations:25", "title": "", "text": "$dp=-\\rho\\,d\\phi$"}
{"_id": "Baum–Connes_conjecture:4", "title": "", "text": "$\\underline{E\\Gamma}$"}
{"_id": "Euler–Mascheroni_constant:41", "title": "", "text": "$\\gamma=1-\\sum_{k=2}^{\\infty}(-1)^{k}\\frac{\\lfloor\\log_{2}k\\rfloor}{k+1}.$"}
{"_id": "Ordinal_arithmetic:1", "title": "", "text": "$\\alpha<\\beta\\Rightarrow\\gamma+\\alpha<\\gamma+\\beta$"}
{"_id": "Aeroacoustics:36", "title": "", "text": "$\\frac{1}{c_{0}^{2}}\\frac{\\partial^{2}p}{\\partial t^{2}}-\\nabla^{2}p=\\rho_{0}\\frac{\\partial^{2}\\hat{T}_{ij}}{\\partial x_{i}\\partial x_{j}},\\quad\\,\\text{where}\\quad\\hat{T}_{ij}=v_{i}v_{j}.$"}
{"_id": "Newton's_theorem_of_revolving_orbits:17", "title": "", "text": "$k=\\sqrt{\\frac{a+b}{am+bn}}$"}
{"_id": "Scapegoat_tree:16", "title": "", "text": "$n/2-1$"}
{"_id": "Statistical_coupling_analysis:1", "title": "", "text": "$P_{i}^{x}=\\frac{N!}{n_{x}!(N-n_{x})!}p_{x}^{n_{x}}(1-p_{x})^{N-n_{x}}$"}
{"_id": "Marginalism:11", "title": "", "text": "$\\frac{\\partial^{2}U}{\\partial g^{2}}<0$"}
{"_id": "List_of_New_Testament_papyri:117", "title": "", "text": "$\\mathfrak{P}^{92}$"}
{"_id": "Classical_Hamiltonian_quaternions:55", "title": "", "text": "$=\\frac{T\\alpha}{T\\beta}(\\cos\\phi+\\epsilon\\sin\\phi)$"}
{"_id": "Corner_transfer_matrix:4", "title": "", "text": "$\\begin{array}[]{cccc}&&\\begin{array}[]{ccccc}\\sigma_{1}^{\\prime}=+1&&&&\\sigma_{1}^{\\prime}=-1\\end{array}\\\\ A&=&\\left[\\begin{array}[]{ccccccc}&&&|\\\\ &A_{+}&&|&&0\\\\ &&&|\\\\ -&-&-&|&-&-&-\\\\ &&&|\\\\ &0&&|&&A_{-}\\\\ &&&|\\end{array}\\right]&\\begin{array}[]{c}\\sigma_{1}=+1\\\\ \\\\ \\\\ \\\\ \\sigma_{1}=-1\\end{array}\\end{array}$"}
{"_id": "Taylor's_theorem:31", "title": "", "text": "$f:\\mathbb{R}\\to\\mathbb{R};\\qquad f(x)=\\begin{cases}e^{-\\frac{1}{x^{2}}}&x>0,\\\\ 0&x\\leq 0.\\end{cases}$"}
{"_id": "Neutral_particle_oscillation:74", "title": "", "text": "$M=\\left(\\begin{matrix}{{M}_{11}}&{{M}_{12}}\\\\ {{M}_{21}}&{{M}_{22}}\\\\ \\end{matrix}\\right)$"}
{"_id": "Level_ancestor_problem:1", "title": "", "text": "$\\ell=\\lfloor\\log_{2}(\\operatorname{depth}(v))\\rfloor$"}
{"_id": "Covariance_and_contravariance_of_vectors:49", "title": "", "text": "$\\begin{aligned}\\displaystyle\\begin{bmatrix}v_{1}\\\\ v_{2}\\end{bmatrix}&\\displaystyle=R^{-1}\\begin{bmatrix}v^{1}\\\\ v^{2}\\end{bmatrix}\\\\ &\\displaystyle=\\begin{bmatrix}4&\\sqrt{2}\\\\ \\sqrt{2}&1\\end{bmatrix}\\begin{bmatrix}v^{1}\\\\ v^{2}\\end{bmatrix}=\\begin{bmatrix}6+2\\sqrt{2}\\\\ 2+3/\\sqrt{2}\\end{bmatrix}\\end{aligned}.$"}
{"_id": "List_of_relativistic_equations:59", "title": "", "text": "$ct^{\\prime}=\\gamma ct-\\gamma\\beta x\\,$"}
{"_id": "Clebsch–Gordan_coefficients_for_SU(3):36", "title": "", "text": "$\\forall x,y\\in G$"}
{"_id": "Algorithms_for_calculating_variance:94", "title": "", "text": "$\\textstyle C_{n}=\\sum_{i=1}^{n}(x_{i}-\\bar{x}_{n})(y_{i}-\\bar{y}_{n})$"}
{"_id": "Metric_tensor:12", "title": "", "text": "$\\begin{bmatrix}du\\\\ dv\\end{bmatrix}=\\begin{bmatrix}\\frac{\\partial u}{\\partial u^{\\prime}}&\\frac{\\partial u}{\\partial v^{\\prime}}\\\\ \\frac{\\partial v}{\\partial u^{\\prime}}&\\frac{\\partial v}{\\partial v^{\\prime}}\\end{bmatrix}\\begin{bmatrix}du^{\\prime}\\\\ dv^{\\prime}\\end{bmatrix}$"}
{"_id": "Logistic_regression:99", "title": "", "text": "$\\operatorname{Pr}(Y_{i}=y_{i}\\mid\\mathbf{X}_{i})={n_{i}\\choose y_{i}}p_{i}^{y_{i}}(1-p_{i})^{n_{i}-y_{i}}={n_{i}\\choose y_{i}}\\left(\\frac{1}{1+e^{-symbol\\beta\\cdot\\mathbf{X}_{i}}}\\right)^{y_{i}}\\left(1-\\frac{1}{1+e^{-symbol\\beta\\cdot\\mathbf{X}_{i}}}\\right)^{n_{i}-y_{i}}$"}
{"_id": "Ruin_theory:28", "title": "", "text": "$e^{-\\delta\\tau}$"}
{"_id": "Stieltjes_constants:8", "title": "", "text": "$\\gamma_{1}\\,=\\,\\frac{\\ln 2}{2}\\sum_{k=2}^{\\infty}\\frac{(-1)^{k}}{k}\\,\\lfloor\\log_{2}{k}\\rfloor\\cdot\\big(2\\log_{2}{k}-\\lfloor\\log_{2}{2k}\\rfloor\\big)$"}
{"_id": "Radiocarbon_dating:1", "title": "", "text": "$\\mathrm{~{}^{14}_{6}C}\\rightarrow\\mathrm{~{}^{14}_{7}N}+e^{-}+\\bar{\\nu}_{e}$"}
{"_id": "Slide_rule:1", "title": "", "text": "$\\log(x/y)=\\log(x)-\\log(y)$"}
{"_id": "Overcompleteness:107", "title": "", "text": "$|c-[c]-\\frac{1}{2}|<\\frac{1}{2}-a$"}
{"_id": "Symbolic_method:2", "title": "", "text": "$\\displaystyle 2\\Delta=(ab)^{2}$"}
{"_id": "Cevian:5", "title": "", "text": "$d=\\sqrt{\\frac{2b^{2}+2c^{2}-a^{2}}{4}}.$"}
{"_id": "Hit-or-miss_transform:5", "title": "", "text": "$A\\odot B=(A\\ominus C)\\cap(A^{c}\\ominus D)$"}
{"_id": "Moran's_I:0", "title": "", "text": "$I=\\frac{N}{\\sum_{i}\\sum_{j}w_{ij}}\\frac{\\sum_{i}\\sum_{j}w_{ij}(X_{i}-\\bar{X})(X_{j}-\\bar{X})}{\\sum_{i}(X_{i}-\\bar{X})^{2}}$"}
{"_id": "Invariant_theory:1", "title": "", "text": "$(g\\cdot f)(x):=f(g^{-1}x)\\qquad\\forall x\\in V,g\\in G,f\\in k[V].$"}
{"_id": "Well-structured_transition_system:19", "title": "", "text": "$t_{1}\\xrightarrow{*}t_{2}$"}
{"_id": "P-adic_exponential_function:0", "title": "", "text": "$\\exp(z)=\\sum_{n=0}^{\\infty}\\frac{z^{n}}{n!}.$"}
{"_id": "Kinetic_scheme:9", "title": "", "text": "$\\frac{d\\vec{P}}{dt}=\\int^{t}_{0}\\mathbf{A}(t-\\tau)\\vec{P}(\\tau)d\\tau$"}
{"_id": "Erdős–Rényi_model:0", "title": "", "text": "$p^{M}(1-p)^{{n\\choose 2}-M}.$"}
{"_id": "Central_differencing_scheme:4", "title": "", "text": "${d\\over dx}(\\rho u\\varphi)={d\\over dx}\\left({d\\varphi\\over dx}\\right)$"}
{"_id": "Exponentiation:0", "title": "", "text": "$b^{n}=\\underbrace{b\\times\\cdots\\times b}_{n}$"}
{"_id": "Feynman_diagram:223", "title": "", "text": "$\\Delta(x)=\\int_{0}^{\\infty}d\\tau\\int DXe^{-\\int_{0}^{\\tau}(\\dot{x}^{2}/2+m^{2})d\\tau^{\\prime}}$"}
{"_id": "Steady_flight:1", "title": "", "text": "$C\\cos{\\mu}+L\\sin{\\mu}+T(\\sin{\\alpha}\\sin{\\mu}+\\cos{\\alpha}\\cos{\\mu}\\sin{\\beta})=\\frac{W}{g}\\frac{(V\\cos{\\gamma})^{2}}{R}\\quad(x_{E}\\,\\text{-}y_{E}\\,\\text{ plane radial direction})$"}
{"_id": "Square_root_of_5:0", "title": "", "text": "$2+\\cfrac{1}{4+\\cfrac{1}{4+\\cfrac{1}{4+\\cfrac{1}{4+\\ddots}}}}$"}
{"_id": "Bose–Hubbard_model:11", "title": "", "text": "$D_{f}=\\frac{L!}{N_{f}!(L-N_{f})!}.$"}
{"_id": "List_of_fractals_by_Hausdorff_dimension:41", "title": "", "text": "$2-\\log_{2}(\\sqrt{2})=\\frac{3}{2}$"}
{"_id": "Algorithmic_inference:45", "title": "", "text": "$1-\\delta/2$"}
{"_id": "RANSAC:16", "title": "", "text": "$1-p=(1-w^{n})^{k}$"}
{"_id": "Conformal_geometric_algebra:25", "title": "", "text": "$\\mathbf{P}=g(\\mathbf{a})-g(\\mathbf{b})$"}
{"_id": "Hyperbolic_angle:3", "title": "", "text": "$\\textstyle\\sinh x=\\sum_{n=0}^{\\infty}\\frac{x^{2n+1}}{(2n+1)!}$"}
{"_id": "Mean_curvature:33", "title": "", "text": "$\\textstyle\\mbox{Hess}~{}(F)=\\begin{pmatrix}\\frac{\\partial^{2}F}{\\partial x^{2}}&\\frac{\\partial^{2}F}{\\partial x\\partial y}&\\frac{\\partial^{2}F}{\\partial x\\partial z}\\\\ \\frac{\\partial^{2}F}{\\partial x\\partial y}&\\frac{\\partial^{2}F}{\\partial y^{2}}&\\frac{\\partial^{2}F}{\\partial y\\partial z}\\\\ \\frac{\\partial^{2}F}{\\partial x\\partial z}&\\frac{\\partial^{2}F}{\\partial y\\partial z}&\\frac{\\partial^{2}F}{\\partial z^{2}}\\end{pmatrix}.$"}
{"_id": "Mode_choice:12", "title": "", "text": "$L^{*}=\\prod_{n=1}^{N}{P_{i}^{Y_{i}}}\\left(1-P_{i}\\right)^{1-Y_{i}}$"}
{"_id": "List_of_representations_of_e:7", "title": "", "text": "$e=2+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{2}{3+\\cfrac{3}{4+\\cfrac{4}{5+\\ddots}}}}}=2+\\cfrac{2}{2+\\cfrac{3}{3+\\cfrac{4}{4+\\cfrac{5}{5+\\cfrac{6}{6+\\ddots\\,}}}}}$"}
{"_id": "Guard_(computer_science):0", "title": "", "text": "$f(x)=\\left\\{\\begin{matrix}1&\\mbox{if }~{}x>0\\\\ 0&\\mbox{otherwise}\\end{matrix}\\right.$"}
{"_id": "Dual_matroid:20", "title": "", "text": "$M\\setminus X=(M^{\\ast}/x)^{\\ast}$"}
{"_id": "Banach_space:112", "title": "", "text": "$\\displaystyle\\forall x,y\\in H:\\quad\\langle y,x\\rangle$"}
{"_id": "Schur_test:28", "title": "", "text": "$\\frac{1}{r}=1-\\Big(\\frac{1}{p}-\\frac{1}{q}\\Big)$"}
{"_id": "Boundary_layer:63", "title": "", "text": "$v_{x}{\\partial c_{A}\\over\\partial x}+v_{y}{\\partial c_{A}\\over\\partial y}=D_{AB}{\\partial^{2}c_{A}\\over\\partial y^{2}}$"}
{"_id": "Convergence_in_measure:10", "title": "", "text": "$k=\\lfloor\\log_{2}n\\rfloor$"}
{"_id": "Cloud_drop_effective_radius:0", "title": "", "text": "$r_{e}=\\dfrac{\\int\\limits_{0}^{\\infty}\\pi\\cdot r^{3}\\cdot n(r)\\,dr}{\\int\\limits_{0}^{\\infty}\\pi\\cdot r^{2}\\cdot n(r)\\,dr}$"}
{"_id": "Xbar_and_s_chart:11", "title": "", "text": "$s=\\sqrt{\\frac{\\sum_{i=1}^{n}{\\left(x_{i}-\\bar{x}\\right)}^{2}}{n-1}}$"}
{"_id": "Otto_cycle:37", "title": "", "text": "$\\mathit{c}_{p}=\\left(\\frac{\\gamma\\mathit{R}}{\\gamma-1}\\right)$"}
{"_id": "Gabor_wavelet:1", "title": "", "text": "$(\\Delta x)^{2}=\\frac{\\int_{-\\infty}^{\\infty}(x-\\mu)^{2}f(x)f^{*}(x)\\,dx}{\\int_{-\\infty}^{\\infty}f(x)f^{*}(x)\\,dx}$"}
{"_id": "Inverse_Galois_problem:70", "title": "", "text": "$x^{n}-\\tfrac{x}{2}-\\tfrac{1}{2}-\\left(s-\\tfrac{1}{2}\\right)(x+1)$"}
{"_id": "Quantum-optical_spectroscopy:6", "title": "", "text": "$\\hat{H}_{\\mathrm{lm}}=-\\sum\\mathcal{F}\\,\\hat{B}\\hat{X}^{\\dagger}+\\mathrm{h.c.}\\,,$"}
{"_id": "Stefan–Boltzmann_law:36", "title": "", "text": "$\\left(\\frac{\\partial u}{\\partial T}\\right)_{V}$"}
{"_id": "Zipf–Mandelbrot_law:6", "title": "", "text": "$H_{N,q,s}=\\sum_{i=1}^{N}\\frac{1}{(i+q)^{s}}$"}
{"_id": "Euclidean_vector:47", "title": "", "text": "$\\begin{bmatrix}u\\\\ v\\\\ w\\\\ \\end{bmatrix}=\\begin{bmatrix}c_{11}&c_{12}&c_{13}\\\\ c_{21}&c_{22}&c_{23}\\\\ c_{31}&c_{32}&c_{33}\\end{bmatrix}\\begin{bmatrix}a_{1}\\\\ a_{2}\\\\ a_{3}\\end{bmatrix}$"}
{"_id": "Langmuir_adsorption_model:40", "title": "", "text": "$\\Omega_{conf}\\,=\\,\\frac{N_{S}!}{N!(N_{S}-N)!}$"}
{"_id": "Lambert_W_function:58", "title": "", "text": "$=\\int_{0}^{\\infty}u^{\\frac{1}{2}}e^{-\\frac{u}{2}}\\mathrm{d}u+\\int_{0}^{\\infty}u^{-\\frac{1}{2}}e^{-\\frac{u}{2}}\\mathrm{d}u$"}
{"_id": "Schur–Horn_theorem:102", "title": "", "text": "$\\Psi(A)(B)=\\mathrm{tr}(iAB)$"}
{"_id": "Elias_delta_coding:1", "title": "", "text": "$\\lfloor\\log_{2}(x)\\rfloor+2\\lfloor\\log_{2}(\\lfloor\\log_{2}(x)\\rfloor+1)\\rfloor+1$"}
{"_id": "Help:Displaying_a_formula:128", "title": "", "text": "$\\stackrel{\\alpha}{\\omega}$"}
{"_id": "Binomial_proportion_confidence_interval:42", "title": "", "text": "$1 –α/ 2$"}
{"_id": "Raychaudhuri_equation:20", "title": "", "text": "$-3/\\theta_{0}$"}
{"_id": "2D_computer_graphics:10", "title": "", "text": "$\\begin{bmatrix}x^{\\prime}\\\\ y^{\\prime}\\\\ \\end{bmatrix}=\\begin{bmatrix}\\cos\\theta&-\\sin\\theta\\\\ \\sin\\theta&\\cos\\theta\\\\ \\end{bmatrix}\\begin{bmatrix}x\\\\ y\\\\ \\end{bmatrix}$"}
{"_id": "Maxwell's_equations:150", "title": "", "text": "$\\nabla\\times\\mathbf{B}-\\frac{1}{c^{2}}\\frac{\\partial\\mathbf{E}}{\\partial t}=\\mu_{0}\\mathbf{J}$"}
{"_id": "Cache-oblivious_algorithm:23", "title": "", "text": "$O(mn)$"}
{"_id": "Pooled_variance:10", "title": "", "text": "$s_{p}^{2}=\\frac{\\sum_{i=1}^{k}(n_{i}-1)s_{i}^{2}}{\\sum_{i=1}^{k}(n_{i}-1)}$"}
{"_id": "Internal_energy:48", "title": "", "text": "$\\pi_{T}=\\left(\\frac{\\partial U}{\\partial V}\\right)_{T}$"}
{"_id": "Coriolis_effect:45", "title": "", "text": "$=\\begin{vmatrix}\\!symbol{i}&\\!symbol{j}&\\!symbol{k}\\\\ 0&0&\\omega\\\\ v\\cos\\alpha&v\\sin\\alpha&\\\\ \\;+\\omega t\\ v\\sin\\alpha&\\;-\\omega t\\ v\\cos\\alpha&0\\end{vmatrix}\\ \\ ,$"}
{"_id": "Vector_bundle:14", "title": "", "text": "$0\\to A\\to B\\to C\\to 0$"}
{"_id": "Operator_product_expansion:12", "title": "", "text": "$A(x)B(y)=\\sum_{i}c_{i}(x-y)^{i}C_{i}(y)$"}
{"_id": "Helmholtz_free_energy:74", "title": "", "text": "$\\left\\langle X\\right\\rangle_{0}$"}
{"_id": "Explained_sum_of_squares:10", "title": "", "text": "$\\begin{aligned}\\displaystyle\\sum_{i=1}^{n}2(\\hat{y}_{i}-\\bar{y})(y_{i}-\\hat{y}_{i})&\\displaystyle=\\sum_{i=1}^{n}2\\hat{b}\\left((y_{i}-\\bar{y})(x_{i}-\\bar{x})-\\hat{b}(x_{i}-\\bar{x})^{2}\\right)\\\\ &\\displaystyle=2\\hat{b}\\left(\\sum_{i=1}^{n}(y_{i}-\\bar{y})(x_{i}-\\bar{x})-\\hat{b}\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}\\right)\\\\ &\\displaystyle=2\\hat{b}\\sum_{i=1}^{n}\\left((y_{i}-\\bar{y})(x_{i}-\\bar{x})-(y_{i}-\\bar{y})(x_{i}-\\bar{x})\\right)\\\\ &\\displaystyle=2\\hat{b}\\cdot 0=0.\\end{aligned}$"}
{"_id": "Constructive_set_theory:2", "title": "", "text": "$\\forall a((\\forall x\\in a\\;\\exists y\\;\\phi(x,y))\\to\\exists b\\;(\\forall x\\in a\\;\\exists y\\in b\\;\\phi(x,y))\\wedge(\\forall y\\in b\\;\\exists x\\in a\\;\\phi(x,y)))$"}
{"_id": "Born–Haber_cycle:3", "title": "", "text": "$\\,\\text{M}+\\,\\text{IE}\\text{M}\\to\\,\\text{M}^{+}+\\,\\text{e}^{-}$"}
{"_id": "Equivalent_impedance_transforms:41", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ 0\\end{bmatrix}=\\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\\\ -R_{2}&R_{2}+R_{3}\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ I_{2}\\end{bmatrix}$"}
{"_id": "Q-difference_polynomial:3", "title": "", "text": "$A(w)e_{q}(zw)=\\sum_{n=0}^{\\infty}\\frac{p_{n}(z)}{[n]_{q}!}w^{n}$"}
{"_id": "Taylor's_theorem:60", "title": "", "text": "$\\begin{aligned}\\displaystyle F^{\\prime}(t)=&\\displaystyle f^{\\prime}(t)+\\big(f^{\\prime\\prime}(t)(x-t)-f^{\\prime}(t)\\big)+\\left(\\frac{f^{(3)}(t)}{2!}(x-t)^{2}-\\frac{f^{(2)}(t)}{1!}(x-t)\\right)+\\cdots\\\\ &\\displaystyle\\cdots+\\left(\\frac{f^{(k+1)}(t)}{k!}(x-t)^{k}-\\frac{f^{(k)}(t)}{(k-1)!}(x-t)^{k-1}\\right)=\\frac{f^{(k+1)}(t)}{k!}(x-t)^{k},\\end{aligned}$"}
{"_id": "Theorems_and_definitions_in_linear_algebra:144", "title": "", "text": "$ax^{2}+bxy+cy^{2}+dx+ey+f=0$"}
{"_id": "Solving_quadratic_equations_with_continued_fractions:6", "title": "", "text": "$x=1+\\cfrac{1}{1+\\left(1+\\cfrac{1}{1+x}\\right)}=1+\\cfrac{1}{2+\\cfrac{1}{1+x}}.\\,$"}
{"_id": "Laws_of_science:44", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}+\\frac{1}{c^{2}}\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Draft:Bock's_Parametrization:5", "title": "", "text": "$G(q,p)=\\frac{\\int_{0}^{p}t^{q-1}e^{-t}\\,\\,\\text{d}t}{\\int_{0}^{\\infty}t^{q-1}e^{-t}\\,\\,\\text{d}t}$"}
{"_id": "Electrical_element:22", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ I_{2}\\end{bmatrix}=\\begin{bmatrix}0&n\\\\ -n&0\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ V_{2}\\end{bmatrix}$"}
{"_id": "Gini_coefficient:18", "title": "", "text": "$G=1-\\frac{1}{\\mu}\\int_{0}^{\\infty}(1-F(y))^{2}dy=\\frac{1}{\\mu}\\int_{0}^{\\infty}F(y)(1-F(y))dy$"}
{"_id": "On_Physical_Lines_of_Force:0", "title": "", "text": "$\\mathbf{\\nabla}\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}+\\underbrace{\\mu_{0}\\epsilon_{0}\\frac{\\partial}{\\partial t}\\mathbf{E}}_{\\mathrm{Maxwell^{\\prime}s\\ term}}$"}
{"_id": "Differential_form:28", "title": "", "text": "$M\\stackrel{f}{\\to}N\\stackrel{\\omega}{\\to}T^{*}N\\stackrel{(Df)^{*}}{\\longrightarrow}T^{*}M.$"}
{"_id": "Algorithmic_inference:4", "title": "", "text": "$S_{\\sigma^{2}}=\\sum_{i=1}^{m}(X_{i}-\\overline{X})^{2},\\,\\text{ where }\\overline{X}=\\frac{S_{\\mu}}{m}$"}
{"_id": "Stirling_numbers_of_the_second_kind:41", "title": "", "text": "$\\left\\{{n\\atop k}\\right\\}=\\sum_{j=1}^{k}(-1)^{k-j}\\frac{j^{n-1}}{(j-1)!(k-j)!}=\\frac{1}{k!}\\sum_{j=0}^{k}(-1)^{k-j}{k\\choose j}j^{n}.$"}
{"_id": "Vector_fields_in_cylindrical_and_spherical_coordinates:2", "title": "", "text": "$\\begin{bmatrix}r\\\\ \\theta\\\\ z\\end{bmatrix}=\\begin{bmatrix}\\sqrt{x^{2}+y^{2}}\\\\ \\operatorname{arctan}(y/x)\\\\ z\\end{bmatrix},\\ \\ \\ 0\\leq\\theta<2\\pi,$"}
{"_id": "Big_O_notation:72", "title": "", "text": "$O(\\log\\log n)\\,$"}
{"_id": "Context-free_language:29", "title": "", "text": "$A\\cap B=\\overline{\\overline{A}\\cup\\overline{B}}$"}
{"_id": "Free_object:37", "title": "", "text": "$U:\\mathbf{C}\\to\\mathbf{Set}$"}
{"_id": "Kinetic_priority_queue:13", "title": "", "text": "$O(\\lambda_{\\delta}(n)\\log n)$"}
{"_id": "P-nuclei:18", "title": "", "text": "$\\nu_{e}+n\\rightarrow e^{-}+p$"}
{"_id": "Eigenvalue_algorithm:157", "title": "", "text": "${\\rm det}\\left(\\beta I-B\\right)=\\beta^{3}-3\\beta-{\\rm det}(B)=0.$"}
{"_id": "Circumconic_and_inconic:2", "title": "", "text": "$\\alpha=\\beta=\\gamma=1/3.$"}
{"_id": "Maxwell–Boltzmann_statistics:30", "title": "", "text": "$\\begin{aligned}\\displaystyle W&\\displaystyle=\\frac{N!}{N_{a}!(N-N_{a})!}\\times\\frac{(N-N_{a})!}{N_{b}!(N-N_{a}-N_{b})!}~{}\\times\\frac{(N-N_{a}-N_{b})!}{N_{c}!(N-N_{a}-N_{b}-N_{c})!}\\times\\ldots\\times\\frac{(N-\\ldots-N_{l})!}{N_{k}!(N-\\ldots-N_{l}-N_{k})!}=\\\\ \\\\ &\\displaystyle=\\frac{N!}{N_{a}!N_{b}!N_{c}!\\ldots N_{k}!(N-\\ldots-N_{l}-N_{k})!}\\end{aligned}$"}
{"_id": "William_Brouncker,_2nd_Viscount_Brouncker:3", "title": "", "text": "$\\frac{4}{\\pi}=1+\\cfrac{1^{2}}{2+\\cfrac{3^{2}}{2+\\cfrac{5^{2}}{2+\\cfrac{7^{2}}{2+\\cfrac{9^{2}}{2+\\ddots}}}}}$"}
{"_id": "Feshbach_resonance:0", "title": "", "text": "$A+B\\rightarrow A^{\\prime}+B^{\\prime}$"}
{"_id": "Gaussian_integral:5", "title": "", "text": "$\\begin{aligned}\\displaystyle\\iint_{\\mathbf{R}^{2}}e^{-(x^{2}+y^{2})}\\,d(x,y)&\\displaystyle=\\int_{0}^{2\\pi}\\int_{0}^{\\infty}e^{-r^{2}}r\\,dr\\,d\\theta\\\\ &\\displaystyle=2\\pi\\int_{0}^{\\infty}re^{-r^{2}}\\,dr\\\\ &\\displaystyle=2\\pi\\int_{-\\infty}^{0}\\tfrac{1}{2}e^{s}\\,ds&&\\displaystyle s=-r^{2}\\\\ &\\displaystyle=\\pi\\int_{-\\infty}^{0}e^{s}\\,ds\\\\ &\\displaystyle=\\pi(e^{0}-e^{-\\infty})\\\\ &\\displaystyle=\\pi,\\end{aligned}$"}
{"_id": "Rogers–Ramanujan_continued_fraction:17", "title": "", "text": "$x=\\left[\\frac{\\sqrt{5}\\,\\eta(5\\tau)}{\\eta(\\tau)}\\right]^{6}$"}
{"_id": "KPP-type_equation:0", "title": "", "text": "$u_{t}-\\alpha u-\\beta u^{2}+\\gamma u^{3}=0$"}
{"_id": "Bernoulli_number:26", "title": "", "text": "$B_{n}=(-1)^{n}B^{\\prime}_{n}$"}
{"_id": "Borel_set:10", "title": "", "text": "$x=a_{0}+\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cfrac{1}{\\ddots\\,}}}}$"}
{"_id": "Recursive_least_squares_filter:100", "title": "", "text": "$=\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)-\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)+d(n)\\mathbf{g}(n)$"}
{"_id": "Whitehead's_point-free_geometry:6", "title": "", "text": "$\\forall z[z<x\\rightarrow z<y]\\rightarrow x\\leq y.$"}
{"_id": "Jacobian_matrix_and_determinant:81", "title": "", "text": "$\\mathbf{J}_{\\mathbf{F}}(r,\\theta,\\varphi)=\\begin{bmatrix}\\dfrac{\\partial x}{\\partial r}&\\dfrac{\\partial x}{\\partial\\theta}&\\dfrac{\\partial x}{\\partial\\varphi}\\\\ \\dfrac{\\partial y}{\\partial r}&\\dfrac{\\partial y}{\\partial\\theta}&\\dfrac{\\partial y}{\\partial\\varphi}\\\\ \\dfrac{\\partial z}{\\partial r}&\\dfrac{\\partial z}{\\partial\\theta}&\\dfrac{\\partial z}{\\partial\\varphi}\\end{bmatrix}=\\begin{bmatrix}\\sin\\theta\\cos\\varphi&r\\cos\\theta\\cos\\varphi&-r\\sin\\theta\\sin\\varphi\\\\ \\sin\\theta\\sin\\varphi&r\\cos\\theta\\sin\\varphi&r\\sin\\theta\\cos\\varphi\\\\ \\cos\\theta&-r\\sin\\theta&0\\end{bmatrix}.$"}
{"_id": "Bias_of_an_estimator:11", "title": "", "text": "$\\sum_{i=1}^{n}(X_{i}-\\overline{X})^{2}$"}
{"_id": "Logarithm_of_a_matrix:4", "title": "", "text": "$AB=e^{\\ln(A)+\\ln(B)}.\\,$"}
{"_id": "Weighted_arithmetic_mean:80", "title": "", "text": "$=\\begin{bmatrix}0.9901&0\\\\ 0&0.9901\\end{bmatrix}\\begin{bmatrix}1\\\\ 1\\end{bmatrix}=\\begin{bmatrix}0.9901\\\\ 0.9901\\end{bmatrix}$"}
{"_id": "Cubic_function:14", "title": "", "text": "$\\begin{aligned}\\displaystyle\\Delta_{0}&\\displaystyle=b^{2}-3ac\\\\ \\displaystyle\\Delta_{1}&\\displaystyle=2b^{3}-9abc+27a^{2}d\\end{aligned}$"}
{"_id": "Catenoid:5", "title": "", "text": "$y(u,v)=-\\cos\\theta\\,\\sinh v\\,\\cos u+\\sin\\theta\\,\\cosh v\\,\\sin u$"}
{"_id": "Discrete_Fourier_transform:44", "title": "", "text": "$\\scriptstyle O(N\\log N)$"}
{"_id": "Legendre_transformation:203", "title": "", "text": "$y=px-f^{\\star}(p)$"}
{"_id": "Eigenvalues_and_eigenvectors:374", "title": "", "text": "$u_{2}=\\begin{bmatrix}0\\\\ 1\\end{bmatrix}$"}
{"_id": "Differential_algebra:19", "title": "", "text": "$x,y\\in A$"}
{"_id": "Cousin_problems:4", "title": "", "text": "$0\\to 2\\pi i\\mathbb{Z}\\to\\mathbf{O}\\xrightarrow{\\exp}\\mathbf{O}^{*}\\to 0$"}
{"_id": "Scattering_parameters:25", "title": "", "text": "$\\begin{pmatrix}b_{1}\\\\ b_{2}\\end{pmatrix}=\\begin{pmatrix}S_{11}&S_{12}\\\\ S_{21}&S_{22}\\end{pmatrix}\\begin{pmatrix}a_{1}\\\\ a_{2}\\end{pmatrix}\\,$"}
{"_id": "Einsteinium:0", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ \\xrightarrow[-2\\ \\beta^{-}]{+\\ 6\\ (n,\\gamma)}\\ ^{244}_{\\ 94}Pu}$"}
{"_id": "Dirac_delta_function:30", "title": "", "text": "$=\\int\\limits_{-\\infty}^{\\infty}f(\\tau)\\delta(\\tau-(t-T))\\,d\\tau$"}
{"_id": "Faulhaber's_formula:19", "title": "", "text": "$B_{1}=-\\frac{1}{2}$"}
{"_id": "Matrix_calculus:36", "title": "", "text": "$\\frac{\\partial\\mathbf{\\phi}(\\mathbf{X})}{\\partial\\mathbf{X}}=\\begin{bmatrix}\\frac{\\partial\\mathbf{\\phi}}{\\partial x_{1,1}}&\\cdots&\\frac{\\partial\\mathbf{\\phi}}{\\partial x_{1,q}}\\\\ \\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial\\mathbf{\\phi}}{\\partial x_{n,1}}&\\cdots&\\frac{\\partial\\mathbf{\\phi}}{\\partial x_{n,q}}\\\\ \\end{bmatrix}$"}
{"_id": "Matrix_multiplication:144", "title": "", "text": "$O(mnp)$"}
{"_id": "Euclidean_algorithm:29", "title": "", "text": "$\\frac{1071}{462}=2+\\cfrac{1}{3+\\cfrac{1}{7}}=[2;3,7]$"}
{"_id": "LCP_array:42", "title": "", "text": "$O(m\\log n)$"}
{"_id": "Kelly_criterion:18", "title": "", "text": "$2^{N}p^{K}(1-p)^{N-K}W\\!.$"}
{"_id": "Fraction_(mathematics):97", "title": "", "text": "$\\tfrac{2}{3}-\\tfrac{1}{2}=\\tfrac{4}{6}-\\tfrac{3}{6}=\\tfrac{1}{6}$"}
{"_id": "Semi-supervised_learning:16", "title": "", "text": "$p(x,y|\\theta)=p(y|\\theta)p(x|y,\\theta)$"}
{"_id": "Abductive_reasoning:62", "title": "", "text": "$x$"}
{"_id": "Superheterodyne_receiver:0", "title": "", "text": "$f_{\\mathrm{img}}=\\begin{cases}f+2f_{\\mathrm{IF}},&\\mbox{if }~{}f_{\\mathrm{LO}}>f\\mbox{ (high side injection)}\\\\ f-2f_{\\mathrm{IF}},&\\mbox{if }~{}f_{\\mathrm{LO}}<f\\mbox{ (low side injection)}\\end{cases}$"}
{"_id": "Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm:2", "title": "", "text": "$O(d^{5}n\\log^{3}B)\\,$"}
{"_id": "Turbulence_modeling:10", "title": "", "text": "$\\nu_{t}=\\left|\\frac{\\partial u}{\\partial y}\\right|l_{m}^{2}$"}
{"_id": "Frobenius_method:12", "title": "", "text": "$=\\sum_{k=0}^{\\infty}\\left[(k+r-1)(k+r)+p(z)(k+r)+q(z)\\right]A_{k}z^{k+r}$"}
{"_id": "Universal_hashing:10", "title": "", "text": "$\\forall x,y\\in U,~{}x\\neq y:~{}~{}\\Pr_{h\\in H}[h(x)=h(y)]\\leq\\frac{1}{m}$"}
{"_id": "Quantum_complex_network:9", "title": "", "text": "$|\\phi\\rangle_{j}$"}
{"_id": "Pseudoholomorphic_curve:21", "title": "", "text": "$2-2g-n$"}
{"_id": "Gauss's_lemma_(number_theory):5", "title": "", "text": "$|x|=\\begin{cases}x&\\mbox{if }~{}1\\leq x\\leq\\frac{p-1}{2},\\\\ p-x&\\mbox{if }~{}\\frac{p+1}{2}\\leq x\\leq p-1.\\end{cases}$"}
{"_id": "Pulley:0", "title": "", "text": "$nT-W=0.$"}
{"_id": "Simplex:31", "title": "", "text": "$-1/n$"}
{"_id": "PFD_allowance:3", "title": "", "text": "$A\\text{pfd}=$"}
{"_id": "Proofs_of_trigonometric_identities:142", "title": "", "text": "$\\cos(\\alpha-\\beta)=\\cos\\alpha\\cos\\beta+\\sin\\alpha\\sin\\beta$"}
{"_id": "Hyperfine_structure:28", "title": "", "text": "$\\scriptstyle{\\underline{\\underline{q}}}$"}
{"_id": "Correlogram:7", "title": "", "text": "$c_{h}=\\frac{1}{N}\\sum_{t=1}^{N-h}\\left(Y_{t}-\\bar{Y}\\right)\\left(Y_{t+h}-\\bar{Y}\\right)$"}
{"_id": "Alternating_series:6", "title": "", "text": "$\\eta(s)=\\sum_{n=1}^{\\infty}{(-1)^{n-1}\\over n^{s}}=\\frac{1}{1^{s}}-\\frac{1}{2^{s}}+\\frac{1}{3^{s}}-\\frac{1}{4^{s}}+\\cdots$"}
{"_id": "Biexciton:40", "title": "", "text": "$V=2(\\frac{1}{r_{12}}-\\frac{1}{r_{1a}}-\\frac{1}{r_{1b}}-\\frac{1}{r_{2a}}-\\frac{1}{r_{2b}}+\\frac{1}{r_{ab}})$"}
{"_id": "Grothendieck_spectral_sequence:47", "title": "", "text": "$I^{p,\\bullet}=J\\oplus K$"}
{"_id": "Universal_joint:41", "title": "", "text": "$\\tan\\gamma_{2}=\\cos\\beta\\,\\tan\\gamma_{1}\\qquad\\tan\\gamma_{4}=\\cos\\beta\\,\\tan\\gamma_{3}$"}
{"_id": "Five_points_determine_a_conic:2", "title": "", "text": "$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$"}
{"_id": "Calibration_curve:4", "title": "", "text": "$=\\sqrt{\\frac{\\sum{(y_{i}-mx_{i}-b)}^{2}}{n-2}}$"}
{"_id": "Asymptotic_theory_(statistics):5", "title": "", "text": "$\\hat{\\theta}_{n}\\ \\xrightarrow{p}\\ \\theta_{0}$"}
{"_id": "Linear_interpolation:9", "title": "", "text": "$R_{T}=f(x)-p(x)\\,\\!$"}
{"_id": "Plurisubharmonic_function:15", "title": "", "text": "$\\lambda_{ij}=\\frac{\\partial^{2}f}{\\partial z_{i}\\partial\\bar{z}_{j}}$"}
{"_id": "Vector_space_model:13", "title": "", "text": "$\\mathrm{sim}(d_{j},q)=\\frac{\\mathbf{d_{j}}\\cdot\\mathbf{q}}{\\left\\|\\mathbf{d_{j}}\\right\\|\\left\\|\\mathbf{q}\\right\\|}=\\frac{\\sum_{i=1}^{N}w_{i,j}w_{i,q}}{\\sqrt{\\sum_{i=1}^{N}w_{i,j}^{2}}\\sqrt{\\sum_{i=1}^{N}w_{i,q}^{2}}}$"}
{"_id": "Force_field_(physics):11", "title": "", "text": "$W=\\phi(b)-\\phi(a)$"}
{"_id": "Representation_theory_of_the_Lorentz_group:571", "title": "", "text": "$a,b,c,α,β,γ,α′,β′,γ′$"}
{"_id": "Generalized_Riemann_hypothesis:0", "title": "", "text": "$L(\\chi,s)=\\sum_{n=1}^{\\infty}\\frac{\\chi(n)}{n^{s}}$"}
{"_id": "Nijenhuis–Richardson_bracket:0", "title": "", "text": "$Alt^{p}(V)=(\\wedge^{p+1}V^{*})\\otimes V$"}
{"_id": "Crystal_base:114", "title": "", "text": "$\\tilde{f}_{i}(b\\otimes b^{\\prime})=\\begin{cases}\\tilde{f}_{i}b\\otimes b^{\\prime},&\\,\\text{if }\\phi_{i}(b)>\\epsilon_{i}(b^{\\prime}),\\\\ b\\otimes\\tilde{f}_{i}b^{\\prime},&\\,\\text{if }\\phi_{i}(b)\\leq\\epsilon_{i}(b^{\\prime}).\\end{cases}$"}
{"_id": "Phonon_scattering:2", "title": "", "text": "$\\frac{1}{\\tau_{C}}=\\frac{1}{\\tau_{U}}+\\frac{1}{\\tau_{M}}+\\frac{1}{\\tau_{B}}+\\frac{1}{\\tau_{ph-e}}$"}
{"_id": "Two-port_network:27", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ I_{2}\\end{bmatrix}=\\begin{bmatrix}h_{11}&h_{12}\\\\ h_{21}&h_{22}\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ V_{2}\\end{bmatrix}$"}
{"_id": "Tanhc_function:2", "title": "", "text": "$\\operatorname{Re}\\left(\\frac{\\tanh\\left(x+iy\\right)}{x+iy}\\right)$"}
{"_id": "Slowly_varying_envelope_approximation:0", "title": "", "text": "$\\nabla^{2}E-\\mu_{0}\\,\\varepsilon_{0}\\,\\frac{\\partial^{2}E}{\\partial t^{2}}=0.$"}
{"_id": "Generalized_method_of_moments:58", "title": "", "text": "$J\\ \\xrightarrow{p}\\ \\infty$"}
{"_id": "YaDICs:7", "title": "", "text": "$NCC(\\mu,\\mathcal{I_{F}},\\mathcal{I_{M}})=\\dfrac{\\sum_{x_{i}\\in\\Omega_{F}}\\left(\\mathcal{I_{F}}(x_{i})-\\overline{\\mathcal{I_{F}}}\\right)\\left(\\mathcal{I_{M}}({T}_{\\mu}(x_{i}))-\\overline{\\mathcal{I_{M}}}\\right)}{\\sqrt{\\sum_{x_{i}\\in\\Omega_{F}}\\left(\\mathcal{I_{F}}(x_{i})-\\overline{\\mathcal{I_{F}}}\\right)^{2}\\sum_{x_{i}\\in\\Omega_{F}}\\left(\\mathcal{I_{M}}({T}_{\\mu}(x_{i}))-\\overline{\\mathcal{I_{M}}}\\right)^{2}}}$"}
{"_id": "Dirichlet_eta_function:53", "title": "", "text": "$\\begin{aligned}\\displaystyle\\Gamma(s)\\eta(s)&\\displaystyle=\\int_{0}^{\\infty}\\frac{x^{s-1}}{e^{x}+1}\\,dx=\\int_{0}^{\\infty}\\int_{0}^{x}\\frac{x^{s-2}}{e^{x}+1}\\,dy\\,dx\\\\ &\\displaystyle=\\int_{0}^{\\infty}\\int_{0}^{\\infty}\\frac{(t+r)^{s-2}}{e^{t+r}+1}{dr}\\,dt=\\int_{0}^{1}\\int_{0}^{1}\\frac{(-\\log(xy))^{s-2}}{1+xy}\\,dx\\,dy.\\end{aligned}$"}
{"_id": "Sufficient_statistic:100", "title": "", "text": "$e^{-n\\lambda}\\lambda^{(x_{1}+x_{2}+\\cdots+x_{n})}\\cdot{1\\over x_{1}!x_{2}!\\cdots x_{n}!}\\,$"}
{"_id": "Arithmetic–geometric_mean:36", "title": "", "text": "$g$"}
{"_id": "Zero_state_response:48", "title": "", "text": "$v_{3}(t)=K_{1}\\int_{t_{0}}^{t}i_{1}(\\tau)d\\tau+K_{2}\\int_{t_{0}}^{t}i_{2}(\\tau)d\\tau,$"}
{"_id": "Maxwell_stress_tensor:11", "title": "", "text": "$\\mathbf{f}+\\epsilon_{0}\\mu_{0}\\frac{\\partial\\mathbf{S}}{\\partial t}\\,=\\nabla\\cdot\\mathbf{\\sigma}$"}
{"_id": "Hayashi_track:17", "title": "", "text": "$=(\\frac{k\\rho C}{\\mu H})^{\\gamma}$"}
{"_id": "Delta_method:7", "title": "", "text": "$X_{n}\\,\\xrightarrow{P}\\,\\theta$"}
{"_id": "Clausen_function:112", "title": "", "text": "$\\,\\sin(x+y)=\\sin x\\cos y+\\cos x\\sin y,\\,$"}
{"_id": "Post-modern_portfolio_theory:0", "title": "", "text": "$d=\\sqrt{\\int_{-\\infty}^{t}(t-r)^{2}f(r)\\,dr}$"}
{"_id": "Pseudo-Zernike_polynomials:15", "title": "", "text": "$R_{3,0}=-4+35r^{3}-60r^{2}+30r$"}
{"_id": "Rectifier_(neural_networks):4", "title": "", "text": "$f(x)=\\begin{cases}x&\\mbox{if }~{}x>0\\\\ 0.01x&\\mbox{otherwise}\\end{cases}$"}
{"_id": "Proofs_of_trigonometric_identities:55", "title": "", "text": "$\\sin(\\alpha-\\beta)=\\sin\\alpha\\cos-\\beta+\\cos\\alpha\\sin-\\beta$"}
{"_id": "Maxwell_relations:29", "title": "", "text": "$\\left(\\frac{\\partial^{2}S}{\\partial y\\partial x}\\right)=\\left(\\frac{\\partial^{2}S}{\\partial x\\partial y}\\right):\\left(\\frac{\\partial^{2}V}{\\partial y\\partial x}\\right)=\\left(\\frac{\\partial^{2}V}{\\partial x\\partial y}\\right)$"}
{"_id": "Mathematical_constants_and_functions:319", "title": "", "text": "$\\textstyle\\frac{3\\sqrt{3}}{4}\\left(1-\\frac{1}{2^{2}}+\\frac{1}{4^{2}}-\\frac{1}{5^{2}}+\\frac{1}{7^{2}}-\\frac{1}{8^{2}}+\\frac{1}{10^{2}}\\pm\\cdots\\right)$"}
{"_id": "Dipole:2", "title": "", "text": "$\\stackrel{\\mathfrak{p}}{}$"}
{"_id": "Joint_entropy:11", "title": "", "text": "$H(X_{1},...,X_{n})=-\\sum_{x_{1}}...\\sum_{x_{n}}P(x_{1},...,x_{n})\\log_{2}[P(x_{1},...,x_{n})]\\!$"}
{"_id": "Finite_water-content_vadose_zone_flow_method:22", "title": "", "text": "$\\left[\\frac{\\partial q}{\\partial\\theta}\\right]_{j}=\\left[\\frac{\\partial z}{\\partial t}\\right]_{j}.$"}
{"_id": "Delta-functor:0", "title": "", "text": "$0\\rightarrow M^{\\prime}\\rightarrow M\\rightarrow M^{\\prime\\prime}\\rightarrow 0$"}
{"_id": "Riemann–Siegel_theta_function:11", "title": "", "text": "$3.530972829\\ldots$"}
{"_id": "Lagrangian_mechanics:101", "title": "", "text": "$\\begin{aligned}\\displaystyle L&\\displaystyle=T-U=\\frac{1}{2}M\\dot{\\mathbf{R}}^{2}+\\left(\\frac{1}{2}\\mu\\dot{\\mathbf{r}}^{2}-U(r)\\right)\\\\ &\\displaystyle=L_{\\mathrm{cm}}+L_{\\mathrm{rel}}\\end{aligned}$"}
{"_id": "Matsubara_frequency:27", "title": "", "text": "$-\\eta n_{\\eta}(\\xi)$"}
{"_id": "Green's_function_for_the_three-variable_Laplace_equation:37", "title": "", "text": "$\\cos\\gamma=\\cos\\theta\\cos\\theta^{\\prime}+\\sin\\theta\\sin\\theta^{\\prime}\\cos(\\varphi-\\varphi^{\\prime}).$"}
{"_id": "Poisson_limit_theorem:2", "title": "", "text": "$\\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}\\rightarrow e^{-\\lambda}\\frac{\\lambda^{k}}{k!}.$"}
{"_id": "Symmetric_tensor:24", "title": "", "text": "$v^{\\odot k}=\\underbrace{v\\odot v\\odot\\cdots\\odot v}_{k\\,\\text{ times}}=\\underbrace{v\\otimes v\\otimes\\cdots\\otimes v}_{k\\,\\text{ times}}=v^{\\otimes k}.$"}
{"_id": "Exponential_family:93", "title": "", "text": "$-1-\\eta$"}
{"_id": "Absolute_magnitude:55", "title": "", "text": "$m_{Moon}=0.25+2.5\\log_{10}{\\left(\\frac{3}{2}0.00257^{2}\\right)}=-12.26\\!\\,$"}
{"_id": "Flux_limiter:12", "title": "", "text": "$\\phi_{cm}(r)=\\left\\{\\begin{array}[]{ll}\\frac{r\\left(3r+1\\right)}{\\left(r+1\\right)^{2}},\\quad r>0,\\quad\\lim_{r\\rightarrow\\infty}\\phi_{cm}(r)=3\\\\ 0\\quad\\quad\\,,\\quad r\\leq 0\\end{array}\\right.$"}
{"_id": "Net_run_rate:16", "title": "", "text": "$\\frac{\\mbox{254}~{}}{\\mbox{147.333}~{}}+\\frac{\\mbox{199}~{}}{\\mbox{147.333}~{}}+\\frac{\\mbox{225}~{}}{\\mbox{147.333}~{}}-\\frac{\\mbox{253}~{}}{\\mbox{150}~{}}-\\frac{\\mbox{110}~{}}{\\mbox{150}~{}}-\\frac{\\mbox{103}~{}}{\\mbox{150}~{}}.$"}
{"_id": "Kubo_formula:26", "title": "", "text": "$\\langle\\rangle_{0}$"}
{"_id": "Transformation_matrix:33", "title": "", "text": "$\\begin{bmatrix}x^{\\prime}\\\\ y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1&0\\\\ k&1\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\end{bmatrix}$"}
{"_id": "Machin-like_formula:9", "title": "", "text": "$\\cos(\\alpha+\\beta)=\\cos\\alpha\\cos\\beta-\\sin\\alpha\\sin\\beta$"}
{"_id": "Indian_mathematics:56", "title": "", "text": "$\\frac{\\pi}{4}=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\cdots$"}
{"_id": "Matrix_(mathematics):44", "title": "", "text": "$H(f)=\\left[\\frac{\\partial^{2}f}{\\partial x_{i}\\,\\partial x_{j}}\\right].$"}
{"_id": "Comparator_applications:0", "title": "", "text": "$V_{\\,\\text{out}}=\\begin{cases}V_{\\,\\text{S}+}&\\,\\text{if }V_{1}>V_{2},\\\\ V_{\\,\\text{S}-}&\\,\\text{if }V_{1}<V_{2},\\\\ 0&\\,\\text{if }V_{1}=V_{2},\\end{cases}$"}
{"_id": "Solving_quadratic_equations_with_continued_fractions:11", "title": "", "text": "$x^{2}+bx+c=0\\,$"}
{"_id": "Golomb_coding:18", "title": "", "text": "$M=\\left\\lfloor\\frac{-1}{\\log_{2}(1-p)}\\right\\rfloor$"}
{"_id": "Karger's_algorithm:86", "title": "", "text": "$O(Tm)=O(n^{2}m\\log n)$"}
{"_id": "Grüneisen_parameter:27", "title": "", "text": "$\\frac{11}{d}-\\frac{1}{2}$"}
{"_id": "Kurtosis:22", "title": "", "text": "$g_{2}=\\frac{m_{4}}{m_{2}^{2}}-3=\\frac{\\tfrac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{4}}{\\left(\\tfrac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}\\right)^{2}}-3$"}
{"_id": "List_of_New_Testament_papyri:66", "title": "", "text": "$\\mathfrak{P}^{42}$"}
{"_id": "Electromagnetic_radiation:14", "title": "", "text": "$\\nabla^{2}\\mathbf{E}=\\mu_{0}\\epsilon_{0}\\frac{\\partial^{2}\\mathbf{E}}{\\partial t^{2}}$"}
{"_id": "Curvilinear_coordinates:26", "title": "", "text": "$\\mathbf{J}=\\begin{bmatrix}\\cfrac{\\partial x_{1}}{\\partial q^{1}}&\\cfrac{\\partial x_{1}}{\\partial q^{2}}&\\cfrac{\\partial x_{1}}{\\partial q^{3}}\\\\ \\cfrac{\\partial x_{2}}{\\partial q^{1}}&\\cfrac{\\partial x_{2}}{\\partial q^{2}}&\\cfrac{\\partial x_{2}}{\\partial q^{3}}\\\\ \\cfrac{\\partial x_{3}}{\\partial q^{1}}&\\cfrac{\\partial x_{3}}{\\partial q^{2}}&\\cfrac{\\partial x_{3}}{\\partial q^{3}}\\\\ \\end{bmatrix},\\quad\\mathbf{J}^{-1}=\\begin{bmatrix}\\cfrac{\\partial q^{1}}{\\partial x_{1}}&\\cfrac{\\partial q^{1}}{\\partial x_{2}}&\\cfrac{\\partial q^{1}}{\\partial x_{3}}\\\\ \\cfrac{\\partial q^{2}}{\\partial x_{1}}&\\cfrac{\\partial q^{2}}{\\partial x_{2}}&\\cfrac{\\partial q^{2}}{\\partial x_{3}}\\\\ \\cfrac{\\partial q^{3}}{\\partial x_{1}}&\\cfrac{\\partial q^{3}}{\\partial x_{2}}&\\cfrac{\\partial q^{3}}{\\partial x_{3}}\\\\ \\end{bmatrix}$"}
{"_id": "Formulas_for_generating_Pythagorean_triples:60", "title": "", "text": "$\\displaystyle\\begin{matrix}+2a+b-c=a_{1}&-2a+2b+2c=b_{1}&-2a+b+3c=c_{1}&\\quad\\to\\left[\\,\\text{ }a_{1},\\,\\text{ }b_{1},\\,\\text{ }c_{1}\\right]\\end{matrix}$"}
{"_id": "Abel–Plana_formula:1", "title": "", "text": "$\\zeta(s,\\alpha)=\\sum_{n=0}^{\\infty}\\frac{1}{(n+\\alpha)^{s}}=\\frac{\\alpha^{1-s}}{s-1}+\\frac{1}{2\\alpha^{s}}+2\\int_{0}^{\\infty}\\frac{\\sin\\left(s\\arctan\\frac{t}{\\alpha}\\right)}{(\\alpha^{2}+t^{2})^{\\frac{s}{2}}}\\frac{dt}{e^{2\\pi t}-1}.$"}
{"_id": "Convergence_of_random_variables:54", "title": "", "text": "$\\scriptstyle X_{n}\\xrightarrow{p}X$"}
{"_id": "List_of_electromagnetism_equations:55", "title": "", "text": "$I_{\\mathrm{rms}}=\\sqrt{\\frac{1}{T}\\int_{0}^{T}\\left[I\\left(t\\right)\\right]^{2}\\mathrm{d}t}\\,\\!$"}
{"_id": "Global-warming_potential:1", "title": "", "text": "$GWP\\left(x\\right)=\\frac{\\int_{0}^{TH}a_{x}\\cdot\\left[x(t)\\right]dt}{\\int_{0}^{TH}a_{r}\\cdot\\left[r(t)\\right]dt}$"}
{"_id": "Digamma_function:11", "title": "", "text": "$\\sum_{n=0}^{\\infty}u_{n}=\\sum_{n=0}^{\\infty}\\frac{p(n)}{q(n)},$"}
{"_id": "Hotelling's_T-squared_distribution:23", "title": "", "text": "${\\mathbf{W}}=\\frac{1}{n-1}\\sum_{i=1}^{n}(\\mathbf{x}_{i}-\\overline{\\mathbf{x}})(\\mathbf{x}_{i}-\\overline{\\mathbf{x}})^{\\prime}$"}
{"_id": "Generalized_continued_fraction:64", "title": "", "text": "$\\log 2=\\log(1+1)=\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{3+\\cfrac{2}{2+\\cfrac{2}{5+\\cfrac{3}{2+\\ddots}}}}}}=\\cfrac{2}{3-\\cfrac{1^{2}}{9-\\cfrac{2^{2}}{15-\\cfrac{3^{2}}{21-\\ddots}}}}$"}
{"_id": "Rectangular_function:0", "title": "", "text": "$\\mathrm{rect}(t)=\\Pi(t)=\\begin{cases}0&\\mbox{if }~{}|t|>\\frac{1}{2}\\\\ \\frac{1}{2}&\\mbox{if }~{}|t|=\\frac{1}{2}\\\\ 1&\\mbox{if }~{}|t|<\\frac{1}{2}.\\\\ \\end{cases}$"}
{"_id": "Brownian_motion:26", "title": "", "text": "${\\left({{N}\\atop{N_{R}}}\\right)}=\\frac{N!}{N_{R}!(N-N_{R})!}$"}
{"_id": "Trigamma_function:23", "title": "", "text": "$z_{1}=-0.4121345\\ldots+i0.5978119\\ldots$"}
{"_id": "Risk_aversion:2", "title": "", "text": "$\\tfrac{1}{2}0+\\tfrac{1}{2}100$"}
{"_id": "Berezinian:4", "title": "", "text": "$X=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}$"}
{"_id": "State_observer:84", "title": "", "text": "$e=H(x)-H(\\hat{x})$"}
{"_id": "Centering_matrix:7", "title": "", "text": "$C_{2}=\\left[\\begin{array}[]{rrr}1&0\\\\ \\\\ 0&1\\end{array}\\right]-\\frac{1}{2}\\left[\\begin{array}[]{rrr}1&1\\\\ \\\\ 1&1\\end{array}\\right]=\\left[\\begin{array}[]{rrr}\\frac{1}{2}&-\\frac{1}{2}\\\\ \\\\ -\\frac{1}{2}&\\frac{1}{2}\\end{array}\\right]$"}
{"_id": "Tobit_model:23", "title": "", "text": "$y_{i}=\\begin{cases}y_{i}^{*}&\\textrm{if}\\;y_{i}^{*}>y_{L}\\\\ y_{L}&\\textrm{if}\\;y_{i}^{*}\\leq y_{L}.\\end{cases}$"}
{"_id": "Kuder–Richardson_Formula_20:1", "title": "", "text": "$\\sigma^{2}_{X}=\\frac{\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}\\,{}}{n}.$"}
{"_id": "Solution_of_triangles:105", "title": "", "text": "$a=\\arccos\\left(\\frac{\\cos\\alpha+\\cos\\beta\\cos\\gamma}{\\sin\\beta\\sin\\gamma}\\right),$"}
{"_id": "Implementation_of_mathematics_in_set_theory:91", "title": "", "text": "$\\forall x,y,z\\,(xRy\\wedge yRz\\rightarrow xRz)$"}
{"_id": "Variance:134", "title": "", "text": "$\\mathbb{C}^{n}$"}
{"_id": "Dirac_spinor:20", "title": "", "text": "$E\\begin{bmatrix}\\phi\\\\ \\chi\\end{bmatrix}=\\begin{bmatrix}m\\mathbf{I}&\\vec{\\sigma}\\vec{p}\\\\ \\vec{\\sigma}\\vec{p}&-m\\mathbf{I}\\end{bmatrix}\\begin{bmatrix}\\phi\\\\ \\chi\\end{bmatrix}\\,$"}
{"_id": "Mereotopology:17", "title": "", "text": "$\\forall x\\exist y[Pyx\\and(Czy\\rightarrow Ozx)\\and\\lnot(Pxy\\and(Czx\\rightarrow Ozy))].$"}
{"_id": "Contact_resistance:0", "title": "", "text": "$r_{c}=\\left\\{\\frac{\\partial V}{\\partial J}\\right\\}_{V=0}$"}
{"_id": "Proof_that_π_is_irrational:58", "title": "", "text": "$\\begin{aligned}\\displaystyle J_{n}\\left(\\frac{\\pi}{2}\\right)&\\displaystyle=\\int_{-\\pi/2}^{\\pi/2}\\left(\\frac{\\pi^{2}}{4}-y^{2}\\right)^{n}\\cos(y)\\,dy\\\\ &\\displaystyle=\\int_{0}^{\\pi}\\left(\\frac{\\pi^{2}}{4}-\\left(y-\\frac{\\pi}{2}\\right)^{2}\\right)^{n}\\cos\\left(y-\\frac{\\pi}{2}\\right)\\,dy\\\\ &\\displaystyle=\\int_{0}^{\\pi}y^{n}(\\pi-y)^{n}\\sin(y)\\,dy\\\\ &\\displaystyle=\\frac{n!}{b^{n}}\\int_{0}^{\\pi}f(x)\\sin(x)\\,dx.\\end{aligned}$"}
{"_id": "Alcubierre_drive:0", "title": "", "text": "$ds^{2}=-\\left(\\alpha^{2}-\\beta_{i}\\beta^{i}\\right)\\,dt^{2}+2\\beta_{i}\\,dx^{i}\\,dt+\\gamma_{ij}\\,dx^{i}\\,dx^{j}$"}
{"_id": "List_of_representations_of_e:8", "title": "", "text": "$e=2+\\cfrac{1}{1+\\cfrac{2}{5+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\ddots\\,}}}}}=1+\\cfrac{2}{1+\\cfrac{1}{6+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\ddots\\,}}}}}$"}
{"_id": "Quantization_(signal_processing):85", "title": "", "text": "$\\mathrm{length}(c_{k})\\approx-\\log_{2}\\left(p_{k}\\right)$"}
{"_id": "Derived_category:0", "title": "", "text": "$\\cdots\\to X^{-1}\\xrightarrow{d^{-1}}X^{0}\\xrightarrow{d^{0}}X^{1}\\xrightarrow{d^{1}}X^{2}\\to\\cdots$"}
{"_id": "Help:Displaying_a_formula:380", "title": "", "text": "$f(x)=\\begin{cases}1&-1\\leq x<0\\\\ \\frac{1}{2}&x=0\\\\ 1-x^{2}&\\,\\text{otherwise}\\end{cases}$"}
{"_id": "Nose_cone_design:29", "title": "", "text": "$\\alpha=\\arctan\\left({R\\over L}\\right)-\\arccos\\left({\\sqrt{L^{2}+R^{2}}\\over 2\\rho}\\right)$"}
{"_id": "Haar_wavelet:32", "title": "", "text": "$H_{2N}=\\begin{bmatrix}H_{N}\\otimes[1,1]\\\\ I_{N}\\otimes[1,-1]\\end{bmatrix}$"}
{"_id": "Half-integer:0", "title": "", "text": "$n+{1\\over 2}$"}
{"_id": "Geometric_spanner:1", "title": "", "text": "$O(MST)$"}
{"_id": "Oxidative_phosphorylation:2", "title": "", "text": "$\\rm Succinate+Q\\rightarrow Fumarate+QH_{2}\\!$"}
{"_id": "Indexed_grammar:20", "title": "", "text": "$U[]\\to\\epsilon$"}
{"_id": "Invariant_subspace:28", "title": "", "text": "$T=\\begin{bmatrix}T_{11}&T_{12}\\\\ 0&T_{22}\\end{bmatrix}$"}
{"_id": "Gauss–Legendre_algorithm:5", "title": "", "text": "$3.140\\dots\\!$"}
{"_id": "Riemann's_differential_equation:8", "title": "", "text": "$\\alpha+\\alpha^{\\prime}+\\beta+\\beta^{\\prime}+\\gamma+\\gamma^{\\prime}=1.$"}
{"_id": "McKay_graph:17", "title": "", "text": "$\\chi_{i}\\xrightarrow{n_{ij}}\\chi_{j}$"}
{"_id": "Prime_zeta_function:25", "title": "", "text": "$-0.493091109\\ldots$"}
{"_id": "Approximations_of_π:219", "title": "", "text": "$\\pi=\\frac{1}{2^{6}}\\sum_{n=0}^{\\infty}\\frac{{(-1)}^{n}}{2^{10n}}\\left(-\\frac{2^{5}}{4n+1}-\\frac{1}{4n+3}+\\frac{2^{8}}{10n+1}-\\frac{2^{6}}{10n+3}-\\frac{2^{2}}{10n+5}-\\frac{2^{2}}{10n+7}+\\frac{1}{10n+9}\\right)\\!$"}
{"_id": "Plancherel_measure:4", "title": "", "text": "$G^{\\wedge}$"}
{"_id": "Coomber's_relationship:0", "title": "", "text": "$p_{i}=\\left(\\frac{\\partial E}{\\partial V}\\right)_{T}\\,$"}
{"_id": "Equations_for_a_falling_body:25", "title": "", "text": "$\\ v_{i}=\\sqrt{\\frac{2GMd}{r^{2}}}$"}
{"_id": "Superelliptic_curve:39", "title": "", "text": "$r_{\\infty}=ms-\\deg(f)$"}
{"_id": "Cahen's_constant:0", "title": "", "text": "$C=\\sum\\frac{(-1)^{i}}{s_{i}-1}=\\frac{1}{1}-\\frac{1}{2}+\\frac{1}{6}-\\frac{1}{42}+\\frac{1}{1806}-\\cdots\\approx 0.64341054629.$"}
{"_id": "Vibration_of_plates:77", "title": "", "text": "$\\begin{aligned}\\displaystyle A_{mn}&\\displaystyle=\\frac{4}{ab}\\int_{0}^{a}\\int_{0}^{b}\\varphi(x_{1},x_{2})\\sin\\frac{m\\pi x_{1}}{a}\\sin\\frac{n\\pi x_{2}}{b}dx_{1}dx_{2}\\\\ \\displaystyle B_{mn}&\\displaystyle=\\frac{4}{ab\\omega_{mn}}\\int_{0}^{a}\\int_{0}^{b}\\psi(x_{1},x_{2})\\sin\\frac{m\\pi x_{1}}{a}\\sin\\frac{n\\pi x_{2}}{b}dx_{1}dx_{2}\\,.\\end{aligned}$"}
{"_id": "Edward_Victor_Appleton:3", "title": "", "text": "$D=h-h^{\\prime}=\\frac{1}{\\frac{1}{\\lambda}-\\frac{1}{\\lambda^{\\prime}}}$"}
{"_id": "Hann_function:1", "title": "", "text": "$w(n)=\\sin^{2}\\left(\\frac{\\pi n}{N-1}\\right)$"}
{"_id": "Table_of_thermodynamic_equations:128", "title": "", "text": "$\\mu_{JT}=\\left(\\frac{\\partial T}{\\partial p}\\right)_{H}$"}
{"_id": "Harmonic_series_(mathematics):3", "title": "", "text": "$\\displaystyle 1\\;\\;+\\;\\;\\frac{1}{2}\\;\\;+\\;\\;\\frac{1}{3}\\,+\\,\\frac{1}{4}\\;\\;+\\;\\;\\frac{1}{5}\\,+\\,\\frac{1}{6}\\,+\\,\\frac{1}{7}\\,+\\,\\frac{1}{8}\\;\\;+\\;\\;\\frac{1}{9}\\,+\\,\\cdots$"}
{"_id": "Draft:Leaving_Certificate_Mathematics_Ordinary_Level:30", "title": "", "text": "$x^{2}+2x+5=0$"}
{"_id": "Mason's_invariant:2", "title": "", "text": "$\\begin{bmatrix}Z^{\\prime}_{11}&Z^{\\prime}_{12}\\\\ Z^{\\prime}_{21}&Z^{\\prime}_{22}\\end{bmatrix}=\\begin{bmatrix}Z_{11}&Z_{12}\\\\ Z_{21}&Z_{22}\\end{bmatrix}^{-1}$"}
{"_id": "Sridhara:1", "title": "", "text": "$4a^{2}x^{2}+4abx+4ac=0$"}
{"_id": "Image_impedance:13", "title": "", "text": "$\\gamma=\\alpha+i\\beta\\,\\!$"}
{"_id": "Saddlepoint_approximation_method:3", "title": "", "text": "$\\hat{F}(x)=\\begin{cases}\\Phi(\\hat{w})+\\phi(\\hat{w})(\\frac{1}{\\hat{w}}-\\frac{1}{\\hat{u}})&\\,\\text{for }x\\neq\\mu\\\\ \\frac{1}{2}+\\frac{K^{\\prime\\prime\\prime}(0)}{6\\sqrt{2\\pi}K^{\\prime\\prime}(0)^{3/2}}&\\,\\text{for }x=\\mu\\end{cases}$"}
{"_id": "Struve_function:14", "title": "", "text": "$n+\\frac{1}{2}$"}
{"_id": "Metric_tensor:75", "title": "", "text": "$D\\varphi=\\begin{bmatrix}\\frac{\\partial\\varphi^{1}}{\\partial x^{1}}&\\frac{\\partial\\varphi^{1}}{\\partial x^{2}}&\\dots&\\frac{\\partial\\varphi^{1}}{\\partial x^{n}}\\\\ \\frac{\\partial\\varphi^{2}}{\\partial x^{1}}&\\frac{\\partial\\varphi^{2}}{\\partial x^{2}}&\\dots&\\frac{\\partial\\varphi^{2}}{\\partial x^{n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial\\varphi^{m}}{\\partial x^{1}}&\\frac{\\partial\\varphi^{m}}{\\partial x^{2}}&\\dots&\\frac{\\partial\\varphi^{m}}{\\partial x^{n}}\\\\ \\end{bmatrix}.$"}
{"_id": "Rate_equation:118", "title": "", "text": "$A+R\\rightarrow\\ C$"}
{"_id": "1_−_2_+_3_−_4_+_⋯:22", "title": "", "text": "$\\frac{1}{2}a_{0}-\\frac{1}{4}\\Delta a_{0}+\\frac{1}{8}\\Delta^{2}a_{0}-\\cdots=\\frac{1}{2}-\\frac{1}{4}.$"}
{"_id": "Self-balancing_binary_search_tree:4", "title": "", "text": "$\\lfloor\\log_{2}(n)\\rfloor$"}
{"_id": "Matrix_multiplication:155", "title": "", "text": "$\\mathbf{C}=\\begin{pmatrix}\\mathbf{C}_{11}&\\mathbf{C}_{12}\\\\ \\mathbf{C}_{21}&\\mathbf{C}_{22}\\\\ \\end{pmatrix}=\\begin{pmatrix}\\mathbf{A}_{11}&\\mathbf{A}_{12}\\\\ \\mathbf{A}_{21}&\\mathbf{A}_{22}\\\\ \\end{pmatrix}\\begin{pmatrix}\\mathbf{B}_{11}&\\mathbf{B}_{12}\\\\ \\mathbf{B}_{21}&\\mathbf{B}_{22}\\\\ \\end{pmatrix}=\\mathbf{A}\\mathbf{B}$"}
{"_id": "Diagram_(category_theory):5", "title": "", "text": "$\\bullet\\overrightarrow{\\to}\\bullet$"}
{"_id": "List_of_New_Testament_papyri:85", "title": "", "text": "$\\mathfrak{P}^{60}$"}
{"_id": "Continued_fraction:56", "title": "", "text": "$\\ a_{0}+\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}}}}$"}
{"_id": "Hollow-cathode_lamp:0", "title": "", "text": "$A+h\\nu\\rightarrow A^{*}$"}
{"_id": "Abel's_summation_formula:8", "title": "", "text": "$A(x)=\\lfloor x\\rfloor\\,$"}
{"_id": "Pulse-Doppler_radar:13", "title": "", "text": "$\\,\\text{Subclutter Visibility}=\\left(\\tfrac{\\,\\text{Dynamic Range}}{\\,\\text{CFAR Detection Threshold}}\\right)$"}
{"_id": "LU_decomposition:40", "title": "", "text": "$L=\\begin{pmatrix}1&&&&&0\\\\ -l_{2,1}&\\ddots&&&&\\\\ &&1&&&\\\\ \\vdots&&-l_{n+1,n}&\\ddots&&\\\\ &&\\vdots&&1&\\\\ -l_{N,1}&&-l_{N,n}&&-l_{N,N-1}&1\\end{pmatrix}.$"}
{"_id": "Ordinal_regression:21", "title": "", "text": "$y=\\begin{cases}1~{}~{}\\,\\text{if}~{}~{}y^{*}\\leq\\theta_{1},\\\\ 2~{}~{}\\,\\text{if}~{}~{}\\theta_{1}<y^{*}\\leq\\theta_{2},\\\\ 3~{}~{}\\,\\text{if}~{}~{}\\theta_{2}<y^{*}\\leq\\theta_{3}\\\\ \\vdots\\\\ K~{}~{}\\,\\text{if}~{}~{}\\theta_{K-1}<y^{*}.\\end{cases}$"}
{"_id": "List_of_convolutions_of_probability_distributions:15", "title": "", "text": "$\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}\\sim\\sigma^{2}\\chi^{2}_{n-1},\\quad$"}
{"_id": "Fisher_transformation:0", "title": "", "text": "$r=\\frac{\\sum^{N}_{i=1}(X_{i}-\\bar{X})(Y_{i}-\\bar{Y})}{\\sqrt{\\sum^{N}_{i=1}(X_{i}-\\bar{X})^{2}}\\sqrt{\\sum^{N}_{i=1}(Y_{i}-\\bar{Y})^{2}}}$"}
{"_id": "Lorentz_transformation:52", "title": "", "text": "$\\,\\text{gyr}[\\mathbf{u},\\mathbf{v}]\\mathbf{w}=\\ominus(\\mathbf{u}\\oplus\\mathbf{v})\\oplus(\\mathbf{u}\\oplus(\\mathbf{v}\\oplus\\mathbf{w}))$"}
{"_id": "Trigamma_function:24", "title": "", "text": "$z_{2}=-1.4455692\\ldots+i0.6992608\\ldots$"}
{"_id": "Noncentral_t-distribution:12", "title": "", "text": "$\\mbox{E}~{}\\left[T^{k}\\right]=\\begin{cases}\\left(\\frac{\\nu}{2}\\right)^{\\frac{k}{2}}\\frac{\\Gamma\\left(\\frac{\\nu-k}{2}\\right)}{\\Gamma\\left(\\frac{\\nu}{2}\\right)}\\mbox{exp}~{}\\left(-\\frac{\\mu^{2}}{2}\\right)\\frac{d^{k}}{d\\mu^{k}}\\mbox{exp}~{}\\left(\\frac{\\mu^{2}}{2}\\right),&\\mbox{if }~{}\\nu>k;\\\\ \\mbox{Does not exist}~{},&\\mbox{if }~{}\\nu\\leq k.\\\\ \\end{cases}$"}
{"_id": "H-infinity_methods_in_control_theory:0", "title": "", "text": "$\\begin{bmatrix}z\\\\ v\\end{bmatrix}=\\mathbf{P}(s)\\,\\begin{bmatrix}w\\\\ u\\end{bmatrix}=\\begin{bmatrix}P_{11}(s)&P_{12}(s)\\\\ P_{21}(s)&P_{22}(s)\\end{bmatrix}\\,\\begin{bmatrix}w\\\\ u\\end{bmatrix}$"}
{"_id": "Stirling_numbers_and_exponential_generating_functions_in_symbolic_combinatorics:31", "title": "", "text": "$\\mathcal{B}=\\mathfrak{P}(\\mathcal{U}\\;\\mathfrak{P}_{\\geq 1}(\\mathcal{Z})).$"}
{"_id": "Line–line_intersection:40", "title": "", "text": "$w=(A^{T}A)^{-1}A^{T}b.$"}
{"_id": "Solar_zenith_angle:1", "title": "", "text": "$\\sin\\alpha_{\\mathrm{s}}=\\cos h\\cos\\delta\\cos\\varphi+\\sin\\delta\\sin\\varphi$"}
{"_id": "Compound_interest:14", "title": "", "text": "$a(n)=e^{\\int_{0}^{n}\\delta_{t}\\,dt}\\ ,$"}
{"_id": "Wirtinger_derivatives:22", "title": "", "text": "$\\left\\{\\begin{aligned}\\displaystyle\\frac{\\partial}{\\partial z_{1}}&\\displaystyle=\\frac{1}{2}\\left(\\frac{\\partial}{\\partial x_{1}}-i\\frac{\\partial}{\\partial y_{1}}\\right)\\\\ &\\displaystyle\\qquad\\qquad\\vdots\\\\ \\displaystyle\\frac{\\partial}{\\partial z_{n}}&\\displaystyle=\\frac{1}{2}\\left(\\frac{\\partial}{\\partial x_{n}}-i\\frac{\\partial}{\\partial y_{n}}\\right)\\\\ \\end{aligned}\\right.\\quad,\\quad$"}
{"_id": "Binary_heap:0", "title": "", "text": "$O(n\\log n)$"}
{"_id": "Continued_fraction:55", "title": "", "text": "$\\ -3+\\cfrac{1}{2+\\cfrac{1}{18}}$"}
{"_id": "Incomplete_information_network_game:35", "title": "", "text": "$\\textstyle\\Q(k;p)={N-2\\choose k-1}p^{(k-1)}(1-p)^{(N-k-1)}$"}
{"_id": "Spherical_harmonics:67", "title": "", "text": "$P_{\\ell}^{-m}=(-1)^{m}\\frac{(\\ell-m)!}{(\\ell+m)!}P_{\\ell}^{m}$"}
{"_id": "FOIL_method:0", "title": "", "text": "$(a+b)(c+d)=\\underbrace{ac}_{\\mathrm{first}}+\\underbrace{ad}_{\\mathrm{outside}}+\\underbrace{bc}_{\\mathrm{inside}}+\\underbrace{bd}_{\\mathrm{last}}$"}
{"_id": "Hodge_dual:43", "title": "", "text": "$\\langle\\cdot,\\cdot\\rangle_{W}$"}
{"_id": "Von_Neumann–Bernays–Gödel_set_theory:10", "title": "", "text": "$\\forall x\\forall y\\exists z\\forall w[w\\in z\\leftrightarrow(w=xw=y)].$"}
{"_id": "Tangent_lines_to_circles:19", "title": "", "text": "$x^{2}+y^{2}=(-r)^{2},$"}
{"_id": "Hundred-dollar,_Hundred-digit_Challenge_problems:10", "title": "", "text": "$||f-p||_{\\infty}$"}
{"_id": "Splay_tree:0", "title": "", "text": "$T_{\\mathrm{amortized}}(m)=O(m\\log n)$"}
{"_id": "System_of_imprimitivity:29", "title": "", "text": "$[w]^{-1}\\begin{bmatrix}s\\\\ t\\end{bmatrix}[w]=\\begin{bmatrix}s\\\\ -ws+t\\end{bmatrix}=\\begin{bmatrix}1&0\\\\ -w&1\\end{bmatrix}\\begin{bmatrix}s\\\\ t\\end{bmatrix}$"}
{"_id": "Multi-index_notation:43", "title": "", "text": "$\\frac{d^{\\alpha_{i}}}{dx_{i}^{\\alpha_{i}}}x_{i}^{\\beta_{i}}=\\frac{\\beta_{i}!}{(\\beta_{i}-\\alpha_{i})!}x_{i}^{\\beta_{i}-\\alpha_{i}}$"}
{"_id": "Davenport–Schmidt_theorem:3", "title": "", "text": "$C=\\left\\{\\begin{array}[]{ c l }C_{0}&\\textrm{if}\\ |\\xi|<1\\\\ C_{0}\\xi^{2}&\\textrm{if}\\ |\\xi|>1.\\end{array}\\right.$"}
{"_id": "Moment_(mathematics):3", "title": "", "text": "$\\mu_{n}=\\int_{-\\infty}^{\\infty}(x-c)^{n}\\,f(x)\\,dx.$"}
{"_id": "Equilibrium_constant:32", "title": "", "text": "$K^{\\ominus}=\\frac{[R]^{\\rho}[S]^{\\sigma}...}{[A]^{\\alpha}[B]^{\\beta}...}\\times\\frac{{\\gamma_{R}}^{\\rho}{\\gamma_{S}}^{\\sigma}...}{{\\gamma_{A}}^{\\alpha}{\\gamma_{B}}^{\\beta}...}\\times\\frac{\\left({c^{\\ominus}_{A}}\\right)^{\\alpha}\\left({c^{\\ominus}_{B}}\\right)^{\\beta}...}{\\left({c^{\\ominus}_{R}}\\right)^{\\rho}\\left({c^{\\ominus}_{S}}\\right)^{\\sigma}...}=Q^{E}\\Gamma C^{0}$"}
{"_id": "Network_simplex_algorithm:2", "title": "", "text": "$O(VE\\log V\\log(VC))$"}
{"_id": "List_of_fractals_by_Hausdorff_dimension:7", "title": "", "text": "$\\frac{-\\log(2)}{\\log\\left(\\displaystyle\\frac{1-\\gamma}{2}\\right)}$"}
{"_id": "Gravitational_lensing_formalism:97", "title": "", "text": "$q_{xy}=\\frac{\\sum(x-\\bar{x})(y-\\bar{y})I(x,y)}{\\sum I(x,y)}$"}
{"_id": "Equioscillation_theorem:5", "title": "", "text": "$||f-g||_{\\infty}$"}
{"_id": "Electric_field_gradient:0", "title": "", "text": "$V_{ij}=\\frac{\\partial^{2}V}{\\partial x_{i}\\partial x_{j}}.$"}
{"_id": "Spinodal_decomposition:19", "title": "", "text": "$\\lambda_{c}=\\sqrt{\\frac{8\\pi^{2}\\kappa}{f^{\\prime\\prime}}}.$"}
{"_id": "Modular_elliptic_curve:2", "title": "", "text": "$L(s,E)=\\sum_{n=1}^{\\infty}\\frac{a_{n}}{n^{s}}.$"}
{"_id": "Euler's_continued_fraction_formula:13", "title": "", "text": "$\\pi=\\cfrac{4}{1+\\cfrac{1^{2}}{2+\\cfrac{3^{2}}{2+\\cfrac{5^{2}}{2+\\cfrac{7^{2}}{2+\\ddots}}}}}.\\,$"}
{"_id": "Linear_independence:19", "title": "", "text": "$A\\Lambda=\\begin{bmatrix}1&-3\\\\ 1&2\\end{bmatrix}\\begin{bmatrix}\\lambda_{1}\\\\ \\lambda_{2}\\end{bmatrix}.\\,\\!$"}
{"_id": "Mean_square_weighted_deviation:6", "title": "", "text": "$\\sigma^{2}=\\frac{\\sum_{i=1}^{N}w_{i}(x_{i}-\\overline{x}^{\\,*})^{2}}{\\sum_{i=1}^{N}w_{i}}$"}
{"_id": "Poynting's_theorem:2", "title": "", "text": "$\\nabla\\cdot\\mathbf{S}+\\epsilon_{0}\\mathbf{E}\\cdot\\frac{\\partial\\mathbf{E}}{\\partial t}+\\frac{\\mathbf{B}}{\\mu_{0}}\\cdot\\frac{\\partial\\mathbf{B}}{\\partial t}+\\mathbf{J}\\cdot\\mathbf{E}=0,$"}
{"_id": "Impedance_analogy:11", "title": "", "text": "$\\begin{bmatrix}v\\\\ F\\end{bmatrix}=\\begin{bmatrix}z_{11}&z_{12}\\\\ z_{21}&z_{22}\\end{bmatrix}\\begin{bmatrix}i\\\\ u\\end{bmatrix}$"}
{"_id": "Rotation_(mathematics):11", "title": "", "text": "$\\begin{bmatrix}x^{\\prime}\\\\ y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}\\cos\\theta&-\\sin\\theta\\\\ \\sin\\theta&\\cos\\theta\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\end{bmatrix}$"}
{"_id": "Jet_bundle:122", "title": "", "text": "$=dx-\\rho(x,u,u_{1})dx+u_{1}du\\,$"}
{"_id": "Order_dimension:7", "title": "", "text": "$O(\\log\\log n)$"}
{"_id": "Jordan_normal_form:50", "title": "", "text": "$\\mathrm{Ran}\\;e_{i}(T)=\\mathrm{Ker}(T-\\lambda_{i})^{\\nu_{i}}.$"}
{"_id": "Kronecker_delta:63", "title": "", "text": "$\\delta=\\frac{1}{4\\pi}\\iint_{R_{st}}\\frac{\\begin{vmatrix}x&y&z\\\\ \\dfrac{\\partial x}{\\partial s}&\\dfrac{\\partial y}{\\partial s}&\\dfrac{\\partial z}{\\partial s}\\\\ \\dfrac{\\partial x}{\\partial t}&\\dfrac{\\partial y}{\\partial t}&\\dfrac{\\partial z}{\\partial t}\\end{vmatrix}}{(x^{2}+y^{2}+z^{2})\\sqrt{x^{2}+y^{2}+z^{2}}}dsdt.$"}
{"_id": "Redfield_equation:22", "title": "", "text": "$\\frac{\\partial}{\\partial t}\\rho_{I}(t)=-\\frac{1}{\\hbar^{2}}\\sum_{m,n}\\int_{0}^{\\infty}d\\tau\\biggl(C_{mn}(\\tau)\\Bigl[S_{m,I}(t),S_{n,I}(t-\\tau)\\rho_{I}(t)\\Bigr]-C_{mn}^{\\ast}(\\tau)\\Bigl[S_{m,I}(t),\\rho_{I}(t)S_{n,I}(t-\\tau)\\Bigr]\\biggr)$"}
{"_id": "Equilibrium_constant:39", "title": "", "text": "$\\mu_{i}=\\left(\\frac{\\partial G}{\\partial N_{i}}\\right)_{T,P}$"}
{"_id": "Hyperbolic_law_of_cosines:2", "title": "", "text": "$\\cos\\alpha=-\\cos\\beta\\cos\\gamma+\\sin\\beta\\sin\\gamma\\cosh\\frac{a}{k},\\,$"}
{"_id": "Magnetic_tension_force:5", "title": "", "text": "$\\mu_{0}\\mathbf{J}=\\nabla\\times\\mathbf{B}$"}
{"_id": "Repeating_decimal:42", "title": "", "text": "$\\displaystyle 7.48181818\\ldots$"}
{"_id": "Cochran's_Q_test:10", "title": "", "text": "$T=k\\left(k-1\\right)\\frac{\\sum\\limits_{j=1}^{k}\\left(X_{\\bullet j}-\\frac{N}{k}\\right)^{2}}{\\sum\\limits_{i=1}^{b}X_{i\\bullet}\\left(k-X_{i\\bullet}\\right)}$"}
{"_id": "Distributed_minimum_spanning_tree:0", "title": "", "text": "$O(V\\log V)$"}
{"_id": "Linear_combination:18", "title": "", "text": "$\\mathbf{R}^{n}$"}
{"_id": "Extended_Kalman_filter:21", "title": "", "text": "$\\begin{aligned}\\displaystyle\\dot{\\hat{\\mathbf{x}}}(t)&\\displaystyle=f\\bigl(\\hat{\\mathbf{x}}(t),\\mathbf{u}(t)\\bigr)+\\mathbf{K}(t)\\Bigl(\\mathbf{z}(t)-h\\bigl(\\hat{\\mathbf{x}}(t)\\bigr)\\Bigr)\\\\ \\displaystyle\\dot{\\mathbf{P}}(t)&\\displaystyle=\\mathbf{F}(t)\\mathbf{P}(t)+\\mathbf{P}(t)\\mathbf{F}(t)^{\\top}-\\mathbf{K}(t)\\mathbf{H}(t)\\mathbf{P}(t)+\\mathbf{Q}(t)\\\\ \\displaystyle\\mathbf{K}(t)&\\displaystyle=\\mathbf{P}(t)\\mathbf{H}(t)^{\\top}\\mathbf{R}(t)^{-1}\\\\ \\displaystyle\\mathbf{F}(t)&\\displaystyle=\\left.\\frac{\\partial f}{\\partial\\mathbf{x}}\\right|_{\\hat{\\mathbf{x}}(t),\\mathbf{u}(t)}\\\\ \\displaystyle\\mathbf{H}(t)&\\displaystyle=\\left.\\frac{\\partial h}{\\partial\\mathbf{x}}\\right|_{\\hat{\\mathbf{x}}(t)}\\end{aligned}$"}
{"_id": "Thematic_vowel:0", "title": "", "text": "$\\underbrace{\\underbrace{\\mathrm{root+suffix}}_{\\mathrm{stem}}+\\mathrm{ending}}_{\\mathrm{word}}$"}
{"_id": "Quantum_finance:1", "title": "", "text": "$C_{0}^{N}=(1+r)^{-N}\\sum_{n=0}^{N}\\frac{N!}{n!(N-n)!}q^{n}{(1-q)}^{N-n}{[S_{0}{(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}$"}
{"_id": "Celestial_coordinate_system:35", "title": "", "text": "$\\sin\\delta=\\sin\\beta\\cos\\varepsilon+\\cos\\beta\\sin\\varepsilon\\sin\\lambda$"}
{"_id": "Mathematical_constants_and_functions:298", "title": "", "text": "$\\tfrac{1}{1+\\tfrac{1}{0+\\tfrac{1}{8+\\tfrac{1}{4+\\tfrac{1}{1+\\tfrac{1}{0+1{/\\cdots}}}}}}}$"}
{"_id": "Word_lists_by_frequency:1", "title": "", "text": "$N=\\left\\lfloor 0.5-\\log_{2}\\left(\\frac{\\,\\text{Frequency of this item}}{\\,\\text{Frequency of most common item}}\\right)\\right\\rfloor$"}
{"_id": "Group_velocity:30", "title": "", "text": "$v_{g}=\\left(\\frac{\\partial(\\operatorname{Re}k)}{\\partial\\omega}\\right)^{-1}.$"}
{"_id": "Stufe_(algebra):9", "title": "", "text": "$0=\\underbrace{1+e_{1}^{2}+\\cdots+e_{n-1}^{2}}_{=:a}+\\underbrace{e_{n}^{2}+\\cdots+e_{s}^{2}}_{=:b}\\;.$"}
{"_id": "Chebyshev_nodes:10", "title": "", "text": "$f(x)-P_{n-1}(x)=\\frac{f^{(n)}(\\xi)}{n!}\\prod_{i=1}^{n}(x-x_{i})$"}
{"_id": "Lie_coalgebra:4", "title": "", "text": "$E\\ \\rightarrow^{\\!\\!\\!\\!\\!\\!d}\\ E\\wedge E\\ \\rightarrow^{\\!\\!\\!\\!\\!\\!d}\\ \\bigwedge^{3}E\\rightarrow^{\\!\\!\\!\\!\\!\\!d}\\ \\dots$"}
{"_id": "Carry-lookahead_adder:8", "title": "", "text": "$P^{\\prime}(A,B)=A\\oplus B$"}
{"_id": "Polar_coordinate_system:39", "title": "", "text": "$\\frac{\\partial u}{\\partial\\varphi}=-\\frac{\\partial u}{\\partial x}r\\sin\\varphi+\\frac{\\partial u}{\\partial y}r\\cos\\varphi=-y\\frac{\\partial u}{\\partial x}+x\\frac{\\partial u}{\\partial y}.$"}
{"_id": "Quantile_function:15", "title": "", "text": "$\\alpha=4p(1-p).\\!$"}
{"_id": "Hydrogen-like_atom:32", "title": "", "text": "$Y_{a,b}=\\begin{cases}(-1)^{b}\\sqrt{\\frac{2a+1}{4\\pi}\\frac{(a-b)!}{(a+b)!}}P_{a}^{b}(\\cos\\theta)e^{ib\\phi}&\\,\\text{if }a>0\\\\ Y_{-a-1,b}&\\,\\text{if }a<0\\end{cases}$"}
{"_id": "Russell's_paradox:2", "title": "", "text": "$\\exists y\\forall x(x\\in y\\iff P(x))$"}
{"_id": "Routing_and_wavelength_assignment:15", "title": "", "text": "$O(m+n\\log n)$"}
{"_id": "FEE_method:78", "title": "", "text": "$O\\left(M(m)\\log^{2}m\\right)=O\\left(M(n)\\log n\\right).\\,$"}
{"_id": "Ramification_group:118", "title": "", "text": "$G^{u}H/H=(G/H)^{u}$"}
{"_id": "Vienna_rectifier:5", "title": "", "text": "$\\phi_{1}=-30^{\\circ}...+30^{\\circ}$"}
{"_id": "Regular_number:4", "title": "", "text": "$O(\\log\\log N)$"}
{"_id": "Quaternion:39", "title": "", "text": "$p\\cdot q=\\textstyle\\frac{1}{2}(p^{*}q+q^{*}p)=\\textstyle\\frac{1}{2}(pq^{*}+qp^{*}).$"}
{"_id": "Point-biserial_correlation_coefficient:3", "title": "", "text": "$s_{n-1}=\\sqrt{\\frac{1}{n-1}\\sum_{i=1}^{n}(X_{i}-\\overline{X})^{2}}.$"}
{"_id": "Plate_theory:19", "title": "", "text": "$\\begin{bmatrix}\\sigma_{11}\\\\ \\sigma_{22}\\\\ \\sigma_{12}\\end{bmatrix}=\\begin{bmatrix}C_{11}&C_{12}&C_{13}\\\\ C_{12}&C_{22}&C_{23}\\\\ C_{13}&C_{23}&C_{33}\\end{bmatrix}\\begin{bmatrix}\\varepsilon_{11}\\\\ \\varepsilon_{22}\\\\ \\varepsilon_{12}\\end{bmatrix}$"}
{"_id": "Mathematical_constants_and_functions:98", "title": "", "text": "$\\sum_{n=1}^{\\infty}\\frac{1}{n{2n\\choose n}}=1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{5}+\\frac{1}{7}-\\frac{1}{8}+\\cdots$"}
{"_id": "Fluid_flow_through_porous_media:7", "title": "", "text": "$Q=\\frac{-\\kappa A}{\\mu}\\left(\\frac{dp}{dx}\\right)$"}
{"_id": "Giant_component:13", "title": "", "text": "$4t-2n$"}
{"_id": "Reduction_potential:29", "title": "", "text": "$-0.05916h/n$"}
{"_id": "Kelvin–Stokes_theorem:95", "title": "", "text": "$\\alpha\\ominus\\beta:=\\alpha\\oplus(\\ominus\\beta)$"}
{"_id": "Tarski–Grothendieck_set_theory:15", "title": "", "text": "$\\forall x\\exists y[x\\in y\\wedge\\forall z\\in y(\\mathcal{P}(z)\\subseteq y\\wedge\\mathcal{P}(z)\\in y)\\wedge\\forall z\\in\\mathcal{P}(y)(\\neg z\\approx y\\to z\\in y)]$"}
{"_id": "Non-standard_calculus:23", "title": "", "text": "$N+\\tfrac{1}{N}$"}
{"_id": "Gospel_of_John:0", "title": "", "text": "$\\mathfrak{P}^{52}$"}
{"_id": "Characteristic_impedance:0", "title": "", "text": "$Z_{0}=\\sqrt{\\frac{R+j\\omega L}{G+j\\omega C}}$"}
{"_id": "Magnetic_field:209", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}+\\mu_{0}\\varepsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t},$"}
{"_id": "Maximum_entropy_probability_distribution:99", "title": "", "text": "$f(k)={n\\choose k}p^{k}(1-p)^{n-k}$"}
{"_id": "Line–sphere_intersection:14", "title": "", "text": "$ad^{2}+bd+c=0$"}
{"_id": "Constructive_set_theory:0", "title": "", "text": "$\\forall A\\;([\\forall x\\in A\\;\\exists y\\;\\phi(x,y)]\\to\\exists B\\;\\forall x\\in A\\;\\exists y\\in B\\;\\phi(x,y))$"}
{"_id": "Neptunium:1", "title": "", "text": "$\\mathrm{{}^{235}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{236}_{\\ 92}U_{m}\\ \\xrightarrow[120\\ ns]{}\\ ^{236}_{\\ 92}U\\ +\\ \\gamma}$"}
{"_id": "Strategic_complements:7", "title": "", "text": "$\\frac{\\partial^{2}\\Pi}{\\partial x_{i}\\partial x_{j}}$"}
{"_id": "LTI_system_theory:109", "title": "", "text": "$\\Pi(t)\\ \\stackrel{\\,\\text{def}}{=}\\ \\begin{cases}1&\\,\\text{if }|t|<\\frac{1}{2},\\\\ 0&\\,\\text{if }|t|>\\frac{1}{2}.\\end{cases}$"}
{"_id": "Twin_paradox:26", "title": "", "text": "$\\Delta t^{2}=\\left[\\int^{\\Delta\\tau}_{0}e^{\\int^{\\bar{\\tau}}_{0}a(\\tau^{\\prime})d\\tau^{\\prime}}\\,d\\bar{\\tau}\\right]\\,\\left[\\int^{\\Delta\\tau}_{0}e^{-\\int^{\\bar{\\tau}}_{0}a(\\tau^{\\prime})d\\tau^{\\prime}}\\,d\\bar{\\tau}\\right],$"}
{"_id": "BKL_singularity:119", "title": "", "text": "$\\overline{\\left[\\left(\\tau_{s}-\\bar{\\tau}_{s}\\right)^{2}\\right]}^{\\frac{1}{2}}=1.4\\sqrt{s},$"}
{"_id": "Deriving_the_Schwarzschild_solution:70", "title": "", "text": "$A(r)B(r)\\,=K$"}
{"_id": "Power_factor:13", "title": "", "text": "$I_{\\mbox{rms}~{}}$"}
{"_id": "Principles_of_grid_generation:11", "title": "", "text": "$\\alpha y_{\\xi\\xi}-2\\beta y_{\\xi\\eta}+\\gamma y_{\\eta\\eta}=-I^{2}(Py_{\\xi}+Qy_{\\eta})$"}
{"_id": "Constraint_algorithm:21", "title": "", "text": "$\\mathbf{J}=\\left(\\begin{array}[]{cccc}\\frac{\\partial\\sigma_{1}}{\\partial\\lambda_{1}}&\\frac{\\partial\\sigma_{1}}{\\partial\\lambda_{2}}&\\dots&\\frac{\\partial\\sigma_{1}}{\\partial\\lambda_{n}}\\\\ \\frac{\\partial\\sigma_{2}}{\\partial\\lambda_{1}}&\\frac{\\partial\\sigma_{2}}{\\partial\\lambda_{2}}&\\dots&\\frac{\\partial\\sigma_{2}}{\\partial\\lambda_{n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial\\sigma_{n}}{\\partial\\lambda_{1}}&\\frac{\\partial\\sigma_{n}}{\\partial\\lambda_{2}}&\\dots&\\frac{\\partial\\sigma_{n}}{\\partial\\lambda_{n}}\\end{array}\\right).$"}
{"_id": "Cover_tree:12", "title": "", "text": "$O(\\eta*\\log{n})$"}
{"_id": "Baker's_theorem:8", "title": "", "text": "$\\beta_{1}\\log\\alpha_{1}+\\cdots+\\beta_{n-1}\\log\\alpha_{n-1}=\\log\\alpha_{n}$"}
{"_id": "Orthogonal_group:200", "title": "", "text": "$\\mathbf{v}−2·B(\\mathbf{v},\\mathbf{u})/Q(\\mathbf{u}) ·\\mathbf{u}$"}
{"_id": "Decoherence-free_subspaces:80", "title": "", "text": "$\\mathbf{\\rho}_{i},\\mathbf{\\rho}_{j}\\in\\mathit{S}\\big(i\\neq j\\big)$"}
{"_id": "Finite_model_theory:2", "title": "", "text": "$\\exists_{x}\\forall_{y}(x\\neq y\\Rightarrow G(x,y)).$"}
{"_id": "Cauchy–Riemann_equations:45", "title": "", "text": "$Df(x,y)=\\begin{bmatrix}\\frac{\\partial u}{\\partial x}&\\frac{\\partial u}{\\partial y}\\\\ \\frac{\\partial v}{\\partial x}&\\frac{\\partial v}{\\partial y}\\end{bmatrix}$"}
{"_id": "McCarthy_91_function:0", "title": "", "text": "$M(n)=\\left\\{\\begin{matrix}n-10,&\\mbox{if }~{}n>100\\mbox{ }\\\\ M(M(n+11)),&\\mbox{if }~{}n\\leq 100\\mbox{ }\\end{matrix}\\right.$"}
{"_id": "Racah_W-coefficient:36", "title": "", "text": "$\\alpha_{1}=a+b+e;\\quad\\beta_{1}=a+b+c+d;$"}
{"_id": "Field_with_one_element:2", "title": "", "text": "$\\frac{n!}{m!(n-m)!}$"}
{"_id": "Unit_vector:39", "title": "", "text": "$\\mathbf{\\hat{r}}=\\sin\\theta\\cos\\varphi\\mathbf{\\hat{x}}+\\sin\\theta\\sin\\varphi\\mathbf{\\hat{y}}+\\cos\\theta\\mathbf{\\hat{z}}$"}
{"_id": "Standardized_mean_of_a_contrast_variable:51", "title": "", "text": "$\\hat{\\lambda}\\text{MM}=\\frac{\\sum_{i=1}^{t}c_{i}\\bar{Y}_{i}}{\\sqrt{\\sum_{i=1}^{t}c_{i}^{2}s_{i}^{2}}}.$"}
{"_id": "Unbiased_estimation_of_standard_deviation:0", "title": "", "text": "$s=\\sqrt{\\frac{1}{n-1}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}}\\,,$"}
{"_id": "Q_factor:21", "title": "", "text": "$e^{-2\\alpha t}$"}
{"_id": "Notation_for_differentiation:27", "title": "", "text": "$\\left(\\frac{\\partial T}{\\partial V}\\right)_{S}$"}
{"_id": "Rule_of_product:0", "title": "", "text": "$\\begin{matrix}&\\underbrace{\\left\\{A,B,C\\right\\}}&&\\underbrace{\\left\\{X,Y\\right\\}}\\\\ \\mathrm{To}\\ \\mathrm{choose}\\ \\mathrm{one}\\ \\mathrm{of}&\\mathrm{these}&\\mathrm{AND}\\ \\mathrm{one}\\ \\mathrm{of}&\\mathrm{these}\\end{matrix}$"}
{"_id": "Multinomial_theorem:13", "title": "", "text": "$\\frac{n!}{k_{1}!k_{2}!\\cdots k_{m-1}!K!}\\frac{K!}{k_{m}!k_{m+1}!}=\\frac{n!}{k_{1}!k_{2}!\\cdots k_{m+1}!}.$"}
{"_id": "List_of_New_Testament_papyri:151", "title": "", "text": "$\\mathfrak{P}^{126}$"}
{"_id": "Pisot–Vijayaraghavan_number:12", "title": "", "text": "$\\alpha=a+\\sqrt{D}\\,\\text{ to }\\alpha^{\\prime}=a-\\sqrt{D}\\,$"}
{"_id": "Machin-like_formula:8", "title": "", "text": "$\\sin(\\alpha+\\beta)=\\sin\\alpha\\cos\\beta+\\cos\\alpha\\sin\\beta$"}
{"_id": "Aeroacoustics:19", "title": "", "text": "$\\frac{\\partial^{2}\\rho}{\\partial t^{2}}-c^{2}_{0}\\nabla^{2}\\rho=\\frac{\\partial^{2}T_{ij}}{\\partial x_{i}\\partial x_{j}},\\quad(*)$"}
{"_id": "Apollos:0", "title": "", "text": "$\\mathfrak{P}$"}
{"_id": "Liouville's_formula:17", "title": "", "text": "$y^{\\prime}=\\underbrace{\\begin{pmatrix}1&-1/x\\\\ 1+x&-1\\end{pmatrix}}_{=\\,A(x)}y$"}
{"_id": "Preclosure_operator:20", "title": "", "text": "$[\\quad]_{\\mbox{seq}}~{}$"}
{"_id": "Lie_algebra_cohomology:34", "title": "", "text": "$0\\rightarrow M\\rightarrow\\mathfrak{h}\\rightarrow\\mathfrak{g}\\rightarrow 0$"}
{"_id": "Tobit_model:9", "title": "", "text": "$y_{i}=\\begin{cases}y_{i}^{*}&\\textrm{if}\\;y_{i}^{*}>0\\\\ 0&\\textrm{if}\\;y_{i}^{*}\\leq 0\\end{cases}$"}
{"_id": "Color-coding:55", "title": "", "text": "$k/2-1$"}
{"_id": "Rifleman's_rule:65", "title": "", "text": "$\\cos(\\theta-\\alpha)=\\cos(\\theta)\\cos(\\alpha)+\\sin(\\theta)\\sin(\\alpha)$"}
{"_id": "Distributed_constraint_optimization:47", "title": "", "text": "$\\mathfrak{P}(\\mathfrak{V})$"}
{"_id": "Heterodyne_detection:7", "title": "", "text": "$+\\underbrace{E_{\\mathrm{sig}}E_{\\mathrm{LO}}\\cos((\\omega_{\\mathrm{sig}}-\\omega_{\\mathrm{LO}})t+\\varphi)}_{beat\\;component}.$"}
{"_id": "Navier–Stokes_equations:19", "title": "", "text": "$0=-\\frac{\\mbox{d}~{}P}{\\mbox{d}~{}x}+\\mu\\left(\\frac{\\mbox{d}~{}^{2}u}{\\mbox{d}~{}y^{2}}\\right)$"}
{"_id": "Hilbert–Poincaré_series:16", "title": "", "text": "$0\\to C^{0}\\stackrel{d_{0}}{\\longrightarrow}C^{1}\\stackrel{d_{1}}{\\longrightarrow}C^{2}\\stackrel{d_{2}}{\\longrightarrow}\\cdots\\stackrel{d_{n-1}}{\\longrightarrow}C^{n}\\longrightarrow 0.$"}
{"_id": "Derived_category:9", "title": "", "text": "$\\mathcal{A}\\stackrel{F}{\\rightarrow}\\mathcal{B}\\stackrel{G}{\\rightarrow}\\mathcal{C},\\,$"}
{"_id": "Small-angle_scattering:13", "title": "", "text": "$p(r)=\\frac{r^{2}}{2\\pi^{2}}\\int_{0}^{\\infty}I(q)\\frac{\\sin qr}{qr}q^{2}dq.$"}
{"_id": "Indian_mathematics:21", "title": "", "text": "$\\ ax^{2}+bx=c$"}
{"_id": "Bochner–Riesz_mean:0", "title": "", "text": "$(\\xi)_{+}=\\begin{cases}\\xi,&\\mbox{if }~{}\\xi>0\\\\ 0,&\\mbox{otherwise}~{}.\\end{cases}$"}
{"_id": "Draft:Portfolio_performance_contributions:48", "title": "", "text": "$r_{t}=\\sum\\limits_{k=1}^{n}\\underbrace{\\left(r_{k,t}^{\\,\\text{base}}+r_{k,t}^{\\,\\text{crcy}}\\right)}_{=r_{k,t}}$"}
{"_id": "Robust_principal_component_analysis:1", "title": "", "text": "$O\\left(mnr^{2}\\log\\frac{1}{\\epsilon}\\right)$"}
{"_id": "Karhunen–Loève_theorem:186", "title": "", "text": "$\\begin{aligned}\\displaystyle\\mathbf{E}[G|H]&\\displaystyle=\\int^{T}_{0}k(t)\\mathbf{E}[x(t)|H]dt=0\\\\ \\displaystyle\\mathbf{E}[G|K]&\\displaystyle=\\int^{T}_{0}k(t)\\mathbf{E}[x(t)|K]dt=\\int^{T}_{0}k(t)S(t)dt\\equiv\\rho\\\\ \\displaystyle\\mathbf{E}[G^{2}|H]&\\displaystyle=\\int^{T}_{0}\\int^{T}_{0}k(t)k(s)R_{N}(t,s)dtds=\\int^{T}_{0}k(t)\\left(\\int^{T}_{0}k(s)R_{N}(t,s)ds\\right)=\\int^{T}_{0}k(t)S(t)dt=\\rho\\\\ \\displaystyle\\,\\text{Var}[G|H]&\\displaystyle=\\mathbf{E}[G^{2}|H]-(\\mathbf{E}[G|H])^{2}=\\rho\\\\ \\displaystyle\\mathbf{E}[G^{2}|K]&\\displaystyle=\\int^{T}_{0}\\int^{T}_{0}k(t)k(s)\\mathbf{E}[x(t)x(s)]dtds=\\int^{T}_{0}\\int^{T}_{0}k(t)k(s)(R_{N}(t,s)+S(t)S(s))dtds=\\rho+\\rho^{2}\\\\ \\displaystyle\\,\\text{Var}[G|K]&\\displaystyle=\\mathbf{E}[G^{2}|K]-(\\mathbf{E}[G|K])^{2}=\\rho+\\rho^{2}-\\rho^{2}=\\rho\\end{aligned}$"}
{"_id": "Cosine_similarity:2", "title": "", "text": "$\\,\\text{similarity}=\\cos(\\theta)={A\\cdot B\\over\\|A\\|\\|B\\|}=\\frac{\\sum\\limits_{i=1}^{n}{A_{i}\\times B_{i}}}{\\sqrt{\\sum\\limits_{i=1}^{n}{(A_{i})^{2}}}\\times\\sqrt{\\sum\\limits_{i=1}^{n}{(B_{i})^{2}}}}$"}
{"_id": "Quantum_entanglement:5", "title": "", "text": "$\\scriptstyle|\\phi\\rangle_{B}$"}
{"_id": "Rapid_shallow_breathing_index:0", "title": "", "text": "$RSBI=\\frac{f}{V_{T}}$"}
{"_id": "Splitting_of_prime_ideals_in_Galois_extensions:2", "title": "", "text": "$pB=(\\prod P_{j})^{e}\\ .$"}
{"_id": "Classical_dichotomy:3", "title": "", "text": "$\\mathbf{J}=\\begin{bmatrix}A&0\\\\ B&C\\\\ \\end{bmatrix}$"}
{"_id": "Freivalds'_algorithm:4", "title": "", "text": "$AB=\\begin{bmatrix}2&3\\\\ 3&4\\end{bmatrix}\\begin{bmatrix}1&0\\\\ 1&2\\end{bmatrix}\\stackrel{?}{=}\\begin{bmatrix}6&5\\\\ 8&7\\end{bmatrix}=C.$"}
{"_id": "Pascal's_simplex:27", "title": "", "text": "$\\wedge^{m}\\subset\\wedge^{m+1}$"}
{"_id": "Lane–Emden_equation:19", "title": "", "text": "$\\displaystyle\\frac{d}{dr}\\left(\\frac{1}{\\rho}\\frac{dP}{dr}\\right)$"}
{"_id": "Unscented_transform:48", "title": "", "text": "$M=\\begin{bmatrix}1.44&0\\\\ 0&2.89\\end{bmatrix}$"}
{"_id": "Z_function:6", "title": "", "text": "$N=\\lfloor u^{2}\\rfloor$"}
{"_id": "Balls_into_bins:5", "title": "", "text": "$O\\left(\\frac{\\log n}{\\log\\log n}\\right)$"}
{"_id": "Spiral_of_Theodorus:6", "title": "", "text": "$\\lim_{k\\to\\infty}c_{2}(k)=-2.157782996659\\ldots.$"}
{"_id": "Cassini_projection:1", "title": "", "text": "$y=\\arctan\\left(\\frac{\\tan(\\phi)}{\\cos(\\lambda)}\\right)$"}
{"_id": "Ideal_lattice_cryptography:92", "title": "", "text": "$O(mn\\log p)=O(n\\log n)$"}
{"_id": "Formulas_for_generating_Pythagorean_triples:45", "title": "", "text": "$\\left[{\\begin{array}[]{*{20}{c}}1&1\\\\ 2&3\\end{array}}\\right]\\leftarrow\\,\\text{parent}$"}
{"_id": "Birnbaum–Saunders_distribution:8", "title": "", "text": "$X=\\frac{1}{2}\\left[\\left(\\frac{T}{\\beta}\\right)^{0.5}-\\left(\\frac{T}{\\beta}\\right)^{-0.5}\\right]$"}
{"_id": "Pivotal_quantity:38", "title": "", "text": "$r=\\frac{\\frac{1}{n-1}\\sum_{i=1}^{n}(X_{i}-\\overline{X})(Y_{i}-\\overline{Y})}{s_{X}s_{Y}}$"}
{"_id": "Blancmange_curve:0", "title": "", "text": "${\\rm blanc}(x)=\\sum_{n=0}^{\\infty}{s(2^{n}x)\\over 2^{n}},$"}
{"_id": "Conditional_expectation:109", "title": "", "text": "$\\operatorname{E}(X\\mid Y=y)=\\int_{\\mathcal{X}}xf_{X\\mid Y}(x\\mid y)\\,dx$"}
{"_id": "Multivariate_probit_model:6", "title": "", "text": "$Y_{1}=\\begin{cases}1&\\,\\text{if }Y^{*}_{1}>0,\\\\ 0&\\,\\text{otherwise},\\end{cases}$"}
{"_id": "Quantum_nonlocality:41", "title": "", "text": "$\\sum_{a,b}P\\left({{a,b}{|}{A,B}}\\right)=1\\quad\\forall{A,B}$"}
{"_id": "TSL_color_space:11", "title": "", "text": "$g=\\begin{cases}-\\sqrt{\\frac{5}{9(x^{2}+1)}}\\cdot S,&\\mbox{if}~{}~{}T>\\frac{1}{2}\\\\ \\sqrt{\\frac{5}{9(x^{2}+1)}}\\cdot S,&\\mbox{if}~{}~{}T<\\frac{1}{2}\\\\ 0,&\\mbox{if}~{}~{}T=0\\\\ \\end{cases}$"}
{"_id": "Kummer's_theorem:2", "title": "", "text": "$\\tfrac{n!}{m!(n-m)!}$"}
{"_id": "Capelli's_identity:130", "title": "", "text": "$X=\\begin{vmatrix}x_{11}&x_{12}&x_{13}&\\cdots&x_{1n}\\\\ x_{12}&x_{22}&x_{23}&\\cdots&x_{2n}\\\\ x_{13}&x_{23}&x_{33}&\\cdots&x_{3n}\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ x_{1n}&x_{2n}&x_{3n}&\\cdots&x_{nn}\\end{vmatrix},D=\\begin{vmatrix}2\\frac{\\partial}{\\partial x_{11}}&\\frac{\\partial}{\\partial x_{12}}&\\frac{\\partial}{\\partial x_{13}}&\\cdots&\\frac{\\partial}{\\partial x_{1n}}\\\\ \\frac{\\partial}{\\partial x_{12}}&2\\frac{\\partial}{\\partial x_{22}}&\\frac{\\partial}{\\partial x_{23}}&\\cdots&\\frac{\\partial}{\\partial x_{2n}}\\\\ \\frac{\\partial}{\\partial x_{13}}&\\frac{\\partial}{\\partial x_{23}}&2\\frac{\\partial}{\\partial x_{33}}&\\cdots&\\frac{\\partial}{\\partial x_{3n}}\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial}{\\partial x_{1n}}&\\frac{\\partial}{\\partial x_{2n}}&\\frac{\\partial}{\\partial x_{3n}}&\\cdots&2\\frac{\\partial}{\\partial x_{nn}}\\end{vmatrix}$"}
{"_id": "Cebeci–Smith_model:1", "title": "", "text": "$\\mu_{t}=\\begin{cases}{\\mu_{t}}\\text{inner}&\\mbox{if }~{}y\\leq y\\text{crossover}\\\\ {\\mu_{t}}\\text{outer}&\\mbox{if }~{}y>y\\text{crossover}\\end{cases}$"}
{"_id": "Prime_number:52", "title": "", "text": "$\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^{s}},$"}
{"_id": "Binomial_type:16", "title": "", "text": "$\\underbrace{a\\diamondsuit\\cdots\\diamondsuit a}_{k\\,\\text{ factors}}.\\,$"}
{"_id": "LU_decomposition:9", "title": "", "text": "$\\begin{bmatrix}4&3\\\\ 6&3\\end{bmatrix}=\\begin{bmatrix}l_{11}&0\\\\ l_{21}&l_{22}\\end{bmatrix}\\begin{bmatrix}u_{11}&u_{12}\\\\ 0&u_{22}\\end{bmatrix}.$"}
{"_id": "Derivative:230", "title": "", "text": "$\\frac{d^{2}y}{dx^{2}}=\\frac{d}{dx}\\left(\\frac{dy}{dx}\\right).$"}
{"_id": "Transmission_Loss_(duct_acoustics):13", "title": "", "text": "$\\begin{bmatrix}\\hat{p}_{i}\\\\ \\hat{q}_{i}\\end{bmatrix}=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}\\begin{bmatrix}\\hat{p}_{o}\\\\ \\hat{q}_{o}\\end{bmatrix}$"}
{"_id": "Hermitian_matrix:25", "title": "", "text": "$AB=BA$"}
{"_id": "10_(number):10", "title": "", "text": "$0.\\overline{6}$"}
{"_id": "Morphological_skeleton:11", "title": "", "text": "$0B=\\{o\\}$"}
{"_id": "Bose–Einstein_statistics:29", "title": "", "text": "$w(n,g)=\\frac{(n+g-1)!}{n!(g-1)!}.$"}
{"_id": "Alternatives_to_general_relativity:239", "title": "", "text": "$-2-2\\gamma$"}
{"_id": "Ambisonics:27", "title": "", "text": "$LB=(2W-X+Y)\\sqrt{8}$"}
{"_id": "Induction-recursion_(type_theory):6", "title": "", "text": "$i:A\\to B\\to D$"}
{"_id": "Euclidean_relation:1", "title": "", "text": "$\\forall a,b,c\\in X\\,(b\\,R\\,a\\land c\\,R\\,a\\to b\\,R\\,c).$"}
{"_id": "Fibonacci_number:10", "title": "", "text": "$\\psi=\\frac{1-\\sqrt{5}}{2}=1-\\varphi=-{1\\over\\varphi}\\approx-0.61803\\,39887\\cdots$"}
{"_id": "Closing_(morphology):0", "title": "", "text": "$A\\bullet B=(A\\oplus B)\\ominus B,\\,$"}
{"_id": "Sliding_mode_control:101", "title": "", "text": "$\\underbrace{\\overbrace{\\sigma^{\\,\\text{T}}}^{\\tfrac{\\partial V}{\\partial\\sigma}}\\overbrace{\\dot{\\sigma}}^{\\tfrac{\\operatorname{d}\\sigma}{\\operatorname{d}t}}}_{\\tfrac{\\operatorname{d}V}{\\operatorname{d}t}}\\leq-\\mu(\\mathord{\\overbrace{\\|\\sigma\\|_{2}}^{\\sqrt{V}}})^{\\alpha}$"}
{"_id": "Morse–Kelley_set_theory:15", "title": "", "text": "$\\forall x\\forall y\\forall s[(\\langle x,s\\rangle\\in F\\and\\langle y,s\\rangle\\in F)\\rightarrow x=y])].$"}
{"_id": "Power-flow_study:18", "title": "", "text": "$J=\\begin{bmatrix}\\dfrac{\\partial\\Delta P}{\\partial\\theta}&\\dfrac{\\partial\\Delta P}{\\partial|V|}\\\\ \\dfrac{\\partial\\Delta Q}{\\partial\\theta}&\\dfrac{\\partial\\Delta Q}{\\partial|V|}\\end{bmatrix}$"}
{"_id": "Plancherel_measure:7", "title": "", "text": "$\\pi\\in G^{\\wedge}$"}
{"_id": "Eisenstein_reciprocity:27", "title": "", "text": "$\\alpha^{m}$"}
{"_id": "Alternatization:21", "title": "", "text": "$\\forall x,y\\in V,$"}
{"_id": "Wilson_loop:9", "title": "", "text": "$\\sigma^{j}$"}
{"_id": "Orthogonal_matrix:14", "title": "", "text": "$\\begin{bmatrix}\\cos(\\alpha)\\cos(\\gamma)-\\sin(\\alpha)\\sin(\\beta)\\sin(\\gamma)&-\\sin(\\alpha)\\cos(\\beta)&-\\cos(\\alpha)\\sin(\\gamma)-\\sin(\\alpha)\\sin(\\beta)\\cos(\\gamma)\\\\ \\cos(\\alpha)\\sin(\\beta)\\sin(\\gamma)+\\sin(\\alpha)\\cos(\\gamma)&\\cos(\\alpha)\\cos(\\beta)&\\cos(\\alpha)\\sin(\\beta)\\cos(\\gamma)-\\sin(\\alpha)\\sin(\\gamma)\\\\ \\cos(\\beta)\\sin(\\gamma)&-\\sin(\\beta)&\\cos(\\beta)\\cos(\\gamma)\\end{bmatrix}$"}
{"_id": "Checking_whether_a_coin_is_fair:4", "title": "", "text": "$f(r|H=h,T=t)=\\frac{(h+t+1)!}{h!\\,\\,t!}\\;r^{h}\\,(1-r)^{t}.\\!$"}
{"_id": "Compressibility_equation:2", "title": "", "text": "$kT\\left(\\frac{\\partial\\rho}{\\partial p}\\right)$"}
{"_id": "Errors-in-variables_models:14", "title": "", "text": "$f_{x^{*}}(x)=\\begin{cases}Ae^{-Be^{Cx}+CDx}(e^{Cx}+E)^{-F},&\\,\\text{if}\\ d>0\\\\ Ae^{-Bx^{2}+Cx}&\\,\\text{if}\\ d=0\\end{cases}$"}
{"_id": "Rule_of_inference:1", "title": "", "text": "$\\underline{A\\quad\\quad\\quad}\\,\\!$"}
{"_id": "Slope:18", "title": "", "text": "$\\,\\text{angle}=\\arctan\\left(\\frac{\\,\\text{slope}}{100\\%}\\right)$"}
{"_id": "Linear_filter:2", "title": "", "text": "$y(t)=\\int_{0}^{T}x(t-\\tau)\\,h(\\tau)\\,d\\tau$"}
{"_id": "Plutonium:1", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ +\\ ^{2}_{1}D\\ \\longrightarrow\\ ^{238}_{\\ 93}Np\\ +\\ 2\\ ^{1}_{0}n\\quad;\\quad^{238}_{\\ 93}Np\\ \\xrightarrow[2.117\\ d]{\\beta^{-}}\\ ^{238}_{\\ 94}Pu}$"}
{"_id": "Magnetization_dynamics:7", "title": "", "text": "$\\mathbf{m}=-\\gamma\\mathbf{L}$"}
{"_id": "Preclosure_operator:1", "title": "", "text": "$[\\quad]_{p}$"}
{"_id": "Kurtosis:29", "title": "", "text": "$=\\frac{(n+1)\\,n\\,(n-1)}{(n-2)\\,(n-3)}\\;\\frac{\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{4}}{\\left(\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}\\right)^{2}}-3\\,\\frac{(n-1)^{2}}{(n-2)\\,(n-3)}$"}
{"_id": "Dilation_(morphology):2", "title": "", "text": "$A\\oplus B=\\{z\\in E\\mid(B^{s})_{z}\\cap A\\neq\\varnothing\\}$"}
{"_id": "Degrees_of_freedom_(statistics):19", "title": "", "text": "$\\frac{\\sqrt{n}(\\bar{X}-\\mu_{0})}{\\sqrt{\\sum\\limits_{i=1}^{n}(X_{i}-\\bar{X})^{2}/(n-1)}}$"}
{"_id": "List_of_New_Testament_papyri:70", "title": "", "text": "$\\mathfrak{P}^{46}$"}
{"_id": "Hydrogen-like_atom:28", "title": "", "text": "$k=\\begin{cases}-j-\\tfrac{1}{2}&\\,\\text{if }j=\\ell+\\tfrac{1}{2}\\\\ j+\\tfrac{1}{2}&\\,\\text{if }j=\\ell-\\tfrac{1}{2}\\end{cases}$"}
{"_id": "Atom_probe:7", "title": "", "text": "$\\frac{m}{n}=-2eV_{1}\\left(\\frac{t}{f}\\right)^{2}$"}
{"_id": "Proof_that_22::7_exceeds_π:38", "title": "", "text": "$\\begin{aligned}\\displaystyle\\frac{x^{4n}(1-x)^{4n}}{2^{2n-2}(1+x^{2})}&\\displaystyle=\\sum_{j=0}^{2n-1}\\frac{(-1)^{j}}{2^{2n-j-2}}x^{4n+j}(1-x)^{4n-2j-2}\\\\ &\\displaystyle\\qquad{}-4\\sum_{j=0}^{3n-1}(-1)^{3n-j}x^{2j}+(-1)^{3n}\\frac{4}{1+x^{2}}.\\qquad(*)\\end{aligned}$"}
{"_id": "Ratio_estimator:2", "title": "", "text": "$r=\\frac{\\bar{y}}{\\bar{x}}=\\frac{\\sum_{i=1}^{n}y}{\\sum_{i=1}^{n}x}$"}
{"_id": "Bessel's_correction:1", "title": "", "text": "$n\\frac{n}{− 1}$"}
{"_id": "Series_(mathematics):57", "title": "", "text": "$\\sum\\limits_{n=1}^{\\infty}{(-1)^{n+1}\\over n}=1-{1\\over 2}+{1\\over 3}-{1\\over 4}+{1\\over 5}-\\cdots$"}
{"_id": "Struve_function:7", "title": "", "text": "$\\mathbf{H}_{\\alpha}(x)=\\frac{2\\left(\\frac{x}{2}\\right)^{\\alpha}}{\\sqrt{\\pi}\\Gamma\\left(\\alpha+\\frac{1}{2}\\right)}\\int_{0}^{\\frac{\\pi}{2}}\\sin(x\\cos\\tau)\\sin^{2\\alpha}(\\tau)d\\tau.$"}
{"_id": "Barnsley_fern:1", "title": "", "text": "$f(x,y)=\\begin{bmatrix}\\ 0.00&\\ 0.00\\\\ 0.00&\\ 0.16\\end{bmatrix}\\begin{bmatrix}\\ x\\\\ y\\end{bmatrix}$"}
{"_id": "Luminosity:20", "title": "", "text": "$m_{\\rm star}=m_{\\odot}-2.5\\log_{10}\\left[\\frac{L_{\\rm star}}{L_{\\odot}}\\left(\\frac{d_{\\odot}}{d_{\\rm star}}\\right)^{2}\\right]$"}
{"_id": "Measurable_function:6", "title": "", "text": "$Y\\stackrel{\\pi}{\\to}X$"}
{"_id": "Bessel_polynomials:8", "title": "", "text": "$\\theta_{n}(x)=\\frac{n!}{(-2)^{n}}\\,L_{n}^{-2n-1}(2x)$"}
{"_id": "Syntactic_monoid:6", "title": "", "text": "$u\\equiv_{S}v\\Leftrightarrow\\forall x,y\\in M(xuy\\in S\\Leftrightarrow xvy\\in S).$"}
{"_id": "Net_tonnage:2", "title": "", "text": "$(\\tfrac{4d}{3D})^{2}$"}
{"_id": "Taylor's_law:21", "title": "", "text": "$s^{2}=np(1-p)$"}
{"_id": "Series_(mathematics):48", "title": "", "text": "$1+{1\\over 2}+{1\\over 3}+{1\\over 4}+{1\\over 5}+\\cdots=\\sum_{n=1}^{\\infty}{1\\over n}.$"}
{"_id": "De_Rham_cohomology:98", "title": "", "text": "$\\omega=d\\alpha+\\delta\\beta+\\gamma$"}
{"_id": "Pseudo-Hadamard_transform:4", "title": "", "text": "$H_{1}=\\begin{bmatrix}2&1\\\\ 1&1\\end{bmatrix}$"}
{"_id": "Random_Fibonacci_sequence:3", "title": "", "text": "$1,1,2,3,1,-2,-3,-5,-2,-3,\\ldots\\,\\text{ for the signs }+,+,+,-,-,+,-,-,\\ldots.$"}
{"_id": "Meridian_arc:11", "title": "", "text": "$\\begin{aligned}\\displaystyle m(\\varphi)&\\displaystyle=\\int_{0}^{\\varphi}M(\\varphi)\\,d\\varphi=a(1-e^{2})\\int_{0}^{\\varphi}\\bigl(1-e^{2}\\sin^{2}\\varphi\\bigr)^{-3/2}\\,d\\varphi.\\end{aligned}$"}
{"_id": "Bailey–Borwein–Plouffe_formula:2", "title": "", "text": "$\\pi=\\sum_{k=0}^{\\infty}\\left[\\frac{1}{16^{k}}\\left(\\frac{4}{8k+1}-\\frac{2}{8k+4}-\\frac{1}{8k+5}-\\frac{1}{8k+6}\\right)\\right]$"}
{"_id": "Ordered_logit:3", "title": "", "text": "$y=\\begin{cases}0&\\,\\text{if }y^{*}\\leq\\mu_{1},\\\\ 1&\\,\\text{if }\\mu_{1}<y^{*}\\leq\\mu_{2},\\\\ 2&\\,\\text{if }\\mu_{2}<y^{*}\\leq\\mu_{3},\\\\ \\vdots\\\\ N&\\,\\text{if }\\mu_{N}<y^{*}.\\end{cases}$"}
{"_id": "Inhomogeneous_electromagnetic_wave_equation:12", "title": "", "text": "$\\nabla^{2}\\varphi+{{\\partial}\\over\\partial t}\\left(\\nabla\\cdot\\mathbf{A}\\right)=-{\\rho\\over\\varepsilon_{0}}\\,,$"}
{"_id": "Neptunium:2", "title": "", "text": "$\\mathrm{{}^{236}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{237}_{\\ 92}U\\ \\xrightarrow[6.75\\ d]{\\beta^{-}}\\ ^{237}_{\\ 93}Np}$"}
{"_id": "Stable_module_category:16", "title": "", "text": "$0\\to X\\to E\\to Y\\to 0\\,$"}
{"_id": "Functional_determinant:19", "title": "", "text": "$\\sum_{n=0}^{\\infty}\\frac{1}{(n+a)}$"}
{"_id": "Quasi-geostrophic_equations:57", "title": "", "text": "${\\nabla({\\partial\\Phi\\over\\partial p})}$"}
{"_id": "Notation_for_differentiation:11", "title": "", "text": "$\\frac{dy}{dx}\\left.{\\!\\!\\frac{}{}}\\right|_{x=a}=\\frac{dy}{dx}(a).$"}
{"_id": "Centroid:25", "title": "", "text": "$C=\\frac{\\int xg(x)\\;dx}{\\int g(x)\\;dx}$"}
{"_id": "Hamiltonian_vector_field:6", "title": "", "text": "$\\mathrm{d}H=\\begin{bmatrix}\\frac{\\partial H}{\\partial q^{i}}\\\\ \\frac{\\partial H}{\\partial p_{i}}\\end{bmatrix}.$"}
{"_id": "Riemann_zeta_function:119", "title": "", "text": "$\\Phi(z,s,q)=\\sum_{k=0}^{\\infty}\\frac{z^{k}}{(k+q)^{s}}\\!$"}
{"_id": "Affine_transformation:37", "title": "", "text": "$\\begin{bmatrix}\\vec{y}\\\\ 1\\end{bmatrix}=\\left[\\begin{array}[]{ccc|c}&A&&\\vec{b}\\\\ 0&\\ldots&0&1\\end{array}\\right]\\begin{bmatrix}\\vec{x}\\\\ 1\\end{bmatrix}$"}
{"_id": "Standard_part_function:23", "title": "", "text": "$\\frac{dy}{dx}=\\mathrm{st}\\left(\\frac{\\Delta y}{\\Delta x}\\right)$"}
{"_id": "Tournament_(graph_theory):24", "title": "", "text": "$1+\\lfloor\\log_{2}n\\rfloor$"}
{"_id": "Completely_multiplicative_function:10", "title": "", "text": "$L(s,a)=\\sum^{\\infty}_{n=1}\\frac{a(n)}{n^{s}}=\\prod_{p}\\biggl(1-\\frac{a(p)}{p^{s}}\\biggr)^{-1},$"}
{"_id": "Tournament_(graph_theory):19", "title": "", "text": "$\\ O(n\\log n)$"}
{"_id": "Bitwise_operation:6", "title": "", "text": "$x=\\lfloor\\log_{2}x\\rfloor+1$"}
{"_id": "Annual_percentage_yield:0", "title": "", "text": "$APY=\\left(1+\\frac{i\\text{nom}}{N}\\right)^{N}-1$"}
{"_id": "Einstein_field_equations:57", "title": "", "text": "$\\frac{d}{dt}\\left(\\frac{dt}{d\\tau}\\right)\\approx 0$"}
{"_id": "Union_(set_theory):14", "title": "", "text": "$A\\cup B=\\left(A^{C}\\cap B^{C}\\right)^{C}$"}
{"_id": "Compound_Poisson_process:14", "title": "", "text": "$\\exp(\\nu)=\\sum_{n=0}^{\\infty}{\\nu^{*n}\\over n!}$"}
{"_id": "Latitude:14", "title": "", "text": "$\\Delta^{1}_{\\rm LAT}=111132.954-559.822\\cos 2\\phi+1.175\\cos 4\\phi$"}
{"_id": "Analytic_set:5", "title": "", "text": "$A=\\{x\\in X|(\\exists y\\in Y)\\langle x,y\\rangle\\in B\\}.$"}
{"_id": "File:GeometricSegment.png:0", "title": "", "text": "$\\frac{1}{2}\\,+\\,\\frac{1}{4}\\,+\\,\\frac{1}{8}\\,+\\,\\frac{1}{16}\\,+\\,\\cdots\\;=\\;1$"}
{"_id": "Standardized_mean_of_a_contrast_variable:11", "title": "", "text": "$\\lambda=\\frac{\\operatorname{E}(V)}{\\operatorname{stdev}(V)}=\\frac{\\sum_{i=1}^{t}c_{i}\\mu_{i}}{\\sqrt{\\,\\text{Var}(\\sum_{i=1}^{t}c_{i}G_{i})}}=\\frac{\\sum_{i=1}^{t}c_{i}\\mu_{i}}{\\sqrt{\\sum_{i=1}^{t}c_{i}^{2}\\sigma_{i}^{2}+2\\sum_{i=1}^{t}\\sum_{j=i}c_{i}c_{j}\\sigma_{ij}}}$"}
{"_id": "Hilbert_transform:45", "title": "", "text": "$1<\\frac{1}{p}+\\frac{1}{r}$"}
{"_id": "Convergence_of_random_variables:84", "title": "", "text": "$Y_{n}\\ \\xrightarrow{p}\\ Y$"}
{"_id": "Two-port_network:58", "title": "", "text": "$\\begin{bmatrix}V_{2}\\\\ I^{\\prime}_{2}\\end{bmatrix}=\\begin{bmatrix}A^{\\prime}&B^{\\prime}\\\\ C^{\\prime}&D^{\\prime}\\end{bmatrix}\\begin{bmatrix}V_{1}\\\\ I_{1}\\end{bmatrix}$"}
{"_id": "Convergent_series:16", "title": "", "text": "${1\\over 1}-{1\\over 2}+{1\\over 4}-{1\\over 8}+{1\\over 16}-{1\\over 32}+\\cdots={2\\over 3}.$"}
{"_id": "Inclusion–exclusion_principle:19", "title": "", "text": "${n\\choose p}=\\frac{n!}{p!(n-p)!}$"}
{"_id": "Genetic_drift:0", "title": "", "text": "$\\frac{1}{2}\\cdot\\frac{1}{2}\\cdot\\frac{1}{2}\\cdot\\frac{1}{2}=\\frac{1}{16}.$"}
{"_id": "Trigonometric_functions:102", "title": "", "text": "$\\cos(x-y)=\\cos x\\cos y+\\sin x\\sin y\\,$"}
{"_id": "Kalman_filter:33", "title": "", "text": "$\\mathbf{G}=\\begin{bmatrix}\\frac{\\Delta t^{2}}{2}\\\\ \\Delta t\\end{bmatrix}$"}
{"_id": "Pi:115", "title": "", "text": "$\\pi=\\frac{4}{1}-\\frac{4}{3}+\\frac{4}{5}-\\frac{4}{7}+\\frac{4}{9}-\\frac{4}{11}+\\frac{4}{13}-\\cdots$"}
{"_id": "Symmetric_difference:35", "title": "", "text": "$A\\triangle B=A^{c}\\triangle B^{c}$"}
{"_id": "List_of_New_Testament_papyri:72", "title": "", "text": "$\\mathfrak{P}^{48}$"}
{"_id": "Binomial_series:27", "title": "", "text": "$(1-x^{2})^{1/3}=1-\\frac{x^{2}}{3}-\\frac{x^{4}}{9}-\\frac{5x^{6}}{81}\\cdots$"}
{"_id": "Skorokhod's_representation_theorem:0", "title": "", "text": "$x_{n}\\xrightarrow{d}\\ \\mu_{n}$"}
{"_id": "Mathematical_constants_and_functions:355", "title": "", "text": "$\\scriptstyle\\zeta^{\\prime}(2)\\,\\,\\text{= Derivative of }\\zeta(2)=-\\sum\\limits_{n=2}^{\\infty}\\frac{\\ln n}{n^{2}}=-0.9375482543\\ldots$"}
{"_id": "S_(set_theory):4", "title": "", "text": "$\\exist r\\forall y[A(y)\\rightarrow Byr]\\rightarrow\\exist x\\forall y[y\\in x\\leftrightarrow A(y)]\\,.$"}
{"_id": "Proofs_involving_ordinary_least_squares:17", "title": "", "text": "$\\begin{aligned}\\displaystyle Var(\\beta_{1})&\\displaystyle=\\frac{\\sigma^{2}}{\\sum_{i=1}^{n}{(x_{i}-\\bar{x})^{2}}}.\\end{aligned}$"}
{"_id": "Pisot–Vijayaraghavan_number:42", "title": "", "text": "$0.0000182\\dots,\\,$"}
{"_id": "List_of_logarithmic_identities:9", "title": "", "text": "$\\log_{b}\\!\\left(\\begin{matrix}\\frac{x}{y}\\end{matrix}\\right)=\\log_{b}(x)-\\log_{b}(y)$"}
{"_id": "Vector_autoregression:5", "title": "", "text": "$\\begin{bmatrix}y_{1,t}\\\\ y_{2,t}\\end{bmatrix}=\\begin{bmatrix}c_{1}\\\\ c_{2}\\end{bmatrix}+\\begin{bmatrix}A_{1,1}&A_{1,2}\\\\ A_{2,1}&A_{2,2}\\end{bmatrix}\\begin{bmatrix}y_{1,t-1}\\\\ y_{2,t-1}\\end{bmatrix}+\\begin{bmatrix}e_{1,t}\\\\ e_{2,t}\\end{bmatrix},$"}
{"_id": "Convergence_of_random_variables:75", "title": "", "text": "$X_{n}\\ \\xrightarrow{p}\\ Y$"}
{"_id": "Attenuation:3", "title": "", "text": "$\\,\\text{Attenuation (dB)}=10\\times\\log_{10}\\left(\\frac{\\,\\text{Input intensity(W)}}{\\,\\text{Output intensity (W)}}\\right)$"}
{"_id": "Product_(chemistry):1", "title": "", "text": "$S+C\\rightarrow P+C$"}
{"_id": "Kendall_rank_correlation_coefficient:18", "title": "", "text": "$O(n\\cdot\\log{n})$"}
{"_id": "Test_functions_for_optimization:90", "title": "", "text": "$\\,\\text{Minimize}=\\begin{cases}f_{1}\\left(x\\right)&=\\begin{cases}-x,&\\,\\text{if }x\\leq 1\\\\ x-2,&\\,\\text{if }1<x\\leq 3\\\\ 4-x,&\\,\\text{if }3<x\\leq 4\\\\ x-4,&\\,\\text{if }x>4\\\\ \\end{cases}\\\\ f_{2}\\left(x\\right)&=\\left(x-5\\right)^{2}\\\\ \\end{cases}$"}
{"_id": "Rencontres_numbers:4", "title": "", "text": "$D_{n,0}=\\left[{n!\\over e}\\right]$"}
{"_id": "Normal_distribution:122", "title": "", "text": "$\\sum_{i=1}^{n}(x_{i}-\\mu)^{2}=\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}+n(\\bar{x}-\\mu)^{2}$"}
{"_id": "Errors-in-variables_models:2", "title": "", "text": "$\\hat{\\beta}=\\frac{\\tfrac{1}{T}\\sum_{t=1}^{T}(x_{t}-\\bar{x})(y_{t}-\\bar{y})}{\\tfrac{1}{T}\\sum_{t=1}^{T}(x_{t}-\\bar{x})^{2}}\\,,$"}
{"_id": "Schnirelmann_density:28", "title": "", "text": "$A\\oplus B=\\mathcal{N}.$"}
{"_id": "Eigenstate_thermalization_hypothesis:46", "title": "", "text": "$\\overline{\\left(A_{t}-\\overline{A}\\right)^{2}}\\equiv\\lim_{\\tau\\to\\infty}\\frac{1}{\\tau}\\int_{0}^{\\tau}\\left(A_{t}-\\overline{A}\\right)^{2}dt,$"}
{"_id": "GEH_statistic:0", "title": "", "text": "$GEH=\\sqrt{\\frac{2(M-C)^{2}}{M+C}}$"}
{"_id": "Fourier_series:8", "title": "", "text": "$c_{n}\\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{cases}\\frac{A_{n}}{2i}e^{i\\phi_{n}}=\\frac{1}{2}(a_{n}-ib_{n})&\\,\\text{for }n>0\\\\ \\frac{1}{2}a_{0}&\\,\\text{for }n=0\\\\ c_{|n|}^{*}&\\,\\text{for }n<0.\\end{cases}$"}
{"_id": "Peano_axioms:8", "title": "", "text": "$\\forall x,y\\in N$"}
{"_id": "Spectral_theory:51", "title": "", "text": "$G(x,z;\\lambda)=\\sum_{i=1}^{n}\\frac{e_{i}(x)f_{i}^{*}(z)}{\\lambda-\\lambda_{i}}.$"}
{"_id": "Pushdown_automaton:135", "title": "", "text": "$\\delta^{{}^{\\prime}}$"}
{"_id": "Counting_single_transferable_votes:8", "title": "", "text": "$\\,\\text{Surplus Transfer Value}=\\left({{\\,\\text{Total value of Candidate's votes}-\\,\\text{Quota}}\\over\\,\\text{Total value of Candidate's votes}}\\right)\\times\\,\\text{Value of each vote}$"}
{"_id": "Padovan_sequence:26", "title": "", "text": "$\\sum_{n=0}^{\\infty}\\frac{P(n)}{\\alpha^{n}}=\\frac{\\alpha^{2}(\\alpha+1)}{\\alpha^{3}-\\alpha-1}.$"}
{"_id": "Mathematical_morphology:16", "title": "", "text": "$A\\bullet B=(A^{c}\\circ B^{s})^{c}$"}
{"_id": "Backstepping:124", "title": "", "text": "$\\begin{cases}\\dot{\\mathbf{x}}=f_{x}(\\mathbf{x})+g_{x}(\\mathbf{x})z_{1}+\\mathord{\\underbrace{\\left(g_{x}(\\mathbf{x})u_{x}(\\mathbf{x})-g_{x}(\\mathbf{x})u_{x}(\\mathbf{x})\\right)}_{0}}\\\\ \\dot{z}_{1}=u_{1}\\end{cases}$"}
{"_id": "Group_extension:3", "title": "", "text": "$1\\to K\\stackrel{i^{\\prime}}{\\rightarrow}G^{\\prime}\\stackrel{\\pi^{\\prime}}{\\rightarrow}H\\rightarrow 1$"}
{"_id": "E7_(mathematics):10", "title": "", "text": "$A\\circ B=(AB+BA)/2$"}
{"_id": "Axiom_of_regularity:0", "title": "", "text": "$\\forall x\\,(x\\neq\\varnothing\\rightarrow\\exists y\\in x\\,(y\\cap x=\\varnothing))$"}
{"_id": "Necklace_(combinatorics):1", "title": "", "text": "$B_{k}(n)=\\begin{cases}{1\\over 2}N_{k}(n)+{1\\over 4}(k+1)k^{n/2}&\\,\\text{if }n\\,\\text{ is even}\\\\ \\\\ {1\\over 2}N_{k}(n)+{1\\over 2}k^{(n+1)/2}&\\,\\text{if }n\\,\\text{ is odd}\\end{cases}$"}
{"_id": "Law_of_cosines:71", "title": "", "text": "$\\cos A=-\\cos B\\cos C+\\sin B\\sin C\\cosh a.\\,$"}
{"_id": "Binding_constant:4", "title": "", "text": "${\\rm R}+{\\rm L}\\to{\\rm RL}$"}
{"_id": "Binary_lambda_calculus:38", "title": "", "text": "$KP(y|x^{\\ast})\\leq KP(x,y)-KP(x)+O(1)$"}
{"_id": "Noncentral_F-distribution:16", "title": "", "text": "$\\operatorname{E}\\left[F\\right]=\\begin{cases}\\frac{\\nu_{2}(\\nu_{1}+\\lambda)}{\\nu_{1}(\\nu_{2}-2)}&\\nu_{2}>2\\\\ \\,\\text{Does not exist}&\\nu_{2}\\leq 2\\\\ \\end{cases}$"}
{"_id": "List_of_rules_of_inference:40", "title": "", "text": "$\\underline{\\lnot\\varphi\\quad\\quad}\\,\\!$"}
{"_id": "Upper-convected_Maxwell_model:3", "title": "", "text": "$\\stackrel{\\nabla}{\\mathbf{T}}$"}
{"_id": "Draft:Exact_couple:35", "title": "", "text": "$0\\to G^{p+1}\\to G^{p}\\to K^{p,*}\\to 0$"}
{"_id": "Loss_of_significance:33", "title": "", "text": "$-4ac$"}
{"_id": "Median_(geometry):16", "title": "", "text": "$m_{b}=\\sqrt{\\frac{2a^{2}+2c^{2}-b^{2}}{4}},$"}
{"_id": "Algebraic_variety:27", "title": "", "text": "$\\wedge^{n}V$"}
{"_id": "Converse_nonimplication:17", "title": "", "text": "${}_{{}_{\\wedge}}\\!$"}
{"_id": "Severi–Brauer_variety:1", "title": "", "text": "$0\\rightarrow\\mathrm{Pic}(X)\\rightarrow\\mathbb{Z}\\stackrel{\\delta}{\\rightarrow}\\mathrm{Br}(K)\\rightarrow\\mathrm{Br}(K)(X)\\rightarrow 0\\ .$"}
{"_id": "Genetic_drift:2", "title": "", "text": "$\\frac{(2N)!}{k!(2N-k)!}p^{k}q^{2N-k}$"}
{"_id": "Duhamel's_principle:24", "title": "", "text": "$=\\int_{-\\infty}^{t}G(t-\\tau)F(\\tau)\\,d\\tau$"}
{"_id": "Clausen_function:155", "title": "", "text": "$\\beta(x)=\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(2k+1)^{x}}$"}
{"_id": "Fermi–Dirac_statistics:101", "title": "", "text": "$w(n_{i},g_{i})=\\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}\\ .$"}
{"_id": "Dimensionless_quantity:30", "title": "", "text": "$\\frac{\\sum_{k=1}^{n}(x_{k}-\\bar{x})(y_{k}-\\bar{y})}{\\sqrt{\\sum_{k=1}^{n}(x_{k}-\\bar{x})^{2}\\sum_{k=1}^{n}(y_{k}-\\bar{y})^{2}}}$"}
{"_id": "Gauss–Legendre_algorithm:6", "title": "", "text": "$3.14159264\\dots\\!$"}
{"_id": "Specific_orbital_energy:33", "title": "", "text": "$2a-R$"}
{"_id": "Sheaf_of_modules:53", "title": "", "text": "$0\\rightarrow F\\rightarrow G\\rightarrow H\\rightarrow 0.$"}
{"_id": "Per_Enflo:34", "title": "", "text": "$\\alpha!=\\Pi_{i=1}^{N}(\\alpha_{i}!)$"}
{"_id": "Gift_wrapping_algorithm:2", "title": "", "text": "$O(n\\log h)$"}
{"_id": "Prosthaphaeresis:0", "title": "", "text": "$\\cos a=\\cos b\\cos c+\\sin b\\sin c\\cos\\alpha$"}
{"_id": "Heavy_traffic_approximation:12", "title": "", "text": "$T_{j}\\xrightarrow{d}T$"}
{"_id": "Logarithm:34", "title": "", "text": "$\\log_{b}\\!\\left(\\frac{x}{y}\\right)=\\log_{b}(x)-\\log_{b}(y)$"}
{"_id": "Kirchhoff's_law_of_thermal_radiation:12", "title": "", "text": "$\\alpha_{\\mathrm{sun}}=\\displaystyle\\frac{\\int_{0}^{\\infty}\\alpha_{\\lambda}I_{\\lambda\\mathrm{sun}}(\\lambda)\\,d\\lambda}{\\int_{0}^{\\infty}I_{\\lambda\\mathrm{sun}}(\\lambda)\\,d\\lambda}$"}
{"_id": "Uniform_integrability:12", "title": "", "text": "$I_{|X|\\geq K}=\\begin{cases}1&\\,\\text{if }|X|\\geq K,\\\\ 0&\\,\\text{if }|X|<K.\\end{cases}$"}
{"_id": "Heat:46", "title": "", "text": "$\\Delta H=Q-P\\,\\Delta V+P\\,\\Delta V$"}
{"_id": "Hopf_lemma:0", "title": "", "text": "$Lu=a_{ij}(x)\\frac{\\partial^{2}u}{\\partial x_{i}\\partial x_{j}}+b_{i}(x)\\frac{\\partial u}{\\partial x_{i}}+c(x)u,\\qquad x\\in\\Omega.$"}
{"_id": "Non-analytic_smooth_function:0", "title": "", "text": "$f(x)=\\begin{cases}\\exp(-1/x)&\\,\\text{if }x>0,\\\\ 0&\\,\\text{if }x\\leq 0,\\end{cases}$"}
{"_id": "Digamma_function:47", "title": "", "text": "$\\psi(x)=\\ln(x)-\\frac{1}{2x}-\\frac{1}{12x^{2}}+\\frac{1}{120x^{4}}-\\frac{1}{252x^{6}}+\\frac{1}{240x^{8}}-\\frac{5}{660x^{10}}+\\frac{691}{32760x^{12}}-\\frac{1}{12x^{14}}+O\\left(\\frac{1}{x^{16}}\\right)$"}
{"_id": "Klee's_measure_problem:2", "title": "", "text": "$O(n^{d-1}\\log n)$"}
{"_id": "Proofs_of_trigonometric_identities:143", "title": "", "text": "$\\cos(\\alpha+\\beta)+\\cos(\\alpha-\\beta)=\\cos\\alpha\\cos\\beta\\ -\\sin\\alpha\\sin\\beta+\\cos\\alpha\\cos\\beta+\\sin\\alpha\\sin\\beta=2\\cos\\alpha\\cos\\beta$"}
{"_id": "Universal_hashing:17", "title": "", "text": "$\\forall x,y\\in U,~{}x\\neq y$"}
{"_id": "Ideal_lattice_cryptography:164", "title": "", "text": "$\\tilde{O}(m)$"}
{"_id": "Ring_of_sets:19", "title": "", "text": "$A\\cup B=(A\\,\\triangle\\,B)\\,\\triangle\\,(A\\cap B)$"}
{"_id": "Coding_theory_approaches_to_nucleic_acid_design:71", "title": "", "text": "$\\mathit{B}=(x^{b_{i}})$"}
{"_id": "Converse_nonimplication:157", "title": "", "text": "$::\\begin{aligned}\\displaystyle(r\\nleftarrow q)\\nleftarrow p&\\displaystyle=r^{\\prime}q\\nleftarrow p\\qquad\\qquad\\qquad~{}~{}~{}~{}\\,\\text{(by definition)}\\\\ &\\displaystyle=(r^{\\prime}q)^{\\prime}p\\qquad\\qquad\\qquad~{}~{}~{}~{}~{}~{}\\,\\text{(by definition)}\\\\ &\\displaystyle=(r+q^{\\prime})p\\qquad\\qquad~{}~{}~{}~{}~{}~{}~{}~{}~{}\\,\\text{(De Morgan's laws)}\\\\ &\\displaystyle=(r+r^{\\prime}q^{\\prime})p\\qquad\\qquad~{}~{}~{}~{}~{}~{}~{}\\,\\text{(Absorption law)}\\\\ &\\displaystyle=rp+r^{\\prime}q^{\\prime}p\\\\ &\\displaystyle=rp+r^{\\prime}(q\\nleftarrow p)\\qquad~{}~{}~{}~{}~{}~{}~{}~{}\\,\\text{(by definition)}\\\\ &\\displaystyle=rp+r\\nleftarrow(q\\nleftarrow p)\\qquad~{}~{}~{}~{}\\,\\text{(by definition)}\\\\ \\end{aligned}$"}
{"_id": "Alpha_decay:0", "title": "", "text": "$\\mathrm{~{}^{238}_{92}U}\\rightarrow\\mathrm{~{}^{234}_{90}Th}+{\\alpha}$"}
{"_id": "Compressibility:3", "title": "", "text": "$c^{2}=\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_{S}$"}
{"_id": "Chern_class:55", "title": "", "text": "$\\ 0\\to E^{\\prime}\\to E\\to E^{\\prime\\prime}\\to 0$"}
{"_id": "Sommerfeld_expansion:23", "title": "", "text": "$I=\\underbrace{\\tau\\int_{-\\infty}^{0}\\frac{H(\\mu+\\tau x)}{e^{x}+1}\\,\\mathrm{d}x}_{I_{1}}+\\underbrace{\\tau\\int_{0}^{\\infty}\\frac{H(\\mu+\\tau x)}{e^{x}+1}\\,\\mathrm{d}x}_{I_{2}}\\,.$"}
{"_id": "Numerical_digit:2", "title": "", "text": "$A+B=C\\,$"}
{"_id": "Axiomatic_design:0", "title": "", "text": "$\\begin{bmatrix}FR_{1}\\\\ FR_{2}\\end{bmatrix}=\\begin{bmatrix}A_{11}&A_{12}\\\\ A_{21}&A_{22}\\end{bmatrix}\\begin{bmatrix}DP_{1}\\\\ DP_{2}\\end{bmatrix}$"}
{"_id": "Effective_mass_(spring–mass_system):31", "title": "", "text": "$=\\int_{0}^{L}\\frac{1}{2}u^{2}\\rho(x)\\,dx$"}
{"_id": "Von_Staudt–Clausen_theorem:12", "title": "", "text": "$\\wp=[\\frac{2n}{p-1}]\\!$"}
{"_id": "Projective_module:0", "title": "", "text": "$0\\rightarrow A\\rightarrow B\\rightarrow P\\rightarrow 0\\,$"}
{"_id": "Two-port_network:86", "title": "", "text": "$\\begin{bmatrix}b_{1}\\\\ b_{2}\\end{bmatrix}=\\begin{bmatrix}S_{11}&S_{12}\\\\ S_{21}&S_{22}\\end{bmatrix}\\begin{bmatrix}a_{1}\\\\ a_{2}\\end{bmatrix}$"}
{"_id": "Selberg_class:3", "title": "", "text": "$L(s,\\Delta)=\\sum_{n=1}^{\\infty}\\frac{a_{n}}{n^{s}}$"}
{"_id": "Haar_wavelet:26", "title": "", "text": "$H_{2}=\\begin{bmatrix}1&1\\\\ 1&-1\\end{bmatrix}.$"}
{"_id": "Convergent_series:10", "title": "", "text": "${1\\over 1}-{1\\over 2}+{1\\over 3}-{1\\over 4}+{1\\over 5}\\cdots=\\ln(2)$"}
{"_id": "Integer:13", "title": "", "text": "$f(x)=\\begin{cases}2|x|,&\\mbox{if }~{}x\\leq 0\\\\ 2x-1,&\\mbox{if }~{}x>0.\\end{cases}$"}
{"_id": "Kepler's_laws_of_planetary_motion:9", "title": "", "text": "$\\frac{1}{r_{\\min}}-\\frac{1}{p}=\\frac{1}{p}-\\frac{1}{r_{\\max}}$"}
{"_id": "Fundamental_plane_(elliptical_galaxies):2", "title": "", "text": "$\\langle I\\rangle_{e}$"}
{"_id": "Ring_(mathematics):243", "title": "", "text": "$+:R\\times R\\to R\\,$"}
{"_id": "Mathematics_of_radio_engineering:15", "title": "", "text": "$\\,q\\overline{q}=-a^{2}+b^{2}+c^{2}+d^{2}$"}
{"_id": "Bootstrapping_(statistics):23", "title": "", "text": "$v_{i}=\\left\\{\\begin{matrix}-1&\\mbox{with prob. }1/2\\\\ 1&\\mbox{with prob. }1/2\\end{matrix}\\right.$"}
{"_id": "Toda_bracket:0", "title": "", "text": "$W\\stackrel{f}{\\ \\to\\ }X\\stackrel{g}{\\ \\to\\ }Y\\stackrel{h}{\\ \\to\\ }Z$"}
{"_id": "Induction_equation:1", "title": "", "text": "$\\vec{\\nabla}\\times\\vec{B}=\\mu_{0}\\vec{J},$"}
{"_id": "Test_Template_Framework:36", "title": "", "text": "$R\\oplus G=(\\,\\text{dom }G\\ntriangleleft R)\\cup G$"}
{"_id": "Fraction_of_variance_unexplained:3", "title": "", "text": "$\\begin{aligned}\\displaystyle SS_{\\rm err}&\\displaystyle=\\sum_{i=1}^{N}\\;(y_{i}-\\widehat{y_{i}})^{2}\\\\ \\displaystyle SS_{\\rm tot}&\\displaystyle=\\sum_{i=1}^{N}\\;(y_{i}-\\bar{y})^{2}\\\\ \\displaystyle SS_{\\rm reg}&\\displaystyle=\\sum_{i=1}^{N}\\;(\\widehat{y_{i}}-\\bar{y})^{2}\\,\\text{ and}\\\\ \\displaystyle\\bar{y}&\\displaystyle=\\frac{1}{N}\\sum{}_{i=1}^{N}\\;y_{i}.\\end{aligned}$"}
{"_id": "Arithmetic_progression:9", "title": "", "text": "$\\left(-\\frac{3}{2}\\right)+\\left(-\\frac{1}{2}\\right)+\\frac{1}{2}=\\frac{3\\left(-\\frac{3}{2}+\\frac{1}{2}\\right)}{2}=-\\frac{3}{2}.$"}
{"_id": "Sieve_of_Eratosthenes:49", "title": "", "text": "$O(n\\log\\log n)$"}
{"_id": "Pell_number:45", "title": "", "text": "$1-1\\sqrt{2}=-0.41421\\ldots$"}
{"_id": "Poker_probability:25", "title": "", "text": "${n\\choose r}={{n!}\\over{r!(n-r)!}}={52\\choose 5}={{52!}\\over{5!(52-5)!}}=2,598,960$"}
{"_id": "Continued_fraction:317", "title": "", "text": "$\\displaystyle\\pi=2+\\cfrac{4}{3+\\cfrac{1\\cdot 3}{4+\\cfrac{3\\cdot 5}{4+\\cfrac{5\\cdot 7}{4+\\ddots}}}}$"}
{"_id": "List_of_formulae_involving_π:98", "title": "", "text": "$\\pi=\\cfrac{4}{1+\\cfrac{1^{2}}{3+\\cfrac{2^{2}}{5+\\cfrac{3^{2}}{7+\\cfrac{4^{2}}{9+\\ddots}}}}}$"}
{"_id": "Bean_machine:1", "title": "", "text": "${n\\choose k}p^{k}(1-p)^{n-k}$"}
{"_id": "Index_of_coincidence:1", "title": "", "text": "$\\mathbf{IC}=\\frac{\\displaystyle\\sum_{i=1}^{c}n_{i}(n_{i}-1)}{N(N-1)/c}$"}
{"_id": "Lerch_zeta_function:0", "title": "", "text": "$L(\\lambda,\\alpha,s)=\\sum_{n=0}^{\\infty}\\frac{\\exp(2\\pi i\\lambda n)}{(n+\\alpha)^{s}}.$"}
{"_id": "Sheaf_cohomology:1", "title": "", "text": "$0\\ \\rightarrow\\mathcal{A}\\ \\stackrel{\\phi}{\\rightarrow}\\ \\mathcal{B}\\ \\stackrel{\\psi}{\\rightarrow}\\ \\mathcal{C}\\ \\rightarrow\\ 0$"}
{"_id": "Analytic_combinatorics:167", "title": "", "text": "$\\mathfrak{P}(\\mathfrak{P}_{\\geq 1}(\\mathcal{Z})).$"}
{"_id": "Generalized_continued_fraction:41", "title": "", "text": "$x=1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\ddots\\,}}}}$"}
{"_id": "Sinhc_function:2", "title": "", "text": "$\\operatorname{Im}\\left(\\frac{\\sinh(x+iy)}{x+iy}\\right)$"}
{"_id": "Hilbert's_thirteenth_problem:0", "title": "", "text": "$x^{7}+ax^{3}+bx^{2}+cx+1=0$"}
{"_id": "Distributed_lag:8", "title": "", "text": "$w_{i}=\\sum_{j=2}^{n}\\frac{a_{j}}{(i+1)^{j}},$"}
{"_id": "Glassy_carbon:0", "title": "", "text": "$\\rm\\stackrel{GCE}{\\rightleftharpoons}$"}
{"_id": "Disjunctive_sequence:0", "title": "", "text": "$0\\ 1\\ 00\\ 01\\ 10\\ 11\\ 000\\ 001\\ldots$"}
{"_id": "Draft:The_study_of_the_momentum_transfer_due_to_fluid_friction_and_heat_transfer_between_a_stream_and_a_solid_object_in_internal_flow,:9", "title": "", "text": "$U=\\frac{D^{2}}{12\\mu}\\left(\\frac{-dP}{dx}\\right)$"}
{"_id": "Spherical_trigonometry:18", "title": "", "text": "$\\cos C=-\\cos A\\,\\cos B+\\sin A\\,\\sin B\\,\\cos c.$"}
{"_id": "Indifference_price:33", "title": "", "text": "$\\beta=-\\frac{1}{2}$"}
{"_id": "Regression_analysis:20", "title": "", "text": "$\\widehat{\\beta_{1}}=\\frac{\\sum(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{\\sum(x_{i}-\\bar{x})^{2}}\\,\\text{ and }\\hat{\\beta_{0}}=\\bar{y}-\\widehat{\\beta_{1}}\\bar{x}$"}
{"_id": "Ring_(mathematics):252", "title": "", "text": "$\\operatorname{pt}\\stackrel{1}{\\to}R$"}
{"_id": "Gauss's_law_for_magnetism:5", "title": "", "text": "$\\nabla\\cdot\\mathbf{B}=\\mu_{0}\\rho_{m}$"}
{"_id": "Mathematical_morphology:0", "title": "", "text": "$\\mathbb{R}^{d}$"}
{"_id": "Gaussian_function:73", "title": "", "text": "$\\log\\left(x\\right)=-\\log\\left(\\frac{1}{x}\\right)$"}
{"_id": "FEE_method:36", "title": "", "text": "$s_{f_{2}}(n)=O\\left(M(n)\\log n\\right).$"}
{"_id": "Neutrino_decoupling:10", "title": "", "text": "$H=\\sqrt{\\frac{8\\pi}{3}G\\rho}$"}
{"_id": "Apéry's_constant:4", "title": "", "text": "$\\zeta(3)=14\\sum_{k=1}^{\\infty}\\frac{1}{k^{3}\\sinh(\\pi k)}-\\frac{11}{2}\\sum_{k=1}^{\\infty}\\frac{1}{k^{3}(e^{2\\pi k}-1)}-\\frac{7}{2}\\sum_{k=1}^{\\infty}\\frac{1}{k^{3}(e^{2\\pi k}+1)}.$"}
{"_id": "Information_geometry:315", "title": "", "text": "$D_{\\gamma}(\\gamma(b)||\\gamma(a))=\\int_{a}^{b}(b-s)g_{\\gamma}(s)ds$"}
{"_id": "Novikov's_condition:4", "title": "", "text": "$\\ Z_{t}\\ =e^{\\int_{0}^{t}X_{s}\\,dW_{s}-\\frac{1}{2}\\int_{0}^{t}X_{s}^{2}\\,ds},\\quad 0\\leq t\\leq T$"}
{"_id": "Self-adjoint_operator:26", "title": "", "text": "$-i\\Gamma$"}
{"_id": "Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm:91", "title": "", "text": "$r=\\lfloor 0.5+\\mu_{2,l}\\rfloor=4$"}
{"_id": "Ratio_estimator:8", "title": "", "text": "$\\theta=\\frac{1}{n}-\\frac{1}{N}$"}
{"_id": "Convolution_power:2", "title": "", "text": "$x^{*n}=\\underbrace{x*x*x*\\cdots*x*x}_{n},\\quad x^{*0}=\\delta_{0}$"}
{"_id": "Generation_time:5", "title": "", "text": "$T=\\frac{\\int_{x=0}^{\\infty}x\\ell(x)m(x)\\,\\mathrm{d}x}{\\int_{x=0}^{\\infty}\\ell(x)m(x)\\,\\mathrm{d}x}$"}
{"_id": "Non-analytic_smooth_function:1", "title": "", "text": "$f^{(n)}(x)=\\begin{cases}\\displaystyle\\frac{p_{n}(x)}{x^{2n}}\\,f(x)&\\,\\text{if }x>0,\\\\ 0&\\,\\text{if }x\\leq 0,\\end{cases}$"}
{"_id": "Lag_operator:15", "title": "", "text": "$[\\ ]_{+}$"}
{"_id": "Tobit_model:33", "title": "", "text": "$y_{3i}=\\begin{cases}y_{3i}^{*}&\\textrm{if}\\;y_{1i}^{*}>0\\\\ 0&\\textrm{if}\\;y_{1i}^{*}\\leq 0.\\end{cases}$"}
{"_id": "Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm:75", "title": "", "text": "$\\mu_{3,1}=\\mu_{3,1}-r=\\frac{14}{3}-5=\\frac{-1}{3}$"}
{"_id": "Help:Displaying_a_formula:381", "title": "", "text": "${}_{p}F_{q}(a_{1},\\dots,a_{p};c_{1},\\dots,c_{q};z)=\\sum_{n=0}^{\\infty}\\frac{(a_{1})_{n}\\cdots(a_{p})_{n}}{(c_{1})_{n}\\cdots(c_{q})_{n}}\\frac{z^{n}}{n!}$"}
{"_id": "Floor_and_ceiling_functions:45", "title": "", "text": "$\\left\\lfloor\\frac{m}{n}\\right\\rfloor+\\left\\lfloor\\frac{2m}{n}\\right\\rfloor+\\dots+\\left\\lfloor\\frac{(n-1)m}{n}\\right\\rfloor=\\left\\lfloor\\frac{n}{m}\\right\\rfloor+\\left\\lfloor\\frac{2n}{m}\\right\\rfloor+\\dots+\\left\\lfloor\\frac{(m-1)n}{m}\\right\\rfloor.$"}
{"_id": "Polymer_electrolyte_membrane_electrolysis:2", "title": "", "text": "$\\Delta H=\\underbrace{\\Delta G}_{\\textrm{elec.}}+\\underbrace{T\\Delta S}_{\\textrm{heat}}$"}
{"_id": "Direct_sum_of_topological_groups:5", "title": "", "text": "$0\\to H\\stackrel{i}{{}\\to{}}G\\stackrel{\\pi}{{}\\to{}}G/H\\to 0$"}
{"_id": "Dynamic_programming:122", "title": "", "text": "$O(n\\log k)$"}
{"_id": "Uniformly_most_powerful_test:43", "title": "", "text": "$\\phi(T)=\\begin{cases}1&\\,\\text{if }T>t_{0}\\\\ 0&\\,\\text{if }T<t_{0}\\end{cases}$"}
{"_id": "Randomized_algorithm:33", "title": "", "text": "$1-\\frac{1}{n}$"}
{"_id": "Ampère's_circuital_law:6", "title": "", "text": "$\\mathbf{\\nabla}\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}$"}
{"_id": "Wave_equation:89", "title": "", "text": "$\\partial^{2}u\\over\\partial t^{2}$"}
{"_id": "Gospel_of_John:2", "title": "", "text": "$\\mathfrak{P}^{75}$"}
{"_id": "Conic_section:97", "title": "", "text": "$\\{x^{2}-y^{2}=0\\}=\\{(x+y)(x-y)=0\\}=\\{x+y=0\\}\\cup\\{x-y=0\\}.\\,$"}
{"_id": "Isodynamic_point:33", "title": "", "text": "$AS^{\\prime}B=ACB-\\pi/3$"}
{"_id": "Natural_logarithm_of_2:40", "title": "", "text": "$\\ln 2=\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{3+\\cfrac{2}{2+\\cfrac{2}{5+\\cfrac{3}{2+\\cfrac{3}{7+\\cfrac{4}{2+\\ddots}}}}}}}}=\\cfrac{2}{3-\\cfrac{1^{2}}{9-\\cfrac{2^{2}}{15-\\cfrac{3^{2}}{21-\\ddots}}}}$"}
{"_id": "List_of_rules_of_inference:25", "title": "", "text": "$\\underline{\\psi\\quad\\quad\\ \\ }\\,\\!$"}
{"_id": "Banach_algebra:0", "title": "", "text": "$\\forall x,y\\in A:\\|x\\,y\\|\\ \\leq\\|x\\|\\,\\|y\\|$"}
{"_id": "Mathematical_constants_and_functions:307", "title": "", "text": "$\\sum_{k=1}^{\\infty}\\frac{(-1)^{k}}{s_{k}-1}=\\frac{1}{1}-\\frac{1}{2}+\\frac{1}{6}-\\frac{1}{42}+\\frac{1}{1806}{\\,\\pm\\cdots}$"}
{"_id": "Clenshaw–Curtis_quadrature:3", "title": "", "text": "$O(N\\log N)$"}
{"_id": "Marchenko–Pastur_distribution:10", "title": "", "text": "$\\mu(A)=\\begin{cases}(1-\\frac{1}{\\lambda})\\mathbf{1}_{0\\in A}+\\nu(A),&\\,\\text{if }\\lambda>1\\\\ \\nu(A),&\\,\\text{if }0\\leq\\lambda\\leq 1,\\end{cases}$"}
{"_id": "Fresnel_integral:24", "title": "", "text": "$\\int_{0}^{\\infty}\\cos t^{2}\\,\\mathrm{d}t=\\int_{0}^{\\infty}\\sin t^{2}\\,\\mathrm{d}t=\\frac{\\sqrt{2\\pi}}{4}=\\sqrt{\\frac{\\pi}{8}}.$"}
{"_id": "Binomial_regression:21", "title": "", "text": "$Y=\\begin{cases}0,&\\mbox{if }~{}Y^{*}>0\\\\ 1,&\\mbox{if }~{}Y^{*}<0.\\end{cases}$"}
{"_id": "Maxwell's_equations:10", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\left(\\mathbf{J}+\\varepsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t}\\right)$"}
{"_id": "Poisson_manifold:95", "title": "", "text": "$\\mathfrak{Poiss}$"}
{"_id": "Spin_representation:103", "title": "", "text": "$\\wedge^{2}V$"}
{"_id": "LU_decomposition:18", "title": "", "text": "$\\begin{bmatrix}4&3\\\\ 6&3\\end{bmatrix}=\\begin{bmatrix}1&0\\\\ 1.5&1\\end{bmatrix}\\begin{bmatrix}4&3\\\\ 0&-1.5\\end{bmatrix}.$"}
{"_id": "AB_magnitude:1", "title": "", "text": "$m\\text{AB}=-\\frac{5}{2}\\log_{10}\\left(\\frac{f_{\\nu}}{\\,\\text{Jy}}\\right)+8.90.$"}
{"_id": "Tidal_tensor:17", "title": "", "text": "$U=-m/\\sqrt{(}x^{2}+y^{2}+z^{2})$"}
{"_id": "Macroemulsion:5", "title": "", "text": "$\\ P={{\\ 1\\over\\Delta R\\sqrt{2\\pi\\,}\\ }\\exp[{\\ (\\ln{R}\\ -\\ln{\\bar{R}}\\ )^{2}\\over\\ 2\\Delta R^{2}\\ \\ }\\ }]$"}
{"_id": "Two-port_network:55", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ I_{1}\\end{bmatrix}=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}\\begin{bmatrix}V_{2}\\\\ -I_{2}\\end{bmatrix}$"}
{"_id": "Sharpe_ratio:0", "title": "", "text": "$S_{a}=\\frac{E[R_{a}-R_{b}]}{\\sigma_{a}}=\\frac{E[R_{a}-R_{b}]}{\\sqrt{\\mathrm{var}[R_{a}-R_{b}]}},$"}
{"_id": "Parametric_derivative:14", "title": "", "text": "$=\\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)$"}
{"_id": "Bochner's_theorem:9", "title": "", "text": "$\\langle\\cdot[e],[e]\\rangle_{f}$"}
{"_id": "Single_transferable_vote:0", "title": "", "text": "$\\mbox{votes needed to win}~{}=\\left({{\\rm\\mbox{valid votes cast}~{}}\\over{\\rm\\mbox{seats to fill}~{}}+1}\\right)+1$"}
{"_id": "Categorification:1", "title": "", "text": "$S^{\\lambda}\\stackrel{\\varphi}{\\to}s_{\\lambda},$"}
{"_id": "Analytic_signal:24", "title": "", "text": "$s_{\\mathrm{a}}(t)=\\begin{cases}e^{j(\\omega t+\\theta)}\\ \\ =\\ e^{j|\\omega|t}\\cdot e^{j\\theta},&\\,\\text{if}\\ \\omega>0,\\\\ e^{-j(\\omega t+\\theta)}=\\ e^{j|\\omega|t}\\cdot e^{-j\\theta},&\\,\\text{if}\\ \\omega<0.\\end{cases}$"}
{"_id": "Group_extension:14", "title": "", "text": "$0\\rightarrow\\mathfrak{a}\\rightarrow\\mathfrak{e}\\rightarrow\\mathfrak{g}\\rightarrow 0$"}
{"_id": "Average:4", "title": "", "text": "$\\frac{2}{\\frac{1}{60}+\\frac{1}{40}}=48$"}
{"_id": "Ideal_(ring_theory):28", "title": "", "text": "$\\forall x,y\\in I:x-y\\in I$"}
{"_id": "Kinetic_sorted_list:0", "title": "", "text": "$O(n\\log m)$"}
{"_id": "Zermelo–Fraenkel_set_theory:1", "title": "", "text": "$\\forall z[z\\in x\\Leftrightarrow z\\in y]\\land\\forall w[x\\in w\\Leftrightarrow y\\in w].$"}
{"_id": "Methods_of_contour_integration:4", "title": "", "text": "$\\Gamma=\\gamma_{1}+\\gamma_{2}+\\cdots+\\gamma_{n}.$"}
{"_id": "Artin_transfer_(group_theory):1324", "title": "", "text": "$H_{2}=\\langle x,G^{\\prime}\\rangle$"}
{"_id": "Field_electron_emission:18", "title": "", "text": "$\\;\\epsilon_{\\mathrm{n}}=\\epsilon-K_{\\mathrm{p}}...........(16)$"}
{"_id": "Beam_diameter:4", "title": "", "text": "$\\bar{x}=\\frac{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y)x\\,dx\\,dy}{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y)\\,dx\\,dy}$"}
{"_id": "Sortino_ratio:6", "title": "", "text": "$DR=\\sqrt{\\int_{-\\infty}^{T}(T-r)^{2}f(r)\\,dr}$"}
{"_id": "Patlak_plot:0", "title": "", "text": "$R(t)=K\\int_{0}^{t}C_{p}(\\tau)\\,d\\tau+V_{0}C_{p}(t)$"}
{"_id": "Fractional_vortices:19", "title": "", "text": "$2π-2φ$"}
{"_id": "Duality_(mathematics):4", "title": "", "text": "$B^{c}\\subseteq A^{c}$"}
{"_id": "Normed_algebra:0", "title": "", "text": "$\\forall x,y\\in A\\qquad\\|xy\\|\\leq\\|x\\|\\|y\\|$"}
{"_id": "Nonimaging_optics:1", "title": "", "text": "$S(\\tau_{B})-S(\\tau_{A})=\\int_{A}^{B}dS=\\int_{\\tau_{A}}^{\\tau_{B}}\\frac{dS}{d\\tau}d\\tau=\\int_{\\tau_{A}}^{\\tau_{B}}\\frac{S(\\tau+d\\tau)-S(\\tau)}{(\\tau+d\\tau)-\\tau}d\\tau$"}
{"_id": "Heterogeneous_random_walk_in_one_dimension:54", "title": "", "text": "$a_{i,0}=\\left[\\frac{n+[L/2]-1}{[L/2]}\\right].$"}
{"_id": "Orthotropic_material:89", "title": "", "text": "$\\underline{\\underline{\\mathsf{S}}}$"}
{"_id": "Normally_distributed_and_uncorrelated_does_not_imply_independent:3", "title": "", "text": "$Y=\\left\\{\\begin{matrix}X&\\,\\text{if }\\left|X\\right|\\leq c\\\\ -X&\\,\\text{if }\\left|X\\right|>c\\end{matrix}\\right.$"}
{"_id": "Quantization_(signal_processing):76", "title": "", "text": "$d_{k}=\\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx$"}
{"_id": "Newton_polynomial:31", "title": "", "text": "$x_{1}=-\\tfrac{3}{4}$"}
{"_id": "Vinculum_(symbol):13", "title": "", "text": "$\\underline{AB}$"}
{"_id": "Quintic_function:37", "title": "", "text": "$x=-1.84208\\dots$"}
{"_id": "Longest_common_subsequence_problem:93", "title": "", "text": "$O((n+r)\\log(n))$"}
{"_id": "Integration_using_parametric_derivatives:0", "title": "", "text": "$\\int_{0}^{\\infty}x^{2}e^{-3x}\\,dx.$"}
{"_id": "Chemical_thermodynamics:13", "title": "", "text": "$dG_{T,P}=-\\sum_{k}\\mathbb{A}_{k}\\,d\\xi_{k}+W^{\\prime}.\\,$"}
{"_id": "Symmetry_of_diatomic_molecules:368", "title": "", "text": "$-13.6eV$"}
{"_id": "Diophantus:5", "title": "", "text": "$ax^{2}+bx=c$"}
{"_id": "NPDGamma_experiment:0", "title": "", "text": "$\\vec{n}+p\\to d+\\gamma$"}
{"_id": "Convex_conjugate:95", "title": "", "text": "$-\\log(x)$"}
{"_id": "Courant_algebroid:22", "title": "", "text": "$\\wedge^{*}A$"}
{"_id": "Delta_method:9", "title": "", "text": "$\\tilde{\\theta}\\,\\xrightarrow{P}\\,\\theta$"}
{"_id": "Markov_chain:13", "title": "", "text": "$\\begin{aligned}\\displaystyle x^{(n+3)}&\\displaystyle=x^{(n+2)}P=\\left(x^{(n+1)}P\\right)P\\\\ \\\\ &\\displaystyle=x^{(n+1)}P^{2}=\\left(x^{(n)}P^{2}\\right)P\\\\ &\\displaystyle=x^{(n)}P^{3}\\\\ &\\displaystyle=\\begin{bmatrix}0&1&0\\end{bmatrix}\\begin{bmatrix}0.9&0.075&0.025\\\\ 0.15&0.8&0.05\\\\ 0.25&0.25&0.5\\end{bmatrix}^{3}\\\\ &\\displaystyle=\\begin{bmatrix}0&1&0\\end{bmatrix}\\begin{bmatrix}0.7745&0.17875&0.04675\\\\ 0.3575&0.56825&0.07425\\\\ 0.4675&0.37125&0.16125\\\\ \\end{bmatrix}\\\\ &\\displaystyle=\\begin{bmatrix}0.3575&0.56825&0.07425\\end{bmatrix}.\\end{aligned}$"}
{"_id": "Cubic_function:10", "title": "", "text": "$ax^{3}+bx^{2}+cx+d=0$"}
{"_id": "Restricted_sumset:8", "title": "", "text": "$n^{\\wedge}A$"}
{"_id": "10_(number):6", "title": "", "text": "$0.\\overline{90}$"}
{"_id": "Combination:1", "title": "", "text": "$\\frac{n!}{k!(n-k)!}$"}
{"_id": "Extension_of_a_topological_group:38", "title": "", "text": "$0\\to G^{\\wedge}\\stackrel{\\pi^{\\wedge}}{\\to}X^{\\wedge}\\stackrel{\\imath^{\\wedge}}{\\to}H^{\\wedge}\\to 0$"}
{"_id": "Watson's_lemma:47", "title": "", "text": "$\\begin{aligned}\\displaystyle(3)\\quad\\int_{0}^{\\delta}e^{-xt}\\phi(t)\\,\\mathrm{d}t&\\displaystyle=\\int_{0}^{\\delta}e^{-xt}t^{\\lambda}g(t)\\,\\mathrm{d}t\\\\ &\\displaystyle=\\sum_{n=0}^{N}\\frac{g^{(n)}(0)}{n!}\\int_{0}^{\\delta}t^{\\lambda+n}e^{-xt}\\,\\mathrm{d}t+\\frac{1}{(N+1)!}\\int_{0}^{\\delta}g^{(N+1)}(t^{*})\\,t^{\\lambda+N+1}e^{-xt}\\,\\mathrm{d}t.\\end{aligned}$"}
{"_id": "Positive_and_negative_parts:0", "title": "", "text": "$f^{+}(x)=\\max(f(x),0)=\\begin{cases}f(x)&\\mbox{ if }~{}f(x)>0\\\\ 0&\\mbox{ otherwise.}\\end{cases}$"}
{"_id": "Estimation_of_covariance_matrices:11", "title": "", "text": "$\\mathcal{L}(\\overline{x},\\Sigma)\\propto\\det(\\Sigma)^{-n/2}\\exp\\left(-{1\\over 2}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{\\mathrm{T}}\\Sigma^{-1}(x_{i}-\\overline{x})\\right),$"}
{"_id": "Wilson–Cowan_model:16", "title": "", "text": "$c_{2}I=c_{1}E-S_{e}^{-1}\\left(\\frac{E}{k_{e}-r_{e}E}\\right)+P$"}
{"_id": "Thorium:0", "title": "", "text": "${}_{\\ 90}^{232}\\mathrm{Th}+\\mathrm{n}\\rightarrow{}_{\\ 90}^{233}\\mathrm{Th}+\\gamma\\ \\xrightarrow{\\beta^{-}}\\ {}_{\\ 91}^{233}\\mathrm{Pa}\\ \\xrightarrow{\\beta^{-}}\\ {}_{\\ 92}^{233}\\mathrm{U}$"}
{"_id": "Schuette–Nesbitt_formula:143", "title": "", "text": "$S_{k}={\\left({{m}\\atop{k}}\\right)}\\frac{(m-k)!}{m!}=\\frac{1}{k!}.$"}
{"_id": "Buffering_agent:4", "title": "", "text": "$\\textrm{pH}_{new}=\\textrm{pK}_{a}+\\log\\left(\\frac{1.5\\textrm{n}}{0.5\\textrm{n}}\\right)$"}
{"_id": "Parabola:43", "title": "", "text": "$ax+by+c=0\\,$"}
{"_id": "Sound_pressure:28", "title": "", "text": "$L_{p_{2}}=L_{p_{1}}+20\\log_{10}\\!\\left(\\frac{r_{1}}{r_{2}}\\right)\\!~{}\\mathrm{dB}.$"}
{"_id": "Monopsony:21", "title": "", "text": "$w=\\begin{cases}w_{min},&\\mbox{if }~{}w_{min}\\geq\\;w(L)\\\\ w(L),&\\mbox{if }~{}w_{min}\\leq\\;w(L)\\end{cases}\\,\\!$"}
{"_id": "Transverse_isotropy:49", "title": "", "text": "$\\begin{bmatrix}\\sigma_{1}\\\\ \\sigma_{2}\\\\ \\sigma_{3}\\\\ \\sigma_{4}\\\\ \\sigma_{5}\\\\ \\sigma_{6}\\end{bmatrix}=\\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\\end{bmatrix}\\begin{bmatrix}\\varepsilon_{1}\\\\ \\varepsilon_{2}\\\\ \\varepsilon_{3}\\\\ \\varepsilon_{4}\\\\ \\varepsilon_{5}\\\\ \\varepsilon_{6}\\end{bmatrix}$"}
{"_id": "Gospel_of_John:1", "title": "", "text": "$\\mathfrak{P}^{90}$"}
{"_id": "Wigner_D-matrix:22", "title": "", "text": "$\\begin{array}[]{lcl}\\hat{\\mathcal{P}}_{1}&=&\\,i\\left({\\cos\\gamma\\over\\sin\\beta}{\\partial\\over\\partial\\alpha}-\\sin\\gamma{\\partial\\over\\partial\\beta}-\\cot\\beta\\cos\\gamma{\\partial\\over\\partial\\gamma}\\right)\\\\ \\hat{\\mathcal{P}}_{2}&=&\\,i\\left(-{\\sin\\gamma\\over\\sin\\beta}{\\partial\\over\\partial\\alpha}-\\cos\\gamma{\\partial\\over\\partial\\beta}+\\cot\\beta\\sin\\gamma{\\partial\\over\\partial\\gamma}\\right)\\\\ \\hat{\\mathcal{P}}_{3}&=&-i{\\partial\\over\\partial\\gamma},\\\\ \\end{array}$"}
{"_id": "Transition_state_theory:3", "title": "", "text": "$k\\propto\\exp\\left(\\frac{-\\Delta^{\\ddagger}G^{\\ominus}}{RT}\\right)$"}
{"_id": "Polite_number:0", "title": "", "text": "$f(n)=n+\\left\\lfloor\\log_{2}\\left(n+\\log_{2}n\\right)\\right\\rfloor.$"}
{"_id": "Net_present_value:12", "title": "", "text": "$\\mathrm{NPV}(i)=\\sum_{t=0}^{N}\\frac{R_{t}}{(1+i)^{t}}$"}
{"_id": "Two-element_Boolean_algebra:6", "title": "", "text": "$A\\cdot B=\\overline{\\overline{A}+\\overline{B}}$"}
{"_id": "Quintic_function:32", "title": "", "text": "$x^{5}-5x+12=0.$"}
{"_id": "Partial_fraction_decomposition:101", "title": "", "text": "$\\frac{1}{18}=\\frac{1}{2}-\\frac{1}{3}-\\frac{1}{3^{2}}.$"}
{"_id": "Shimizu_L-function:0", "title": "", "text": "$L(M,V,s)=\\sum_{\\mu\\in\\{M-0\\}/V}\\frac{\\operatorname{sign}N(\\mu)}{|N(\\mu)|^{s}}$"}
{"_id": "Proofs_of_trigonometric_identities:76", "title": "", "text": "$\\cos(\\alpha-\\beta)=\\cos\\alpha\\cos\\beta+\\sin\\alpha\\sin\\beta\\,$"}
{"_id": "Numeric_precision_in_Microsoft_Excel:15", "title": "", "text": "$ax^{2}+bx+c=0\\ .$"}
{"_id": "Standard_deviation:20", "title": "", "text": "$\\hat{\\sigma}=\\sqrt{\\frac{1}{N-1.5}\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}},$"}
{"_id": "Kullback–Leibler_divergence:34", "title": "", "text": "$P_{n}\\xrightarrow{D}Q$"}
{"_id": "Orthotropic_material:57", "title": "", "text": "$\\underline{\\underline{\\mathbf{A}}}$"}
{"_id": "142857_(number):0", "title": "", "text": "$0.\\overline{142857}$"}
{"_id": "Heat_capacity:46", "title": "", "text": "$C_{i,m}=\\left(\\frac{\\partial C}{\\partial n}\\right)$"}
{"_id": "Maxwell_stress_tensor:3", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}+\\mu_{0}\\epsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Saint-Venant's_compatibility_condition:7", "title": "", "text": "$W_{ijkl}=\\frac{\\partial^{2}F_{ij}}{\\partial x_{k}\\partial x_{l}}+\\frac{\\partial^{2}F_{kl}}{\\partial x_{i}\\partial x_{j}}-\\frac{\\partial^{2}F_{il}}{\\partial x_{j}\\partial x_{k}}-\\frac{\\partial^{2}F_{jk}}{\\partial x_{i}\\partial x_{l}}$"}
{"_id": "Triangle_inequality:46", "title": "", "text": "$\\alpha=\\pi/2-\\gamma,\\ \\mathrm{while}\\ \\beta=\\pi/2-\\gamma/2\\ ,$"}
{"_id": "Multiple_time_dimensions:0", "title": "", "text": "$(\\underbrace{+,\\cdots,+}_{k},\\underbrace{-,\\cdots,-}_{n})$"}
{"_id": "Spherical_law_of_cosines:18", "title": "", "text": "$\\cos(A)=-\\cos(B)\\cos(C)+\\sin(B)\\sin(C)\\cos(a)\\,$"}
{"_id": "Characteristic_polynomial:16", "title": "", "text": "$BA=A^{-1}(AB)A.$"}
{"_id": "Casus_irreducibilis:3", "title": "", "text": "$ax^{3}+bx^{2}+cx+d=0\\,$"}
{"_id": "Risk_aversion:6", "title": "", "text": "$A(c)=\\alpha$"}
{"_id": "Belevitch's_theorem:2", "title": "", "text": "$\\mathbf{S}(p)=\\begin{bmatrix}s_{11}&s_{12}\\\\ s_{21}&s_{22}\\end{bmatrix}$"}
{"_id": "MA_plot:0", "title": "", "text": "$M=\\log_{2}(R/G)=\\log_{2}(R)-\\log_{2}(G)$"}
{"_id": "S_(set_theory):3", "title": "", "text": "$\\forall r\\forall x[Fxr\\leftrightarrow[\\forall y(y\\in x\\rightarrow Byr)\\and\\lnot Bxr]]\\,.$"}
{"_id": "Approximations_of_π:203", "title": "", "text": "$\\pi=\\sum_{n=0}^{\\infty}\\left(\\frac{4}{8n+1}-\\frac{2}{8n+4}-\\frac{1}{8n+5}-\\frac{1}{8n+6}\\right)\\left(\\frac{1}{16}\\right)^{n}\\!$"}
{"_id": "Fluctuation-dissipation_theorem:7", "title": "", "text": "$\\langle x\\rangle_{0}$"}
{"_id": "Balance_puzzle:22", "title": "", "text": "$\\mathrm{0}.$"}
{"_id": "Mathematical_constants_and_functions:94", "title": "", "text": "$:\\;\\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0$"}
{"_id": "Equivalent_isotropically_radiated_power:6", "title": "", "text": "$\\,\\text{dBW}=10\\log\\left(\\frac{\\,\\text{power out}}{1\\,\\mathrm{W}}\\right)$"}
{"_id": "Dedekind_domain:71", "title": "", "text": "$M=T\\oplus P$"}
{"_id": "Dual_impedance:18", "title": "", "text": "$\\frac{1}{Z_{1}}+\\frac{1}{Z_{2}}$"}
{"_id": "MIDI_Tuning_Standard:0", "title": "", "text": "$d=69+12\\log_{2}\\left(\\frac{f}{440\\ \\mathrm{Hz}}\\right).\\,$"}
{"_id": "Xbar_and_s_chart:2", "title": "", "text": "$\\bar{s}_{i}=\\sqrt{\\frac{\\sum_{j=1}^{n}\\left(x_{ij}-\\bar{x}_{i}\\right)^{2}}{n-1}}$"}
{"_id": "Zimm–Bragg_model:19", "title": "", "text": "$\\left\\langle i\\right\\rangle=\\left(\\frac{s}{q}\\right)\\frac{dq}{ds}$"}
{"_id": "Patent_Box:0", "title": "", "text": "$\\mathbf{P}\\mathbf{B}*\\left(\\frac{MR-PBR}{MR}\\right)$"}
{"_id": "Method_of_steepest_descent:92", "title": "", "text": "$\\mathcal{I}_{j}=\\frac{2}{\\sqrt{-\\mu_{j}}\\sqrt{\\lambda}}\\int_{0}^{\\infty}e^{-\\frac{\\xi^{2}}{2}}d\\xi=\\sqrt{\\frac{2\\pi}{\\lambda}}(-\\mu_{j})^{-\\frac{1}{2}}.$"}
{"_id": "Estimation_of_covariance_matrices:28", "title": "", "text": "$\\sum_{i=1}^{n}(X_{i}-\\overline{X})(X_{i}-\\overline{X})^{\\mathrm{T}}\\sim W_{p}(\\Sigma,n-1).$"}
{"_id": "Steering_law:5", "title": "", "text": "$=\\lim_{N\\to\\infty}Nb\\log_{2}\\left(\\frac{A}{NW}+1\\right)$"}
{"_id": "Stein's_method:87", "title": "", "text": "$||f^{\\prime}||_{\\infty}$"}
{"_id": "Epidemic_model:44", "title": "", "text": "$\\tfrac{dS}{dT}=B-\\beta SI-\\mu S+\\gamma I$"}
{"_id": "Matrix_(mathematics):6", "title": "", "text": "$\\underline{\\underline{A}}$"}
{"_id": "David_Shmoys:51", "title": "", "text": "$O(\\log{k}\\ \\log{\\log{k}})$"}
{"_id": "Assured_Clear_Distance_Ahead:11", "title": "", "text": "$F_{total}=\\mu mg\\cos{\\theta}+mg\\sin{\\theta}$"}
{"_id": "Binomial_coefficient:19", "title": "", "text": "${\\left({{n}\\atop{k}}\\right)}=\\begin{cases}n^{\\underline{k}}/k!&\\,\\text{if }\\ k\\leq\\frac{n}{2}\\\\ n^{\\underline{n-k}}/(n-k)!&\\,\\text{if }\\ k>\\frac{n}{2}\\end{cases}.$"}
{"_id": "Length_of_a_module:1", "title": "", "text": "$0\\rightarrow L\\rightarrow M\\rightarrow N\\rightarrow 0$"}
{"_id": "Financial_ratio:7", "title": "", "text": "$\\left(\\frac{\\mbox{Net Income}~{}}{\\mbox{Net Sales}~{}}\\right)\\left(\\frac{\\mbox{Net Sales}~{}}{\\mbox{Total Assets}~{}}\\right)$"}
{"_id": "Continuous_mapping_theorem:3", "title": "", "text": "$X_{n}\\xrightarrow{d}X$"}
{"_id": "Breusch–Pagan_test:9", "title": "", "text": "$e_{i}^{2}=\\gamma_{1}+\\gamma_{2}z_{2i}+\\dots+\\gamma_{p}z_{pi}+\\eta_{i}.$"}
{"_id": "Hadamard_matrix:6", "title": "", "text": "$H_{2}=\\begin{bmatrix}1&1\\\\ 1&-1\\end{bmatrix},$"}
{"_id": "Twin-lead:0", "title": "", "text": "$Z=\\sqrt{{R+j\\omega L}\\over{G+j\\omega C}}$"}
{"_id": "Mathematical_constant:56", "title": "", "text": "$G=\\left.\\begin{matrix}3\\underbrace{\\uparrow\\ldots\\uparrow}3\\\\ \\underbrace{\\vdots}\\\\ 3\\uparrow\\uparrow\\uparrow\\uparrow 3\\end{matrix}\\right\\}\\,\\text{64 layers}$"}
{"_id": "Window_function:72", "title": "", "text": "$w_{0}(n)=\\mathrm{sinc}\\left(\\frac{2n}{N-1}\\right)\\,$"}
{"_id": "Koszul_complex:4", "title": "", "text": "$0\\to R\\xrightarrow{\\ d_{2}\\ }R^{2}\\xrightarrow{\\ d_{1}\\ }R\\to 0,$"}
{"_id": "Coshc_function:2", "title": "", "text": "$\\operatorname{Im}\\left(\\frac{\\cosh(x+iy)}{x+iy}\\right)$"}
{"_id": "Essential_matrix:50", "title": "", "text": "$2\\mathbf{E}\\mathbf{E}^{T}\\mathbf{E}-\\operatorname{tr}(\\mathbf{E}\\mathbf{E}^{T})\\mathbf{E}=0.$"}
{"_id": "Derivations_of_the_Lorentz_transformations:89", "title": "", "text": "$\\begin{bmatrix}t^{\\prime}\\\\ -vt^{\\prime}\\end{bmatrix}=\\begin{bmatrix}\\gamma&\\delta\\\\ \\beta&\\alpha\\end{bmatrix}\\begin{bmatrix}t\\\\ 0\\end{bmatrix},$"}
{"_id": "Clairaut's_equation:0", "title": "", "text": "$y(x)=x\\frac{dy}{dx}+f\\left(\\frac{dy}{dx}\\right).$"}
{"_id": "Perceptron:2", "title": "", "text": "$f(x)=\\begin{cases}1&\\,\\text{if }w\\cdot x+b>0\\\\ 0&\\,\\text{otherwise}\\end{cases}$"}
{"_id": "Expectation_value_(quantum_mechanics):4", "title": "", "text": "$\\langle A\\rangle_{\\sigma}$"}
{"_id": "Commutative_non-associative_magmas:15", "title": "", "text": "$x\\oplus y=(x+y)/2$"}
{"_id": "Splitting_lemma:10", "title": "", "text": "$0\\rightarrow A\\stackrel{q}{\\longrightarrow}B\\stackrel{r}{\\longrightarrow}C\\rightarrow 0\\,$"}
{"_id": "Primitive_element_theorem:23", "title": "", "text": "$\\gamma=\\alpha+c\\beta$"}
{"_id": "Glossary_of_Lie_algebras:9", "title": "", "text": "$\\forall x,y\\in A$"}
{"_id": "Combination:20", "title": "", "text": "${\\left({{n}\\atop{k}}\\right)}=\\frac{n!}{k!(n-k)!},$"}
{"_id": "Thorium_fuel_cycle:1", "title": "", "text": "$\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow{}_{\\ 90}^{233}\\mathrm{Th}\\xrightarrow{\\beta^{-}}{}_{\\ 91}^{233}\\mathrm{Pa}\\xrightarrow{\\beta^{-}}{}_{\\ 92}^{233}\\mathrm{U}+\\mathrm{n}\\rightarrow{}_{\\ 92}^{232}\\mathrm{U}+2\\mathrm{n}$"}
{"_id": "Hopf_invariant:61", "title": "", "text": "$V^{\\infty}\\wedge V^{\\infty}\\wedge Y\\wedge Y$"}
{"_id": "Golden_ratio:63", "title": "", "text": "$\\lfloor n/2-1\\rfloor=m$"}
{"_id": "Boomerang_attack:3", "title": "", "text": "$P^{\\prime}=P\\oplus\\Delta$"}
{"_id": "Geometric_algebra:30", "title": "", "text": "$\\langle A\\rangle_{r}$"}
{"_id": "Inverse_element:45", "title": "", "text": "$x=(A^{T}A)^{-1}A^{T}b.$"}
{"_id": "Multivariate_kernel_density_estimation:25", "title": "", "text": "$\\hat{{H}}_{\\operatorname{PI}}=\\begin{bmatrix}0.052&0.510\\\\ 0.510&8.882\\end{bmatrix}.$"}
{"_id": "Skewness:7", "title": "", "text": "$b_{1}=\\frac{m_{3}}{s^{3}}=\\frac{\\tfrac{1}{n}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{3}}{\\left[\\tfrac{1}{n-1}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}\\right]^{3/2}}\\ ,$"}
{"_id": "Let_expression:22", "title": "", "text": "$\\iff x=y\\and(L\\ z)[x:=y]$"}
{"_id": "Fraction_(mathematics):136", "title": "", "text": "$\\frac{1+\\tfrac{1}{x}}{1-\\tfrac{1}{x}}$"}
{"_id": "Incidence_algebra:14", "title": "", "text": "$\\mu(\\sigma,\\tau)=(-1)^{n-r}(2!)^{r_{3}}(3!)^{r_{4}}\\cdots((n-1)!)^{r_{n}}$"}
{"_id": "Proof_that_22::7_exceeds_π:48", "title": "", "text": "$\\frac{1}{4}\\int_{0}^{1}\\frac{x^{8}(1-x)^{8}}{1+x^{2}}\\,dx=\\pi-\\frac{47\\,171}{15\\,015}$"}
{"_id": "Causal_filter:9", "title": "", "text": "$f(t)=\\int_{0}^{\\infty}h(\\tau)s(t-\\tau)\\,d\\tau$"}
{"_id": "Q-analog:7", "title": "", "text": "$e_{q}^{x}=\\sum_{n=0}^{\\infty}\\frac{x^{n}}{[n]_{q}!}.$"}
{"_id": "Einstein_relation_(kinetic_theory):20", "title": "", "text": "$J_{\\mathrm{diffusion}}(x)=-D\\frac{d\\rho}{dx}$"}
{"_id": "Square_root_of_2:1", "title": "", "text": "$1+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\ddots}}}}$"}
{"_id": "Gerolamo_Cardano:0", "title": "", "text": "$ax^{3}+bx+c=0$"}
{"_id": "Binomial_proportion_confidence_interval:23", "title": "", "text": "$\\frac{\\hat{p}+\\frac{1}{2n}z^{2}}{1+\\frac{1}{n}z^{2}}$"}
{"_id": "Wright_Omega_function:7", "title": "", "text": "$\\omega(z)=\\sum_{n=0}^{+\\infty}\\frac{q_{n}(\\omega_{a})}{(1+\\omega_{a})^{2n-1}}\\frac{(z-a)^{n}}{n!}$"}
{"_id": "Greeks_(finance):29", "title": "", "text": "$\\frac{\\partial^{2}V}{\\partial S\\partial\\sigma}$"}
{"_id": "Ramanujan's_master_theorem:10", "title": "", "text": "$\\zeta(s,a)=\\sum_{n=0}^{\\infty}\\frac{1}{(n+a)^{s}}\\!$"}
{"_id": "Student's_t-test:16", "title": "", "text": "$SE_{\\widehat{\\beta}}=\\frac{\\sqrt{\\frac{1}{n-2}\\sum_{i=1}^{n}(y_{i}-\\widehat{y}_{i})^{2}}}{\\sqrt{\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}}}$"}
{"_id": "Simple_linear_regression:107", "title": "", "text": "$\\displaystyle\\frac{\\partial\\,\\mathrm{SSE}\\left(\\hat{\\alpha},\\hat{\\beta}\\right)}{\\partial\\hat{\\beta}}=-2\\sum_{i=1}^{n}\\left[\\left(y_{i}-\\bar{y}\\right)-\\hat{\\beta}\\left(x_{i}-\\bar{x}\\right)\\right]\\left(x_{i}-\\bar{x}\\right)=0$"}
{"_id": "String_searching_algorithm:2", "title": "", "text": "$O(m)$"}
{"_id": "Ridge_detection:95", "title": "", "text": "$\\frac{\\partial^{2}f}{\\partial\\sigma^{2}}<0$"}
{"_id": "Transfer_principle:35", "title": "", "text": "$\\forall n\\in\\mathbb{N}\\ \\exists A\\subseteq\\mathbb{N}\\ \\forall x\\in\\mathbb{N}\\ [x\\in A\\,\\text{ iff }x\\leq n].$"}
{"_id": "Beam_propagation_method:4", "title": "", "text": "$\\frac{\\partial^{2}(A(x,y))}{\\partial y^{2}}=0$"}
{"_id": "Hysteresis:1", "title": "", "text": "$Y(t)=\\chi\\text{i}X(t)+\\int_{0}^{\\infty}\\Phi\\text{d}(\\tau)X(t-\\tau)\\,\\mathrm{d}\\tau,$"}
{"_id": "Salinon:1", "title": "", "text": "$AB=\\frac{1}{2}\\pi\\left(2x+y\\right)^{2}$"}
{"_id": "Interval_arithmetic:157", "title": "", "text": "$\\left([-1,1]+\\frac{1}{2}\\right)^{2}-\\frac{1}{4}=\\left[-\\frac{1}{2},\\frac{3}{2}\\right]^{2}-\\frac{1}{4}=\\left[0,\\frac{9}{4}\\right]-\\frac{1}{4}=\\left[-\\frac{1}{4},2\\right]$"}
{"_id": "Pitot_tube:1", "title": "", "text": "$u=\\sqrt{\\frac{2(p_{t}-p_{s})}{\\rho}}$"}
{"_id": "Chebyshev_polynomials:111", "title": "", "text": "$T_{j}(x)U_{k}(x)=\\begin{cases}\\tfrac{1}{2}\\left(U_{j+k}(x)+U_{k-j}(x)\\right),&\\,\\text{if }k\\geq j-1.\\\\ \\tfrac{1}{2}\\left(U_{j+k}(x)-U_{j-k-2}(x)\\right),&\\,\\text{if }k\\leq j-2.\\end{cases}$"}
{"_id": "List_of_zeta_functions:0", "title": "", "text": "$\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^{s}}.$"}
{"_id": "Algebraic_function:14", "title": "", "text": "$q(x)y-p(x)=0.$"}
{"_id": "Tilting_theory:8", "title": "", "text": "$0\\to U\\to M\\to V\\to 0$"}
{"_id": "Lambert_W_function:61", "title": "", "text": "$=2\\sqrt{2}\\int_{0}^{\\infty}w^{\\frac{1}{2}}e^{-w}\\mathrm{d}w+\\sqrt{2}\\int_{0}^{\\infty}w^{-\\frac{1}{2}}e^{-w}\\mathrm{d}w$"}
{"_id": "Hurwitz_zeta_function:0", "title": "", "text": "$\\zeta(s,q)=\\sum_{n=0}^{\\infty}\\frac{1}{(q+n)^{s}}.$"}
{"_id": "Ordered_probit:3", "title": "", "text": "$y=\\begin{cases}0~{}~{}\\,\\text{if}~{}~{}y^{*}\\leq 0,\\\\ 1~{}~{}\\,\\text{if}~{}~{}0<y^{*}\\leq\\mu_{1},\\\\ 2~{}~{}\\,\\text{if}~{}~{}\\mu_{1}<y^{*}\\leq\\mu_{2}\\\\ \\vdots\\\\ N~{}~{}\\,\\text{if}~{}~{}\\mu_{N-1}<y^{*}.\\end{cases}$"}
{"_id": "Moore–Penrose_pseudoinverse:184", "title": "", "text": "$A_{0}A=(A_{0}A)^{*}$"}
{"_id": "Unscrupulous_diner's_dilemma:2", "title": "", "text": "$\\frac{1}{n}x+\\frac{1}{n}l$"}
{"_id": "Gauss's_continued_fraction:67", "title": "", "text": "$e=2+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{4+{}\\ddots}}}}}$"}
{"_id": "Schuette–Nesbitt_formula:33", "title": "", "text": "$\\begin{aligned}\\displaystyle(*)&\\displaystyle=\\sum_{k=l}^{n}(-1)^{k-l}\\frac{n!}{k!\\,(n-k)!}\\,\\frac{k!}{l!\\,(k-l)!}\\\\ &\\displaystyle=\\underbrace{\\frac{n!}{l!\\,(n-l)!}}_{={\\left({{n}\\atop{l}}\\right)}}\\underbrace{\\sum_{k=l}^{n}(-1)^{k-l}\\frac{(n-l)!}{(n-k)!\\,(k-l)!}}_{=:\\,(**)}\\\\ \\end{aligned}$"}
{"_id": "Extended_periodic_table:0", "title": "", "text": "$\\,{}^{254}_{99}\\mathrm{Es}+\\,^{48}_{20}\\mathrm{Ca}\\to\\,^{302}_{119}\\mathrm{Uue}^{*}$"}
{"_id": "Separation_of_variables:4", "title": "", "text": "$\\frac{1}{h(y)}\\frac{dy}{dx}=g(x),$"}
{"_id": "Gravitational_instanton:1", "title": "", "text": "$\\sigma_{2}=\\cos\\psi\\,d\\theta+\\sin\\psi\\sin\\theta\\,d\\phi$"}
{"_id": "Matrix_calculus:298", "title": "", "text": "$\\frac{\\partial^{2}\\mathbf{U}^{-1}}{\\partial x\\partial y}=$"}
{"_id": "Contour_set:17", "title": "", "text": "$\\left\\{y~{}\\backepsilon~{}(y\\succcurlyeq x)\\land\\lnot(x\\succcurlyeq y)\\right\\}$"}
{"_id": "Dual_matroid:21", "title": "", "text": "$M/X=(M^{\\ast}\\setminus x)^{\\ast}$"}
{"_id": "Photosynthetically_active_radiation:2", "title": "", "text": "$\\eta_{\\mathrm{PAR}}(T)=\\frac{\\int_{\\lambda_{1}}^{\\lambda_{2}}B(\\lambda,T)\\,d\\lambda}{\\int_{0}^{\\infty}B(\\lambda,T)\\,d\\lambda},$"}
{"_id": "Homology_(mathematics):47", "title": "", "text": "$0\\rightarrow A\\rightarrow B\\rightarrow C\\rightarrow 0$"}
{"_id": "Hypoexponential_distribution:15", "title": "", "text": "$\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}$"}
{"_id": "Stern–Volmer_relationship:1", "title": "", "text": "$\\mathrm{A}^{*}+\\mathrm{Q}\\rightarrow\\mathrm{A}+\\mathrm{Q}^{*}$"}
{"_id": "Luminous_efficacy:0", "title": "", "text": "$\\frac{\\int^{\\infty}_{0}y_{\\lambda}J_{\\lambda}d\\lambda}{\\int^{\\infty}_{0}J_{\\lambda}d\\lambda},$"}
{"_id": "Error_function:71", "title": "", "text": "$E_{n}(x)=\\frac{n!}{\\sqrt{\\pi}}\\int_{0}^{x}e^{-t^{n}}\\,\\mathrm{d}t=\\frac{n!}{\\sqrt{\\pi}}\\sum_{p=0}^{\\infty}(-1)^{p}\\frac{x^{np+1}}{(np+1)p!}\\,.$"}
{"_id": "Discrete_Morse_theory:63", "title": "", "text": "$A\\stackrel{\\rho}{\\to}A^{\\prime}$"}
{"_id": "Hotelling's_T-squared_distribution:33", "title": "", "text": "${\\mathbf{W}}=\\frac{\\sum_{i=1}^{n_{x}}(\\mathbf{x}_{i}-\\overline{\\mathbf{x}})(\\mathbf{x}_{i}-\\overline{\\mathbf{x}})^{\\prime}+\\sum_{i=1}^{n_{y}}(\\mathbf{y}_{i}-\\overline{\\mathbf{y}})(\\mathbf{y}_{i}-\\overline{\\mathbf{y}})^{\\prime}}{n_{x}+n_{y}-2}$"}
{"_id": "Cutler's_bar_notation:1", "title": "", "text": "$\\begin{matrix}&\\underbrace{a^{a^{{}^{.\\,^{.\\,^{.\\,^{a}}}}}}}&\\\\ &b\\mbox{ copies of }~{}a\\end{matrix}$"}
{"_id": "1_−_2_+_3_−_4_+_⋯:24", "title": "", "text": "$\\sum_{k=0}^{\\infty}a_{k}=\\sum_{k=0}^{\\infty}(-1)^{k}(k+1),$"}
{"_id": "Ellipse:58", "title": "", "text": "$B^{2}-4AC<0.$"}
{"_id": "Extraneous_and_missing_solutions:5", "title": "", "text": "$x^{2}+4x+4=0$"}
{"_id": "Complexification_(Lie_group):300", "title": "", "text": "$\\displaystyle{\\mathfrak{g}=\\mathfrak{k}\\oplus\\mathfrak{p}}$"}
{"_id": "Final_functor:3", "title": "", "text": "$\\{*\\}\\xrightarrow{d}D$"}
{"_id": "Algebra:10", "title": "", "text": "$ax^{2}+bx+c=0,$"}
{"_id": "Generalized_hypergeometric_function:4", "title": "", "text": "$A(n)=1$"}
{"_id": "Biot–Tolstoy–Medwin_diffraction_model:0", "title": "", "text": "$p(t)=\\int_{0}^{\\infty}h(\\tau)q(t-\\tau)\\,d\\tau$"}
{"_id": "Axiom_schema_of_replacement:1", "title": "", "text": "$\\forall w_{1},\\ldots,w_{n}\\,[(\\forall x\\,\\exists\\,y\\phi(x,y,w_{1},\\ldots,w_{n}))\\Rightarrow\\forall A\\,\\exists B\\,\\forall x\\in A\\,\\exists y\\in B\\,\\phi(x,y,w_{1},\\ldots,w_{n})]$"}
{"_id": "Kostant's_convexity_theorem:23", "title": "", "text": "$\\displaystyle{M\\rightarrow\\mathfrak{k}^{*}\\rightarrow\\mathfrak{t}^{*}.}$"}
{"_id": "Coiflet:1", "title": "", "text": "$N/3-1$"}
{"_id": "Cake_number:0", "title": "", "text": "${n\\choose k}=\\frac{n!}{k!\\,(n-k)!},$"}
{"_id": "Quantum_finite_automata:59", "title": "", "text": "$P=\\begin{bmatrix}1&0\\\\ 0&0\\end{bmatrix}$"}
{"_id": "Conditional_probability_distribution:6", "title": "", "text": "$f_{Y}(y\\mid X=x)f_{X}(x)=f_{X,Y}(x,y)=f_{X}(x\\mid Y=y)f_{Y}(y).$"}
{"_id": "Alpha_helix:0", "title": "", "text": "$i+4\\rightarrow i$"}
{"_id": "Coherent_states:91", "title": "", "text": "$\\langle n\\rangle_{\\,\\text{th}}$"}
{"_id": "Apollonian_circles:6", "title": "", "text": "$\\displaystyle\\alpha(x^{2}+y^{2})-2\\beta x-2\\gamma y+\\delta=0,$"}
{"_id": "Product_(chemistry):2", "title": "", "text": "$S+E\\rightarrow P+E$"}
{"_id": "Parabolic_partial_differential_equation:1", "title": "", "text": "$B^{2}-AC=0.$"}
{"_id": "Closed-form_expression:3", "title": "", "text": "$f(x)=\\sum_{i=0}^{\\infty}{x\\over 2^{i}}$"}
{"_id": "Rate_of_return:9", "title": "", "text": "$1+R=(1+r)^{t}$"}
{"_id": "1::2_−_1::4_+_1::8_−_1::16_+_⋯:1", "title": "", "text": "$1-\\frac{1}{2}-\\frac{1}{4}+\\frac{1}{8}-\\frac{1}{16}+\\cdots=\\frac{1}{3}.$"}
{"_id": "Electromagnetic_radiation:7", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\epsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t}\\qquad\\quad\\ (4)$"}
{"_id": "Geometric_algebra:0", "title": "", "text": "$A(BC)=(AB)C$"}
{"_id": "Floor_and_ceiling_functions:84", "title": "", "text": "$\\gamma=\\sum_{k=2}^{\\infty}(-1)^{k}\\frac{\\left\\lfloor\\log_{2}k\\right\\rfloor}{k}=\\tfrac{1}{2}-\\tfrac{1}{3}+2\\left(\\tfrac{1}{4}-\\tfrac{1}{5}+\\tfrac{1}{6}-\\tfrac{1}{7}\\right)+3\\left(\\tfrac{1}{8}-\\dots-\\tfrac{1}{15}\\right)+\\dots$"}
{"_id": "Biot–Savart_law:35", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}$"}
{"_id": "Semi-invariant_of_a_quiver:9", "title": "", "text": "$1\\xrightarrow{\\ \\ \\alpha\\ }2$"}
{"_id": "Artificial_neuron:3", "title": "", "text": "$y=\\begin{cases}1&\\,\\text{if }u\\geq\\theta\\\\ 0&\\,\\text{if }u<\\theta\\end{cases}$"}
{"_id": "Neptunium:0", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{239}_{\\ 92}U\\ \\xrightarrow[23\\ min]{\\beta^{-}}\\ ^{239}_{\\ 93}Np\\ \\xrightarrow[2.355\\ days]{\\beta^{-}}\\ ^{239}_{\\ 94}Pu}$"}
{"_id": "Transformation_matrix:30", "title": "", "text": "$\\begin{bmatrix}x^{\\prime}\\\\ y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1&k\\\\ 0&1\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\end{bmatrix}$"}
{"_id": "Centering_matrix:6", "title": "", "text": "$C_{1}=\\begin{bmatrix}0\\end{bmatrix}$"}
{"_id": "Josephus_problem:83", "title": "", "text": "$O(k\\log n)$"}
{"_id": "Light-front_quantization_applications:75", "title": "", "text": "$\\pi^{+}\\to W^{*}\\to\\mu^{+}\\nu_{\\mu}$"}
{"_id": "Albertson_conjecture:0", "title": "", "text": "$\\textrm{cr}(K_{n})=\\frac{1}{4}\\left\\lfloor\\frac{n}{2}\\right\\rfloor\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor\\left\\lfloor\\frac{n-2}{2}\\right\\rfloor\\left\\lfloor\\frac{n-3}{2}\\right\\rfloor.$"}
{"_id": "Probabilistic_automaton:18", "title": "", "text": "$q\\stackrel{a}{\\rightarrow}q^{\\prime}$"}
{"_id": "Legendre_chi_function:0", "title": "", "text": "$\\chi_{\\nu}(z)=\\sum_{k=0}^{\\infty}\\frac{z^{2k+1}}{(2k+1)^{\\nu}}.$"}
{"_id": "Static_forces_and_virtual-particle_exchange:147", "title": "", "text": "${\\mathit{l}\\over\\mathit{l}^{*}}={1\\over 2}$"}
{"_id": "Spherical_harmonics:111", "title": "", "text": "$\\cos(\\theta^{\\prime}-\\theta)=\\cos\\theta^{\\prime}\\cos\\theta+\\sin\\theta\\sin\\theta^{\\prime}$"}
{"_id": "Fresnel_zone:0", "title": "", "text": "$F_{n}=\\sqrt{\\frac{n\\lambda d_{1}d_{2}}{d_{1}+d_{2}}}$"}
{"_id": "Abelian_category:0", "title": "", "text": "$0\\to M^{\\prime}\\to M\\to M^{\\prime\\prime}\\to 0$"}
{"_id": "Cylinder_(geometry):41", "title": "", "text": "$A^{2}+B^{2}=R^{2}\\,$"}
{"_id": "Ackermann_set_theory:23", "title": "", "text": "$x\\in V\\land\\exists y(y\\in x)\\rightarrow\\exists y(y\\in x\\land\\lnot\\exists z(z\\in y\\land z\\in x)).$"}
{"_id": "Table_of_thermodynamic_equations:135", "title": "", "text": "$C_{p}=\\left(\\frac{\\partial H}{\\partial T}\\right)_{p}$"}
{"_id": "Force-free_magnetic_field:6", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{j}$"}
{"_id": "Brownian_model_of_financial_markets:37", "title": "", "text": "$A(\\cdot)=0$"}
{"_id": "Esscher_transform:3", "title": "", "text": "$\\,\\frac{e^{hk}p^{k}(1-p)^{n-k}}{1-p+pe^{h}}$"}
{"_id": "Skorokhod's_representation_theorem:1", "title": "", "text": "$x\\xrightarrow{d}\\ \\mu$"}
{"_id": "Loop_quantum_gravity:218", "title": "", "text": "$J=\\Big|{\\partial x^{a}\\over\\partial x^{{}^{\\prime}b}}\\Big|$"}
{"_id": "Brownian_motion:27", "title": "", "text": "$P_{N}(N_{R})=\\frac{N!}{2^{N}N_{R}!(N-N_{R})!}$"}
{"_id": "AB_magnitude:7", "title": "", "text": "$\\lambda\\text{piv}=\\sqrt{\\frac{\\int e(\\lambda)\\lambda d\\lambda}{\\int e(\\lambda)\\lambda^{-1}d\\lambda}}$"}
{"_id": "Least_mean_squares_filter:97", "title": "", "text": "$\\mathbf{\\delta}=\\hat{\\mathbf{h}}(n)-\\mathbf{h}(n)$"}
{"_id": "Epsilon-induction:1", "title": "", "text": "$\\forall x\\Big(\\forall y(y\\in x\\rightarrow P[y])\\rightarrow P[x]\\Big)\\rightarrow\\forall x\\,P[x]$"}
{"_id": "Ptolemy's_theorem:23", "title": "", "text": "$\\sin(\\alpha+\\beta)\\sin(\\beta+\\gamma)=\\sin\\alpha\\sin\\gamma+\\sin\\beta\\sin(\\alpha+\\beta+\\gamma)$"}
{"_id": "Periodic_points_of_complex_quadratic_mappings:38", "title": "", "text": "$\\ Ax^{2}+Bx+C=0$"}
{"_id": "Actinide:0", "title": "", "text": "$\\mathrm{{}^{238}_{92}U+{}^{1}_{0}n\\ \\xrightarrow{\\ }\\ {}^{239}_{92}U\\ \\xrightarrow[23.5\\ min]{\\beta^{-}}\\ {}^{239}_{93}Np\\ \\xrightarrow[2.3\\ days]{\\beta^{-}}\\ {}^{239}_{94}Pu\\ \\xrightarrow[2.4\\cdot 10^{4}\\ years]{\\alpha}}$"}
{"_id": "Variety_(universal_algebra):6", "title": "", "text": "$U\\colon A\\to{Set}$"}
{"_id": "Notation_for_differentiation:9", "title": "", "text": "$\\frac{d\\Bigl(\\frac{d\\left(\\frac{dy}{dx}\\right)}{dx}\\Bigr)}{dx}=\\left(\\frac{d}{dx}\\right)^{3}\\bigl(f(x)\\bigr)$"}
{"_id": "Hessian_matrix:5", "title": "", "text": "$H=\\begin{bmatrix}\\dfrac{\\partial^{2}f}{\\partial x_{1}^{2}}&\\dfrac{\\partial^{2}f}{\\partial x_{1}\\,\\partial x_{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{1}\\,\\partial x_{n}}\\\\ \\dfrac{\\partial^{2}f}{\\partial x_{2}\\,\\partial x_{1}}&\\dfrac{\\partial^{2}f}{\\partial x_{2}^{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{2}\\,\\partial x_{n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\dfrac{\\partial^{2}f}{\\partial x_{n}\\,\\partial x_{1}}&\\dfrac{\\partial^{2}f}{\\partial x_{n}\\,\\partial x_{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{n}^{2}}\\end{bmatrix}.$"}
{"_id": "Point_set_triangulation:2", "title": "", "text": "$2n-h-2$"}
{"_id": "Special_values_of_L-functions:0", "title": "", "text": "$1\\,-\\,\\frac{1}{3}\\,+\\,\\frac{1}{5}\\,-\\,\\frac{1}{7}\\,+\\,\\frac{1}{9}\\,-\\,\\cdots\\;=\\;\\frac{\\pi}{4},\\!$"}
{"_id": "Henderson–Hasselbalch_equation:0", "title": "", "text": "$\\mathrm{pH}=\\mathrm{p}K_{\\mathrm{a}}+\\log_{10}\\left(\\frac{[\\mathrm{A}^{-}]}{[\\mathrm{HA}]}\\right)$"}
{"_id": "Minakshisundaram–Pleijel_zeta_function:3", "title": "", "text": "$Z(P,Q,s)=\\sum_{n=1}^{\\infty}\\frac{f_{n}(P)f_{n}(Q)}{\\lambda_{n}^{s}}$"}
{"_id": "Littlewood–Paley_theory:4", "title": "", "text": "$g(f)(x)=(\\int_{0}^{\\infty}|\\nabla u(x,y)|^{2}ydy)^{\\frac{1}{2}}$"}
{"_id": "Operator_theory:1", "title": "", "text": "$A^{*}A=(A^{*}A)^{\\frac{1}{2}}(A^{*}A)^{\\frac{1}{2}},$"}
{"_id": "Mobility_analogy:11", "title": "", "text": "$\\begin{bmatrix}i\\\\ u\\end{bmatrix}=\\begin{bmatrix}y_{11}&y_{12}\\\\ y_{21}&y_{22}\\end{bmatrix}\\begin{bmatrix}v\\\\ F\\end{bmatrix}$"}
{"_id": "Monotone_likelihood_ratio:6", "title": "", "text": "$\\frac{\\partial}{\\partial x}\\left(\\frac{f(x)}{g(x)}\\right)\\geq 0$"}
{"_id": "Shannon_wavelet:1", "title": "", "text": "$\\prod(x):=\\begin{cases}1,&\\mbox{if }~{}{|x|\\leq 1/2},\\\\ 0&\\mbox{if }~{}\\mbox{otherwise}~{}.\\\\ \\end{cases}$"}
{"_id": "Abel–Ruffini_theorem:2", "title": "", "text": "$\\textstyle{ax^{2}+bx+c=0,a\\neq 0}$"}
{"_id": "Abel_transform:4", "title": "", "text": "$\\int_{-\\infty}^{\\infty}f(r)\\,dx=2\\int_{0}^{\\infty}f(r)\\,dx,\\;\\;r=\\sqrt{x^{2}+y^{2}}.$"}
{"_id": "Opacity_(optics):12", "title": "", "text": "$\\frac{1}{\\kappa}=\\frac{\\int_{0}^{\\infty}\\kappa_{\\nu}^{-1}u(\\nu,T)d\\nu}{\\int_{0}^{\\infty}u(\\nu,T)d\\nu}$"}
{"_id": "Thomas_precession:87", "title": "", "text": "$B(\\mathbf{u})B(−\\mathbf{u})=1$"}
{"_id": "Silver_ratio:0", "title": "", "text": "$2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{\\ddots}}}}$"}
{"_id": "Sliding_mode_control:204", "title": "", "text": "$\\begin{cases}\\dot{\\mathbf{x}}=A\\mathbf{x}+B\\mathbf{u}\\\\ y=\\begin{bmatrix}1&0&0&\\cdots&\\end{bmatrix}\\mathbf{x}=x_{1}\\end{cases}$"}
{"_id": "ACE_Encrypt:18", "title": "", "text": "$x,y\\in A^{\\ast}$"}
{"_id": "Bell_polynomials:15", "title": "", "text": "$\\displaystyle\\underbrace{x\\diamondsuit\\cdots\\diamondsuit x}_{k\\ \\mathrm{factors}}.\\,$"}
{"_id": "State-space_representation:85", "title": "", "text": "$\\left(A+BK\\left(I-DK\\right)^{-1}C\\right)$"}
{"_id": "Triangle_center:0", "title": "", "text": "$\\begin{cases}\\\\ \\\\ \\end{cases}$"}
{"_id": "Leibniz's_notation:62", "title": "", "text": "$\\frac{d^{2}y}{dx^{2}}\\,=\\,\\frac{d}{dx}\\left(\\frac{dy}{dx}\\right).$"}
{"_id": "Omnibus_test:3", "title": "", "text": "$F=\\frac{{\\displaystyle\\sum_{i=1}^{n}\\left(\\widehat{y_{i}}-\\bar{y}\\right)^{2}}/{k}}{{\\displaystyle{\\sum_{j=1}^{k}}{\\sum_{i=1}^{n_{j}}}\\left(y_{ij}-\\widehat{y_{i}}\\right)^{2}}/{(n-k-1)}}$"}
{"_id": "Leibniz's_notation:63", "title": "", "text": "$\\frac{d\\left(\\frac{d\\left(\\frac{d\\left(f(x)\\right)}{dx}\\right)}{dx}\\right)}{dx}\\,,$"}
{"_id": "Binary_search_algorithm:0", "title": "", "text": "$\\lfloor\\log_{2}N\\rfloor$"}
{"_id": "Logan_plot:11", "title": "", "text": "$\\int_{0}^{t}\\mathrm{ROI}(\\tau)\\,d\\tau=(-\\mathbf{U}_{n}^{T}\\mathbf{K}^{-1}\\mathbf{Q}+V_{p})\\int_{0}^{t}C_{p}(\\tau)\\,d\\tau+\\mathbf{U}_{n}^{T}\\mathbf{K}^{-1}\\mathbf{A}$"}
{"_id": "Replica_trick:27", "title": "", "text": "$\\exp y=1+y+{1\\over 2!}y^{2}+{1\\over 3!}y^{3}+\\dots=\\lim_{N\\to\\infty}\\sum_{r=0}^{N}{N!\\over r!(N-r)!}({y\\over N})^{r}=\\lim_{N\\to\\infty}(1+{y\\over N})^{N}.$"}
{"_id": "Implementation_of_mathematics_in_set_theory:116", "title": "", "text": "$\\forall x,y,z\\,\\left(xFy\\wedge xFz\\to y=z\\right)$"}
{"_id": "Qualitative_variation:33", "title": "", "text": "$I_{B}=\\frac{\\log(N!)-\\sum_{i=1}^{K}(\\log(n_{i}!))}{N}$"}
{"_id": "Geometry_index:2", "title": "", "text": "$\\tau_{5}=\\frac{\\beta-\\alpha}{60^{\\circ}}\\approx-0.01667\\alpha+0.01667\\beta$"}
{"_id": "Space-time_block_coding_based_transmit_diversity:3", "title": "", "text": "$\\begin{bmatrix}x_{1}\\\\ x_{2}^{*}\\end{bmatrix}=\\begin{bmatrix}h_{1}&-h_{2}\\\\ h_{2}^{*}&h_{1}^{*}\\end{bmatrix}\\begin{bmatrix}S_{1}\\\\ S_{2}^{*}\\end{bmatrix}+\\begin{bmatrix}n_{1}\\\\ n_{2}^{*}\\end{bmatrix}$"}
{"_id": "Computationally_bounded_adversary:12", "title": "", "text": "$Y^{\\prime}=Y\\oplus N$"}
{"_id": "Lattice_problem:44", "title": "", "text": "$\\epsilon=(\\log\\log n)^{c}$"}
{"_id": "Airy_wave_theory:103", "title": "", "text": "$\\begin{aligned}\\displaystyle\\mathbb{S}&\\displaystyle=\\,\\begin{pmatrix}S_{xx}&S_{xy}\\\\ S_{yx}&S_{yy}\\end{pmatrix}\\,=\\,\\mathbb{I}\\,\\left(\\frac{c_{g}}{c_{p}}-\\frac{1}{2}\\right)\\,E\\,+\\,\\frac{1}{k^{2}}\\,\\begin{pmatrix}k_{x}\\,k_{x}&k_{x}\\,k_{y}\\\\ k_{y}\\,k_{x}&k_{y}\\,k_{y}\\end{pmatrix}\\,\\frac{c_{g}}{c_{p}}\\,E,\\\\ \\displaystyle\\mathbb{I}&\\displaystyle=\\,\\begin{pmatrix}1&0\\\\ 0&1\\end{pmatrix}\\quad\\,\\text{and}\\\\ \\displaystyle\\nabla symbol{U}&\\displaystyle=\\,\\begin{pmatrix}\\displaystyle\\frac{\\partial U_{x}}{\\partial x}&\\displaystyle\\frac{\\partial U_{y}}{\\partial x}\\\\ \\displaystyle\\frac{\\partial U_{x}}{\\partial y}&\\displaystyle\\frac{\\partial U_{y}}{\\partial y}\\end{pmatrix},\\end{aligned}$"}
{"_id": "Convolution:2", "title": "", "text": "$=\\int_{-\\infty}^{\\infty}f(t-\\tau)\\,g(\\tau)\\,d\\tau.$"}
{"_id": "Two-port_network:84", "title": "", "text": "$\\gamma=\\alpha+i\\beta$"}
{"_id": "Goertzel_algorithm:61", "title": "", "text": "$O(KNM)$"}
{"_id": "Infinitesimal_strain_theory:13", "title": "", "text": "$\\left[\\begin{matrix}\\varepsilon_{xx}&\\varepsilon_{xy}&\\varepsilon_{xz}\\\\ \\varepsilon_{yx}&\\varepsilon_{yy}&\\varepsilon_{yz}\\\\ \\varepsilon_{zx}&\\varepsilon_{zy}&\\varepsilon_{zz}\\\\ \\end{matrix}\\right]=\\left[\\begin{matrix}\\frac{\\partial u_{x}}{\\partial x}&\\frac{1}{2}\\left(\\frac{\\partial u_{x}}{\\partial y}+\\frac{\\partial u_{y}}{\\partial x}\\right)&\\frac{1}{2}\\left(\\frac{\\partial u_{x}}{\\partial z}+\\frac{\\partial u_{z}}{\\partial x}\\right)\\\\ \\frac{1}{2}\\left(\\frac{\\partial u_{y}}{\\partial x}+\\frac{\\partial u_{x}}{\\partial y}\\right)&\\frac{\\partial u_{y}}{\\partial y}&\\frac{1}{2}\\left(\\frac{\\partial u_{y}}{\\partial z}+\\frac{\\partial u_{z}}{\\partial y}\\right)\\\\ \\frac{1}{2}\\left(\\frac{\\partial u_{z}}{\\partial x}+\\frac{\\partial u_{x}}{\\partial z}\\right)&\\frac{1}{2}\\left(\\frac{\\partial u_{z}}{\\partial y}+\\frac{\\partial u_{y}}{\\partial z}\\right)&\\frac{\\partial u_{z}}{\\partial z}\\\\ \\end{matrix}\\right]\\,\\!$"}
{"_id": "Copula_(probability_theory):99", "title": "", "text": "$-\\log(t)\\,$"}
{"_id": "Egyptian_Mathematical_Leather_Roll:28", "title": "", "text": "$\\frac{1}{25}+\\frac{1}{15}+\\frac{1}{75}+\\frac{1}{200}=\\frac{1}{8}$"}
{"_id": "Skew_and_direct_sums_of_permutations:15", "title": "", "text": "$\\pi\\oplus\\sigma=\\operatorname{rev}(\\operatorname{rev}(\\sigma)\\ominus\\operatorname{rev}(\\pi))$"}
{"_id": "Tribimaximal_mixing:0", "title": "", "text": "$\\begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\\\ |U_{\\mu 1}|^{2}&|U_{\\mu 2}|^{2}&|U_{\\mu 3}|^{2}\\\\ |U_{\\tau 1}|^{2}&|U_{\\tau 2}|^{2}&|U_{\\tau 3}|^{2}\\end{bmatrix}=\\begin{bmatrix}\\frac{2}{3}&\\frac{1}{3}&0\\\\ \\frac{1}{6}&\\frac{1}{3}&\\frac{1}{2}\\\\ \\frac{1}{6}&\\frac{1}{3}&\\frac{1}{2}\\end{bmatrix}.$"}
{"_id": "Scapegoat_tree:17", "title": "", "text": "$O(\\log{n})$"}
{"_id": "Jordan_normal_form:15", "title": "", "text": "$\\;(A-4I)^{2}p_{4}=0.$"}
{"_id": "Heaviside_step_function:2", "title": "", "text": "$H[n]=\\begin{cases}0,&n<0,\\\\ 1/2,&n=0,\\\\ 1,&n>0,\\end{cases}$"}
{"_id": "Sparse_distributed_memory:11", "title": "", "text": "$\\mathbf{a}_{i}=\\begin{cases}1&\\,\\text{if }w_{i}>0,\\\\ 0&\\,\\text{if }w_{i}<0.\\end{cases}$"}
{"_id": "Fourier_series:48", "title": "", "text": "$\\begin{bmatrix}\\dfrac{\\partial x_{1}}{\\partial x}&\\dfrac{\\partial x_{1}}{\\partial y}&\\dfrac{\\partial x_{1}}{\\partial z}\\\\ \\dfrac{\\partial x_{2}}{\\partial x}&\\dfrac{\\partial x_{2}}{\\partial y}&\\dfrac{\\partial x_{2}}{\\partial z}\\\\ \\dfrac{\\partial x_{3}}{\\partial x}&\\dfrac{\\partial x_{3}}{\\partial y}&\\dfrac{\\partial x_{3}}{\\partial z}\\end{bmatrix}$"}
{"_id": "Feynman–Kac_formula:21", "title": "", "text": "$e^{-\\int_{0}^{t}V(x(\\tau))\\,d\\tau}$"}
{"_id": "Braess'_paradox:3", "title": "", "text": "$A=B=2000$"}
{"_id": "Analytic_continuation:14", "title": "", "text": "$g=\\left(1,0,1,-\\frac{1}{2},\\frac{1}{3},-\\frac{1}{4},\\frac{1}{5},-\\frac{1}{6},\\cdots\\right)$"}
{"_id": "Indian_mathematics:61", "title": "", "text": "$\\frac{\\pi}{4}=\\frac{3}{4}+\\frac{1}{3^{3}-3}-\\frac{1}{5^{3}-5}+\\frac{1}{7^{3}-7}-\\cdots$"}
{"_id": "Eigenvalues_and_eigenvectors:294", "title": "", "text": "$A=\\begin{bmatrix}4&1\\\\ 6&3\\end{bmatrix}$"}
{"_id": "List_of_logic_systems:82", "title": "", "text": "$B\\to(A\\lor B)$"}
{"_id": "Quantum_Fourier_transform:28", "title": "", "text": "$[0.x_{1}\\ldots x_{m}]=\\sum_{k=1}^{m}x_{k}2^{-k}.$"}
{"_id": "Vickrey_auction:18", "title": "", "text": "$\\Omega\\left(\\frac{1}{np}\\right)$"}
{"_id": "Berkelium:12", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ +\\ ^{10}_{\\ 5}B\\ \\longrightarrow\\ ^{242}_{\\ 97}Bk\\ +\\ 6\\ ^{1}_{0}n\\quad;\\quad^{232}_{\\ 90}Th\\ +\\ ^{15}_{\\ 7}N\\ \\longrightarrow\\ ^{242}_{\\ 97}Bk\\ +\\ 5\\ ^{1}_{0}n}$"}
{"_id": "Brenke–Chihara_polynomials:0", "title": "", "text": "$A(w)B(xw)=\\sum_{n=0}^{\\infty}P_{n}(x)w^{n}.$"}
{"_id": "AB_magnitude:6", "title": "", "text": "$\\lambda\\text{piv}=\\sqrt{\\frac{\\int e(\\lambda)d\\lambda}{\\int e(\\lambda)\\lambda^{-2}d\\lambda}}$"}
{"_id": "Draft:List_of_shape_topics_in_various_fields:16", "title": "", "text": "$\\textstyle{f(x)=\\sum_{n=0}^{\\infty}{s(2^{n}x)\\over 2^{n}}}$"}
{"_id": "Bijection,_injection_and_surjection:2", "title": "", "text": "$\\forall x,y\\in A,x\\neq y\\Rightarrow f(x)\\neq f(y).$"}
{"_id": "Thermal_fluctuations:63", "title": "", "text": "$T\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}$"}
{"_id": "11_(number):18", "title": "", "text": "$0.\\overline{72}$"}
{"_id": "Mean_absolute_difference:7", "title": "", "text": "$RMD(S)=\\frac{\\sum_{i=1}^{n}\\sum_{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\\sum_{i=1}^{n}y_{i}}$"}
{"_id": "Pythagorean_triple:45", "title": "", "text": "$A=\\begin{bmatrix}\\alpha&\\beta\\\\ \\gamma&\\delta\\end{bmatrix}$"}
{"_id": "Adequality:29", "title": "", "text": "$\\scriptstyle b-2\\,x-e\\;\\sim\\;0.$"}
{"_id": "Determination_of_equilibrium_constants:57", "title": "", "text": "$\\left(\\frac{\\partial E}{\\partial v}\\right)_{k}$"}
{"_id": "Rodion_Kuzmin:1", "title": "", "text": "$\\Delta_{n}(s)=\\mathbb{P}\\left\\{x_{n}\\leq s\\right\\}-\\log_{2}(1+s),$"}
{"_id": "Total_order:9", "title": "", "text": "$x,y\\in A_{1}$"}
{"_id": "Reflections_of_signals_on_conducting_lines:61", "title": "", "text": "$\\theta=\\begin{cases}\\pi-2\\arctan\\frac{X_{\\mathrm{L}}}{R_{\\mathrm{0}}}&\\mbox{if}~{}{X_{\\mathrm{L}}}>0\\\\ -\\pi-2\\arctan\\frac{X_{\\mathrm{L}}}{R_{\\mathrm{0}}}&\\mbox{if }~{}{X_{\\mathrm{L}}}<0\\\\ \\end{cases}$"}
{"_id": "Automorphic_factor:5", "title": "", "text": "$\\gamma=\\left[\\begin{matrix}a&b\\\\ c&d\\end{matrix}\\right]$"}
{"_id": "Einsteinium:14", "title": "", "text": "$\\,{}^{254}_{99}\\mathrm{Es}+\\,^{48}_{20}\\mathrm{Ca}\\to\\,^{302}_{119}\\mathrm{Uue}^{*}\\to\\ \\ no\\ atoms$"}
{"_id": "Differential_geometry_of_surfaces:60", "title": "", "text": "$\\cos c=\\cos a\\,\\cos b+\\sin a\\,\\sin b\\,\\cos\\gamma.$"}
{"_id": "Minimum_mean_square_error:144", "title": "", "text": "$W=(A^{T}A)^{-1}A^{T}$"}
{"_id": "Random_permutation_statistics:260", "title": "", "text": "$\\mathfrak{P}(-\\mathcal{Z}+\\mathcal{V}\\mathcal{Z}+\\mathfrak{C}_{1}(\\mathcal{Z})+\\mathcal{U}\\mathfrak{C}_{2}(\\mathcal{Z})+\\mathcal{U}^{2}\\mathfrak{C}_{3}(\\mathcal{Z})+\\mathcal{U}^{3}\\mathfrak{C}_{4}(\\mathcal{Z})+\\cdots)$"}
{"_id": "Sinc_filter:8", "title": "", "text": "$H_{HPF}(f)=1-\\mathrm{rect}\\left(\\frac{f}{2B_{H}}\\right).$"}
{"_id": "Srinivasa_Ramanujan:8", "title": "", "text": "$e^{\\pi\\sqrt{58}}=396^{4}-104.000000177\\dots.$"}
{"_id": "Quotient_space_(linear_algebra):0", "title": "", "text": "$V=U\\oplus W$"}
{"_id": "Conway–Maxwell–Poisson_distribution:3", "title": "", "text": "$Z(\\lambda,\\nu)=\\sum_{j=0}^{\\infty}\\frac{\\lambda^{j}}{(j!)^{\\nu}}.$"}
{"_id": "Whitehead's_point-free_geometry:4", "title": "", "text": "$x<y\\rightarrow\\exists z[x<z<y].$"}
{"_id": "Avalanche_photodiode:0", "title": "", "text": "$M=\\frac{1}{1-\\int_{0}^{L}\\alpha(x)\\,dx}$"}
{"_id": "Transcendental_number:9", "title": "", "text": "${1+\\cfrac{1}{2+\\cfrac{1}{3+\\cfrac{1}{4+\\cfrac{1}{5+\\cfrac{1}{6+\\ddots}}}}}}$"}
{"_id": "Rydberg_atom:3", "title": "", "text": "$e^{-}+A\\rightarrow A^{*}+e^{-}$"}
{"_id": "Mirsky's_theorem:0", "title": "", "text": "$1+\\lfloor\\log_{2}N\\rfloor$"}
{"_id": "Hermite_number:2", "title": "", "text": "$H_{2}=-2\\,$"}
{"_id": "Perron's_formula:1", "title": "", "text": "$g(s)=\\sum_{n=1}^{\\infty}\\frac{a(n)}{n^{s}}$"}
{"_id": "Draft:Leaving_Certificate_Mathematics_Ordinary_Level:21", "title": "", "text": "$\\frac{1}{2}-\\frac{1}{4}=\\frac{1}{4}$"}
{"_id": "Einstein_solid:1", "title": "", "text": "$C_{V}=\\left({\\partial U\\over\\partial T}\\right)_{V}.$"}
{"_id": "Sintering:0", "title": "", "text": "$E_{n}/E=(D/d)^{3.4}$"}
{"_id": "Beam_diameter:3", "title": "", "text": "$D4\\sigma=4\\sigma=4\\sqrt{\\frac{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y)(x-\\bar{x})^{2}\\,dx\\,dy}{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y)\\,dx\\,dy}}$"}
{"_id": "Identity_(mathematics):11", "title": "", "text": "$\\log_{b}\\!\\left(\\frac{x}{y}\\right)=\\log_{b}(x)-\\log_{b}(y)\\,$"}
{"_id": "Inductance:26", "title": "", "text": "$=\\frac{\\mu_{0}r^{2}N^{2}\\pi}{l}\\left(1-\\frac{8w}{3\\pi}+\\frac{w^{2}}{2}-\\frac{w^{4}}{4}+\\frac{5w^{6}}{16}-\\frac{35w^{8}}{64}+...\\right)$"}
{"_id": "Traveling_wave_reactor:0", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U+\\,^{1}_{0}n\\;\\rightarrow\\;^{239}_{\\ 92}U\\;\\rightarrow\\;^{239}_{\\ 93}Np+\\beta\\;\\rightarrow\\;^{239}_{\\ 94}Pu+\\beta}$"}
{"_id": "Hunt–McIlroy_algorithm:2", "title": "", "text": "$O(mn\\log m)$"}
{"_id": "Rational_pricing:2", "title": "", "text": "$P_{0}=\\sum_{t=1}^{T}\\frac{C_{t}}{(1+r_{t})^{t}}$"}
{"_id": "Quadratic_set:17", "title": "", "text": "$\\mathfrak{P}_{n}$"}
{"_id": "Extended_Boolean_model:9", "title": "", "text": "$q_{and}=k_{1}\\land^{p}k_{2}\\land^{p}....\\land^{p}k_{t}$"}
{"_id": "Sheaf_of_modules:18", "title": "", "text": "$\\wedge^{k}F$"}
{"_id": "Sahlqvist_formula:22", "title": "", "text": "$\\forall x[\\forall y(Rxy\\rightarrow\\exists z[Ryz])\\rightarrow\\exists y(Rxy\\wedge\\forall z[Ryz\\rightarrow z=y])]$"}
{"_id": "Structure_(mathematical_logic):42", "title": "", "text": "$\\langle B\\rangle_{\\mathcal{A}}$"}
{"_id": "Interior_reconstruction:90", "title": "", "text": "$\\begin{bmatrix}f_{1}\\\\ g_{1}\\end{bmatrix}=\\begin{bmatrix}A&B\\\\ C&D\\end{bmatrix}\\begin{bmatrix}0\\\\ y_{0}\\end{bmatrix}$"}
{"_id": "Semi-continuity:2", "title": "", "text": "$f(x)=\\begin{cases}1&\\mbox{if }~{}x<1,\\\\ 2&\\mbox{if }~{}x=1,\\\\ 1/2&\\mbox{if }~{}x>1,\\end{cases}$"}
{"_id": "Dirichlet_series:29", "title": "", "text": "$F(s)=\\sum_{n=1}^{\\infty}\\frac{f(n)}{n^{s}}$"}
{"_id": "Derived_category:3", "title": "", "text": "$0\\rightarrow X\\rightarrow Y\\rightarrow Z\\rightarrow 0$"}
{"_id": "Orbit_modeling:65", "title": "", "text": "$\\Bigg[{\\partial{V}\\over{\\partial{\\mathbf{r}}}}\\Bigg]_{0}$"}
{"_id": "Fokker–Planck_equation:61", "title": "", "text": "$dX_{t}=-aX_{t}dt+\\sigma dW_{t}$"}
{"_id": "Sample_mean_and_sample_covariance:36", "title": "", "text": "$q_{jk}=\\frac{\\sum_{i=1}^{N}w_{i}}{\\left(\\sum_{i=1}^{N}w_{i}\\right)^{2}-\\sum_{i=1}^{N}w_{i}^{2}}\\sum_{i=1}^{N}w_{i}\\left(x_{ij}-\\bar{x}_{j}\\right)\\left(x_{ik}-\\bar{x}_{k}\\right).$"}
{"_id": "Transmission_line:28", "title": "", "text": "$Z_{0}=\\sqrt{\\frac{R+j\\omega L}{G+j\\omega C}}.$"}
{"_id": "Isodynamic_point:30", "title": "", "text": "$ASB=ACB+\\pi/3$"}
{"_id": "Bidirectional_associative_memory:2", "title": "", "text": "$M=\\left[{\\begin{array}[]{*{10}c}2&0&0&-2\\\\ 0&-2&2&0\\\\ 2&0&0&-2\\\\ -2&0&0&2\\\\ 0&2&-2&0\\\\ -2&0&0&2\\\\ \\end{array}}\\right]$"}
{"_id": "Functional_completeness:13", "title": "", "text": "$\\ A\\vee B:=(A\\rightarrow B)\\rightarrow B.$"}
{"_id": "Zero_divisor:82", "title": "", "text": "$M\\stackrel{a}{\\to}M$"}
{"_id": "Binomial_coefficient:197", "title": "", "text": "$\\begin{aligned}\\displaystyle{\\left({{-n}\\atop{k}}\\right)}&\\displaystyle=\\frac{-n\\cdot-(n+1)\\dots-(n+k-2)\\cdot-(n+k-1)}{k!}\\\\ &\\displaystyle=(-1)^{k}\\;\\frac{n\\cdot(n+1)\\cdot(n+2)\\cdots(n+k-1)}{k!}\\\\ &\\displaystyle=(-1)^{k}{\\left({{n+k-1}\\atop{k}}\\right)}\\\\ &\\displaystyle=(-1)^{k}\\left(\\!\\!{\\left({{n}\\atop{k}}\\right)}\\!\\!\\right)\\;.\\end{aligned}$"}
{"_id": "Root-mean-square_deviation:5", "title": "", "text": "$\\operatorname{RMSD}=\\sqrt{\\frac{\\sum_{t=1}^{n}(\\hat{y}_{t}-y)^{2}}{n}}.$"}
{"_id": "Pell_number:67", "title": "", "text": "$19601-13860\\sqrt{2}=0.00002\\ldots$"}
{"_id": "Brahmagupta's_formula:16", "title": "", "text": "$\\cos C=-\\cos A$"}
{"_id": "Multinomial_theorem:23", "title": "", "text": "${N\\choose n_{1}}{N-n_{1}\\choose n_{2}}{N-n_{1}-n_{2}\\choose n_{3}}...=\\frac{N!}{(N-n_{1})!n_{1}!}\\frac{(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}\\frac{(N-n_{1}-n_{2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}....$"}
{"_id": "Poynting's_theorem:3", "title": "", "text": "$\\epsilon_{0}\\mathbf{E}\\cdot\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Models_of_DNA_evolution:80", "title": "", "text": "$P_{ij}(\\nu)=\\left\\{\\begin{array}[]{cc}{1\\over 4}+{3\\over 4}e^{-4\\nu/3}&\\mbox{ if }~{}i=j\\\\ {1\\over 4}-{1\\over 4}e^{-4\\nu/3}&\\mbox{ if }~{}i\\neq j\\end{array}\\right.$"}
{"_id": "Glossary_of_module_theory:79", "title": "", "text": "$0\\to I\\to L\\to L^{\\prime}\\to 0$"}
{"_id": "Abductive_reasoning:81", "title": "", "text": "$p(x)=a(y)p(x|y)+a(\\overline{y})p(x|\\overline{y})$"}
{"_id": "Newton's_method:20", "title": "", "text": "$R_{1}=\\frac{1}{2!}f^{\\prime\\prime}(\\xi_{n})(\\alpha-x_{n})^{2}\\,,$"}
{"_id": "Hessian_matrix:37", "title": "", "text": "$H(f,g)=\\begin{bmatrix}0&\\dfrac{\\partial g}{\\partial x_{1}}&\\dfrac{\\partial g}{\\partial x_{2}}&\\cdots&\\dfrac{\\partial g}{\\partial x_{n}}\\\\ \\dfrac{\\partial g}{\\partial x_{1}}&\\dfrac{\\partial^{2}f}{\\partial x_{1}^{2}}&\\dfrac{\\partial^{2}f}{\\partial x_{1}\\,\\partial x_{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{1}\\,\\partial x_{n}}\\\\ \\dfrac{\\partial g}{\\partial x_{2}}&\\dfrac{\\partial^{2}f}{\\partial x_{2}\\,\\partial x_{1}}&\\dfrac{\\partial^{2}f}{\\partial x_{2}^{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{2}\\,\\partial x_{n}}\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ \\dfrac{\\partial g}{\\partial x_{n}}&\\dfrac{\\partial^{2}f}{\\partial x_{n}\\,\\partial x_{1}}&\\dfrac{\\partial^{2}f}{\\partial x_{n}\\,\\partial x_{2}}&\\cdots&\\dfrac{\\partial^{2}f}{\\partial x_{n}^{2}}\\end{bmatrix}$"}
{"_id": "Continued_fraction:57", "title": "", "text": "$\\ \\cfrac{1}{15+\\cfrac{1}{1+\\cfrac{1}{102}}}$"}
{"_id": "Binary_heap:1", "title": "", "text": "$O(\\log n)$"}
{"_id": "Smale's_problems:3", "title": "", "text": "$N^{O(\\log\\log N)}$"}
{"_id": "Boolean_algebra:9", "title": "", "text": "$x\\oplus y=(x\\vee y)\\wedge\\neg{(x\\wedge y)}$"}
{"_id": "Bernoulli_trial:21", "title": "", "text": "$q=1-p=1-\\tfrac{1}{2}=\\tfrac{1}{2}$"}
{"_id": "Trigamma_function:5", "title": "", "text": "$\\psi_{1}(z)=\\sum_{n=0}^{\\infty}\\frac{1}{(z+n)^{2}},$"}
{"_id": "Stern–Brocot_tree:4", "title": "", "text": "$\\frac{23}{16}=1+\\cfrac{1}{2+\\cfrac{1}{3+\\frac{1}{2}}}=[1;2,3,2],$"}
{"_id": "Propagation_constant:1", "title": "", "text": "$\\gamma=\\alpha+i\\beta\\,$"}
{"_id": "Binomial_proportion_confidence_interval:68", "title": "", "text": "$1-\\alpha/2$"}
{"_id": "Parameterized_post-Newtonian_formalism:29", "title": "", "text": "$\\zeta_{2}=2\\beta+2\\beta_{2}-3\\gamma-1$"}
{"_id": "J_integral:20", "title": "", "text": "$\\sigma_{jk}\\cfrac{\\partial\\epsilon_{jk}}{\\partial x_{1}}=\\sigma_{jk}\\cfrac{\\partial^{2}u_{j}}{\\partial x_{1}\\partial x_{k}}$"}
{"_id": "Elasticity_coefficient:29", "title": "", "text": "$\\mathbf{\\varepsilon}=\\begin{bmatrix}\\dfrac{\\partial v_{1}}{\\partial S_{1}}&\\cdots&\\dfrac{\\partial v_{1}}{\\partial S_{m}}\\\\ \\vdots&\\ddots&\\vdots\\\\ \\dfrac{\\partial v_{n}}{\\partial S_{1}}&\\cdots&\\dfrac{\\partial v_{n}}{\\partial S_{m}}\\end{bmatrix}.$"}
{"_id": "Excess_chemical_potential:6", "title": "", "text": "$\\mu_{a}=\\left(\\frac{\\partial G}{\\partial N_{a}}\\right)_{PTN}$"}
{"_id": "Reduction_(mathematics):0", "title": "", "text": "$\\begin{bmatrix}K_{11}&K_{12}\\\\ K_{21}&K_{22}\\end{bmatrix}\\begin{bmatrix}x_{1}\\\\ x_{2}\\end{bmatrix}=\\begin{bmatrix}F_{1}\\\\ F_{2}\\end{bmatrix}$"}
{"_id": "Ogden_(hyperelastic_model):2", "title": "", "text": "$W\\left(\\lambda_{1},\\lambda_{2},\\lambda_{3}\\right)=\\sum_{p=1}^{N}\\frac{\\mu_{p}}{\\alpha_{p}}\\left(\\lambda_{1}^{\\alpha_{p}}+\\lambda_{2}^{\\alpha_{p}}+\\lambda_{3}^{\\alpha_{p}}-3\\right)$"}
{"_id": "Monic_polynomial:1", "title": "", "text": "$\\ ax^{2}+bx+c=0$"}
{"_id": "Phase-comparison_monopulse:0", "title": "", "text": "$\\theta=\\sin^{-1}\\left(\\frac{\\lambda\\ \\Delta\\Phi}{2\\pi d}\\right)$"}
{"_id": "F-test_of_equality_of_variances:1", "title": "", "text": "$S_{X}^{2}=\\frac{1}{n-1}\\sum_{i=1}^{n}\\left(X_{i}-\\overline{X}\\right)^{2}\\,\\text{ and }S_{Y}^{2}=\\frac{1}{m-1}\\sum_{i=1}^{m}\\left(Y_{i}-\\overline{Y}\\right)^{2}$"}
{"_id": "Ext_functor:8", "title": "", "text": "$0\\rightarrow B\\rightarrow E^{\\prime}\\rightarrow A\\rightarrow 0$"}
{"_id": "Mathematical_constants_and_functions:388", "title": "", "text": "$\\underset{\\,\\text{For }x=1/2\\qquad\\qquad}{\\sum_{n=0}^{\\infty}\\frac{(\\!-1\\!)^{n}\\,x^{2n+1}}{2n+1}=\\frac{1}{2}{-}\\frac{1}{3\\!\\cdot\\!2^{3}}{+}\\frac{1}{5\\!\\cdot\\!2^{5}}{-}\\frac{1}{7\\!\\cdot\\!2^{7}}{+}\\cdots}$"}
{"_id": "Triangle:9", "title": "", "text": "$\\cos^{2}\\alpha+\\cos^{2}\\beta+\\cos^{2}\\gamma+2\\cos(\\alpha)\\cos(\\beta)\\cos(\\gamma)=1,$"}
{"_id": "Ring_of_sets:16", "title": "", "text": "$A\\,\\triangle\\,B=(A\\setminus B)\\cup(B\\setminus A)$"}
{"_id": "Harmony_search:17", "title": "", "text": "$\\mathbf{x}^{{}^{\\prime}}$"}
{"_id": "Strong_antichain:0", "title": "", "text": "$\\forall x,y\\in A\\;[x\\neq y\\rightarrow\\neg\\exists z\\in X\\;[z\\leq x\\land z\\leq y]].$"}
{"_id": "C-minimal_theory:3", "title": "", "text": "$\\forall xy\\,[x\\neq y\\rightarrow\\exists z\\neq y\\,C(x;yz)].$"}
{"_id": "Fourier_transform:202", "title": "", "text": "$f(t)=2\\int_{0}^{\\infty}\\int_{-\\infty}^{\\infty}f(\\tau)\\cos 2\\pi\\lambda(\\tau-t)d\\tau d\\lambda.$"}
{"_id": "Bounded_quantifier:16", "title": "", "text": "$\\forall x\\in t\\ (\\phi)\\Leftrightarrow\\forall x(x\\in t\\rightarrow\\phi)$"}
{"_id": "Reciprocity_(network_science):6", "title": "", "text": "$\\rho\\equiv\\frac{\\sum_{i\\neq j}(a_{ij}-\\bar{a})(a_{ji}-\\bar{a})}{\\sum_{i\\neq j}(a_{ij}-\\bar{a})^{2}}$"}
{"_id": "Linear_subspace:25", "title": "", "text": "$U+W$"}
{"_id": "Asymptote:10", "title": "", "text": "$f(x)=\\begin{cases}\\frac{1}{x}&\\mbox{if }~{}x>0,\\\\ 5&\\mbox{if }~{}x\\leq 0.\\end{cases}$"}
{"_id": "Random_permutation_statistics:79", "title": "", "text": "$H_{2n}-H_{n}\\sim\\log 2-\\frac{1}{4n}+\\frac{1}{16n^{2}}-\\frac{1}{128n^{4}}+\\frac{1}{256n^{6}}-\\frac{17}{4096n^{8}}+\\cdots,$"}
{"_id": "Cohomology:5", "title": "", "text": "$0\\rightarrow\\ker h\\rightarrow H^{n}(C;G)\\stackrel{h}{\\rightarrow}\\,\\text{Hom}(H_{n}(C),G)\\rightarrow 0.$"}
{"_id": "Derivation_of_the_Routh_array:92", "title": "", "text": "$-j\\infty\\,$"}
{"_id": "Electromagnetic_reverberation_chamber:77", "title": "", "text": "$\\zeta=\\begin{cases}1&\\mbox{if }~{}n\\neq 0\\\\ 1/2&\\mbox{if }~{}n=0\\end{cases},\\quad\\xi=\\begin{cases}1&\\mbox{if }~{}m\\neq 0\\\\ 1/2&\\mbox{if }~{}m=0\\end{cases}$"}
{"_id": "Repeating_decimal:1", "title": "", "text": "$0.333…$"}
{"_id": "Internal_rate_of_return:6", "title": "", "text": "$\\mathrm{NPV}=\\sum_{n=0}^{N}\\frac{C_{n}}{(1+r)^{n}}=0$"}
{"_id": "Semilinear_transformation:3", "title": "", "text": "$\\lambda^{\\theta}$"}
{"_id": "Levinson_recursion:10", "title": "", "text": "$\\mathbf{T}^{n}\\begin{bmatrix}0\\\\ \\vec{b}^{n-1}\\\\ \\end{bmatrix}=\\begin{bmatrix}t_{0}&\\dots&t_{-n+2}&t_{-n+1}\\\\ \\vdots&&&\\\\ t_{n-2}&&\\mathbf{T}^{n-1}&\\\\ t_{n-1}&&&\\end{bmatrix}\\begin{bmatrix}\\\\ 0\\\\ \\\\ \\vec{b}^{n-1}\\\\ \\\\ \\end{bmatrix}=\\begin{bmatrix}\\epsilon_{b}^{n}\\\\ 0\\\\ \\vdots\\\\ 0\\\\ 1\\end{bmatrix}.$"}
{"_id": "Seminormal_subgroup:3", "title": "", "text": "$AB=G$"}
{"_id": "Hurwitz_zeta_function:60", "title": "", "text": "$\\Phi(z,s,q)=\\sum_{k=0}^{\\infty}\\frac{z^{k}}{(k+q)^{s}}$"}
{"_id": "Pell_number:13", "title": "", "text": "$\\sqrt{2}=1+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\ddots\\,}}}}}.$"}
{"_id": "Mathematical_morphology:24", "title": "", "text": "$A\\oplus B=(A^{c}\\ominus B^{s})^{c}$"}
{"_id": "Pulse-Doppler_signal_processing:3", "title": "", "text": "$Peak\\ Criteria\\begin{cases}\\mathrm{\\left(\\frac{\\Delta Amplitude}{\\Delta Frequency}\\right)Cell(n-1)<0}\\\\ \\mathrm{\\left(\\frac{\\Delta Amplitude}{\\Delta Frequency}\\right)Cell(n+1)>0}\\end{cases}$"}
{"_id": "Chromatographic_response_function:6", "title": "", "text": "$s_{m}=\\sqrt{\\frac{\\sum(\\Delta hR_{Fi}-\\Delta hR_{Ft})^{2}}{n+1}}$"}
{"_id": "Algorithms_for_calculating_variance:12", "title": "", "text": "$s^{2}=\\displaystyle\\frac{\\sum_{i=1}^{n}(x_{i}-K)^{2}-(\\sum_{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}.\\!$"}
{"_id": "Lady_tasting_tea:2", "title": "", "text": "$\\frac{8!}{4!(8-4)!}=70$"}
{"_id": "Closed-form_expression:0", "title": "", "text": "$ax^{2}+bx+c=0,\\,$"}
{"_id": "Eigenvalues_and_eigenvectors_of_the_second_derivative:18", "title": "", "text": "$v_{i,j}=\\begin{cases}n^{-\\frac{1}{2}}&j=1\\\\ \\sqrt{\\frac{2}{n}}\\cos(\\frac{\\pi(j-1)(i-\\frac{1}{2})}{n})&otherwise\\end{cases}$"}
{"_id": "Composite_laminates:14", "title": "", "text": "$\\begin{bmatrix}\\mathbf{N}\\\\ \\mathbf{M}\\end{bmatrix}=\\begin{bmatrix}\\mathbf{A}&\\mathbf{B}\\\\ \\mathbf{B}&\\mathbf{D}\\end{bmatrix}\\begin{bmatrix}\\varepsilon^{0}\\\\ \\kappa\\end{bmatrix}$"}
{"_id": "Juggler_sequence:0", "title": "", "text": "$a_{k+1}=\\begin{cases}\\left\\lfloor a_{k}^{\\frac{1}{2}}\\right\\rfloor,&\\mbox{if }~{}a_{k}\\mbox{ is even}\\\\ \\\\ \\left\\lfloor a_{k}^{\\frac{3}{2}}\\right\\rfloor,&\\mbox{if }~{}a_{k}\\mbox{ is odd}~{}.\\end{cases}$"}
{"_id": "List_of_rules_of_inference:10", "title": "", "text": "$\\underline{\\lnot\\varphi}\\,\\!$"}
{"_id": "Stress_resultants:10", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ V_{2}\\end{bmatrix}=\\int_{-t/2}^{t/2}\\begin{bmatrix}\\sigma_{13}\\\\ \\sigma_{23}\\end{bmatrix}\\,dx_{3}\\,.$"}
{"_id": "Explained_sum_of_squares:9", "title": "", "text": "$\\hat{b}=\\frac{\\sum_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}},$"}
{"_id": "Sufficient_statistic:107", "title": "", "text": "$\\begin{aligned}\\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\\displaystyle=(2\\pi\\sigma^{2})^{-n\\over 2}\\,e^{{-1\\over 2\\sigma^{2}}(\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}+n(\\theta-\\overline{x})^{2})}&\\displaystyle=(2\\pi\\sigma^{2})^{-n\\over 2}\\,e^{{-1\\over 2\\sigma^{2}}\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}}\\,e^{{-n\\over 2\\sigma^{2}}(\\theta-\\overline{x})^{2}}.\\end{aligned}$"}
{"_id": "Yigu_yanduan:20", "title": "", "text": "$1.96x^{2}+19.6x=$"}
{"_id": "Simple_linear_regression:109", "title": "", "text": "$\\displaystyle\\hat{\\beta}=\\frac{}{}\\frac{\\sum_{i=1}^{n}\\left(y_{i}-\\bar{y}\\right)\\left(x_{i}-\\bar{x}\\right)}{\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^{2}}=\\frac{Cov\\left(x,y\\right)}{Var\\left(x\\right)}$"}
{"_id": "Effective_nuclear_charge:4", "title": "", "text": "$\\langle r\\rangle_{\\rm H}$"}
{"_id": "Itō_calculus:31", "title": "", "text": "$\\dot{y}=\\frac{\\partial y}{\\partial x_{j}}\\dot{x}_{j}+\\frac{1}{2}\\frac{\\partial^{2}y}{\\partial x_{k}\\,\\partial x_{l}}g_{km}g_{ml}.$"}
{"_id": "Koszul_cohomology:0", "title": "", "text": "$\\wedge^{p+1}M_{q-1}\\rightarrow\\wedge^{p}M_{q}\\rightarrow\\wedge^{p-1}M_{q+1}$"}
{"_id": "Extended_periodic_table:4", "title": "", "text": "$\\,{}^{238}_{92}\\mathrm{U}+\\,^{64}_{28}\\mathrm{Ni}\\to\\,^{302}_{120}\\mathrm{Ubn}^{*}\\to\\ \\mathit{fission\\ only}$"}
{"_id": "General_set_theory:0", "title": "", "text": "$\\forall x\\forall y[\\forall z[z\\in x\\leftrightarrow z\\in y]\\rightarrow x=y].$"}
{"_id": "Haversine_formula:8", "title": "", "text": "$\\cos(c)=\\cos(a)\\cos(b)+\\sin(a)\\sin(b)\\cos(C).\\,$"}
{"_id": "Matrix_calculus:85", "title": "", "text": "$\\frac{\\partial y}{\\partial\\mathbf{X}}=\\begin{bmatrix}\\frac{\\partial y}{\\partial x_{11}}&\\frac{\\partial y}{\\partial x_{12}}&\\cdots&\\frac{\\partial y}{\\partial x_{1q}}\\\\ \\frac{\\partial y}{\\partial x_{21}}&\\frac{\\partial y}{\\partial x_{22}}&\\cdots&\\frac{\\partial y}{\\partial x_{2q}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial y}{\\partial x_{p1}}&\\frac{\\partial y}{\\partial x_{p2}}&\\cdots&\\frac{\\partial y}{\\partial x_{pq}}\\\\ \\end{bmatrix}.$"}
{"_id": "Surface_reconstruction:8", "title": "", "text": "$G=\\begin{pmatrix}G_{11}&G_{12}\\\\ G_{21}&G_{22}\\end{pmatrix}$"}
{"_id": "Stufe_(algebra):39", "title": "", "text": "$-1-S$"}
{"_id": "Padé_table:12", "title": "", "text": "$\\frac{1+{\\scriptstyle\\frac{1}{2}}z}{1-{\\scriptstyle\\frac{1}{2}}z}$"}
{"_id": "Schur_complement:3", "title": "", "text": "$\\begin{aligned}\\displaystyle ML&\\displaystyle=\\left[\\begin{matrix}A&B\\\\ C&D\\end{matrix}\\right]\\left[\\begin{matrix}I_{p}&0\\\\ -D^{-1}C&I_{q}\\end{matrix}\\right]=\\left[\\begin{matrix}A-BD^{-1}C&B\\\\ 0&D\\end{matrix}\\right]\\\\ &\\displaystyle=\\left[\\begin{matrix}I_{p}&BD^{-1}\\\\ 0&I_{q}\\end{matrix}\\right]\\left[\\begin{matrix}A-BD^{-1}C&0\\\\ 0&D\\end{matrix}\\right].\\end{aligned}$"}
{"_id": "Polar_curve:6", "title": "", "text": "$H(f)=\\begin{bmatrix}\\frac{\\partial^{2}f}{\\partial x^{2}}&\\frac{\\partial^{2}f}{\\partial x\\,\\partial y}&\\frac{\\partial^{2}f}{\\partial x\\,\\partial z}\\\\ \\\\ \\frac{\\partial^{2}f}{\\partial y\\,\\partial x}&\\frac{\\partial^{2}f}{\\partial y^{2}}&\\frac{\\partial^{2}f}{\\partial y\\,\\partial z}\\\\ \\\\ \\frac{\\partial^{2}f}{\\partial z\\,\\partial x}&\\frac{\\partial^{2}f}{\\partial z\\,\\partial y}&\\frac{\\partial^{2}f}{\\partial z^{2}}\\end{bmatrix},$"}
{"_id": "4_(number):7", "title": "", "text": "$0.\\overline{571428}$"}
{"_id": "Leibniz_formula_for_π:1", "title": "", "text": "$1\\,-\\,\\frac{1}{3}\\,+\\,\\frac{1}{5}\\,-\\,\\frac{1}{7}\\,+\\,\\frac{1}{9}\\,-\\,\\cdots\\;=\\;\\frac{\\pi}{4}.\\!$"}
{"_id": "Degrees_of_freedom_(statistics):32", "title": "", "text": "$\\,\\text{SSE}=\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}+\\sum_{i=1}^{n}(Y_{i}-\\bar{Y})^{2}+\\sum_{i=1}^{n}(Z_{i}-\\bar{Z})^{2}$"}
{"_id": "Real_closed_field:8", "title": "", "text": "$2^{2^{\\cdot^{\\cdot^{\\cdot^{n}}}}}$"}
{"_id": "Importance_sampling:36", "title": "", "text": "$P(k_{t}=k)={K\\choose k}p_{t}^{k}(1-p_{t})^{K-k},\\,\\quad\\quad k=0,1,\\dots,K.$"}
{"_id": "Gasification:0", "title": "", "text": "${\\rm C}+{\\rm O}_{2}\\rightarrow{\\rm CO}_{2}$"}
{"_id": "Two-port_network:4", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ V_{2}\\end{bmatrix}=\\begin{bmatrix}z_{11}&z_{12}\\\\ z_{21}&z_{22}\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ I_{2}\\end{bmatrix}$"}
{"_id": "Checking_whether_a_coin_is_fair:15", "title": "", "text": "$s_{p}=\\sqrt{\\frac{p\\,(1-p)}{n}}$"}
{"_id": "Prothrombin_time:0", "title": "", "text": "$\\,\\text{INR}=\\left(\\frac{\\,\\text{PT}\\text{test}}{\\,\\text{PT}\\text{normal}}\\right)\\text{ISI}$"}
{"_id": "Hook_length_formula:126", "title": "", "text": "$\\frac{n!}{l_{1}!l_{2}!\\cdots l_{k}!}\\prod_{i<j}(l_{i}-l_{j})$"}
{"_id": "Fluid_mechanics:28", "title": "", "text": "$\\tau_{ij}=\\mu\\left(\\frac{\\partial v_{i}}{\\partial x_{j}}+\\frac{\\partial v_{j}}{\\partial x_{i}}-\\frac{2}{3}\\delta_{ij}\\nabla\\cdot\\mathbf{v}\\right)+\\kappa\\delta_{ij}\\nabla\\cdot\\mathbf{v}$"}
{"_id": "Coalgebra:3", "title": "", "text": "$\\epsilon(X^{n})=\\begin{cases}1&\\mbox{if }~{}n=0\\\\ 0&\\mbox{if }~{}n>0\\end{cases}$"}
{"_id": "Multivariate_normal_distribution:111", "title": "", "text": "$\\displaystyle\\widehat{symbol\\Sigma}={1\\over n}\\sum_{j=1}^{n}\\left(\\mathbf{x}_{j}-\\bar{\\mathbf{x}}\\right)\\left(\\mathbf{x}_{j}-\\bar{\\mathbf{x}}\\right)^{T}$"}
{"_id": "Multiset:32", "title": "", "text": "$\\mathbf{1}_{A}(x)=\\begin{cases}1&\\,\\text{if }x\\in A,\\\\ 0&\\,\\text{if }x\\notin A.\\end{cases}$"}
{"_id": "Exact_category:26", "title": "", "text": "$0\\to M^{\\prime}\\xrightarrow{f}M\\to M^{\\prime\\prime}\\to 0,$"}
{"_id": "Ext_functor:18", "title": "", "text": "$0\\rightarrow B\\rightarrow Y\\rightarrow A\\rightarrow 0$"}
{"_id": "Perifocal_coordinate_system:17", "title": "", "text": "$\\begin{aligned}\\displaystyle p_{i}&\\displaystyle=\\cos\\Omega\\cos\\omega-\\sin\\Omega\\cos i\\sin\\omega\\\\ \\displaystyle p_{j}&\\displaystyle=\\sin\\Omega\\cos\\omega+\\cos\\Omega\\cos i\\sin\\omega\\\\ \\displaystyle p_{k}&\\displaystyle=\\sin i\\sin\\omega\\\\ \\displaystyle q_{i}&\\displaystyle=-\\cos\\Omega\\sin\\omega-\\sin\\Omega\\cos i\\cos\\omega\\\\ \\displaystyle q_{j}&\\displaystyle=-\\sin\\Omega\\sin\\omega+\\cos\\Omega\\cos i\\cos\\omega\\\\ \\displaystyle q_{k}&\\displaystyle=\\sin i\\cos\\omega\\\\ \\displaystyle w_{i}&\\displaystyle=\\sin i\\sin\\Omega\\\\ \\displaystyle w_{j}&\\displaystyle=-\\sin i\\cos\\Omega\\\\ \\displaystyle w_{k}&\\displaystyle=\\cos i\\end{aligned}$"}
{"_id": "Grassmannian:123", "title": "", "text": "$\\psi:\\mathbf{Gr}(r,V)\\to\\mathbf{P}\\left(\\wedge^{r}V\\right).$"}
{"_id": "Casting_out_nines:9", "title": "", "text": "$\\underline{+\\bcancel{3}20\\bcancel{6}}$"}
{"_id": "Dedekind_eta_function:31", "title": "", "text": "$j_{2A}\\Big(\\tfrac{\\sqrt{-58}}{2}\\Big)=396^{4},\\qquad\\quad e^{\\pi\\sqrt{58}}\\approx 396^{4}-104.00000017\\dots$"}
{"_id": "Harmonic_series_(mathematics):16", "title": "", "text": "$1\\,-\\,\\frac{1}{2}\\,+\\,\\frac{1}{3}\\,-\\,\\frac{1}{4}\\,+\\,\\frac{1}{5}\\,-\\,\\cdots\\;=\\;\\ln 2.$"}
{"_id": "Exponentiation:18", "title": "", "text": "$b^{x}=\\lim_{r(\\in\\mathbb{Q})\\to x}b^{r}\\quad(b\\in\\mathbb{R}^{+},\\,x\\in\\mathbb{R})$"}
{"_id": "Sufficient_statistic:106", "title": "", "text": "$\\sum_{i=1}^{n}(x_{i}-\\overline{x})(\\theta-\\overline{x})=0$"}
{"_id": "Vector_spherical_harmonics:53", "title": "", "text": "$\\nabla\\times\\hat{\\mathbf{B}}=\\mu_{0}\\hat{\\mathbf{J}}+\\mathrm{i}\\mu_{0}\\varepsilon_{0}\\omega\\hat{\\mathbf{E}}\\quad\\Rightarrow\\quad-\\frac{B^{r}}{r}+\\frac{\\mathrm{d}B^{(1)}}{\\mathrm{d}r}+\\frac{B^{(1)}}{r}=\\mu_{0}J+\\mathrm{i}\\omega\\mu_{0}\\varepsilon_{0}E$"}
{"_id": "Range_of_a_projectile:13", "title": "", "text": "$\\sin(x+y)=\\sin x\\,\\cos y\\ +\\ \\sin y\\,\\cos x$"}
{"_id": "Radioactive_displacement_law_of_Fajans_and_Soddy:0", "title": "", "text": "${}^{238}_{92}\\,\\text{U}\\to{}^{234}_{90}\\,\\text{Th}$"}
{"_id": "Trilinear_coordinates:12", "title": "", "text": "$y\\cos B-z\\cos C=0.$"}
{"_id": "Supply_(economics):10", "title": "", "text": "$\\left(\\tfrac{\\Delta Q}{\\Delta P}\\right)\\times\\tfrac{P}{Q}$"}
{"_id": "Cissoid:2", "title": "", "text": "$r=f_{2}(\\theta)-f_{1}(\\theta)$"}
{"_id": "Almost_periodic_function:28", "title": "", "text": "$x(t)=\\frac{1}{2}a_{0}(t)\\ +\\ \\sum_{n=1}^{\\infty}\\left[a_{n}(t)\\cos\\left(2\\pi n\\int_{0}^{t}f_{0}(\\tau)\\,d\\tau\\right)-b_{n}(t)\\sin\\left(2\\pi n\\int_{0}^{t}f_{0}(\\tau)\\,d\\tau\\right)\\right]$"}
{"_id": "Quantum_algorithm_for_linear_systems_of_equations:74", "title": "", "text": "$O(Ns\\kappa)$"}
{"_id": "Hilbert_matrix:1", "title": "", "text": "$H=\\begin{bmatrix}1&\\frac{1}{2}&\\frac{1}{3}&\\frac{1}{4}&\\frac{1}{5}\\\\ \\frac{1}{2}&\\frac{1}{3}&\\frac{1}{4}&\\frac{1}{5}&\\frac{1}{6}\\\\ \\frac{1}{3}&\\frac{1}{4}&\\frac{1}{5}&\\frac{1}{6}&\\frac{1}{7}\\\\ \\frac{1}{4}&\\frac{1}{5}&\\frac{1}{6}&\\frac{1}{7}&\\frac{1}{8}\\\\ \\frac{1}{5}&\\frac{1}{6}&\\frac{1}{7}&\\frac{1}{8}&\\frac{1}{9}\\end{bmatrix}.$"}
{"_id": "Hypergeometric_function:0", "title": "", "text": "${}_{2}F_{1}(a,b;c;z)=\\sum_{n=0}^{\\infty}\\frac{(a)_{n}(b)_{n}}{(c)_{n}}\\frac{z^{n}}{n!}.$"}
{"_id": "List_of_fractals_by_Hausdorff_dimension:42", "title": "", "text": "$\\displaystyle f(x)=\\sum_{k=1}^{\\infty}\\frac{\\sin(2^{k}x)}{\\sqrt{2}^{k}}$"}
{"_id": "Heun_function:1", "title": "", "text": "$\\epsilon=\\alpha+\\beta-\\gamma-\\delta+1$"}
{"_id": "Goertzel_algorithm:70", "title": "", "text": "$O(KN\\log N)$"}
{"_id": "Capillary_surface:11", "title": "", "text": "$\\begin{aligned}\\displaystyle\\sigma_{ij}&\\displaystyle=-\\begin{pmatrix}p&0&0\\\\ 0&p&0\\\\ 0&0&p\\end{pmatrix}+\\mu\\begin{pmatrix}2\\frac{\\partial u}{\\partial x}&\\frac{\\partial u}{\\partial y}+\\frac{\\partial v}{\\partial x}&\\frac{\\partial u}{\\partial z}+\\frac{\\partial w}{\\partial x}\\\\ \\frac{\\partial v}{\\partial x}+\\frac{\\partial u}{\\partial y}&2\\frac{\\partial v}{\\partial y}&\\frac{\\partial v}{\\partial z}+\\frac{\\partial w}{\\partial y}\\\\ \\frac{\\partial w}{\\partial x}+\\frac{\\partial u}{\\partial z}&\\frac{\\partial w}{\\partial y}+\\frac{\\partial v}{\\partial z}&2\\frac{\\partial w}{\\partial z}\\end{pmatrix}\\\\ &\\displaystyle=-pI+\\mu(\\nabla\\mathbf{v}+(\\nabla\\mathbf{v})^{T})\\end{aligned}$"}
{"_id": "Return_period:13", "title": "", "text": "$\\frac{n!}{(n-r)!r!}\\mu^{r}(1-\\mu)^{n-r}\\rightarrow e^{-\\lambda}\\frac{\\lambda^{r}}{r!}.$"}
{"_id": "Time_in_physics:10", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\varepsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t}=\\frac{1}{c^{2}}\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Faraday's_laws_of_electrolysis:6", "title": "", "text": "$Q=\\int_{0}^{t}I(\\tau)\\ d\\tau$"}
{"_id": "Gluon_field_strength_tensor:28", "title": "", "text": "$\\mathcal{L}=-\\frac{1}{2}\\mathrm{tr}\\left(G_{\\alpha\\beta}G^{\\alpha\\beta}\\right)+\\bar{\\psi}\\left(iD_{\\mu}\\right)\\gamma^{\\mu}\\psi$"}
{"_id": "Solution_of_triangles:71", "title": "", "text": "$c=\\arccos\\left(\\cos a\\cos b+\\sin a\\sin b\\cos\\gamma\\right)$"}
{"_id": "List_of_rules_of_inference:43", "title": "", "text": "$\\underline{\\lnot\\psi\\quad\\quad}\\,\\!$"}
{"_id": "S-duality:27", "title": "", "text": "$-1/\\tau$"}
{"_id": "Harmonic_number:32", "title": "", "text": "$H_{\\frac{1}{4}}=4-\\tfrac{\\pi}{2}-3\\ln{2}$"}
{"_id": "Transmission_line:53", "title": "", "text": "$-\\omega\\delta$"}
{"_id": "Protactinium:0", "title": "", "text": "$\\mathrm{{}^{232}_{\\ 90}Th\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{233}_{\\ 90}Th\\ \\xrightarrow[22.3\\ min]{\\beta^{-}}\\ ^{233}_{\\ 91}Pa\\ \\xrightarrow[26.967\\ d]{\\beta^{-}}\\ ^{233}_{\\ 92}U}$"}
{"_id": "Alternating_current:30", "title": "", "text": "$V_{\\mathrm{rms}}=\\sqrt{\\frac{1}{T}\\int_{0}^{T}{[v(t)]^{2}dt}}.$"}
{"_id": "Logic_gate:11", "title": "", "text": "$\\overline{A\\oplus B}$"}
{"_id": "Allan_variance:130", "title": "", "text": "$\\sigma_{y}^{2}(\\tau)=\\int_{0}^{\\infty}S_{y}(f)\\frac{2\\sin^{4}\\pi\\tau f}{(\\pi\\tau f)^{2}}\\,df$"}
{"_id": "Blossom_algorithm:0", "title": "", "text": "$M_{1}=M\\oplus P=(M\\setminus P)\\cup(P\\setminus M)$"}
{"_id": "Collectively_exhaustive_events:0", "title": "", "text": "$A\\cup B=S$"}
{"_id": "Revitalizant:0", "title": "", "text": "$\\mathrm{nMe+mC\\rightarrow Me_{n}C_{m}},$"}
{"_id": "Tensor_product:118", "title": "", "text": "$V^{\\otimes n}\\;\\overset{\\mathrm{def}}{=}\\;\\underbrace{V\\otimes\\cdots\\otimes V}_{n}.$"}
{"_id": "Rule_of_succession:7", "title": "", "text": "$f(p)={(n+1)!\\over s!(n-s)!}p^{s}(1-p)^{n-s}.$"}
{"_id": "First-order_logic:124", "title": "", "text": "$\\forall x\\forall y[\\forall z(z\\in x\\Leftrightarrow z\\in y)\\Rightarrow\\forall z(x\\in z\\Leftrightarrow y\\in z)]$"}
{"_id": "Wrapped_exponential_distribution:0", "title": "", "text": "$\\frac{1-e^{-\\lambda\\theta}}{1-e^{-2\\pi\\lambda}}$"}
{"_id": "Koszul_complex:7", "title": "", "text": "$d_{1}=\\begin{bmatrix}x&y\\\\ \\end{bmatrix}$"}
{"_id": "Fisher_information:32", "title": "", "text": "$\\frac{\\partial\\Sigma}{\\partial\\theta_{m}}=\\begin{bmatrix}\\frac{\\partial\\Sigma_{1,1}}{\\partial\\theta_{m}}&\\frac{\\partial\\Sigma_{1,2}}{\\partial\\theta_{m}}&\\cdots&\\frac{\\partial\\Sigma_{1,N}}{\\partial\\theta_{m}}\\\\ \\\\ \\frac{\\partial\\Sigma_{2,1}}{\\partial\\theta_{m}}&\\frac{\\partial\\Sigma_{2,2}}{\\partial\\theta_{m}}&\\cdots&\\frac{\\partial\\Sigma_{2,N}}{\\partial\\theta_{m}}\\\\ \\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\\\ \\frac{\\partial\\Sigma_{N,1}}{\\partial\\theta_{m}}&\\frac{\\partial\\Sigma_{N,2}}{\\partial\\theta_{m}}&\\cdots&\\frac{\\partial\\Sigma_{N,N}}{\\partial\\theta_{m}}\\end{bmatrix}.$"}
{"_id": "Binomial_theorem:51", "title": "", "text": "${n\\choose k}=\\frac{n!}{k!\\,(n-k)!}$"}
{"_id": "E_(mathematical_constant):154", "title": "", "text": "$e=2+\\cfrac{1}{1+\\cfrac{1}{\\mathbf{2}+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{\\mathbf{4}+\\cfrac{1}{1+\\cfrac{1}{1+\\ddots}}}}}}}=1+\\cfrac{1}{\\mathbf{0}+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{\\mathbf{2}+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{\\mathbf{4}+\\cfrac{1}{1+\\cfrac{1}{1+\\ddots}}}}}}}}}.$"}
{"_id": "Lab_color_space:6", "title": "", "text": "$f^{-1}(t)=\\begin{cases}t^{3}&\\,\\text{if }t>\\tfrac{6}{29}\\\\ 3\\left(\\tfrac{6}{29}\\right)^{2}\\left(t-\\tfrac{4}{29}\\right)&\\,\\text{otherwise}\\end{cases}$"}
{"_id": "Completing_the_square:30", "title": "", "text": "$x^{2}-10x+18=0.\\,\\!$"}
{"_id": "Local_case-control_sampling:27", "title": "", "text": "$1+\\frac{1}{c}$"}
{"_id": "Stokes_wave:28", "title": "", "text": "$\\Phi(x,z,t)=\\beta x-\\gamma t+\\varphi(x-ct,z),$"}
{"_id": "Divided_power_structure:8", "title": "", "text": "$x,y\\in I$"}
{"_id": "Mathematical_coincidence:29", "title": "", "text": "$(\\pi+20)^{i}=-0.9999999992\\ldots-i\\cdot 0.000039\\ldots\\approx-1$"}
{"_id": "Alpha_process:7", "title": "", "text": "$\\mathrm{{}_{20}^{40}Ca}+\\mathrm{{}_{2}^{4}He}\\rightarrow\\mathrm{{}_{22}^{44}Ti}+\\gamma$"}
{"_id": "Cubic_function:212", "title": "", "text": "$x^{3}+bx^{2}+cx+d=0$"}
{"_id": "Trigonometric_functions:84", "title": "", "text": "$\\,\\ -\\sin x$"}
{"_id": "Compressed_suffix_array:2", "title": "", "text": "$O(n\\,{\\log|\\Sigma|})$"}
{"_id": "Quantum_complex_network:8", "title": "", "text": "$|\\phi\\rangle_{i}$"}
{"_id": "ST_type_theory:13", "title": "", "text": "$\\forall x,y[x\\neq y\\rightarrow[xRy\\vee yRx]]$"}
{"_id": "Endogeneity_(econometrics):3", "title": "", "text": "$y_{i}=\\alpha+\\beta x_{i}+\\gamma z_{i}+u_{i}$"}
{"_id": "Convex_conjugate:101", "title": "", "text": "$\\begin{cases}x^{*}\\log(x^{*})-x^{*}&\\,\\text{if }x^{*}>0\\\\ 0&\\,\\text{if }x^{*}=0\\end{cases}$"}
{"_id": "Modularity_theorem:1", "title": "", "text": "$L(E,s)=\\sum_{n=1}^{\\infty}\\frac{a_{n}}{n^{s}}.$"}
{"_id": "Ricci_curvature:54", "title": "", "text": "$\\kappa=\\wedge^{n}\\Omega_{X}.$"}
{"_id": "Admittance_parameters:3", "title": "", "text": "${I_{1}\\choose I_{2}}=\\begin{pmatrix}Y_{11}&Y_{12}\\\\ Y_{21}&Y_{22}\\end{pmatrix}{V_{1}\\choose V_{2}}$"}
{"_id": "Projection_(linear_algebra):4", "title": "", "text": "$P^{2}=\\begin{bmatrix}0&0\\\\ \\alpha&1\\end{bmatrix}\\begin{bmatrix}0&0\\\\ \\alpha&1\\end{bmatrix}=\\begin{bmatrix}0&0\\\\ \\alpha&1\\end{bmatrix}=P.$"}
{"_id": "Fractional_coordinates:6", "title": "", "text": "$\\mathbf{\\begin{bmatrix}\\hat{a}\\\\ \\hat{b}\\\\ \\hat{c}\\\\ \\end{bmatrix}=\\begin{bmatrix}\\frac{1}{a}&-\\frac{\\cos(\\gamma)}{a\\sin(\\gamma)}&\\frac{\\cos(\\alpha)\\cos(\\gamma)-\\cos(\\beta)}{av\\sin(\\gamma)}\\\\ 0&\\frac{1}{b\\sin(\\gamma)}&\\frac{\\cos(\\beta)\\cos(\\gamma)-\\cos(\\alpha)}{bv\\sin(\\gamma)}\\\\ 0&0&\\frac{\\sin(\\gamma)}{cv}\\\\ \\end{bmatrix}}\\begin{bmatrix}x\\\\ y\\\\ z\\\\ \\end{bmatrix}$"}
{"_id": "Multivariate_normal_distribution:101", "title": "", "text": "$\\widehat{symbol\\Sigma}={1\\over n}\\sum_{i=1}^{n}({\\mathbf{x}}_{i}-\\overline{\\mathbf{x}})({\\mathbf{x}}_{i}-\\overline{\\mathbf{x}})^{T}$"}
{"_id": "Exchange_interaction:25", "title": "", "text": "$\\tfrac{1}{2}(0-\\tfrac{6}{4})=-\\tfrac{3}{4}$"}
{"_id": "Riemann_zeta_function:117", "title": "", "text": "$\\zeta(s,q)=\\sum_{k=0}^{\\infty}\\frac{1}{(k+q)^{s}}$"}
{"_id": "Hyperfine_structure:30", "title": "", "text": "$q_{ij}=\\frac{\\partial^{2}V}{\\partial x_{i}\\partial x_{j}}.$"}
{"_id": "Linkage_disequilibrium:150", "title": "", "text": "$\\frac{1}{w}=\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}+\\frac{1}{d+1}$"}
{"_id": "Cotton_tensor:4", "title": "", "text": "$\\widetilde{\\Gamma}^{\\alpha}_{\\beta\\gamma}=\\Gamma^{\\alpha}_{\\beta\\gamma}+S^{\\alpha}_{\\beta\\gamma}$"}
{"_id": "Gabor_wavelet:6", "title": "", "text": "$(\\Delta k)^{2}=\\frac{\\int_{-\\infty}^{\\infty}(k-k_{0})^{2}F(k)F^{*}(k)\\,dk}{\\int_{-\\infty}^{\\infty}F(k)F^{*}(k)\\,dk}$"}
{"_id": "Bessel_function:5", "title": "", "text": "$J_{\\alpha}(x)=\\frac{1}{\\pi}\\int_{0}^{\\pi}\\cos(\\alpha\\tau-x\\sin\\tau)\\,d\\tau-\\frac{\\sin(\\alpha\\pi)}{\\pi}\\int_{0}^{\\infty}e^{-x\\sinh(t)-\\alpha t}\\,dt.$"}
{"_id": "Leibniz's_notation:24", "title": "", "text": "$f^{\\prime}(x)={\\rm st}\\Bigg(\\frac{\\Delta y}{\\Delta x}\\Bigg)$"}
{"_id": "Planarity:13", "title": "", "text": "$O(L^{2}\\log L)$"}
{"_id": "Sommerfeld_expansion:27", "title": "", "text": "$I_{1}=\\tau\\int_{0}^{\\infty}H(\\mu-\\tau x)\\,\\mathrm{d}x-\\tau\\int_{0}^{\\infty}\\frac{H(\\mu-\\tau x)}{e^{x}+1}\\,\\mathrm{d}x\\,$"}
{"_id": "Reaction_rate:33", "title": "", "text": "$A+B\\rightleftharpoons|A\\cdots B|^{\\ddagger}\\rightarrow P$"}
{"_id": "Equation_solving:3", "title": "", "text": "$2x^{5}-5x^{4}-x^{3}-7x^{2}+2x+3=0\\,$"}
{"_id": "Axiom_schema_of_replacement:2", "title": "", "text": "$\\forall w_{1},\\ldots,w_{n}\\,\\forall A\\,\\exists B\\,\\forall x\\in A\\,[\\exists y\\phi(x,y,w_{1},\\ldots,w_{n})\\Rightarrow\\exists y\\in B\\,\\phi(x,y,w_{1},\\ldots,w_{n})]$"}
{"_id": "Weibel_instability:30", "title": "", "text": "$\\nabla\\times\\mathbf{B_{1}}=\\mu_{0}\\mathbf{J_{1}}-i\\omega\\epsilon_{0}\\mu_{0}\\mathbf{E_{1}}$"}
{"_id": "M::M::∞_queue:3", "title": "", "text": "$\\frac{1}{\\lambda}\\sum_{i>c}\\frac{c!}{i!}\\left(\\frac{\\lambda}{\\mu}\\right)^{i-c}$"}
{"_id": "TRIAC:1", "title": "", "text": "$\\left(\\frac{\\operatorname{d}v}{\\operatorname{d}t}\\right)_{s}$"}
{"_id": "Subadditivity:1", "title": "", "text": "$\\forall x,y\\in A,f(x+y)\\leq f(x)+f(y).$"}
{"_id": "Chi-squared_distribution:74", "title": "", "text": "$\\sum_{i=1}^{n}(X_{i}-\\bar{X})^{2}\\sim\\sigma^{2}\\chi^{2}_{n-1}$"}
{"_id": "Preference_ranking_organization_method_for_enrichment_evaluation:43", "title": "", "text": "$\\begin{array}[]{cc}P_{j}(d_{j})=\\left\\{\\begin{array}[]{lll}0&\\,\\text{if}&|d_{j}|\\leq q_{j}\\\\ \\\\ \\frac{1}{2}&\\,\\text{if}&q_{j}<|d_{j}|\\leq p_{j}\\\\ \\\\ 1&\\,\\text{if}&|d_{j}|>p_{j}\\\\ \\end{array}\\right.\\end{array}$"}
{"_id": "Universal_parabolic_constant:8", "title": "", "text": "$\\begin{aligned}\\displaystyle d\\text{avg}&\\displaystyle:=8\\int_{0}^{1\\over 2}\\int_{0}^{x}\\sqrt{x^{2}+y^{2}}\\,dy\\,dx\\\\ &\\displaystyle=8\\int_{0}^{1\\over 2}{1\\over 2}x^{2}(\\ln(1+\\sqrt{2})+\\sqrt{2})\\,dx\\\\ &\\displaystyle=4P\\int_{0}^{1\\over 2}x^{2}\\,dx\\\\ &\\displaystyle={P\\over 6}.\\end{aligned}$"}
{"_id": "Proportional_approval_voting:1", "title": "", "text": "$\\frac{c!}{s!(c-s)!}$"}
{"_id": "Electrical_element:23", "title": "", "text": "$\\begin{bmatrix}V_{1}\\\\ V_{2}\\end{bmatrix}=\\begin{bmatrix}0&-r\\\\ r&0\\end{bmatrix}\\begin{bmatrix}I_{1}\\\\ I_{2}\\end{bmatrix}$"}
{"_id": "Golden_ratio:5", "title": "", "text": "$1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\ddots}}}}$"}
{"_id": "Arbitrarily_varying_channel:35", "title": "", "text": "$\\textstyle U:X\\rightarrow S$"}
{"_id": "Pairing_function:10", "title": "", "text": "$x,y\\in\\mathbb{N}$"}
{"_id": "Euler's_equations_(rigid_body_dynamics):12", "title": "", "text": "$\\left(\\frac{d\\mathbf{L}}{dt}\\right)_{\\mathrm{relative}}$"}
{"_id": "Group_testing:104", "title": "", "text": "$O(d\\log{n})$"}
{"_id": "Geoneutrino:0", "title": "", "text": "$\\begin{array}[]{rcl}{}_{~{}92}^{238}\\,\\text{U}&\\longrightarrow&{}_{~{}82}^{206}\\,\\text{Pb}+8\\alpha+6e^{-}+6\\bar{\\nu}_{e}+51.698\\,\\,\\text{MeV}\\\\ {}_{~{}92}^{235}\\,\\text{U}&\\longrightarrow&{}_{~{}82}^{207}\\,\\text{Pb}+7\\alpha+4e^{-}+4\\bar{\\nu}_{e}+46.402\\,\\,\\text{MeV}\\\\ {}_{~{}90}^{232}\\,\\text{Th}&\\longrightarrow&{}_{~{}82}^{208}\\,\\text{Pb}+6\\alpha+4e^{-}+4\\bar{\\nu}_{e}+42.652\\,\\,\\text{MeV}\\\\ {}_{19}^{40}\\,\\text{K}&\\stackrel{89.3\\,\\%}{\\longrightarrow}&{}_{20}^{40}\\,\\text{Ca}+e^{-}+\\bar{\\nu}_{e}+1.311\\,\\,\\text{MeV}\\\\ {}_{19}^{40}\\,\\text{K}+e^{-}&\\stackrel{10.7\\,\\%}{\\longrightarrow}&{}_{18}^{40}\\,\\text{Ar}+\\nu_{e}+1.505\\,\\,\\text{MeV}\\end{array}$"}
{"_id": "Magnetic_circular_dichroism:4", "title": "", "text": "$A=-\\log(\\frac{V_{dc}}{V_{ex}})$"}
{"_id": "Estimation_of_covariance_matrices:27", "title": "", "text": "${S\\over n}={1\\over n}\\sum_{i=1}^{n}(X_{i}-\\overline{X})(X_{i}-\\overline{X})^{\\mathrm{T}}$"}
{"_id": "Slow_manifold:34", "title": "", "text": "$\\frac{da}{dt}=-2a^{2}b\\quad\\,\\text{and }\\frac{db}{dt}=0.$"}
{"_id": "Homogeneous_distribution:6", "title": "", "text": "$x_{+}^{\\alpha}=\\begin{cases}x^{\\alpha}&\\,\\text{if }x>0\\\\ 0&\\,\\text{otherwise}\\end{cases}$"}
{"_id": "Morse–Kelley_set_theory:20", "title": "", "text": "$\\exists y[My\\and\\varnothing\\in y\\and\\forall z(z\\in y\\rightarrow\\exists x[x\\in y\\and\\forall w(w\\in x\\leftrightarrow[w=zw\\in z])])].$"}
{"_id": "Pulse_wave_velocity:23", "title": "", "text": "$\\mathrm{PWV}=\\sqrt{\\dfrac{\\operatorname{d}\\!P\\cdot V}{\\rho\\cdot\\operatorname{d}\\!V}}$"}
{"_id": "Monic_polynomial:7", "title": "", "text": "$\\ 2x^{2}+3x+1=0$"}
{"_id": "Sine-Gordon_equation:7", "title": "", "text": "$\\cos(\\varphi)=\\sum_{n=0}^{\\infty}\\frac{(-\\varphi^{2})^{n}}{(2n)!},$"}
{"_id": "Split-radix_FFT_algorithm:34", "title": "", "text": "$N/4-1$"}
{"_id": "Intersection_(set_theory):1", "title": "", "text": "$A\\cap B=\\varnothing$"}
{"_id": "Liouville_function:2", "title": "", "text": "$\\frac{\\zeta(2s)}{\\zeta(s)}=\\sum_{n=1}^{\\infty}\\frac{\\lambda(n)}{n^{s}}.$"}
{"_id": "A._H._Lightstone:4", "title": "", "text": "$0.\\underbrace{999\\ldots 9}_{H}\\,$"}
{"_id": "Engel_expansion:14", "title": "", "text": "$\\frac{1}{n}=\\sum_{r=1}^{\\infty}\\frac{1}{(n+1)^{r}}.$"}
{"_id": "List_of_New_Testament_papyri:97", "title": "", "text": "$\\mathfrak{P}^{72}$"}
{"_id": "Gyrovector_space:14", "title": "", "text": "$\\mathrm{gyr}[\\mathbf{u},\\mathbf{v}]\\mathbf{w}=\\ominus(\\mathbf{u}\\oplus\\mathbf{v})\\oplus(\\mathbf{u}\\oplus(\\mathbf{v}\\oplus\\mathbf{w}))$"}
{"_id": "Karp–Lipton_theorem:38", "title": "", "text": "$z\\in L\\implies\\Pr\\nolimits_{x}[\\exists y.\\phi(x,y,z)]\\geq\\tfrac{2}{3}$"}
{"_id": "Celestial_coordinate_system:31", "title": "", "text": "$\\tan\\lambda={\\sin\\alpha\\cos\\varepsilon+\\tan\\delta\\sin\\varepsilon\\over\\cos\\alpha};\\qquad\\qquad\\begin{cases}\\cos\\beta\\sin\\lambda=\\cos\\delta\\sin\\alpha\\cos\\varepsilon+\\sin\\delta\\sin\\varepsilon;\\\\ \\cos\\beta\\cos\\lambda=\\cos\\delta\\cos\\alpha.\\end{cases}$"}
{"_id": "Standard_part_function:6", "title": "", "text": "${}^{\\ast}\\mathbb{R}$"}
{"_id": "Extended_periodic_table:8", "title": "", "text": "$\\,{}^{238}_{92}\\mathrm{U}+\\,^{66}_{30}\\mathrm{Zn}\\to\\,^{304}_{122}\\mathrm{Ubb}^{*}\\to\\ \\mbox{no atoms}~{}.$"}
{"_id": "Scintillator:1", "title": "", "text": "$T_{0}+T_{0}\\rightarrow S^{*}+S_{0}+\\,\\text{phonons}$"}
{"_id": "Prime_integer_topology:2", "title": "", "text": "$\\mathfrak{P}=\\{U_{p}(b)\\,|\\,p,b\\in\\mathbf{Z}^{+},p\\,\\text{ is prime}\\}$"}
{"_id": "Cardinal_characteristic_of_the_continuum:91", "title": "", "text": "${}^{\\omega}\\omega$"}
{"_id": "Common_integrals_in_quantum_field_theory:32", "title": "", "text": "$A=\\begin{bmatrix}a&c\\\\ c&b\\end{bmatrix}$"}
{"_id": "Mathematical_descriptions_of_the_electromagnetic_field:13", "title": "", "text": "$\\mathbf{\\nabla}\\cdot\\mathbf{A}^{\\prime}=-\\mu_{0}\\varepsilon_{0}\\frac{\\partial\\varphi^{\\prime}}{\\partial t}$"}
{"_id": "Kernel_method:26", "title": "", "text": "$\\langle\\cdot,\\cdot\\rangle_{\\mathcal{V}}$"}
{"_id": "Stein's_method:100", "title": "", "text": "$A_{i}=B_{i}=\\{i\\}$"}
{"_id": "Bernstein's_constant:0", "title": "", "text": "$\\cfrac{1}{3+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{3+\\cfrac{1}{9+\\ddots}}}}}$"}
{"_id": "Gospel_of_John:3", "title": "", "text": "$\\mathfrak{P}^{66}$"}
{"_id": "Order_statistic:26", "title": "", "text": "${n!\\over(k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\\,du+O(du^{2}),$"}
{"_id": "Pigeonhole_principle:12", "title": "", "text": "$\\left\\lfloor\\frac{(n-1)}{m}\\right\\rfloor+1=\\left\\lfloor\\frac{(7-1)}{4}\\right\\rfloor+1=\\left\\lfloor\\frac{6}{4}\\right\\rfloor+1=1+1=2$"}
{"_id": "Cayley's_Ω_process:0", "title": "", "text": "$\\Omega=\\begin{vmatrix}\\frac{\\partial}{\\partial x_{11}}&\\cdots&\\frac{\\partial}{\\partial x_{1n}}\\\\ \\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial}{\\partial x_{n1}}&\\cdots&\\frac{\\partial}{\\partial x_{nn}}\\end{vmatrix}.$"}
{"_id": "Characteristic_(algebra):1", "title": "", "text": "$\\underbrace{a+\\cdots+a}_{n\\,\\text{ summands}}=0$"}
{"_id": "Lender_of_last_resort:0", "title": "", "text": "$M=\\quad\\left[\\frac{1+\\frac{C}{D}}{\\frac{C}{D}+\\frac{R}{D}}\\right]B$"}
{"_id": "Baker–Campbell–Hausdorff_formula:15", "title": "", "text": "$Δ(s)=s⊗1 + 1⊗s$"}
{"_id": "Vincent's_theorem:2", "title": "", "text": "$a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cfrac{1}{\\ddots}}}$"}
{"_id": "Tom_Sanders_(mathematician):0", "title": "", "text": "$O\\left(\\frac{N(\\log\\log N)^{5}}{\\log N}\\right)$"}
{"_id": "Clausen_function:107", "title": "", "text": "$\\,\\text{Cl}_{2m}\\left(\\frac{q\\pi}{p}\\right)=$"}
{"_id": "Identical_particles:30", "title": "", "text": "$|x_{1}x_{2}\\cdots x_{N};S\\rangle=\\frac{\\prod_{j}n_{j}!}{N!}\\sum_{p}|x_{p(1)}\\rangle|x_{p(2)}\\rangle\\cdots|x_{p(N)}\\rangle$"}
{"_id": "Padua_points:51", "title": "", "text": "$w_{\\mathbf{\\xi}}=\\frac{1}{n(n+1)}\\cdot\\begin{cases}\\frac{1}{2}\\,\\text{ if }\\mathbf{\\xi}\\,\\text{ is a vertex point}\\\\ 1\\,\\text{ if }\\mathbf{\\xi}\\,\\text{ is an edge point}\\\\ 2\\,\\text{ if }\\mathbf{\\xi}\\,\\text{ is an interior point.}\\end{cases}$"}
{"_id": "Multimodal_distribution:13", "title": "", "text": "$\\nu_{r}=\\int(x-\\mu)^{r}f(x)\\,dx$"}
{"_id": "Frobenius_solution_to_the_hypergeometric_equation:53", "title": "", "text": "$y_{2}=\\left.\\frac{\\partial y_{b}}{\\partial c}\\right|_{c=0}$"}
{"_id": "Complex_geometry:9", "title": "", "text": "$0\\to\\mathbb{Z}\\to\\mathcal{O}\\to\\mathcal{O}^{*}\\to 0$"}
{"_id": "Histogram:18", "title": "", "text": "$\\textstyle v=\\frac{1}{k}\\sum_{i=1}^{k}(m_{i}-\\bar{m})^{2}$"}
{"_id": "Omega_ratio:0", "title": "", "text": "$\\Omega(r)=\\frac{\\int_{r}^{\\infty}(1-F(x))\\,dx}{\\int_{-\\infty}^{r}F(x)dx}$"}
{"_id": "Ring_(mathematics):251", "title": "", "text": "$R\\stackrel{i}{\\to}R$"}
{"_id": "Laplace_distribution:8", "title": "", "text": "$\\mu_{r}^{\\prime}=\\bigg({\\frac{1}{2}}\\bigg)\\sum_{k=0}^{r}\\bigg[{\\frac{r!}{k!(r-k)!}}b^{k}\\mu^{(r-k)}k!\\{1+(-1)^{k}\\}\\bigg]$"}
{"_id": "Carmichael_function:7", "title": "", "text": "$\\lambda(n)=\\begin{cases}\\;\\;\\varphi(n)&\\mbox{if }~{}n=2,3,4,5,6,7,9,10,11,13,14,17,19,22,23,25,26,27,29\\dots\\\\ \\tfrac{1}{2}\\varphi(n)&\\,\\text{if }n=8,16,32,64,128,256\\dots\\end{cases}$"}
{"_id": "Exponential_family:209", "title": "", "text": "$\\frac{n!}{\\prod_{i=1}^{k}x_{i}!}$"}
{"_id": "Sample_mean_and_sample_covariance:11", "title": "", "text": "$\\mathbf{Q}={1\\over{N-1}}\\sum_{i=1}^{N}(\\mathbf{x}_{i}-\\mathbf{\\bar{x}})(\\mathbf{x}_{i}-\\mathbf{\\bar{x}})^{\\mathrm{T}},$"}
{"_id": "Optic_equation:4", "title": "", "text": "$\\frac{1}{IA^{2}}+\\frac{1}{IC^{2}}=\\frac{1}{IB^{2}}+\\frac{1}{ID^{2}}=\\frac{1}{r^{2}}.$"}
{"_id": "Stress_(mechanics):24", "title": "", "text": "$\\begin{bmatrix}T_{1}\\\\ T_{2}\\\\ T_{3}\\end{bmatrix}=\\begin{bmatrix}\\sigma_{11}&\\sigma_{21}&\\sigma_{31}\\\\ \\sigma_{12}&\\sigma_{22}&\\sigma_{32}\\\\ \\sigma_{13}&\\sigma_{23}&\\sigma_{33}\\end{bmatrix}\\begin{bmatrix}n_{1}\\\\ n_{2}\\\\ n_{3}\\end{bmatrix}$"}
{"_id": "Lie_point_symmetry:61", "title": "", "text": "$\\left\\{\\begin{array}[]{l}f_{1}(Z)=(1-cP)+bQ+1,\\\\ f_{2}(Z)=aP-lQ+1.\\end{array}\\right.$"}
{"_id": "Step_function:9", "title": "", "text": "$\\chi_{A}(x)=\\begin{cases}1&\\mbox{if }~{}x\\in A,\\\\ 0&\\mbox{if }~{}x\\notin A.\\\\ \\end{cases}$"}
{"_id": "Studentized_residual:10", "title": "", "text": "$\\operatorname{var}(\\widehat{\\varepsilon}_{i})=\\sigma^{2}\\left(1-\\frac{1}{n}-\\frac{(x_{i}-\\bar{x})^{2}}{\\sum_{j=1}^{n}(x_{j}-\\bar{x})^{2}}\\right).$"}
{"_id": "True_anomaly:7", "title": "", "text": "$2\\pi-\\nu$"}
{"_id": "Big_O_notation:78", "title": "", "text": "$\\log^{*}(n)=\\begin{cases}0,&\\,\\text{if }n\\leq 1\\\\ 1+\\log^{*}(\\log n),&\\,\\text{if }n>1\\end{cases}$"}
{"_id": "Galois_theory:0", "title": "", "text": "$x^{2}-4x+1=0.$"}
{"_id": "Hilbert_series_and_Hilbert_polynomial:18", "title": "", "text": "$0\\;\\rightarrow\\;A\\;\\rightarrow\\;B\\;\\rightarrow\\;C\\;\\rightarrow\\;0$"}
{"_id": "Pitchfork_bifurcation:16", "title": "", "text": "$\\begin{array}[]{lll}\\displaystyle\\frac{\\partial f}{\\partial x}(0,r_{o})=0,&\\displaystyle\\frac{\\partial^{2}f}{\\partial x^{2}}(0,r_{o})=0,&\\displaystyle\\frac{\\partial^{3}f}{\\partial x^{3}}(0,r_{o})\\neq 0,\\\\ \\displaystyle\\frac{\\partial f}{\\partial r}(0,r_{o})=0,&\\displaystyle\\frac{\\partial^{2}f}{\\partial r\\partial x}(0,r_{o})\\neq 0.\\end{array}$"}
{"_id": "Flow_graph_(mathematics):1", "title": "", "text": "$\\begin{bmatrix}c_{11}&c_{12}&c_{13}\\\\ c_{21}&c_{22}&c_{23}\\\\ c_{31}&c_{32}&c_{33}\\end{bmatrix}\\begin{bmatrix}x_{1}\\\\ x_{2}\\\\ x_{3}\\end{bmatrix}=\\begin{bmatrix}y_{1}\\\\ y_{2}\\\\ y_{3}\\end{bmatrix}$"}
{"_id": "Gauss–Kuzmin_distribution:4", "title": "", "text": "$\\Delta_{n}(s)=\\mathbb{P}\\left\\{x_{n}\\leq s\\right\\}-\\log_{2}(1+s)$"}
{"_id": "Wallis_product:7", "title": "", "text": "$\\begin{aligned}\\displaystyle\\Rightarrow I(n)&\\displaystyle=\\int_{0}^{\\pi}\\sin^{n}xdx=\\int_{0}^{\\pi}udv=uv|_{x=0}^{x=\\pi}-\\int_{0}^{\\pi}vdu\\\\ &\\displaystyle=-\\sin^{n-1}x\\cos x|_{x=0}^{x=\\pi}-\\int_{0}^{\\pi}-\\cos x(n-1)\\sin^{n-2}x\\cos xdx\\\\ &\\displaystyle=0-(n-1)\\int_{0}^{\\pi}-\\cos^{2}x\\sin^{n-2}xdx,n>1\\\\ &\\displaystyle=(n-1)\\int_{0}^{\\pi}(1-\\sin^{2}x)\\sin^{n-2}xdx\\\\ &\\displaystyle=(n-1)\\int_{0}^{\\pi}\\sin^{n-2}xdx-(n-1)\\int_{0}^{\\pi}\\sin^{n}xdx\\\\ &\\displaystyle=(n-1)I(n-2)-(n-1)I(n)\\\\ &\\displaystyle=\\frac{n-1}{n}I(n-2)\\\\ \\displaystyle\\Rightarrow\\frac{I(n)}{I(n-2)}&\\displaystyle=\\frac{n-1}{n}\\\\ \\displaystyle\\Rightarrow\\frac{I(2n-1)}{I(2n+1)}&\\displaystyle=\\frac{2n+1}{2n}\\end{aligned}$"}
{"_id": "Z-transform:45", "title": "", "text": "$-z\\frac{dX(z)}{dz}$"}
{"_id": "Kerala_school_of_astronomy_and_mathematics:15", "title": "", "text": "$\\frac{\\pi}{4}=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\ldots$"}
{"_id": "Cartesian_product:13", "title": "", "text": "$(A\\times B)^{c}=(A^{c}\\times B^{c})\\cup(A^{c}\\times B)\\cup(A\\times B^{c}).$"}
{"_id": "Monic_polynomial:4", "title": "", "text": "$2x^{2}+3x+1=0$"}
{"_id": "Riemann–Siegel_theta_function:12", "title": "", "text": "$\\theta^{\\prime}(0)=-\\frac{\\ln\\pi+\\gamma+\\pi/2+3\\ln 2}{2}=-2.6860917\\ldots$"}
{"_id": "Uranium:1", "title": "", "text": "${}_{\\ 92}^{238}\\mathrm{U}+\\mathrm{n}\\rightarrow{}_{\\ 92}^{239}\\mathrm{U}+\\gamma\\xrightarrow{\\beta^{-}}{}_{\\ 93}^{239}\\mathrm{Np}\\xrightarrow{\\beta^{-}}{}_{\\ 94}^{239}\\mathrm{Pu}$"}
{"_id": "Legendre_transformation:75", "title": "", "text": "$f^{*}(x^{*})=\\begin{cases}2x^{*}-4,&x^{*}<4\\\\ \\frac{{x^{*}}^{2}}{4},&4\\leqslant x^{*}\\leqslant 6,\\\\ 3x^{*}-9,&x^{*}>6\\end{cases}$"}
{"_id": "Axiom_of_adjunction:0", "title": "", "text": "$\\forall x\\,\\forall y\\,\\exists w\\,\\forall z\\,[z\\in w\\leftrightarrow(z\\in xz=y)].$"}
{"_id": "Tanc_function:2", "title": "", "text": "$\\operatorname{Re}\\left(\\frac{\\tan\\left(x+iy\\right)}{x+iy}\\right)$"}
{"_id": "Lagrange's_identity_(boundary_value_problem):22", "title": "", "text": "$=\\frac{d}{dx}\\left[p(x)\\left(u\\frac{dv}{dx}-v\\frac{du}{dx}\\right)\\right],$"}
{"_id": "Sufficient_statistic:105", "title": "", "text": "$\\begin{aligned}\\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\\displaystyle=\\prod_{i=1}^{n}\\tfrac{1}{\\sqrt{2\\pi\\sigma^{2}}}\\,e^{-(x_{i}-\\theta)^{2}/(2\\sigma^{2})}=(2\\pi\\sigma^{2})^{-n/2}\\,e^{-\\sum_{i=1}^{n}(x_{i}-\\theta)^{2}/(2\\sigma^{2})}\\\\ &\\displaystyle=(2\\pi\\sigma^{2})^{-n/2}\\,e^{-\\sum_{i=1}^{n}((x_{i}-\\overline{x})-(\\theta-\\overline{x}))^{2}/(2\\sigma^{2})}\\\\ &\\displaystyle=(2\\pi\\sigma^{2})^{-n/2}\\,\\exp\\left({-1\\over 2\\sigma^{2}}\\left(\\sum_{i=1}^{n}(x_{i}-\\overline{x})^{2}+\\sum_{i=1}^{n}(\\theta-\\overline{x})^{2}-2\\sum_{i=1}^{n}(x_{i}-\\overline{x})(\\theta-\\overline{x})\\right)\\right).\\end{aligned}$"}
{"_id": "Aqua_regia:0", "title": "", "text": "$\\rightleftharpoons$"}
{"_id": "Difference_polynomials:13", "title": "", "text": "$A(w)=1$"}
{"_id": "Clausen_function:59", "title": "", "text": "$\\cos(x-y)=\\cos x\\cos y-\\sin x\\sin y$"}
{"_id": "Fermium:0", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ \\xrightarrow{+\\ 15n,7\\beta^{-}}\\ ^{253}_{\\ 99}Es}$"}
{"_id": "Forward_volatility:13", "title": "", "text": "$\\sigma_{t,T}=\\sqrt{\\frac{T\\sigma_{0,T}^{2}-t\\sigma_{0,t}^{2}}{T-t}}$"}
{"_id": "Simplicial_commutative_ring:13", "title": "", "text": "$S^{1}\\wedge S^{1}\\to S^{1}\\wedge S^{1}$"}
{"_id": "Associated_Legendre_polynomials:58", "title": "", "text": "$(-1)^{s-m-w}\\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}$"}
{"_id": "Rydberg_atom:5", "title": "", "text": "$A+\\gamma\\rightarrow A^{*}$"}
{"_id": "Zermelo–Fraenkel_set_theory:46", "title": "", "text": "$\\forall x\\exists y\\forall z[z\\subseteq x\\Rightarrow z\\in y].$"}
{"_id": "Ordinal_arithmetic:2", "title": "", "text": "$\\alpha<\\beta\\Rightarrow\\alpha+\\gamma\\leq\\beta+\\gamma$"}
{"_id": "Zero_object_(algebra):10", "title": "", "text": "$\\begin{bmatrix}\\\\ \\end{bmatrix}$"}
{"_id": "De_Morgan's_laws:60", "title": "", "text": "$(A\\cup B)^{c}=A^{c}\\cap B^{c}$"}
{"_id": "Algebraic_K-theory:0", "title": "", "text": "$0\\to V^{\\prime}\\to V\\to V^{\\prime\\prime}\\to 0,$"}
{"_id": "Multiplicity_function_for_N_noninteracting_spins:0", "title": "", "text": "$W(n,N)={N\\choose n}={{N!}\\over{n!(N-n)!}}$"}
{"_id": "Negative_binomial_distribution:11", "title": "", "text": "$f(k;r,p)=\\frac{\\Gamma(k+r)}{k!\\cdot\\Gamma(r)}p^{k}(1-p)^{r}=\\frac{\\lambda^{k}}{k!}\\cdot\\frac{\\Gamma(r+k)}{\\Gamma(r)\\;(r+\\lambda)^{k}}\\cdot\\frac{1}{\\left(1+\\frac{\\lambda}{r}\\right)^{r}}$"}
{"_id": "Horrocks–Mumford_bundle:0", "title": "", "text": "$\\wedge^{2}F$"}
{"_id": "General_frame:3", "title": "", "text": "$\\Box A=\\{x\\in F;\\,\\forall y\\in F\\,(x\\,R\\,y\\to y\\in A)\\}$"}
{"_id": "List_of_derivatives_and_integrals_in_alternative_calculi:28", "title": "", "text": "$1+\\frac{1}{x}\\,$"}
{"_id": "Reynolds_stress:19", "title": "", "text": "$\\rho\\frac{Du_{i}}{Dt}=-\\frac{\\partial p}{\\partial x_{i}}+\\mu\\left(\\frac{\\partial^{2}u_{i}}{\\partial x_{j}\\partial x_{j}}\\right),$"}
{"_id": "Opacity_(optics):17", "title": "", "text": "$\\frac{1}{\\kappa}=\\frac{\\int_{0}^{\\infty}(\\kappa_{\\nu,{\\rm es}}+\\kappa_{\\nu,{\\rm ff}})^{-1}u(\\nu,T)d\\nu}{\\int_{0}^{\\infty}u(\\nu,T)d\\nu}$"}
{"_id": "Extent_of_reaction:6", "title": "", "text": "$\\Delta_{r}H=\\left(\\frac{\\partial H}{\\partial\\xi}\\right)_{p,T}$"}
{"_id": "Haar_wavelet:1", "title": "", "text": "$\\psi(t)=\\begin{cases}1&0\\leq t<\\frac{1}{2},\\\\ -1&\\frac{1}{2}\\leq t<1,\\\\ 0&\\mbox{otherwise.}\\end{cases}$"}
{"_id": "Dot_(diacritic):1", "title": "", "text": "$0.\\dot{3}$"}
{"_id": "Landen's_transformation:19", "title": "", "text": "$\\begin{aligned}\\displaystyle I&\\displaystyle=\\int_{0}^{\\infty}\\frac{1}{\\sqrt{(x^{2}+a^{2})(x^{2}+b^{2})}}\\,dx\\\\ &\\displaystyle=\\int_{-\\infty}^{\\infty}\\frac{1}{2\\sqrt{\\left(t^{2}+\\left(\\frac{a+b}{2}\\right)^{2}\\right)(t^{2}+ab)}}\\,dt\\\\ &\\displaystyle=\\int_{0}^{\\infty}\\frac{1}{\\sqrt{\\left(t^{2}+\\left(\\frac{a+b}{2}\\right)^{2}\\right)\\left(t^{2}+\\left(\\sqrt{ab}\\right)^{2}\\right)}}\\,dt\\end{aligned}$"}
{"_id": "Quadratic_formula:100", "title": "", "text": "$-1,$"}
{"_id": "Jacobson_density_theorem:88", "title": "", "text": "$A(z)=z•r$"}
{"_id": "Orthonormality:7", "title": "", "text": "$\\cos\\theta_{1}\\cos\\theta_{2}+\\sin\\theta_{1}\\sin\\theta_{2}=0$"}
{"_id": "(a,b,0)_class_of_distributions:13", "title": "", "text": "${\\left({{n}\\atop{k}}\\right)}p^{k}(1-p)^{n-k}$"}
{"_id": "Mean_shift:34", "title": "", "text": "$F(x)=\\begin{cases}1&\\,\\text{if}\\ \\|x\\|\\leq\\lambda\\\\ 0&\\,\\text{if}\\ \\|x\\|>\\lambda\\\\ \\end{cases}$"}
{"_id": "Quadratic_integer:19", "title": "", "text": "$\\omega=\\begin{cases}\\sqrt{D}&\\mbox{if }~{}D\\equiv 2,3\\;\\;(\\mathop{{\\rm mod}}4)\\\\ {{1+\\sqrt{D}}\\over 2}&\\mbox{if }~{}D\\equiv 1\\;\\;(\\mathop{{\\rm mod}}4)\\end{cases}$"}
{"_id": "Size_effect_on_structural_strength:36", "title": "", "text": "$-1/m$"}
{"_id": "Alternatization:20", "title": "", "text": "$\\forall x,y\\in S,\\quad\\alpha(x,y)+\\alpha(y,x)=0$"}
{"_id": "Tseitin_transformation:10", "title": "", "text": "$C=A\\oplus B$"}
{"_id": "Kinetic_priority_queue:10", "title": "", "text": "$O(m\\log n\\log\\log n)$"}
{"_id": "Graham's_number:6", "title": "", "text": "$\\left.\\begin{matrix}G&=&3\\underbrace{\\uparrow\\uparrow\\cdots\\cdots\\cdots\\cdots\\cdots\\uparrow}3\\\\ &&3\\underbrace{\\uparrow\\uparrow\\cdots\\cdots\\cdots\\cdots\\uparrow}3\\\\ &&\\underbrace{\\qquad\\;\\;\\vdots\\qquad\\;\\;}\\\\ &&3\\underbrace{\\uparrow\\uparrow\\cdots\\cdot\\cdot\\uparrow}3\\\\ &&3\\uparrow\\uparrow\\uparrow\\uparrow 3\\end{matrix}\\right\\}\\,\\text{64 layers}$"}
{"_id": "Binomial_theorem:62", "title": "", "text": "$\\frac{n!}{k!\\,(n-k)!}$"}
{"_id": "12_(number):5", "title": "", "text": "$\\mathrm{0.\\overline{923076}}$"}
{"_id": "Stochastic_processes_and_boundary_value_problems:1", "title": "", "text": "$L=\\sum_{i=1}^{n}b_{i}(x)\\frac{\\partial}{\\partial x_{i}}+\\sum_{i,j=1}^{n}a_{ij}(x)\\frac{\\partial^{2}}{\\partial x_{i}\\,\\partial x_{j}},$"}
{"_id": "Celestial_coordinate_system:39", "title": "", "text": "$\\sin a=\\sin\\phi_{o}\\sin\\delta+\\cos\\phi_{o}\\cos\\delta\\cos h$"}
{"_id": "Lambda_lifting:229", "title": "", "text": "$\\operatorname{build-param-lists}[o\\ x\\ y,D,V,[]]\\and D[g]=[x,\\_,\\_]::[o,\\_,\\_]::[y,\\_,\\_]::[]$"}
{"_id": "Particular_values_of_Riemann_zeta_function:40", "title": "", "text": "$\\zeta^{\\prime}(-1)=\\frac{1}{12}-\\ln A\\approx-0.1654211437\\ldots$"}
{"_id": "Finite_volume_method_for_unsteady_flow:37", "title": "", "text": "${a_{P}}^{0}=[\\frac{a_{E}+a_{W}}{2}]$"}
{"_id": "Linear_logic:31", "title": "", "text": "$A\\equiv B\\quad=\\quad(A\\multimap B)\\&(B\\multimap A)$"}
{"_id": "Electromagnetic_tensor:9", "title": "", "text": "$\\nabla\\cdot\\mathbf{E}=\\frac{\\rho}{\\epsilon_{0}},\\quad\\nabla\\times\\mathbf{B}-\\frac{1}{c^{2}}\\frac{\\partial\\mathbf{E}}{\\partial t}=\\mu_{0}\\mathbf{J}$"}
{"_id": "State_observer:95", "title": "", "text": "$\\dot{e}_{2}=h_{3}(\\hat{x})-m_{2}(\\hat{x})\\operatorname{sgn}(e_{2})$"}
{"_id": "Special_relativity:34", "title": "", "text": "$\\begin{pmatrix}\\frac{1}{c}\\frac{\\partial\\phi}{\\partial t^{\\prime}}&\\frac{\\partial\\phi}{\\partial x^{\\prime}}&\\frac{\\partial\\phi}{\\partial y^{\\prime}}&\\frac{\\partial\\phi}{\\partial z^{\\prime}}\\end{pmatrix}=\\begin{pmatrix}\\frac{1}{c}\\frac{\\partial\\phi}{\\partial t}&\\frac{\\partial\\phi}{\\partial x}&\\frac{\\partial\\phi}{\\partial y}&\\frac{\\partial\\phi}{\\partial z}\\end{pmatrix}\\begin{pmatrix}\\gamma&-\\beta\\gamma&0&0\\\\ -\\beta\\gamma&\\gamma&0&0\\\\ 0&0&1&0\\\\ 0&0&0&1\\end{pmatrix}\\,.$"}
{"_id": "Regression_toward_the_mean:3", "title": "", "text": "$\\displaystyle\\hat{\\beta}=\\frac{\\sum_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{\\sum_{i=1}^{n}(x_{i}-\\bar{x})^{2}}=\\frac{\\overline{xy}-\\bar{x}\\bar{y}}{\\overline{x^{2}}-\\bar{x}^{2}}=\\frac{\\operatorname{Cov}[x,y]}{\\operatorname{Var}[x]}=r_{xy}\\frac{s_{y}}{s_{x}},$"}
{"_id": "Observed_information:6", "title": "", "text": "$=-\\left.\\left(\\begin{array}[]{cccc}\\tfrac{\\partial^{2}}{\\partial\\theta_{1}^{2}}&\\tfrac{\\partial^{2}}{\\partial\\theta_{1}\\partial\\theta_{2}}&\\cdots&\\tfrac{\\partial^{2}}{\\partial\\theta_{1}\\partial\\theta_{n}}\\\\ \\tfrac{\\partial^{2}}{\\partial\\theta_{2}\\partial\\theta_{1}}&\\tfrac{\\partial^{2}}{\\partial\\theta_{2}^{2}}&\\cdots&\\tfrac{\\partial^{2}}{\\partial\\theta_{2}\\partial\\theta_{n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\tfrac{\\partial^{2}}{\\partial\\theta_{n}\\partial\\theta_{1}}&\\tfrac{\\partial^{2}}{\\partial\\theta_{n}\\partial\\theta_{2}}&\\cdots&\\tfrac{\\partial^{2}}{\\partial\\theta_{n}^{2}}\\\\ \\end{array}\\right)\\ell(\\theta)\\right|_{\\theta=\\theta^{*}}$"}
{"_id": "Dirichlet_character:2", "title": "", "text": "$L(s,\\chi)=\\sum_{n=1}^{\\infty}\\frac{\\chi(n)}{n^{s}}$"}
{"_id": "General_set_theory:4", "title": "", "text": "$\\forall z\\exists y\\forall x[x\\in y\\leftrightarrow(x\\in z\\land\\phi(x))].$"}
{"_id": "Sliding_mode_control:84", "title": "", "text": "$\\mathord{\\underbrace{D^{+}\\Bigl(\\mathord{\\underbrace{2\\mathord{\\overbrace{\\sqrt{V}}^{{}\\propto\\|\\sigma\\|_{2}}}}_{W}}\\Bigr)}_{D^{+}W\\,\\triangleq\\,\\mathord{\\,\\text{Upper right-hand }\\dot{W}}}}=\\frac{1}{\\sqrt{V}}\\frac{\\operatorname{d}V}{\\operatorname{d}t}\\leq-\\mu$"}
{"_id": "Principalization_(algebra):67", "title": "", "text": "$D_{\\mathfrak{P}}$"}
{"_id": "Dynamo_theory:18", "title": "", "text": "$B=(\\rho\\Omega/\\sigma)^{1/2}$"}
{"_id": "Regular_embedding:7", "title": "", "text": "$U\\overset{j}{\\to}V\\overset{g}{\\to}Y$"}
{"_id": "Nth_root:12", "title": "", "text": "$\\sqrt[5]{-2}\\,=-1.148698354\\ldots$"}
{"_id": "Complex_number:97", "title": "", "text": "$=(ac-bd)+(bc+ad)i$"}
{"_id": "Vincenty's_formulae:33", "title": "", "text": "$\\alpha_{2}=\\arctan\\left(\\frac{\\sin\\alpha}{-\\sin U_{1}\\sin\\sigma+\\cos U_{1}\\cos\\sigma\\cos\\alpha_{1}}\\right)\\,$"}
{"_id": "Set_(mathematics):0", "title": "", "text": "$A\\,\\Delta\\,B=(A\\setminus B)\\cup(B\\setminus A).$"}
{"_id": "Triangle:1", "title": "", "text": "$\\underbrace{\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad}$"}
{"_id": "Isotopes_of_beryllium:3", "title": "", "text": "$\\mathrm{{}^{7}_{4}Be}+\\mathrm{e{}^{-}}\\ \\xrightarrow{\\ \\mathrm{53.22d}}\\ \\mathrm{{}^{7}_{3}Li}$"}
{"_id": "No-cloning_theorem:10", "title": "", "text": "$|\\psi\\rangle_{A}$"}
{"_id": "Cylinder_(geometry):16", "title": "", "text": "$=\\int_{0}^{h}\\pi r^{2}\\,dy$"}
{"_id": "Actinium:6", "title": "", "text": "$\\mathrm{{}^{226}_{\\ 88}Ra\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{227}_{\\ 88}Ra\\ \\xrightarrow[42.2\\ min]{\\beta^{-}}\\ ^{227}_{\\ 89}Ac}$"}
{"_id": "Correlation_and_dependence:1", "title": "", "text": "$r_{xy}=\\frac{\\sum\\limits_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{(n-1)s_{x}s_{y}}=\\frac{\\sum\\limits_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{\\sqrt{\\sum\\limits_{i=1}^{n}(x_{i}-\\bar{x})^{2}\\sum\\limits_{i=1}^{n}(y_{i}-\\bar{y})^{2}}},$"}
{"_id": "Parabola:137", "title": "", "text": "$\\therefore x^{2}+R^{2}-2Ry+y^{2}=R^{2}$"}
{"_id": "Quartic_interaction:58", "title": "", "text": "$v=\\sqrt{\\frac{2\\mu_{0}^{2}}{3\\lambda}}$"}
{"_id": "Power_(physics):4", "title": "", "text": "$W_{C}=U(B)-U(A),$"}
{"_id": "Convergent_series:9", "title": "", "text": "${1\\over 1}+{1\\over 2}+{1\\over 3}+{1\\over 4}+{1\\over 5}+{1\\over 6}+\\cdots\\rightarrow\\infty.$"}
{"_id": "Van_'t_Hoff_factor:10", "title": "", "text": "$i=1-(1-\\frac{1}{2})\\alpha=1-\\frac{\\alpha}{2}$"}
{"_id": "List_of_formulae_involving_π:36", "title": "", "text": "$\\sum_{k=0}^{\\infty}\\frac{1}{16^{k}}\\left(\\frac{4}{8k+1}-\\frac{2}{8k+4}-\\frac{1}{8k+5}-\\frac{1}{8k+6}\\right)=\\pi\\!$"}
{"_id": "Hitchin_functional:25", "title": "", "text": "$\\wedge^{3}(\\mathbb{R}^{n})$"}
{"_id": "Metrical_task_system:19", "title": "", "text": "$O(\\log^{2}n\\log\\log n)$"}
{"_id": "Rate_equation:104", "title": "", "text": "$\\left[B\\right]=\\left\\{\\begin{array}[]{*{35}l}\\left[A\\right]_{0}\\frac{k_{1}}{k_{2}-k_{1}}\\left(e^{-k_{1}t}-e^{-k_{2}t}\\right)+\\left[B\\right]_{0}e^{-k_{2}t}&k_{1}\\neq k_{2}\\\\ \\left[A\\right]_{0}k_{1}te^{-k_{1}t}+\\left[B\\right]_{0}e^{-k_{1}t}&\\,\\text{otherwise}\\\\ \\end{array}\\right.$"}
{"_id": "Mandelbrot_set:100", "title": "", "text": "$z^{2}+c\\to z$"}
{"_id": "Maximum_disjoint_set:26", "title": "", "text": "$O(\\log{\\log{n}})$"}
{"_id": "Theta_function:75", "title": "", "text": "$\\widehat{\\theta}_{F}(z)=\\sum_{k=0}^{\\infty}R_{F}(k)\\exp(2\\pi ikz),$"}
{"_id": "Original_proof_of_Gödel's_completeness_theorem:100", "title": "", "text": "$(\\forall x)Eq(x,x)\\wedge(\\forall x,y,z)[Eq(x,y)\\rightarrow(Eq(x,z)\\rightarrow Eq(y,z))]$"}
{"_id": "Qualitative_variation:178", "title": "", "text": "$-\\log_{10}\\left(\\frac{1+2d}{365}\\right)$"}
{"_id": "Differential_geometry_of_surfaces:55", "title": "", "text": "$=\\alpha+\\beta+\\gamma-\\pi,$"}
{"_id": "Adaptive-additive_algorithm:5", "title": "", "text": "$I_{n}^{f}=\\left(A_{n}^{f}\\right)^{2},$"}
{"_id": "Bessel's_correction:34", "title": "", "text": "$\\begin{aligned}\\displaystyle\\sum_{i=1}^{n}\\left(x_{i}-\\overline{x}\\right)^{2}&\\displaystyle=\\sum_{i=1}^{n}\\left(x_{i}-\\frac{1}{n}\\sum_{j=1}^{n}x_{j}\\right)^{2}\\\\ &\\displaystyle=\\sum_{i=1}^{n}x_{i}^{2}-n\\left(\\frac{1}{n}\\sum_{j=1}^{n}x_{j}\\right)^{2}\\\\ &\\displaystyle=\\sum_{i=1}^{n}x_{i}^{2}-n\\overline{x}^{2}\\end{aligned}$"}
{"_id": "Path_integrals_in_polymer_science:68", "title": "", "text": "$\\textstyle d\\nu=\\left(\\frac{dR}{d\\nu}\\right)^{-1}$"}
{"_id": "Magnus_expansion:32", "title": "", "text": "$\\Omega_{1}=\\int_{0}^{t}A(\\tau)d\\tau$"}
{"_id": "Infinity_(philosophy):1", "title": "", "text": "$\\forall n\\in\\mathbb{Z}(\\exists m\\in\\mathbb{Z}[m>n\\wedge P(m)])$"}
{"_id": "Electromagnetic_field:4", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{J}+\\mu_{0}\\varepsilon_{0}\\frac{\\partial\\mathbf{E}}{\\partial t}$"}
{"_id": "Separation_principle:9", "title": "", "text": "$\\begin{bmatrix}\\dot{x}\\\\ \\dot{e}\\\\ \\end{bmatrix}=\\begin{bmatrix}A-BK&BK\\\\ 0&A-LC\\\\ \\end{bmatrix}\\begin{bmatrix}x\\\\ e\\\\ \\end{bmatrix}.$"}
{"_id": "Boltzmann_machine:24", "title": "", "text": "$\\frac{\\Delta E_{i}}{T}=\\ln\\left(\\frac{p\\text{i=on}}{1-p\\text{i=on}}\\right)$"}
{"_id": "BKL_singularity:76", "title": "", "text": "$\\chi=\\alpha-\\beta=k\\ln\\xi+\\mathrm{const},\\,$"}
{"_id": "Tetration:19", "title": "", "text": "${\\ \\atop{\\ }}{{\\underbrace{a^{a^{\\cdot^{\\cdot^{a}}}}}}\\atop n}$"}
{"_id": "Elementary_algebra:116", "title": "", "text": "$ax^{2}+bx+c=0$"}
{"_id": "Binomial_regression:6", "title": "", "text": "$Y_{n}=\\begin{cases}1,&\\,\\text{if }U_{n}>0,\\\\ 0,&\\,\\text{if }U_{n}\\leq 0\\end{cases}$"}
{"_id": "Binomial_distribution:26", "title": "", "text": "$\\operatorname{Var}[X]=np(1-p).$"}
{"_id": "Estimation_of_covariance_matrices:38", "title": "", "text": "$\\sum_{i=1}^{n}(x_{i}-\\mu)(x_{i}-\\mu)^{\\mathrm{T}}=\\sum_{i=1}^{n}(x_{i}-\\bar{x})(x_{i}-\\bar{x})^{\\mathrm{T}}=S$"}
{"_id": "Ramsey's_theorem:78", "title": "", "text": "$|C_{k}|\\leq c^{\\frac{k!}{n!(k-n)!}}$"}
{"_id": "Proof_that_22::7_exceeds_π:11", "title": "", "text": "${22\\over 7}-{1\\over 630}<\\pi<{22\\over 7}-{1\\over 1260},$"}
{"_id": "Block_matrix:14", "title": "", "text": "$\\mathbf{P}=\\begin{bmatrix}\\mathbf{P}_{11}&\\mathbf{P}_{12}\\\\ \\mathbf{P}_{21}&\\mathbf{P}_{22}\\end{bmatrix}.$"}
{"_id": "Bridgman's_thermodynamic_equations:0", "title": "", "text": "$C_{P}=\\left(\\frac{\\partial H}{\\partial T}\\right)_{P}$"}
{"_id": "Degree_diameter_problem:2", "title": "", "text": "$M_{d,k}=\\begin{cases}1+d\\frac{(d-1)^{k}-1}{d-2}&\\,\\text{ if }d>2\\\\ 2k+1&\\,\\text{ if }d=2\\end{cases}$"}
{"_id": "Derivations_of_the_Lorentz_transformations:86", "title": "", "text": "$\\begin{bmatrix}t^{\\prime}\\\\ x^{\\prime}\\end{bmatrix}=\\begin{bmatrix}\\gamma&\\delta\\\\ \\beta&\\alpha\\end{bmatrix}\\begin{bmatrix}t\\\\ x\\end{bmatrix},$"}
{"_id": "Logic_redundancy:10", "title": "", "text": "$ABCD=1110$"}
{"_id": "Multivariate_probit_model:7", "title": "", "text": "$Y_{2}=\\begin{cases}1&\\,\\text{if }Y^{*}_{2}>0,\\\\ 0&\\,\\text{otherwise},\\end{cases}$"}
{"_id": "Internal_energy:38", "title": "", "text": "$T\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}$"}
{"_id": "Square_root_of_3:0", "title": "", "text": "$1+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\ddots}}}}}$"}
{"_id": "1_+_2_+_3_+_4_+_⋯:14", "title": "", "text": "$c=-\\frac{1}{6}\\cdot\\frac{1}{2!}=-\\frac{1}{12}.$"}
{"_id": "Beal's_conjecture:18", "title": "", "text": "$A^{x}+B^{y}=C^{z}$"}
{"_id": "Relations_between_heat_capacities:24", "title": "", "text": "$\\left(\\frac{\\partial S}{\\partial V}\\right)_{T}$"}
{"_id": "Riemann_series_theorem:32", "title": "", "text": "${1}+{1\\over 3}+\\cdots+{1\\over 2a-1}-{1\\over 2}-{1\\over 4}-\\cdots-{1\\over 2b}+{1\\over 2a+1}+\\cdots+{1\\over 4a-1}-{1\\over 2b+2}-\\cdots$"}
{"_id": "Pascal's_triangle:5", "title": "", "text": "${\\textstyle\\left({{n}\\atop{r}}\\right)}=\\tfrac{n!}{r!(n-r)!}$"}
{"_id": "Nowcast_(Air_Quality_Index):1", "title": "", "text": "$w=\\begin{cases}w^{*}&\\mbox{if }~{}w^{*}>\\frac{1}{2},\\\\ \\frac{1}{2}&\\mbox{if }~{}w^{*}\\leq\\frac{1}{2}.\\\\ \\end{cases}$"}
{"_id": "List_of_formulae_involving_π:97", "title": "", "text": "$\\pi={3+\\cfrac{1^{2}}{6+\\cfrac{3^{2}}{6+\\cfrac{5^{2}}{6+\\cfrac{7^{2}}{6+\\ddots\\,}}}}}$"}
{"_id": "Proofs_of_trigonometric_identities:64", "title": "", "text": "$\\sin(\\alpha+\\beta)=\\sin\\alpha\\cos\\beta+\\sin\\beta\\cos\\alpha$"}
{"_id": "Uncertainty_coefficient:1", "title": "", "text": "$H(X|Y)=-\\sum_{x,~{}y}P_{X,Y}(x,~{}y)\\log P_{X|Y}(x|y).$"}
{"_id": "Principle_of_maximum_entropy:17", "title": "", "text": "$\\begin{array}[]{rcl}\\frac{1}{N}\\log W&=&\\frac{1}{N}\\log\\frac{N!}{n_{1}!\\,n_{2}!\\,\\cdots\\,n_{m}!}\\\\ \\\\ &=&\\frac{1}{N}\\log\\frac{N!}{(Np_{1})!\\,(Np_{2})!\\,\\cdots\\,(Np_{m})!}\\\\ \\\\ &=&\\frac{1}{N}\\left(\\log N!-\\sum_{i=1}^{m}\\log((Np_{i})!)\\right).\\end{array}$"}
{"_id": "Tanhc_function:1", "title": "", "text": "$\\operatorname{Im}\\left(\\frac{\\tanh(x+iy)}{x+iy}\\right)$"}
{"_id": "Thorium_fuel_cycle:0", "title": "", "text": "$\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow{}_{\\ 90}^{233}\\mathrm{Th}\\xrightarrow{\\beta^{-}}{}_{\\ 91}^{233}\\mathrm{Pa}\\xrightarrow{\\beta^{-}}{}_{\\ 92}^{233}\\mathrm{U}$"}
{"_id": "Algebraic_solution:1", "title": "", "text": "$ax^{2}+bx+c=0\\,$"}
{"_id": "Sone:1", "title": "", "text": "$L_{N}=40+10\\log_{2}(N)$"}
{"_id": "Envelope_(mathematics):73", "title": "", "text": "$t^{2}-2tx+y(x)=0.$"}
{"_id": "Lambda_calculus_definition:18", "title": "", "text": "$x\\neq z\\to M[x:=y][z]=M[z]$"}
{"_id": "General_Leibniz_rule:1", "title": "", "text": "${n\\choose k}={n!\\over k!(n-k)!}$"}
{"_id": "Spherical_trigonometry:17", "title": "", "text": "$\\cos B=-\\cos C\\,\\cos A+\\sin C\\,\\sin A\\,\\cos b,$"}
{"_id": "Formal_concept_analysis:0", "title": "", "text": "$A(o)=o^{\\prime}$"}
{"_id": "Methods_of_contour_integration:53", "title": "", "text": "$\\int_{C}{\\sqrt{z}\\over z^{2}+6z+8}\\,dz=2\\int_{0}^{\\infty}{\\sqrt{x}\\over x^{2}+6x+8}\\,dx.$"}
{"_id": "Mechanical_filter:6", "title": "", "text": "$\\begin{bmatrix}V\\\\ F\\end{bmatrix}=\\begin{bmatrix}z_{11}&z_{12}\\\\ z_{21}&z_{22}\\end{bmatrix}\\begin{bmatrix}I\\\\ v\\end{bmatrix}$"}
{"_id": "Gumbel_distribution:12", "title": "", "text": "$\\operatorname{E}(X)=\\mu+\\gamma\\beta,$"}
{"_id": "Reduction_of_order:5", "title": "", "text": "$a\\lambda^{2}+b\\lambda+c=0\\;$"}
{"_id": "Bell_polynomials:1", "title": "", "text": "$=\\sum{n!\\over j_{1}!j_{2}!\\cdots j_{n-k+1}!}\\left({x_{1}\\over 1!}\\right)^{j_{1}}\\left({x_{2}\\over 2!}\\right)^{j_{2}}\\cdots\\left({x_{n-k+1}\\over(n-k+1)!}\\right)^{j_{n-k+1}},$"}
{"_id": "Planck_units:53", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\frac{1}{c^{2}}\\left(\\frac{1}{\\epsilon_{0}}\\mathbf{J}+\\frac{\\partial\\mathbf{E}}{\\partial t}\\right)$"}
{"_id": "Pseudo-order:0", "title": "", "text": "$\\forall x,y:\\neg\\;(x<y\\;\\wedge\\;y<x)$"}
{"_id": "Binomial_coefficient:1", "title": "", "text": "$\\tfrac{n!}{k!\\,(n-k)!}$"}
{"_id": "Dynamic_programming:75", "title": "", "text": "$O(nk\\log k)$"}
{"_id": "Construction_of_the_real_numbers:9", "title": "", "text": "$x,y\\in\\,\\textbf{Q}$"}
{"_id": "Invariant_estimator:106", "title": "", "text": "$\\delta(x)=\\frac{\\int_{-\\infty}^{\\infty}{\\theta f(x_{1}-\\theta,\\dots,x_{n}-\\theta)d\\theta}}{\\int_{-\\infty}^{\\infty}{f(x_{1}-\\theta,\\dots,x_{n}-\\theta)d\\theta}}.$"}
{"_id": "Symmetric_difference:28", "title": "", "text": "$A\\,\\cup\\,B=(A\\,\\triangle\\,B)\\,\\triangle\\,(A\\cap B)$"}
{"_id": "Goldbach–Euler_theorem:7", "title": "", "text": "$x-1-\\frac{1}{2}-\\frac{1}{4}-\\frac{1}{5}-\\frac{1}{6}-\\frac{1}{9}-\\cdots=1$"}
{"_id": "Weyl_scalar:12", "title": "", "text": "$\\Psi_{1}=D\\beta-\\delta\\varepsilon-(\\alpha+\\pi)\\sigma-(\\bar{\\rho}-\\bar{\\varepsilon})\\beta+(\\mu+\\gamma)\\kappa+(\\bar{\\alpha}-\\bar{\\pi})\\varepsilon\\,,$"}
{"_id": "BKL_singularity:38", "title": "", "text": "$\\alpha_{\\tau\\tau}=-\\frac{1}{2}\\lambda^{2}e^{4\\alpha},\\ \\beta_{\\tau\\tau}=\\gamma_{\\tau\\tau}=\\frac{1}{2}\\lambda^{2}e^{4\\alpha}$"}
{"_id": "Particular_values_of_Riemann_zeta_function:29", "title": "", "text": "$\\zeta(-2n)=0.\\,$"}
{"_id": "List_of_formulae_involving_π:42", "title": "", "text": "$\\sum_{n=0}^{\\infty}{\\left(\\frac{(-1)^{n}}{2n+1}\\right)}^{1}=\\frac{1}{1}-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\frac{1}{9}-\\cdots=\\arctan{1}=\\frac{\\pi}{4}\\!$"}
{"_id": "Random_variable:15", "title": "", "text": "$f_{Y}(y)=\\begin{cases}\\tfrac{1}{2},&\\,\\text{if }y=1,\\\\ \\\\ \\tfrac{1}{2},&\\,\\text{if }y=0,\\\\ \\end{cases}$"}
{"_id": "K-means_clustering:5", "title": "", "text": "$O(n^{dk+1}\\log{n})$"}
{"_id": "Principalization_(algebra):161", "title": "", "text": "$\\mathfrak{P}\\mid\\mathfrak{p}$"}
{"_id": "Lieb's_square_ice_constant:0", "title": "", "text": "$1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{5+\\cfrac{1}{1+\\cfrac{1}{4+\\ddots}}}}}$"}
{"_id": "Reachability:31", "title": "", "text": "$O(n\\log{n})$"}
{"_id": "Attribute_Hierarchy_Method:1", "title": "", "text": "$R=(A+I)^{n}$"}
{"_id": "Haynsworth_inertia_additivity_formula:1", "title": "", "text": "$H=\\begin{bmatrix}H_{11}&H_{12}\\\\ H_{12}^{\\ast}&H_{22}\\end{bmatrix}$"}
{"_id": "Aberration_of_light:47", "title": "", "text": "$n$"}
{"_id": "Assortativity:27", "title": "", "text": "$r(\\alpha,\\beta)=\\frac{\\sum_{i}(j^{\\alpha}_{i}-\\bar{j^{\\alpha}})(k^{\\beta}_{i}-\\bar{k^{\\beta}})}{\\sqrt{\\sum_{i}(j^{\\alpha}_{i}-\\bar{j^{\\alpha}})^{2}}\\sqrt{\\sum_{i}(k^{\\beta}_{i}-\\bar{k^{\\beta}})^{2}}}.$"}
{"_id": "Golomb_coding:32", "title": "", "text": "$\\underbrace{111\\cdots 111}_{N}0$"}
{"_id": "Binomial_distribution:9", "title": "", "text": "${n\\choose k}=\\frac{n!}{k!(n-k)!}$"}
{"_id": "RA_plot:0", "title": "", "text": "$a=\\begin{cases}a+\\epsilon,&\\,\\text{if }a=0\\\\ a,&\\,\\text{if }a>0\\end{cases}$"}
{"_id": "Elias_gamma_coding:1", "title": "", "text": "$2\\lfloor\\log_{2}(x)\\rfloor+1$"}
{"_id": "Braid_group:22", "title": "", "text": "$1 + 1 − 1 + 1 + 1=3$"}
{"_id": "Trimaximal_mixing:4", "title": "", "text": "$U=\\begin{bmatrix}\\frac{1}{\\sqrt{3}}&\\frac{1}{\\sqrt{3}}&\\frac{1}{\\sqrt{3}}\\\\ \\frac{\\omega}{\\sqrt{3}}&\\frac{1}{\\sqrt{3}}&\\frac{\\bar{\\omega}}{\\sqrt{3}}\\\\ \\frac{\\bar{\\omega}}{\\sqrt{3}}&\\frac{1}{\\sqrt{3}}&\\frac{\\omega}{\\sqrt{3}}\\end{bmatrix}\\Rightarrow(|U_{i\\alpha}|^{2})=\\begin{bmatrix}\\frac{1}{3}&\\frac{1}{3}&\\frac{1}{3}\\\\ \\frac{1}{3}&\\frac{1}{3}&\\frac{1}{3}\\\\ \\frac{1}{3}&\\frac{1}{3}&\\frac{1}{3}\\end{bmatrix}$"}
{"_id": "3-sphere:9", "title": "", "text": "$N$"}
{"_id": "Square_root:116", "title": "", "text": "$\\sqrt{11}=3+\\cfrac{1}{3+\\cfrac{1}{6+\\cfrac{1}{3+\\cfrac{1}{6+\\cfrac{1}{3+\\ddots}}}}}$"}
{"_id": "Stress_functions:5", "title": "", "text": "$\\sigma_{x}=\\frac{\\partial^{2}\\Phi_{yy}}{\\partial z\\partial z}+\\frac{\\partial^{2}\\Phi_{zz}}{\\partial y\\partial y}-2\\frac{\\partial^{2}\\Phi_{yz}}{\\partial y\\partial z}$"}
{"_id": "Line_element:8", "title": "", "text": "$h_{i}=\\left|\\frac{\\partial{r}}{\\partial q^{i}}\\right|$"}
{"_id": "Exterior_algebra:115", "title": "", "text": "$0\\rightarrow U\\rightarrow V\\rightarrow W\\rightarrow 0$"}
{"_id": "Riemann–Siegel_theta_function:9", "title": "", "text": "$\\pm 17.8455995405\\ldots$"}
{"_id": "Brocard_points:1", "title": "", "text": "$\\cot\\omega=\\cot\\alpha+\\cot\\beta+\\cot\\gamma.\\,$"}
{"_id": "Mathematical_constants_and_functions:70", "title": "", "text": "$\\prod_{n=1}^{\\infty}\\left(\\frac{4n^{2}}{4n^{2}-1}\\right)=\\frac{2}{1}\\cdot\\frac{2}{3}\\cdot\\frac{4}{3}\\cdot\\frac{4}{5}\\cdot\\frac{6}{5}\\cdot\\frac{6}{7}\\cdot\\frac{8}{7}\\cdot\\frac{8}{9}\\cdots$"}
{"_id": "Bernoulli_number:112", "title": "", "text": "$\\scriptstyle B_{1}=-{1\\over 2}$"}
{"_id": "Numerical_methods_for_ordinary_differential_equations:8", "title": "", "text": "$-Ay$"}
{"_id": "Arithmetic_function:47", "title": "", "text": "$\\lambda(n)=\\begin{cases}\\;\\;\\phi(n)&\\,\\text{if }n=2,3,4,5,7,9,11,13,17,19,23,25,27,\\dots\\\\ \\tfrac{1}{2}\\phi(n)&\\,\\text{if }n=8,16,32,64,\\dots\\end{cases}$"}
{"_id": "Trigonometry_in_Galois_fields:40", "title": "", "text": "$\\mathcal{|}a|=\\begin{cases}a,&\\textrm{if }a^{(p-1)/2}\\equiv 1\\bmod{p},\\\\ -a,&\\textrm{if }a^{(p-1)/2}\\equiv-1\\bmod{p}.\\end{cases}$"}
{"_id": "Generalized_continued_fraction:71", "title": "", "text": "$\\pi=\\cfrac{4}{1+\\cfrac{1^{2}}{3+\\cfrac{2^{2}}{5+\\cfrac{3^{2}}{7+\\ddots}}}}=4-1+\\frac{1}{6}-\\frac{1}{34}+\\frac{16}{3145}-\\frac{4}{4551}+\\frac{1}{6601}-\\frac{1}{38341}+-\\cdots$"}
{"_id": "Polylogarithm:101", "title": "", "text": "$-1/\\phi\\,$"}
{"_id": "Madelung_constant:14", "title": "", "text": "$M=-6+12/\\sqrt{2}-8/\\sqrt{3}+6/2-24/\\sqrt{5}+\\cdots=-1.74756\\dots.$"}
{"_id": "Geoneutrino:5", "title": "", "text": "$n+p\\rightarrow d+\\gamma$"}
{"_id": "Pidduck_polynomials:0", "title": "", "text": "$\\displaystyle\\sum_{n}\\frac{s_{n}(x)}{n!}t^{n}=\\left(\\frac{1+t}{1-t}\\right)^{x}(1-t)^{-1}$"}
{"_id": "Trigonometric_functions:70", "title": "", "text": "$\\cos\\left(x-y\\right)=\\cos x\\cos y+\\sin x\\sin y.\\,$"}
{"_id": "Alternatives_to_general_relativity:238", "title": "", "text": "$-4-4\\gamma$"}
{"_id": "Airfoil:30", "title": "", "text": "$\\ C_{M}=-0.5\\pi(A_{0}+A_{1}-A_{2}/2)$"}
{"_id": "Magnetohydrodynamics:14", "title": "", "text": "$\\mu_{0}\\mathbf{J}=\\nabla\\times\\mathbf{B}.$"}
{"_id": "(a,b,0)_class_of_distributions:27", "title": "", "text": "$\\frac{\\Gamma(r+k)}{k!\\,\\Gamma(r)}\\,p^{r}\\,(1-p)^{k}\\,$"}
{"_id": "Riemannian_connection_on_a_surface:81", "title": "", "text": "$d\\alpha=\\beta\\wedge\\gamma,\\,\\,d\\beta=\\gamma\\wedge\\alpha,\\,\\,d\\gamma=\\alpha\\wedge\\beta.$"}
{"_id": "Catalan's_constant:16", "title": "", "text": "$\\Phi(z,s,\\alpha)=\\sum_{n=0}^{\\infty}\\frac{z^{n}}{(n+\\alpha)^{s}}.$"}
{"_id": "Complex_base_systems:86", "title": "", "text": "$-1+\\mathrm{i}$"}
{"_id": "Fanning_friction_factor:28", "title": "", "text": "$B=\\left(\\frac{37530}{Re}\\right)^{16}$"}
{"_id": "Arf_invariant:31", "title": "", "text": "$\\mu(x+y)+\\mu(x)+\\mu(y)\\equiv\\lambda(x,y)\\;\\;(\\mathop{{\\rm mod}}2)\\;\\forall\\,x,y\\in H_{k}(M;\\mathbb{Z}_{2})$"}
{"_id": "Fock_space:49", "title": "", "text": "$|\\Psi\\rangle_{-}$"}
{"_id": "Short-time_Fourier_transform:3", "title": "", "text": "$\\int_{-\\infty}^{\\infty}w(\\tau)\\,d\\tau=1.$"}
{"_id": "Baguenaudier:0", "title": "", "text": "$a(n)=\\begin{cases}\\frac{2^{n+1}-2}{3},&\\,\\text{when }n\\,\\text{ is even,}\\\\ \\frac{2^{n+1}-1}{3},&\\,\\text{when }n\\,\\text{ is odd.}\\end{cases}$"}
{"_id": "Bresenham's_line_algorithm:16", "title": "", "text": "$\\begin{aligned}\\displaystyle y&\\displaystyle=mx+b\\\\ \\displaystyle y&\\displaystyle=\\frac{(\\Delta y)}{(\\Delta x)}x+b\\\\ \\displaystyle(\\Delta x)y&\\displaystyle=(\\Delta y)x+(\\Delta x)b\\\\ \\displaystyle 0&\\displaystyle=(\\Delta y)x-(\\Delta x)y+(\\Delta x)b\\end{aligned}$"}
{"_id": "Symmetrical_components:6", "title": "", "text": "$\\begin{aligned}\\displaystyle V_{abc}&\\displaystyle=\\begin{bmatrix}V_{0}\\\\ V_{0}\\\\ V_{0}\\end{bmatrix}+\\begin{bmatrix}V_{1}\\\\ \\alpha^{2}V_{1}\\\\ \\alpha V_{1}\\end{bmatrix}+\\begin{bmatrix}V_{2}\\\\ \\alpha V_{2}\\\\ \\alpha^{2}V_{2}\\end{bmatrix}\\\\ &\\displaystyle=\\begin{bmatrix}1&1&1\\\\ 1&\\alpha^{2}&\\alpha\\\\ 1&\\alpha&\\alpha^{2}\\end{bmatrix}\\begin{bmatrix}V_{0}\\\\ V_{1}\\\\ V_{2}\\end{bmatrix}\\\\ &\\displaystyle=\\,\\textbf{A}V_{012}\\end{aligned}$"}
{"_id": "Metric_tensor:10", "title": "", "text": "$J=\\begin{bmatrix}\\frac{\\partial u}{\\partial u^{\\prime}}&\\frac{\\partial u}{\\partial v^{\\prime}}\\\\ \\frac{\\partial v}{\\partial u^{\\prime}}&\\frac{\\partial v}{\\partial v^{\\prime}}\\end{bmatrix}.$"}
{"_id": "Sahlqvist_formula:11", "title": "", "text": "$\\forall x\\forall y\\forall z[(Rxy\\land Ryz)\\rightarrow Rxz]$"}
{"_id": "Mass_noun:0", "title": "", "text": "$\\forall X\\subseteq U_{p}[CUM_{p}(X)\\Leftrightarrow\\exists x,y[X(x)\\,\\wedge\\,X(y)\\,\\wedge\\,\\neg(x=y)]\\;\\wedge\\;\\forall x,y[X(x)\\,\\wedge\\,X(y)\\Rightarrow X(x\\,\\oplus\\,y)]]$"}
{"_id": "Linear_least_squares_(mathematics):22", "title": "", "text": "$-0.7,$"}
{"_id": "Rigid_rotor:75", "title": "", "text": "$\\mathbf{R}(\\alpha,\\beta,\\gamma)=\\begin{pmatrix}\\cos\\alpha&-\\sin\\alpha&0\\\\ \\sin\\alpha&\\cos\\alpha&0\\\\ 0&0&1\\end{pmatrix}\\begin{pmatrix}\\cos\\beta&0&\\sin\\beta\\\\ 0&1&0\\\\ -\\sin\\beta&0&\\cos\\beta\\\\ \\end{pmatrix}\\begin{pmatrix}\\cos\\gamma&-\\sin\\gamma&0\\\\ \\sin\\gamma&\\cos\\gamma&0\\\\ 0&0&1\\end{pmatrix}$"}
{"_id": "W_state:13", "title": "", "text": "$A\\cup B=\\{1,...,N\\}$"}
{"_id": "Potential_flow:28", "title": "", "text": "$a^{2}=\\left[\\frac{\\partial p}{\\partial\\rho}\\right]_{S}.$"}
{"_id": "Linear_seismic_inversion:33", "title": "", "text": "$\\mathbf{A}=\\begin{bmatrix}\\frac{\\partial F_{1}(\\vec{q})}{\\partial p_{1}}&\\frac{\\partial F_{1}(\\vec{q})}{\\partial p_{2}}&\\cdots&\\frac{\\partial F_{1}(\\vec{q})}{\\partial p_{n}}\\\\ \\frac{\\partial F_{2}(\\vec{q})}{\\partial p_{1}}&\\cdots&\\frac{\\partial F_{2}(\\vec{q})}{\\partial p_{n-1}}&\\frac{\\partial F_{2}(\\vec{q})}{\\partial p_{n}}\\\\ \\vdots&\\frac{\\partial F_{j}(\\vec{q})}{\\partial p_{i}}&\\vdots&\\vdots\\\\ \\frac{\\partial F_{m}(\\vec{q})}{\\partial p_{1}}&\\frac{\\partial F_{m}(\\vec{q})}{\\partial p_{2}}&\\cdots&\\frac{\\partial F_{m}(\\vec{q})}{\\partial p_{n}}\\\\ \\end{bmatrix}\\!$"}
{"_id": "Schanuel's_lemma:6", "title": "", "text": "$0\\rightarrow K\\rightarrow X\\rightarrow P^{\\prime}\\rightarrow 0,$"}
{"_id": "K-space_(functional_analysis):1", "title": "", "text": "$0\\rightarrow\\mathbb{R}\\rightarrow X\\rightarrow V\\rightarrow 0.\\,\\!$"}
{"_id": "Bijection,_injection_and_surjection:1", "title": "", "text": "$\\forall x,y\\in A,f(x)=f(y)\\Rightarrow x=y.$"}
{"_id": "Molecularity:1", "title": "", "text": "$A+B\\to AB^{*}$"}
{"_id": "Quartic_function:143", "title": "", "text": "$x^{4}+px^{2}+qx+r=0$"}
{"_id": "Method_of_analytic_tableaux:158", "title": "", "text": "$\\forall x,y.P(x,y)\\vee Q(f(x))$"}
{"_id": "Symmetry_of_diatomic_molecules:126", "title": "", "text": "$-\\Lambda\\hbar$"}
{"_id": "Tachyphylaxis:0", "title": "", "text": "$A\\xrightarrow{\\ \\ p\\ \\ }B$"}
{"_id": "Atiyah_algebroid:1", "title": "", "text": "$0\\to VP\\to TP\\xrightarrow{d\\pi}\\pi^{*}TM\\to 0$"}
{"_id": "Kendall's_W:2", "title": "", "text": "$S=\\sum_{i=1}^{n}(R_{i}-\\bar{R})^{2},$"}
{"_id": "Classical_orthogonal_polynomials:194", "title": "", "text": "$-3x\\,$"}
{"_id": "Invariant_of_a_binary_form:0", "title": "", "text": "$H(f)=\\begin{bmatrix}\\frac{\\partial^{2}f}{\\partial x^{2}}&\\frac{\\partial^{2}f}{\\partial x\\,\\partial y}\\\\ \\frac{\\partial^{2}f}{\\partial y\\,\\partial x}&\\frac{\\partial^{2}f}{\\partial y^{2}}\\end{bmatrix}.$"}
{"_id": "Real_line:25", "title": "", "text": "$A=R\\oplus V,$"}
{"_id": "Heavy_traffic_approximation:13", "title": "", "text": "$S_{j}\\xrightarrow{d}S$"}
{"_id": "Hyperbolic_angle:2", "title": "", "text": "$\\textstyle\\cosh x=\\sum_{n=0}^{\\infty}\\frac{x^{2n}}{(2n)!}$"}
{"_id": "Gibbs_paradox:71", "title": "", "text": "$=\\left(\\ell_{A}\\right)^{n_{A}}\\left(\\ell_{B}\\right)^{n_{B}}\\left(\\underbrace{\\frac{N!}{n_{A}!n_{B}!}}_{combination}\\right)\\left(\\underbrace{\\frac{n_{A}\\pi^{n_{A}/2}}{(n_{A}/2)!}(2E_{A})^{\\frac{n_{A}-1}{2}}}_{n_{A}-sphere}\\right)\\left(\\underbrace{\\frac{n_{B}\\pi^{n_{B}/2}}{(n_{B}/2)!}(2E_{B})^{\\frac{n_{B}-1}{2}}}_{n_{B}-sphere}\\right)$"}
{"_id": "K-factor_(fire_protection):1", "title": "", "text": "$LPM/\\sqrt{\\,}\\text{bar}$"}
{"_id": "Lorenz_curve:8", "title": "", "text": "$L(F)=\\frac{\\int_{0}^{F}x(F_{1})\\,dF_{1}}{\\int_{0}^{1}x(F_{1})\\,dF_{1}}$"}
{"_id": "Absolute_value:96", "title": "", "text": "$|a|=\\begin{cases}a,&\\mbox{if }~{}a\\geq 0\\\\ -a,&\\mbox{if }~{}a\\leq 0\\end{cases}\\;$"}
{"_id": "Meijer_G-function:58", "title": "", "text": "$g(s)=\\sqrt{2/\\pi}\\int_{0}^{\\infty}(st)^{1/2}\\,K_{\\nu}(st)\\,f(t)\\;dt,$"}
{"_id": "Magnetic_reconnection:0", "title": "", "text": "$\\nabla\\times\\mathbf{B}=\\mu\\mathbf{J}+\\mu\\epsilon\\frac{\\partial\\mathbf{E}}{\\partial t}.$"}
{"_id": "DLVO_theory:28", "title": "", "text": "$\\gamma=\\tanh\\left(\\frac{ze\\psi_{0}}{4kT}\\right)$"}
{"_id": "HSAB_theory:2", "title": "", "text": "$\\mu=\\left(\\frac{\\partial E}{\\partial N}\\right)_{Z},$"}
{"_id": "Periodic_points_of_complex_quadratic_mappings:92", "title": "", "text": "$B=\\alpha_{1}\\alpha_{2}+\\alpha_{1}\\beta_{1}+\\alpha_{1}\\beta_{2}+\\alpha_{2}\\beta_{1}+\\alpha_{2}\\beta_{2}+\\beta_{1}\\beta_{2}\\,$"}
{"_id": "Americium:2", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ \\xrightarrow{(n,\\gamma)}\\ ^{239}_{\\ 92}U\\ \\xrightarrow[23.5\\ min]{\\beta^{-}}\\ ^{239}_{\\ 93}Np\\ \\xrightarrow[2.3565\\ d]{\\beta^{-}}\\ ^{239}_{\\ 94}Pu}$"}
{"_id": "Stretched_exponential_function:1", "title": "", "text": "$\\langle\\tau\\rangle\\equiv\\int_{0}^{\\infty}dt\\,e^{-\\left({t/\\tau_{K}}\\right)^{\\beta}}={\\tau_{K}\\over\\beta}\\Gamma({1\\over\\beta})$"}
{"_id": "Grünwald–Letnikov_derivative:3", "title": "", "text": "$=\\lim_{h\\to 0}\\frac{f(x+2h)-2f(x+h)+f(x)}{h^{2}},$"}
{"_id": "Group_(mathematics):2", "title": "", "text": "$\\underbrace{a\\cdots a}_{n\\,\\text{ factors}},$"}
{"_id": "Trapezoid:12", "title": "", "text": "$p=\\sqrt{\\frac{ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}},$"}
{"_id": "Exponential_function:18", "title": "", "text": "$e^{2}=1+\\cfrac{4}{0+\\cfrac{2^{2}}{6+\\cfrac{2^{2}}{10+\\cfrac{2^{2}}{14+\\ddots\\,}}}}=7+\\cfrac{2}{5+\\cfrac{1}{7+\\cfrac{1}{9+\\cfrac{1}{11+\\ddots\\,}}}}$"}
{"_id": "Help:Displaying_a_formula:138", "title": "", "text": "$\\underbrace{a+b+\\cdots+z}_{26}$"}
{"_id": "Luby_transform_code:4", "title": "", "text": "$\\begin{aligned}\\displaystyle\\mathrm{P}\\{d=1\\}&\\displaystyle=\\frac{1}{n}\\\\ \\displaystyle\\mathrm{P}\\{d=k\\}&\\displaystyle=\\frac{1}{k(k-1)}\\qquad(k=2,3,\\dots,n).\\end{aligned}$"}
{"_id": "Boundary_current:16", "title": "", "text": "$\\alpha=\\left(\\frac{D}{R}\\right)\\left(\\frac{\\partial f}{\\partial y}\\right)$"}
{"_id": "Vertical_electrical_sounding:3", "title": "", "text": "$k=\\frac{2\\pi}{\\frac{1}{r_{AM}}-\\frac{1}{r_{BM}}-\\frac{1}{r_{AN}}+\\frac{1}{r_{BN}}}$"}
{"_id": "Hierarchical_matrix:3", "title": "", "text": "$O(nk\\,\\log(n))$"}
{"_id": "Coherent_effects_in_semiconductor_optics:44", "title": "", "text": "$\\mathrm{i}\\hbar\\frac{\\partial}{\\partial t}n^{c}_{\\mathbf{k}}=(\\Omega_{\\mathbf{k}}^{\\star}\\,p_{\\mathbf{k}}-\\Omega_{\\mathbf{k}}\\,p_{\\mathbf{k}}^{\\star})+\\mathrm{i}\\hbar\\frac{\\partial}{\\partial t}n^{c}_{\\mathbf{k}}|_{\\,\\text{corr}}\\,,$"}
{"_id": "Rewriting:23", "title": "", "text": "$x\\stackrel{*}{\\rightarrow}z\\stackrel{*}{\\leftarrow}y$"}
{"_id": "An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances:0", "title": "", "text": "$f(p)=\\frac{(n+1)!}{k!(n-k)!}p^{k}(1-p)^{n-k}\\,\\text{ for }0\\leq p\\leq 1$"}
{"_id": "Fitts's_law:1", "title": "", "text": "$\\,\\text{IP}=\\Bigg(\\frac{ID}{MT}\\Bigg)$"}
{"_id": "Strategic_complements:19", "title": "", "text": "$\\begin{bmatrix}\\dfrac{dx_{1}^{*}}{dp_{1}}\\\\ \\dfrac{dx_{2}^{*}}{dp_{1}}\\\\ \\dfrac{dy_{2}^{*}}{dp_{1}}\\end{bmatrix}=\\begin{bmatrix}\\dfrac{\\partial^{2}U_{x}}{\\partial x_{1}\\partial x_{1}}&\\dfrac{\\partial^{2}U_{x}}{\\partial x_{1}\\partial x_{2}}&\\dfrac{\\partial^{2}U_{x}}{\\partial x_{1}\\partial y_{2}}\\\\ \\dfrac{\\partial^{2}U_{x}}{\\partial x_{1}\\partial x_{2}}&\\dfrac{\\partial^{2}U_{x}}{\\partial x_{2}\\partial x_{2}}&\\dfrac{\\partial^{2}U_{x}}{\\partial y_{2}\\partial x_{2}}\\\\ \\dfrac{\\partial^{2}U_{y}}{\\partial x_{1}\\partial y_{2}}&\\dfrac{\\partial^{2}U_{y}}{\\partial x_{2}\\partial y_{2}}&\\dfrac{\\partial^{2}U_{y}}{\\partial y_{2}\\partial y_{2}}\\end{bmatrix}^{-1}\\begin{bmatrix}-\\dfrac{\\partial^{2}U_{x}}{\\partial p_{1}\\partial x_{1}}\\\\ -\\dfrac{\\partial^{2}U_{x}}{\\partial p_{1}\\partial x_{2}}\\\\ -\\dfrac{\\partial^{2}U_{y}}{\\partial p_{1}\\partial y_{2}}\\end{bmatrix}$"}
{"_id": "Unbounded_operator:55", "title": "", "text": "$\\langle Tx\\mid y\\rangle_{2}=\\left\\langle x\\mid T^{*}y\\right\\rangle_{1},\\qquad x\\in D(T).$"}
{"_id": "Birch–Murnaghan_equation_of_state:2", "title": "", "text": "$B_{0}^{\\prime}=\\left(\\frac{\\partial B}{\\partial P}\\right)_{P=0}$"}
{"_id": "Gravitational_lensing_formalism:95", "title": "", "text": "$q_{xx}=\\frac{\\sum(x-\\bar{x})^{2}I(x,y)}{\\sum I(x,y)}$"}
{"_id": "Plutonium-239:1", "title": "", "text": "$\\mathrm{{}^{238}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow\\ ^{239}_{\\ 92}U\\ \\xrightarrow[23.5\\ min]{\\beta^{-}}\\ ^{239}_{\\ 93}Np\\ \\xrightarrow[2.3565\\ d]{\\beta^{-}}\\ ^{239}_{\\ 94}Pu}$"}
{"_id": "Harmonic_number:7", "title": "", "text": "$H_{n}=\\frac{1}{n!}\\left[{n+1\\atop 2}\\right].$"}
{"_id": "Logarithm:28", "title": "", "text": "$\\log_{2}\\!\\left(\\frac{1}{2}\\right)=-1,\\,$"}
{"_id": "Angular_velocity:26", "title": "", "text": "$symbol\\omega=(\\dot{\\alpha}\\sin\\beta\\sin\\gamma+\\dot{\\beta}\\cos\\gamma)I+(\\dot{\\alpha}\\sin\\beta\\cos\\gamma-\\dot{\\beta}\\sin\\gamma)J+(\\dot{\\alpha}\\cos\\beta+\\dot{\\gamma})K$"}
{"_id": "Natural_logarithm_of_2:38", "title": "", "text": "$\\ln 2=\\frac{1}{1}-\\frac{1}{2}+\\frac{1}{3}-\\frac{1}{4}+\\frac{1}{5}-\\cdots.$"}
{"_id": "Adjoint_functors:39", "title": "", "text": "$-\\otimes A$"}
{"_id": "Material_failure_theory:7", "title": "", "text": "$\\sigma=\\sqrt{\\cfrac{2E\\gamma}{\\pi a}}$"}
{"_id": "Electric-field_screening:9", "title": "", "text": "$k_{0}\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{3e^{2}\\rho}{2\\epsilon_{0}E_{F}}}=\\sqrt{\\frac{me^{2}k_{f}}{\\epsilon_{0}\\pi^{2}\\hbar^{2}}}$"}
{"_id": "Lieb–Thirring_inequality:22", "title": "", "text": "$-\\mathrm{i}\\nabla$"}
{"_id": "Residence_time_distribution:15", "title": "", "text": "$\\sigma^{2}=\\int_{0}^{\\infty}(t-\\bar{t})^{2}\\cdot E(t)\\,dt$"}
{"_id": "Confluent_hypergeometric_function:8", "title": "", "text": "$M(a,b,z)=\\sum_{n=0}^{\\infty}\\frac{a^{(n)}z^{n}}{b^{(n)}n!}={}_{1}F_{1}(a;b;z),$"}
{"_id": "Electromagnetic_radiation:33", "title": "", "text": "$\\nabla\\times\\mathbf{E}=\\hat{\\mathbf{k}}\\times\\mathbf{E}_{0}f^{\\prime}\\left(\\hat{\\mathbf{k}}\\cdot\\mathbf{x}-c_{0}t\\right)=-\\frac{\\partial\\mathbf{B}}{\\partial t}$"}
{"_id": "Polarity_(international_relations):0", "title": "", "text": "$\\,\\text{Concentration}_{t}=\\sqrt{\\frac{\\sum_{i=1}^{N_{t}}(S_{it})^{2}-\\frac{1}{N_{t}}}{1-\\frac{1}{N_{t}}}}$"}
{"_id": "Gauss–Kuzmin–Wirsing_operator:17", "title": "", "text": "$t_{n}=\\sum_{m=0}^{\\infty}\\frac{G_{mn}}{(m+1)(m+2)}.$"}
{"_id": "Cartesian_tensor:55", "title": "", "text": "$\\begin{pmatrix}\\bar{x}_{1}\\\\ \\bar{x}_{2}\\\\ \\bar{x}_{3}\\end{pmatrix}=\\begin{pmatrix}\\frac{\\partial\\bar{x}_{1}}{\\partial x_{1}}&\\frac{\\partial\\bar{x}_{1}}{\\partial x_{2}}&\\frac{\\partial\\bar{x}_{1}}{\\partial x_{3}}\\\\ \\frac{\\partial\\bar{x}_{2}}{\\partial x_{1}}&\\frac{\\partial\\bar{x}_{2}}{\\partial x_{2}}&\\frac{\\partial\\bar{x}_{2}}{\\partial x_{3}}\\\\ \\frac{\\partial\\bar{x}_{3}}{\\partial x_{1}}&\\frac{\\partial\\bar{x}_{3}}{\\partial x_{2}}&\\frac{\\partial\\bar{x}_{3}}{\\partial x_{3}}\\end{pmatrix}\\begin{pmatrix}x_{1}\\\\ x_{2}\\\\ x_{3}\\end{pmatrix}$"}
{"_id": "Cox's_theorem:2", "title": "", "text": "$y\\,f\\left({f(z)\\over y}\\right)=z\\,f\\left({f(y)\\over z}\\right)$"}
{"_id": "List_of_rules_of_inference:120", "title": "", "text": "$\\begin{aligned}\\displaystyle p\\vee q\\\\ \\displaystyle\\neg p\\\\ \\displaystyle\\therefore\\overline{q\\quad\\quad\\quad}\\\\ \\end{aligned}$"}
{"_id": "Discretization_of_Navier–Stokes_equations:1", "title": "", "text": "$\\iiint_{V}\\left[\\frac{\\partial u_{i}}{\\partial t}+\\frac{\\partial u_{i}u_{j}}{\\partial x_{j}}\\right]dV=\\iiint_{V}\\left[-\\frac{\\partial P}{\\partial x_{i}}+\\nu\\frac{\\partial^{2}u_{i}}{\\partial x_{j}\\partial x_{j}}+f_{i}\\right]dV$"}
{"_id": "Randomized_algorithm:35", "title": "", "text": "$O(mn)=O(n^{3}\\log n)$"}
{"_id": "Partial_derivative:38", "title": "", "text": "$\\frac{\\partial^{2}f}{\\partial y\\,\\partial x}=\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial x}\\right)=(f_{x})_{y}=f_{xy}=\\partial_{yx}f.$"}
{"_id": "Lucas–Kanade_method:12", "title": "", "text": "$\\mathrm{v}=(A^{T}A)^{-1}A^{T}b$"}
{"_id": "Spherical_trigonometry:16", "title": "", "text": "$\\cos A=-\\cos B\\,\\cos C+\\sin B\\,\\sin C\\,\\cos a,$"}
{"_id": "Power_series_solution_of_differential_equations:33", "title": "", "text": "$c_{2}=-\\frac{1}{6}\\quad;\\quad c_{3}=-\\frac{1}{108}\\quad;\\quad c_{4}=\\frac{7}{3240}\\quad;\\quad c_{5}=-\\frac{19}{48600}\\ \\dots$"}
{"_id": "Lie_algebra_representation:81", "title": "", "text": "$l_{x}(y)=xy,x\\in\\mathfrak{g},y\\in U(\\mathfrak{g})$"}
{"_id": "Derivations_of_the_Lorentz_transformations:87", "title": "", "text": "$\\begin{bmatrix}t^{\\prime}\\\\ 0\\end{bmatrix}=\\begin{bmatrix}\\gamma&\\delta\\\\ \\beta&\\alpha\\end{bmatrix}\\begin{bmatrix}t\\\\ vt\\end{bmatrix},$"}
{"_id": "Gaussian_integral:35", "title": "", "text": "$\\int x^{k_{1}}\\cdots x^{k_{2N}}\\,\\exp\\left(-\\frac{1}{2}\\sum_{i,j=1}^{n}A_{ij}x_{i}x_{j}\\right)\\,d^{n}x=\\sqrt{\\frac{(2\\pi)^{n}}{\\det A}}\\,\\frac{1}{2^{N}N!}\\,\\sum_{\\sigma\\in S_{2N}}(A^{-1})^{k_{\\sigma(1)}k_{\\sigma(2)}}\\cdots(A^{-1})^{k_{\\sigma(2N-1)}k_{\\sigma(2N)}}$"}
{"_id": "Quintic_function:104", "title": "", "text": "$x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\\,$"}
{"_id": "Bravais_lattice:1", "title": "", "text": "$abc\\sqrt{1-\\cos^{2}\\alpha-\\cos^{2}\\beta-\\cos^{2}\\gamma+2\\cos\\alpha\\cos\\beta\\cos\\gamma}$"}
{"_id": "Affirmative_conclusion_from_a_negative_premise:0", "title": "", "text": "$A\\cap B=\\emptyset$"}
{"_id": "Inverse_beta_decay:1", "title": "", "text": "$e^{-}+p\\to\\nu_{e}+n$"}
{"_id": "Khinchin's_constant:0", "title": "", "text": "$x=a_{0}+\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cfrac{1}{\\ddots}}}}\\;$"}
{"_id": "Ptolemy's_theorem:26", "title": "", "text": "$\\cos(x+y)=\\cos{x}\\cos{y}-\\sin{x}\\sin{y}$"}
{"_id": "Sophie_Germain:6", "title": "", "text": "$θ =kn+ 1$"}
{"_id": "Partial_fraction_decomposition:89", "title": "", "text": "$P-Q_{i}A_{i}=O((x-\\lambda_{i})^{\\nu_{i}})\\qquad$"}
{"_id": "Jaccard_index:26", "title": "", "text": "$J(f,g)=\\frac{\\int\\min(f,g)d\\mu}{\\int\\max(f,g)d\\mu},$"}
{"_id": "Calibration_curve:2", "title": "", "text": "$s_{x}=\\frac{s_{y}}{|m|}\\sqrt{\\frac{1}{n}+\\frac{1}{k}+\\frac{(y_{unk}-\\bar{y})^{2}}{m^{2}\\sum{(x_{i}-\\bar{x})^{2}}}}$"}
{"_id": "Regular_category:2", "title": "", "text": "$0\\to R\\xrightarrow{r-s}X\\to Y\\to 0$"}
{"_id": "Impedance_parameters:3", "title": "", "text": "${V_{1}\\choose V_{2}}=\\begin{pmatrix}Z_{11}&Z_{12}\\\\ Z_{21}&Z_{22}\\end{pmatrix}{I_{1}\\choose I_{2}}$"}
{"_id": "List_of_representations_of_e:17", "title": "", "text": "$e=\\sum_{k=0}^{\\infty}\\frac{3-4k^{2}}{(2k+1)!}$"}
{"_id": "Intraclass_correlation:0", "title": "", "text": "$r=\\frac{1}{Ns^{2}}\\sum_{n=1}^{N}(x_{n,1}-\\bar{x})(x_{n,2}-\\bar{x})$"}
{"_id": "Collaborative_filtering:6", "title": "", "text": "$\\operatorname{simil}(x,y)=\\frac{\\sum\\limits_{i\\in I_{xy}}(r_{x,i}-\\bar{r_{x}})(r_{y,i}-\\bar{r_{y}})}{\\sqrt{\\sum\\limits_{i\\in I_{xy}}(r_{x,i}-\\bar{r_{x}})^{2}\\sum\\limits_{i\\in I_{xy}}(r_{y,i}-\\bar{r_{y}})^{2}}}$"}
{"_id": "Solving_quadratic_equations_with_continued_fractions:7", "title": "", "text": "$x=1+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\cfrac{1}{2+\\ddots}}}}}=\\sqrt{2}.\\,$"}