first500formulae.jso

[{
  "math_input": \mathrm{Ai}(x)
 },
 {
  "math_input": \beth_{d-1}(|\alpha+\omega|^{2^{\aleph_0}})
 },
 {
  "math_input": c^2 = a^2 + b^2 - 2ab\cos(\gamma)\,
 },
 {
  "math_input": W(2, k) > 2^k/k^\varepsilon
 },
 {
  "math_input": b\in\mathbb{N},>1
 },
 {
  "math_input": h=-1
 },
 {
  "math_input":  \Pr\left ( \frac{\sum_{i=1}^n X_i }{n} - 1 \ge \frac{1}{n} \right) \le \frac{ 7 }{ 8 }. 
 },
 {
  "math_input": (X,\Sigma)
 },
 {
  "math_input": \log \colon \mathbb{C}^* \to \mathbb{C}
 },
 {
  "math_input": 1-\left[\frac{15}{16}\right]^{16} \,=\, 64.39%
 },
 {
  "math_input": W = \mathbb R \times (\mathbb R^*)^k \times S^2( (\mathbb R^*)^k) \times \cdots \times S^{m} ( (\mathbb R^*)^k)
 },
 {
  "math_input": \tan\frac{3\pi}{20}=\tan 27^\circ=\sqrt5-1-\sqrt{5-2\sqrt5}\,
 },
 {
  "math_input": 
\chi(T) = T^{2g} + a_1T^{2g-1} + \cdots + a_gT^g + \cdots + a_1q^{g-1}T + q^g,

 },
 {
  "math_input": p = { E \over c } = { hf \over c } = { h \over \lambda }. 
 },
 {
  "math_input": \psi\to e^{i\gamma_{d+1}\alpha(x)}\psi\,
 },
 {
  "math_input": e \Delta\rho \simeq \epsilon_0 k_0^2 \Delta\phi
 },
 {
  "math_input": h_a(n) \,\!
 },
 {
  "math_input": f_c(z) = z^2 + c
 },
 {
  "math_input": H \rightarrow G/N \times G'/N'
 },
 {
  "math_input": A_\mu(x_i)
 },
 {
  "math_input":  ds^2 = g_{00} \, dt^2 + g_{jk} \, dx^j \, dx^k,\;\; j,\; k \in \{1, 2, 3\} 
 },
 {
  "math_input": 
\bar{h}(s,i;L)=\prod_{c=1}^i\sum_{k_c=2+k_{c-1}}^{L-1-2(i-c)}\bar{f}_{k_c}(s)

 },
 {
  "math_input": - f_i^{(k)}  = \frac {(x^{(k)}-x_i)} {p_i} (x^{(k+1)}- x^{(k)}) + \frac {(y^{(k)}-y_i)} {p_i} (y^{(k+1)}- y^{(k)})\;+
 },
 {
  "math_input": 
   \mu = 2C_1~\sum_{i=1}^5 i\,\alpha_i~\beta^{i-1}~I_1^{i-1} \,.
 
 },
 {
  "math_input": f:\mathcal{H}_g \rightarrow V
 },
 {
  "math_input": q(\mathbf{\pi}) \prod_{k=1}^K q(\mathbf{\mu}_k,\mathbf{\Lambda}_k)
 },
 {
  "math_input": R(X_1, \ldots, X_{n})
 },
 {
  "math_input": 
P_{ij}(f)=\frac{ A_{ij}(f)}
{\sqrt{\mathbf{a}^{*}_j(f)\mathbf{a}_j(f)}}

 },
 {
  "math_input": A(x, y)\,dx + B(x, y)\,dy
 },
 {
  "math_input": C_1=0.\,
 },
 {
  "math_input": \frac{\$\text{40m}}{\$\text{30m}} = 1 \frac{1}{3} \approx 1.33
 },
 {
  "math_input": \int u \, dv=uv-\int v \, du.\!
 },
 {
  "math_input": f_2(z)= 1/z \!
 },
 {
  "math_input": \Psi(w,v)=w^\alpha \cdot v = \sum_{i=1}^n w_i^q v_i
 },
 {
  "math_input": v_i = \frac{\partial \Phi}{\partial x_i}
 },
 {
  "math_input": \forall x \, \forall y \, P(x,y) \Leftrightarrow \forall y \, \forall x \, P(x,y)
 },
 {
  "math_input": \mathrm{resultant}(p, T)=0
 },
 {
  "math_input": P(X_i=a)
 },
 {
  "math_input": H= \frac {\delta^*}{\theta}
 },
 {
  "math_input": p = c \cdot u \cdot \rho
 },
 {
  "math_input": \textstyle P ( A \Delta f^{-1}(B) ) = 0. 
 },
 {
  "math_input": p = 1\; \text{GeV}/c = \frac{(1 \times 10^{9}) \cdot (1.60217646 \times 10^{-19} \; \text{C}) \cdot \text{V}}{(2.99792458 \times 10^{8}\; \text{m}/\text{s})} = 5.344286 \times 10^{-19}\; \text{kg}{\cdot}\text{m}/\text{s}.
 },
 {
  "math_input": \alpha(x)
 },
 {
  "math_input": \,2
 },
 {
  "math_input": \nabla T = \omega\otimes T. \, 
 },
 {
  "math_input": f(\lambda x + (1 - \lambda)y)>\min\big(f(x),f(y)\big)
 },
 {
  "math_input": r_{k} = \frac{B_{0} - B_{k}}{B^{*} - B_{0}}
 },
 {
  "math_input": \mathfrak{H} =
\begin{pmatrix}
Z_\infty & - \gamma_1 \gamma_2 \\
1        & - z_\infty
\end{pmatrix}, \;\;
Z_\infty = \gamma_1 + \gamma_2 - z_\infty.

 },
 {
  "math_input": \ x(\tau)
 },
 {
  "math_input": Y_{8}^{6}(\theta,\varphi)={1\over 128}\sqrt{7293\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)
 },
 {
  "math_input": \|u\|=\sqrt{(u|u)}.
 },
 {
  "math_input":  C_T',
 },
 {
  "math_input": \frac{\Box p}p
 },
 {
  "math_input": (a+bi) (c+di) = (ac-bd) + (bc+ad)i.\ 
 },
 {
  "math_input": \Delta\circ N=1
 },
 {
  "math_input": V=5 (Y/19.77)^{0.426}=1.4 Y^{0.426}
 },
 {
  "math_input": \textbf{r}_2-\textbf{r}_3
 },
 {
  "math_input":  \left| \int_{C_L} \frac{f(z)}{5-z} dz \right| \le
2 \pi \rho \frac{\rho^{3/4} (3+1/1000)^{1/4}}{5-1/1000}
\in \mathcal{O} \left( \rho^{7/4} \right) \rightarrow 0.
 },
 {
  "math_input": H=H_e + H_h +V(r_e -r_h)
 },
 {
  "math_input": f \in \mathcal{O}(g)
 },
 {
  "math_input": N_{dopant}
 },
 {
  "math_input": E \cup F
 },
 {
  "math_input": e^{i\mathbf{k \cdot r_{12}}}
 },
 {
  "math_input": f(x) = x^2 - x + 2
 },
 {
  "math_input": e^{c(\ln n)^\alpha(\ln\ln n)^{1-\alpha}}
 },
 {
  "math_input": \frac{1}{\ln p}
 },
 {
  "math_input":  \mbox{E} =\frac{\sqrt{1.64 \cdot N} \cdot  \sqrt{ 120\cdot \pi}}{2\cdot \sqrt{\pi}\cdot d} 

 \approx 7\cdot\frac{ \sqrt{N}}{d}
 },
 {
  "math_input":  \text{Spec }B 
 },
 {
  "math_input": 
J_{\alpha} = 
\int_{0}^{\infty}  \frac{dx}{\left( x + b^{2} \right) \sqrt{\left( x + a^{2} \right)^{3}}}

 },
 {
  "math_input": U_{11} - U_{21}
 },
 {
  "math_input": |{\Psi}\rangle=\sum_{i_1,i_2,\alpha_1,\alpha_2}\Gamma^{[1]i_1}_{\alpha_1}\lambda^{[1]}_{\alpha_1}\Gamma^{[2]i_2}_{\alpha_1\alpha_2}\lambda^{[2]}_{{\alpha}_2}|{i_1i_2}\rangle|{\Phi^{[3..N]}_{\alpha_2}}\rangle
 },
 {
  "math_input": e = O( n^{2/3} m^{2/3} + n + m )
 },
 {
  "math_input": 
R^{m}_{\ell}(-\mathbf{r}) = (-1)^{\ell} R^{m}_{\ell}(\mathbf{r}) . 

 },
 {
  "math_input": \sqrt{\frac{N}{2}}
 },
 {
  "math_input": (t_2,t_1,F_{t_1,t_0}(p)) \in D(X)
 },
 {
  "math_input": \scriptstyle (m\mid k)
 },
 {
  "math_input": n!! - 1
 },
 {
  "math_input": \vec{b} \equiv \vec{B}/B
 },
 {
  "math_input": \mathit{MPC} = \frac{\Delta C}{\Delta Y}
 },
 {
  "math_input": (r,\theta_r,\phi_r)
 },
 {
  "math_input": 
L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,

 },
 {
  "math_input": \sqrt{\det g}\mathcal{D}\Sigma.
 },
 {
  "math_input": (x',y') = (x,y) A + b\,
 },
 {
  "math_input": \tilde{k}\,
 },
 {
  "math_input": \frac{\mathbf{T}(s+\Delta{s})-\mathbf{T}(s)}{\Delta{s}}=-\mathbf{q}(s). 
 },
 {
  "math_input": 34^2
 },
 {
  "math_input": \; P(s_i)
 },
 {
  "math_input": 
\rho_{x^{n}\left(  m\right)  }=\rho_{x_{1}\left(  m\right)  }\otimes
\cdots\otimes\rho_{x_{n}\left(  m\right)  }.

 },
 {
  "math_input":   \mathbf{A}\mathbf{B} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}  + \mathbf{A} \wedge \mathbf{B}.  
 },
 {
  "math_input": \lim \sup _{\alpha} (n_{\alpha}/m_{\alpha}) < r
 },
 {
  "math_input":  \vec{A} = \frac{B}{2}(x\hat{y} - y\hat{x})
 },
 {
  "math_input": 
\begin{align}
q &= q \left(p + 2 q + r\right)\\
&= q p + 2 q^2 + q r\\
&= q^2 + q (p + r) + q^2\\
&= q^2 + q (p + r) + p r\\
&= \left(p + q\right) \left(q + r\right)\\
&= q_1
\end{align}

 },
 {
  "math_input": z*x\le y
 },
 {
  "math_input": \psi(\hat{\alpha}) - \psi(\hat{\alpha} + \hat{\beta})= \ln \hat{G}_X
 },
 {
  "math_input": \lim_\alpha \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t<n\}}.
 },
 {
  "math_input": a_{12}\,dx\wedge dy + a_{13}\,dx\wedge dz + a_{23}\,dy\wedge dz;
 },
 {
  "math_input": \rho(-X)
 },
 {
  "math_input":  \frac{d}{dx}\left( \log_c x\right) = {1 \over x \ln c} , \qquad c > 0, c \ne 1
 },
 {
  "math_input": \textstyle x+C_{i}
 },
 {
  "math_input":  \frac{1}{2}\left( 1 \right) + \frac{1}{2}\left( \frac{1}{2} \right) + \cdots =\sum_{n=0}^{\infty} \frac{1}{2} \left( \frac{1}{2} \right) ^n 
 },
 {
  "math_input":  \{T_a, T_b\} = \frac{4}{3}\delta_{ab} + 2\sum_{c=1}^8{d_{abc} T_c} \,
 },
 {
  "math_input": w_ i(E)=0
 },
 {
  "math_input": \kappa( \cdot, \cdot)
 },
 {
  "math_input":  K = \mathbb{Q} 
 },
 {
  "math_input":  \exists  y \leq n\colon (prf(y,neg(sub(k)))=0 
 },
 {
  "math_input": x_{k+1} = ((n-1)x_k +A/x_k^{n-1})/n
 },
 {
  "math_input": \theta = n \times 137.508^\circ,
 },
 {
  "math_input": \chi_{\text{e}} \mathbf E=N\alpha\mathbf E_{\text{local}}
 },
 {
  "math_input": B^\prime=-(n_b-n_\bar{b})
 },
 {
  "math_input": RD = \min\left(\sqrt{{RD_0}^2 + c^2 t},350\right)
 },
 {
  "math_input": \displaystyle{g^\prime=\begin{pmatrix} a & b \\ c & d \end{pmatrix},}
 },
 {
  "math_input": 
  \frac{1}{(p+1)\left(b^2-4 a\,c\right) \left(c\,d^2-b\,d\,e+a\,e^2\right)}\,\cdot

 },
 {
  "math_input": (n+1)!
 },
 {
  "math_input": s_V(\mathcal{R})
 },
 {
  "math_input": |f(s)g(s)| \le \frac{|f(s)|^p}p + \frac{|g(s)|^q}q,\qquad s\in S.
 },
 {
  "math_input": c_{\rm s}
 },
 {
  "math_input": X,Y,Z
 },
 {
  "math_input": S^k \,
 },
 {
  "math_input": \mathbf{a}_{\mathrm{average}} = \frac{\Delta\mathbf{v}}{\Delta t} 
 },
 {
  "math_input":  \sum_i {}^\phi{V}_i= q V - (q-1)\sum_i V_i \,
 },
 {
  "math_input": WL
 },
 {
  "math_input": \ell(m)
 },
 {
  "math_input": \Gamma = \{ \gamma_i \}_{i =1}^m
 },
 {
  "math_input": id_\tau
 },
 {
  "math_input":  \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{I} \cdot \boldsymbol{\omega} 
 },
 {
  "math_input": x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\cdots}}}
 },
 {
  "math_input": \left( \frac{2}{3} \right) ^3 \times 2^2
 },
 {
  "math_input": \nabla_v f(p)
 },
 {
  "math_input":  Pr\left[\overline{A_1} \wedge \ldots \wedge \overline{A_n}\right]
 },
 {
  "math_input": \mathrm{Poi}\left(\frac{C(23, 2)}{365}\right) =\mathrm{Poi}\left(\frac{253}{365}\right) \approx \mathrm{Poi}(0.6932)
 },
 {
  "math_input": bx-x^2
 },
 {
  "math_input": \texttt{find}
 },
 {
  "math_input":  \mathbf{E}(z,t) = e^{-z / \delta_{skin} } \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})
 },
 {
  "math_input": 
\begin{bmatrix}
0&1&0&1&0&0&0&0&0\\
0&0&1&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0\\
0&0&0&0&1&0&1&0&1\\
0&0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0
\end{bmatrix}

 },
 {
  "math_input": f(x)=  \begin{cases} 
\frac{\nu}{x} \left \{ F_{\nu+2,\mu} \left (x\sqrt{1+\frac{2}{\nu}} \right ) - F_{\nu,\mu}(x)\right \}, &\mbox{if } x\neq 0; \\
\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu} \Gamma(\frac{\nu}{2})} \exp\left (-\frac{\mu^2}{2}\right), &\mbox{if } x=0.
\end{cases}
 },
 {
  "math_input": \Sigma \chi(n)\,
 },
 {
  "math_input": \omega_{k}
 },
 {
  "math_input": f_i(\beta)
 },
 {
  "math_input": \mathbf{m}_1
 },
 {
  "math_input": (-3n,5+5n)
 },
 {
  "math_input":  \operatorname{Weight}(\sigma) = \prod_{i=1}^n a_{i,\sigma(i)}.
 },
 {
  "math_input": \hat{U}^{\dagger}\hat{U} = I,
 },
 {
  "math_input": C\ell(E) = F(E) \times_\rho C\ell_n\mathbb R
 },
 {
  "math_input": \theta=\zeta_n^{a_{g,n}}
 },
 {
  "math_input": \cos a \cos b = \frac{\cos(a - b) + \cos(a + b)}{2}
 },
 {
  "math_input": A/4\ell_\text{P}^2
 },
 {
  "math_input": r_{ij}
 },
 {
  "math_input": P\{N(B)=n\}=\frac{(\lambda|B|)^n}{m!}e^{-\lambda|B|}

 },
 {
  "math_input":  
\mu_k(A-A_k)<\epsilon,~\forall k\geq N.

 },
 {
  "math_input": \Leftrightarrow P(B|A) \ = \ P(B)
 },
 {
  "math_input": \mathcal{C} = \{ \mathbf{q} \in \mathbb{R}^N \}\,,
 },
 {
  "math_input": \vec r (t)
 },
 {
  "math_input":  Z = \sum_{j} g_j \cdot \mathrm{e}^{- \beta E_j}
 },
 {
  "math_input": \tau = \int_{E_{th}}^{E'} dE'' \frac{1}{E''} \frac{D(E'')}{\overline{\xi} \left[ D(E'') {B_g}^2 + \Sigma_t(E') \right]}
 },
 {
  "math_input": \displaystyle{H=f-P(f_{\overline{z}})}
 },
 {
  "math_input": 
V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) }

 },
 {
  "math_input": f_1,\dots,f_{2^n} : \{0,1\}^k\to \{0,1\}
 },
 {
  "math_input": -0.073843+(-0.085588)+3*(0.0086648)=-0.1334366
 },
 {
  "math_input": \theta_{k}-\theta_{k-1}
 },
 {
  "math_input": s_0(1-s_0)
 },
 {
  "math_input": \frac{e^x}{x^x}\,
 },
 {
  "math_input": \frac{\langle E \rangle}{A} = 
\lim_{s\to 0} \frac{\langle E(s) \rangle}{A} = 
-\frac {\hbar c \pi^{2}}{6a^{3}} \zeta (-3).
 },
 {
  "math_input": q_2 = 1+\frac{k+1}{6N}+\frac{k^2}{6N^2}. 
 },
 {
  "math_input": <k> = pN(N-1)
 },
 {
  "math_input":  {v} \,
 },
 {
  "math_input":  \langle 0 | R\phi(x)\phi(y) + \phi(y)R\phi(x)|0\rangle = 0 \, 
 },
 {
  "math_input": S'
 },
 {
  "math_input": \text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).
 },
 {
  "math_input": x=\frac{X-X_0}{\lambda}
 },
 {
  "math_input": Y_1,Y_2,Y_3
 },
 {
  "math_input": 2v_c \sin(\alpha + \beta) = c (\cos(\alpha - \beta) - \cos (\alpha + \beta)).\,
 },
 {
  "math_input":  e(\mathbf{p},u)
 },
 {
  "math_input": p_1 = p_2
 },
 {
  "math_input": P_G(3)
 },
 {
  "math_input": f_i^{(k)}
 },
 {
  "math_input": \Pr(X \leq \mu - \sigma) \geq \frac{ 1 }{ 2 } 
 },
 {
  "math_input": x_r(\theta_r(t))
 },
 {
  "math_input": D(G,H) = \sum_{i=1}^{29} | F_i(G) - F_i(H) |
 },
 {
  "math_input": f(a) = {1 \over 2\pi i} \oint_C {f(z) \over z-a}\, dz. 
 },
 {
  "math_input": \gamma-
 },
 {
  "math_input": C(x_j,x_k)
 },
 {
  "math_input": (gu)h = (gh^{-1})u
 },
 {
  "math_input": \Delta^0_n,
 },
 {
  "math_input": rK=D_K[F(K,L)]*K\,
 },
 {
  "math_input": e(E) \cup e(E) = e(E \oplus E) = e(E \otimes \mathbb{C}) = c_r(E \otimes \mathbb{C}) \in H^{2r}(X, \mathbb{Z}).
 },
 {
  "math_input": p \mid m_i
 },
 {
  "math_input": \! t
 },
 {
  "math_input": \! E_\mathrm{h} / a_0 
 },
 {
  "math_input": 2\omega
 },
 {
  "math_input": d_{z^2}
= Y_2^0
= \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{- x^2 - y^2 + 2 z^2}{r^2}

 },
 {
  "math_input":  E_\text{k} = E_t + E_\text{r} \, 
 },
 {
  "math_input": ((-g)(T^{\mu \nu} + t_{LL}^{\mu \nu}))_{,\mu} = 0 
 },
 {
  "math_input":  \sum_{g \in G} f(g) g
 },
 {
  "math_input":  \longrightarrow 
 },
 {
  "math_input": \begin{pmatrix}
-i & i\\
0 & i
\end{pmatrix}
 },
 {
  "math_input": v(t) = \int_{t_0}^{t} i(\tau) d\tau.\,
 },
 {
  "math_input": \frac{v^{2}}{2c^{2}}\approx 10 ^{-10}
 },
 {
  "math_input": \delta^\prime_0 \Omega^\prime_0 = \left ( \delta_0^{-1} + k^2 + kx - 1 \right ) \delta_0 \Omega_0.
 },
 {
  "math_input": \|Df\|_{\infty,U}\le K
 },
 {
  "math_input": (z_0,\dots,z_n)
 },
 {
  "math_input": \lambda(L(B)) \leq d
 },
 {
  "math_input": A,A^2, A^4,...,A^{2^L}
 },
 {
  "math_input": R_A=R/A=5R/3
 },
 {
  "math_input": \left(\frac{\partial \mathbf{u}}{\partial x}\right)^{\rm T}
 },
 {
  "math_input": (\sigma_x, \tau_{yx})
 },
 {
  "math_input": PFB = \frac{(3200)(FC)}{(FW)(MC)}
 },
 {
  "math_input": L\left(C\right) \leq L\left(T\right)
 },
 {
  "math_input":  P^{\, a} {}_{\, ;\tau} = (q/m)\,F^{\,ab}P_b
 },
 {
  "math_input": v = \frac{c}{n}
 },
 {
  "math_input": \left\{\begin{matrix}ax+by&={\color{red}e}\\ cx + dy&= {\color{red}f}\end{matrix}\right.\ 
 },
 {
  "math_input": P(\Lambda(X)\leq \eta|H_0)=\alpha 
 },
 {
  "math_input": \text{bind}\colon A^{?} \to (A \to B^{?}) \to B^{?} = a \mapsto f \mapsto \begin{cases} \text{Nothing} & \text{if} \ a = \text{Nothing}\\ f \, a' & \text{if} \ a = \text{Just} \, a' \end{cases}
 },
 {
  "math_input": \Delta \tau = \sqrt{\frac{\Delta s^2}{c^2}},\, \Delta s^2 > 0
 },
 {
  "math_input":  f^{*} = \frac{bp - q}{b} = \frac{p(b + 1) - 1}{b}, \! 
 },
 {
  "math_input": \Lambda(d_k)
 },
 {
  "math_input": \alpha_{\tau\tau}-\beta_{\tau\tau}=e^{4\beta}-e^{4\alpha},\,
 },
 {
  "math_input": h=C{{\left[ \frac{k_{v}^{3}{{\rho }_{v}}g\left( {{\rho }_{L}}-{{\rho }_{v}} \right)\left( {{h}_{fg}}+0.4{{c}_{pv}}\left( {{T}_{s}}-{{T}_{sat}} \right) \right)}{{{D}_{o}}{{\mu }_{v}}\left( {{T}_{s}}-{{T}_{sat}} \right)} \right]}^{{}^{1}\!\!\diagup\!\!{}_{4}\;}}
 },
 {
  "math_input": f:M \mapsto N
 },
 {
  "math_input": F=\overline{(A \wedge B) \vee (C \wedge D)}
 },
 {
  "math_input": a^2+c^2=b^2.\quad
 },
 {
  "math_input": SU_{\mu}(2) = (C(SU_{\mu}(2),u)
 },
 {
  "math_input": \mathrm{2\ Squares\ of\ Land} =(\frac{\mathrm{77\ acres}}{\mathrm{3\ Squares\ of\ Land}}) \cdot 2\ Squares\ of\ Land\ = 50.82\ acres 
 },
 {
  "math_input": P(v_i^k = 1|h) = \frac{\exp(a_i^k + \Sigma_j h_j W_{ij}^k)} {\Sigma_{k=1}^K \exp(a_i^k + \Sigma_j h_j W_{ij}^k)}
 },
 {
  "math_input": \sigma_y^2(\tau) = \frac{2\pi^2\tau}{3}h_{-2}
 },
 {
  "math_input": 
  \mathbf{E} = \xi \exp[i(kx - \omega t)] \mathbf{\hat{x}}

 },
 {
  "math_input": (-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = Z_ien_{i0}\nabla\cdot\vec E 
 },
 {
  "math_input": \varphi = 2\cos{\pi\over 5} = \frac{1+\sqrt 5}{2}\qquad\xi = 2\sin{\pi\over 5} = \sqrt{\frac{5-\sqrt 5}{2}} = 5^{1/4}\varphi^{-1/2}.
 },
 {
  "math_input": \scriptstyle M_A(H)
 },
 {
  "math_input": N / \Gamma 
 },
 {
  "math_input": \langle x, y \rangle\ M\ N = M\ x\ y\ N
 },
 {
  "math_input":  X(z) \ 
 },
 {
  "math_input": f_{k,i}
 },
 {
  "math_input": \begin{align} {z \choose k} = \frac{1}{k!}\sum_{i=0}^k z^i s_{k,i}&=\sum_{i=0}^k (z- z_0)^i \sum_{j=i}^k {z_0 \choose j-i} \frac{s_{k+i-j,i}}{(k+i-j)!} \\ &=\sum_{i=0}^k (z-z_0)^i \sum_{j=i}^k z_0^{j-i} {j \choose i} \frac{s_{k,j}}{k!}.\end{align}
 },
 {
  "math_input": C_j^n
 },
 {
  "math_input": \rho: S \times X \rightarrow \{0,1\}
 },
 {
  "math_input": u\equiv\frac{r}{\alpha^2}
 },
 {
  "math_input": (a_1,\ b_1,\ c_1,\ d_1) + (a_2,\ b_2,\ c_2,\ d_2) = (a_1 + a_2,\ b_1 + b_2,\ c_1 + c_2,\ d_1 + d_2).
 },
 {
  "math_input": P(y)\,dy + Q(x)\,dx =0\,\!
 },
 {
  "math_input": =2^2\cdot5\cdot17\cdot3719
 },
 {
  "math_input": A=\frac{2}{3}bh
 },
 {
  "math_input": \displaystyle{K=\begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}.}
 },
 {
  "math_input": \Delta_{\mathrm{adv}}(x-y)
 },
 {
  "math_input": \langle M, N \rangle = \lambda z.\, z M N
 },
 {
  "math_input":  R_{s\ normal} = \sqrt{ \frac{\omega \mu_0} {2 \sigma} }
 },
 {
  "math_input": \!\,\tfrac 12
 },
 {
  "math_input": s, h' \models P
 },
 {
  "math_input":  ([\mathbf{t}]_{\times})^{T} = \mathbf{V} \, (\mathbf{W} \, \mathbf{\Sigma})^{T} \, \mathbf{V}^{T} = - \mathbf{V} \, \mathbf{W} \, \mathbf{\Sigma} \, \mathbf{V}^{T} = - [\mathbf{t}]_{\times} 
 },
 {
  "math_input": \lambda (\lambda 1 (1 ((\lambda 1 1) (\lambda \lambda \lambda 1 (\lambda \lambda 1) ((\lambda 4 4 1 ((\lambda 1 1) (\lambda 2 (1 1)))) (\lambda \lambda \lambda \lambda 1 3 (2 (6 4))))) (\lambda \lambda \lambda 4 (1 3))))) (\lambda \lambda 1 (\lambda \lambda 2) 2)
 },
 {
  "math_input": \begin{align}
 Area &{}= \frac{1}{2} * base * height \\
      &{}= \frac{1}{2} * 2 \pi r * r \\
      &{}= \pi r^2
\end{align}
 },
 {
  "math_input": \frac{1,310,000\ \mathrm{N}}{(2,430\ \mathrm{kg})(9.807\ \mathrm{m/s^2})}=54.97
 },
 {
  "math_input": \phi_1,\phi_2,\phi_3 
 },
 {
  "math_input":  a _{i}
 },
 {
  "math_input": T_r = {T \over T_c}
 },
 {
  "math_input": \exp(-e^{-(x-\mu)/\beta})\!
 },
 {
  "math_input": \Pi\,
 },
 {
  "math_input":   g(E) = \sum_{F \subseteq E} M(F), \forall E \subseteq X .
 },
 {
  "math_input": h={{k_w}\over{D_H}}Nu
 },
 {
  "math_input": \forall x \Big(\forall y (y \in x \rightarrow P[y]) \rightarrow P[x]\Big) \rightarrow \forall x \, P[x]
 },
 {
  "math_input": z^p\overline{z}^q.
 },
 {
  "math_input": v_{(G; c)}(\{1,3\})=23
 },
 {
  "math_input": \rho _{\alpha +} ^{i_0 } \ge A_{\alpha + }^{\sigma (i_0 )} 
 },
 {
  "math_input":  \phi_1 = -30^\circ...+30^\circ
 },
 {
  "math_input": \binom Sk\,
 },
 {
  "math_input": \begin{cases}  
   1 & (e^{-p})\mbox{ no disaster} \\
   1-b & (1-e^{-p})\mbox{ disaster} \\
\end{cases}
 },
 {
  "math_input": \mathbb{U} \mathbb{V} \mathbb{W} \mathbb{X} \mathbb{Y} \mathbb{Z} \!
 },
 {
  "math_input":  f'(x) = n\left((1+x)^{n-1}-1\right)\ge 0 
 },
 {
  "math_input": y = \psi^{-1}(x)
 },
 {
  "math_input": x^{q^{2}}\neq x_{\bar{q}}
 },
 {
  "math_input": \log_2 (1-p) + 1-R
 },
 {
  "math_input": \gamma_I
 },
 {
  "math_input": T \rightarrow \infty
 },
 {
  "math_input": 1928 = [43, 36]_{44}
 },
 {
  "math_input": \ \beta =  \pi - tan^{-1}(\frac{1}{10}) - tan^{-1}(L/D) 
 },
 {
  "math_input":  \vec X(n) = \{ X_d(n) \}, d = 1..D.
 },
 {
  "math_input": dU=TdS-PdV + \sum_i \mu_i dN_i.\,
 },
 {
  "math_input":  S \subseteq [n] 
 },
 {
  "math_input": T_c
 },
 {
  "math_input": 
 \operatorname{E}(X) = \int_0^\infty \int_0^x \;dt\; dF(x) = \int_0^\infty \int_t^\infty dF(x) \;dt = \int_0^\infty (1-F(t)) \;dt.

 },
 {
  "math_input": \mathbb{CFM}_I(R)
 },
 {
  "math_input": t^{\mbox{th}}
 },
 {
  "math_input":  \phi \circ \sigma_{t} = \phi 
 },
 {
  "math_input": \Phi(\phi)
 },
 {
  "math_input": \begin{array}{lll}
\omega_1^1 = 0, & \omega_1^2 = {}^\mathrm{N}\omega^\mathrm{E}_3, & \omega_1^3 = -{}^\mathrm{N}\omega^\mathrm{E}_2, \\
\omega_2^1 = -{}^\mathrm{N}\omega^\mathrm{E}_3, & \omega_2^2 = 0, & \omega_2^3 = {}^\mathrm{N}\omega^\mathrm{E}_1, \\
\omega_3^1 = {}^\mathrm{N}\omega^\mathrm{E}_2, & \omega_3^2 = -{}^\mathrm{N}\omega^\mathrm{E}_1 & \omega_3^3 = 0.
\end{array}
 },
 {
  "math_input": \ p_i 
 },
 {
  "math_input": \ P_2= x_2P^*_2f_{2,M}\,
 },
 {
  "math_input": m = n \sqrt{2}
 },
 {
  "math_input": r_n = (1/2) - x_n h_n
 },
 {
  "math_input": dn_1
 },
 {
  "math_input": P_1(X)=P(X)/(X-\alpha_1)
 },
 {
  "math_input": (S; \wedge, \vee)
 },
 {
  "math_input": H^*_GX,
 },
 {
  "math_input": r_m = r_c ( 1 - t ) \, 
 },
 {
  "math_input": S = -k_B\,\sum_i p_i \ln \,p_i,
 },
 {
  "math_input": 
   U = \frac{1}{2} \int_0^a \int_{-b/2}^{b/2}D\left\{\left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2}\right)^2 +
     2(1-\nu)\left[\left(\frac{\partial^2 w}{\partial x \partial y}\right)^2 - \frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}\right]
     \right\}\text{d}x\text{d}y

 },
 {
  "math_input": n < \lambda \leq n+p
 },
 {
  "math_input": \theta_k(z) = \sum_{\gamma\in\Gamma^*} (cz+d)^{-2k}H\left(\frac{az+b}{cz+d}\right)
 },
 {
  "math_input": \tfrac{\vec x_{n+1}-\vec x_n}{\Delta t}
 },
 {
  "math_input": \phi _1 , \phi _2 , \dots , \phi _{n-1} \,
 },
 {
  "math_input": V_0 = 0,1 
 },
 {
  "math_input": \int_{-\infty}^0 f(x)\,\mathrm{d}x=\pm\infty
 },
 {
  "math_input": a_\text{Red}
 },
 {
  "math_input": (A\rightarrow B,A\rightarrow C \Rightarrow B\rightarrow C)
 },
 {
  "math_input": \|f_{\theta}-f_{\theta'}\|_{L_1}\geq \alpha,\,
 },
 {
  "math_input":  V_1 = k_1 [E_{1T}], 
 },
 {
  "math_input": n \geq n_0 
 },
 {
  "math_input": \{a_n\} \subset G
 },
 {
  "math_input": \|f\|_{L^{p,\infty}(X,\mu)}^p = \sup_{t>0}\left(t^p\mu\left\{x\mid |f(x)|>t\right\}\right).
 },
 {
  "math_input": {R^{2} = {1-{\textit{VAR}_\text{err} \over \textit{VAR}_\text{tot}}}}
 },
 {
  "math_input": = \frac{k}{n}.
 },
 {
  "math_input":  {\alpha \choose k} = \frac{(-1)^k} {\Gamma(-\alpha)k^ {1+\alpha} } \,(1+o(1)), \quad\text{as }k\to\infty. \qquad\qquad(4)
 },
 {
  "math_input":  dT^{2} - \frac{r_{s}}{r} dR^{2}- r^{2}d\Omega^{2}
 },
 {
  "math_input": 
\frac{d\mathbf{\hat{z}}}{dt} = \Omega K \mathbf{\hat{z}}

 },
 {
  "math_input": \sigma^2_N = \frac{(N-1) \, \sigma^2_{N-1} + (x_N - \bar x_{N-1})(x_N - \bar x_{N})}{N}.
 },
 {
  "math_input": (\psi'(\theta))^2/I(\theta)
 },
 {
  "math_input": \Delta\ W_{ij}(n) = \gamma\ \Delta\ W_{ij}(n-1) \Delta\ R(n) + r_i(n) 
 },
 {
  "math_input": \scriptstyle\Phi(x,t):=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right)
 },
 {
  "math_input": \kappa = v \frac{\mu \Delta x}{\Delta P}
 },
 {
  "math_input": x_i(\mathbf{w}, y) = \frac{\partial c (\mathbf{w}, y)}{ \partial w_i}
 },
 {
  "math_input": d^* = \sup_{y^* \in Y^*} \{-f^*(A^*y^*) - g^*(-y^*)\}
 },
 {
  "math_input": \alpha_c : S(c,c)\to T(c,c)
 },
 {
  "math_input": n = \prod_{i=1}^r p_i^{a_i}
 },
 {
  "math_input": H(j \omega) = \mathcal{F}\{h(t)\}
 },
 {
  "math_input": X_3
 },
 {
  "math_input": {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}=0
 },
 {
  "math_input": 
ds^{2} = d\tau^{2} - \frac{r_{g}}{r} d\rho^{2}
- r^{2}(d\theta^{2} +\sin^{2}\theta
d\phi^{2})

 },
 {
  "math_input": e_{ij} = \mathbf{e}_i\cdot\mathbf{e}_j.
 },
 {
  "math_input":  {S_3 \over S_2} = {{16\over15} \div {135\over128}} 
 },
 {
  "math_input": \pi/4
 },
 {
  "math_input": N=O(n)
 },
 {
  "math_input": (S^0, S^1,\dots)
 },
 {
  "math_input": \scriptstyle(\lnot u)\Rightarrow v
 },
 {
  "math_input": (x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}.

 },
 {
  "math_input": \,Q
 },
 {
  "math_input": t_a = t+\frac{|\mathbf r - \mathbf r'|}{c}
 },
 {
  "math_input":  D_x = \frac{1}{i} \frac{\partial}{\partial x}. \,
 },
 {
  "math_input": 
\left( \frac{dr}{d\tau} \right)^{2} = 
\frac{E^2}{m^2 c^2} - c^{2} + \frac{ r_{s} c^2}{r} - 
\frac{h^2}{ r^2 } + \frac{ r_{s} h^2 }{ r^3 }

 },
 {
  "math_input": I_{\mathrm{center}} = \frac{m L^2}{12} \,\!
 },
 {
  "math_input":  h(-,Z) = d\Delta
 },
 {
  "math_input":  \displaystyle M(f) = \sup_{x\in D} \mu(f'(x)).
 },
 {
  "math_input": [b_\alpha^\dagger,b_\beta^\dagger]=[b_\alpha,b_\beta]=0,\quad [b_\alpha,b_\beta^\dagger]=\delta_{\alpha\beta}.
 },
 {
  "math_input": \left(\frac{1 + \sqrt{1-\beta^2}}{2}\right) T
 },
 {
  "math_input":   
D = O^T A O =  \begin{bmatrix}
 \lambda_{-}&0\\ 0 & \lambda_{+}
\end{bmatrix} 
 
    
 },
 {
  "math_input": f( B_1, B_2, \ldots, B_m)\subset B
 },
 {
  "math_input":  \nabla \cdot ( A \nabla u ) = 0 
 },
 {
  "math_input": P_{a\le p\le b} (t) = \int\limits_a^b d p \, |\Phi(p,t)|^2 \,,
 },
 {
  "math_input":  \operatorname{Var}(X \mid X>a) = \sigma^2[1-\delta(\alpha)],\!
 },
 {
  "math_input": X_t=Y_t
 },
 {
  "math_input": (\tfrac{q^*}{p})=1,
 },
 {
  "math_input": \int_{\mathbf{R}^d}(f*g)(x) \, dx=\left(\int_{\mathbf{R}^d}f(x) \, dx\right)\left(\int_{\mathbf{R}^d}g(x) \, dx\right).
 },
 {
  "math_input": R(\hat{n},\phi) \equiv e^{-\frac{i}{\hbar}\phi \mathbf{J}\cdot \hat{n}}
 },
 {
  "math_input":  { X_t = \mathrm{e}^{-t} W_{\mathrm{e}^{2t}} } 
 },
 {
  "math_input": \ [A]_t = -kt + [A]_0
 },
 {
  "math_input": j:N \subset \partial W
 },
 {
  "math_input":  [x_t - x^{*}] = A[x_{t-1}-x^{*}]. \, 
 },
 {
  "math_input": \text{WH}:\{0,1\}^n\to\{0,1\}^{2^n}
 },
 {
  "math_input": \lim_{n \to \infty}(\sum_{k=0}^n \frac{{i x}^k}{k!} (\frac{n!}{n^k (n-k)!} - 1)) = 0
 },
 {
  "math_input": \mathbf{aaaaaa}\,\xrightarrow[\;H\;]{}\,\mathrm{281DAF40}\,\xrightarrow[\;R\;]{}\,\mathrm{sgfnyd}\,\xrightarrow[\;H\;]{}\,\mathrm{920ECF10}\,\xrightarrow[\;R\;]{}\,\mathbf{kiebgt}
 },
 {
  "math_input": \beta_k=\frac{\partial S}{\partial\alpha_k},\quad k=1,2 \cdots N 
 },
 {
  "math_input": \left[\begin{array}{l,l}  s&t\\u&v \end{array}\right]
 },
 {
  "math_input": q^{42}
 },
 {
  "math_input": SSE=\sum_{i=1}^N e_i^2. \, 
 },
 {
  "math_input": [f,2f]
 },
 {
  "math_input": \theta_1 < \theta_2
 },
 {
  "math_input": R_{k,l}
 },
 {
  "math_input": \alpha(d) \le \left(\sqrt{3/2} + \varepsilon\right)^d
 },
 {
  "math_input":  
\begin{cases}
N_j\left(U^\left(n\right)\right)=\Gamma_{jk}U_k^\left(n\right)-U_j^\left(n\right)  \\
M_j\left(U^\left(n\right)\right)=p_i~a_{ijkl}\frac{\partial U_k^\left(n\right)}{\partial x_l}+
\rho^{-1}\frac{\partial }{\partial x_i}\left(\rho~a_{ijkl}~p_l U_k^\left(n\right)\right)  \\
L_j\left(U^\left(n\right)\right)=\rho^{-1}\frac{\partial }{\partial x_i}\left(\rho~a_{ijkl} \frac{\partial U_k^\left(n\right)} {\partial x_l} \right)
\end{cases}

 },
 {
  "math_input": c = {r \over {1-(1+r)^{-N}}} P_0
 },
 {
  "math_input": \mathcal{X}(S(z;u))=\mathcal{X}(u)+z\ 
 },
 {
  "math_input": P_{em} = \frac{3R_r^{'}I_r^{'2}n_r}{sn_s}
 },
 {
  "math_input": \dot q^\mathrm{T}
 },
 {
  "math_input": dE_\theta(t+\textstyle{{r\over c}})=\displaystyle{-d\ell j\omega \over 4\pi\varepsilon_\circ c^2} {\sin\theta \over r} e^{j\omega t}\,
 },
 {
  "math_input": \mathfrak{P}^{51}
 },
 {
  "math_input": \gamma=3\Omega/4\ ,
 },
 {
  "math_input":    f^{\mu} = - 8\pi  { G \over { 3 c^4   }   } \left (  {A \over 2} T_{\alpha \beta}  + {B \over 2} T \eta_{\alpha \beta} \right ) \left ( \delta^{\mu}_{\nu} + u^{\mu} u_{\nu} \right )  u^{\alpha} x^{\nu} u^{\beta} 
 },
 {
  "math_input": = \sum_{k=1}^{d} \left(\dot v_k \   + \sum_{j=1}^{d} \sum_{i=1}^{d}v_j{\Gamma^k}_{ij}\dot q_i    \right)\boldsymbol{e_k}  \ . 
 },
 {
  "math_input": \qquad{\it (Comp1)} \quad \frac{\displaystyle M \ \rightarrow
\ M'} {\displaystyle M\|N \ \rightarrow \ M'\|N}; \qquad \qquad {\it (Comp2)}
\quad \frac{\displaystyle M \ \rightarrow \ M'\qquad\displaystyle N
\ \rightarrow \ N'} {\displaystyle M\|N \ \rightarrow \ M'\|N'}
 },
 {
  "math_input": \exp\left(\sum_{n=1}^\infty {a_n \over n!} x^n \right)
= \sum_{n=0}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n.
 },
 {
  "math_input": (\cdot,\,\cdot)
 },
 {
  "math_input": F_0=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}
 },
 {
  "math_input":  \frac{D_g u_g}{Dt} - f_{0}v_a - \beta y v_g = 0 
 },
 {
  "math_input": (\alpha_j - \alpha_i)
 },
 {
  "math_input": S \subset L\,
 },
 {
  "math_input": \frac{2}{1/60+1/40}=48.
 },
 {
  "math_input": \mathbf{v}_{A\text{ relative to }B} = \mathbf{v} - \mathbf{w}
 },
 {
  "math_input": \operatorname{E} (X_t)=\operatorname{E} (c)+\varphi\operatorname{E} (X_{t-1})+\operatorname{E}(\varepsilon_t),

 },
 {
  "math_input": \overline{z} = z \!\ 
 },
 {
  "math_input": I_c
 },
 {
  "math_input": \frac{9}{10^1} + \frac{9}{10^2} + \frac{9}{10^3} + \cdots = \sum_{n=1}^{\infty}\frac{9}{10^n}
 },
 {
  "math_input": \sum_{n=0}^\infty c_n(z-\alpha)^n
 },
 {
  "math_input": \, A \mapsto M\alpha(A)M^{-1} ,
 },
 {
  "math_input": 
E\bar{X}_A = \mu_{HA}\frac{p_{HA}}{p_{HA}+p_{LA}} + \mu_{LA}\frac{p_{LA}}{p_{HA}+p_{LA}},

 },
 {
  "math_input": \alpha^L
 },
 {
  "math_input":  {d^2 \bar h^i \over ds^2} + 2 \Gamma^i_j {d \bar h^i \over ds} + {d \Gamma^i_j \over ds} \bar h^j + \Gamma^i_j \Gamma^j_k \bar h^k + \bar R^i_j \bar h^j = 0 
 },
 {
  "math_input": \tbinom24
 },
 {
  "math_input": U=\frac{U_*}{\kappa}ln\left(\frac{z}{z_0}\right)
 },
 {
  "math_input": \part_0^2f_p(x)=\left(\Delta-m^2\right) f_p(x)
 },
 {
  "math_input": det(A)\ne 0
 },
 {
  "math_input": i^2= -1
 },
 {
  "math_input": \scriptstyle R_\text{out}
 },
 {
  "math_input": f(x_i) = \sum_{f=1}^n c_j \mathbf K_{ij}
 },
 {
  "math_input": \sqrt{M/3}
 },
 {
  "math_input": (a_n)_{n\in\N} \times (b_n)_{n\in\N} = \left( \sum_{k=0}^n a_k b_{n-k} \right)_{n\in\N}.
 },
 {
  "math_input": C=\{C_k^i\}
 },
 {
  "math_input": 
M(X) = \left( {\begin{array}{*{20}c}
   \mu   \\
   \Sigma   \\
\end{array}} \right)

 },
 {
  "math_input": \sigma_\text{l}
 },
 {
  "math_input":  \frac{\partial F \left( u   \left( t \right) \right)}{ \partial u}.  
 },
 {
  "math_input":  \ddot{r} = \frac{1}{2} \, \frac{d}{dr} \left( (E^2-V) \, (1+m/r)^4 \right) 
 },
 {
  "math_input":  f,g_1,\ldots,g_n\in H
 },
 {
  "math_input": 
(0, 653, 1854, 4063) \rightarrow
(653, 1201, 2209, 4063) \rightarrow
(548, 1008, 1854, 3410) \rightarrow

 },
 {
  "math_input":  (\sqrt{2},1); \quad (-\sqrt{2},1); \quad (\sqrt{2},-1); \quad (-\sqrt{2},-1); \quad (0,\sqrt{3}); \quad (0,-\sqrt{3}). 
 },
 {
  "math_input":  (X_4,X_2,X_9) \, 
 },
 {
  "math_input": \rho(\lambda) = \frac{G_{\mathrm{refl}}(\lambda)}{G_{\mathrm{incid}}(\lambda)}
 },
 {
  "math_input": \frac{b^2}{\sqrt{a^2-b^2}}
 },
 {
  "math_input": A \leq_{F} B
 },
 {
  "math_input": u(x,\dot{x})
 },
 {
  "math_input": \|x\|_H = \sqrt{\langle x, x \rangle},
 },
 {
  "math_input": c r^n \in I^n
 },
 {
  "math_input": (x-c_2)^2
 },
 {
  "math_input": R\mathcal S(\mathcal F \ast \mathcal G)  = R\mathcal S(\mathcal F) \otimes R\mathcal S(\mathcal G)
 },
 {
  "math_input": \int_a^b f(x,\alpha)\;dx=\phi(\alpha).\,
 },
 {
  "math_input": p_\varepsilon (x,t) = 0\text{ for }x \in \partial\Omega_a
 },
 {
  "math_input": \Gamma_{\infty}
 },
 {
  "math_input": \hbar {\mathbf k'}
 },
 {
  "math_input": v_e = \sqrt{\;\frac{T\;R}{M}\cdot\frac{2\;k}{k-1}\cdot\bigg[ 1-(p_e/p)^{(k-1)/k}\bigg]} 
 },
 {
  "math_input": n_2^2\sigma_2^2-2\sigma_2n_2^2\sigma_\mathrm{n}+n_2^2\lambda=0\,\!
 },
 {
  "math_input": \mathrm{SO}(2)
 },
 {
  "math_input": f(X)=1+X+X^2+\dots
 },
 {
  "math_input": N=g^{\mu\nu}K_\mu K_\nu\;
 },
 {
  "math_input": f(b)-f(a)\geq f(x_n+0)-f(x_1-0)=\sum_{i=1}^n [f(x_i+0)-f(x_i-0)]+
 },
 {
  "math_input":  y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a 
 },
 {
  "math_input": \exp(\lambda (e^{t} - 1))
 },
 {
  "math_input": d^2=4*x*b_{7}*c_{12}^2=
 },
 {
  "math_input": \underline{x} \in \R^n
 },
 {
  "math_input": c_2 = 2.04901523, \,\!
 },
 {
  "math_input": F_{SR} = -p_{SR}c_{R}A_{\odot}r_{\oplus\odot}
 },
 {
  "math_input": \{\psi_l \}
 },
 {
  "math_input": \{| \phi_i \rangle\}
 },
 {
  "math_input": s_a(t).\,
 },
 {
  "math_input": L(w)
 },
 {
  "math_input": \mathrm{core}_2
 },
 {
  "math_input": 
a = \frac{a^4+b^4+c^4+a^2b^2+b^2c^2+c^2a^2}{\left( a^2+b^2+c^2 \right)^2} \Delta

 },
 {
  "math_input": \mathcal{E}(\exp)=\{0\}
 },
 {
  "math_input": e^{(1)}_i = a_i
 },
 {
  "math_input": \sigma = 0, \sigma = 0.2, \sigma = 0.4, \sigma = 0.6, \sigma = 0.8, \sigma = 1
 },
 {
  "math_input": D = \prod_{i=1}^K d_i.
 },
 {
  "math_input": \begin{align}
1-\theta+\theta e^u &= \theta \left (\frac{1}{\theta} - 1 + e^u \right) \\
& = \theta \left ( -\frac{b}{a} + e^u \right ) \\
& > 0 && \theta > 0, \quad \frac{b}{a} <0
\end{align}
 },
 {
  "math_input":   B_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{y}_{x\gamma}p^{y}_{\gamma x} }{ E_0-E_{\gamma} },  
 },
 {
  "math_input": ji=-k
 },
 {
  "math_input": \sigma_{mk}
 },
 {
  "math_input": {\scriptstyle \ast:G\times H\rightarrow H}
 },
 {
  "math_input": \mathcal{O}(x_1,\ldots,x_n)
 },
 {
  "math_input": \operatorname{P}(X\leq m) = \operatorname{P}(X\geq m)=\int_{-\infty}^m f(x)\, dx=\frac{1}{2}.\,\!
 },
 {
  "math_input": Y = \beta T_8 + I X
 },
 {
  "math_input":  \frac{g[m=L]}{\sum_{q=1}^M D[q,q]} \ge 0.9\, 
 },
 {
  "math_input": 
\psi_0 |0\rangle + \int_x \psi_1(x) |1;x\rangle + \int_{x_1x_2} \psi_2(x_1,x_2)|2;x_1 x_2\rangle + \ldots
\,
 },
 {
  "math_input": F = GHB
 },
 {
  "math_input": \scriptstyle k_i
 },
 {
  "math_input": k_0 \in (K_0 \cap K_\pm)
 },
 {
  "math_input": n_{2}=\sum\limits_{\alpha_l=1}^{\chi_c} (c_{\alpha_{{{l-1}}}\alpha_{l}})^2\cdot({\lambda'}^{[l]}_{\alpha_l})^2=\sum\limits_{\alpha_l=1}^{\chi_c}(c_{\alpha_{{{l-1}}}\alpha_{l}})^2\frac{(\lambda^{[l]}_{\alpha_l})^2}{R} = \frac{S_1}{R}
 },
 {
  "math_input":  \mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ 
                               0 & \mbox{ if } 0 \notin A.
\end{cases}
 },
 {
  "math_input": \theta = 90^\circ
 },
 {
  "math_input": 
J:X\to (X'_\beta)'_\beta.

 },
 {
  "math_input": \ell _{({M},\varphi )}({\bar x},{\bar y})=\sum _{p=(x,y)\atop x\le {\bar x}, y>\bar y }\mu\big(p\big)+\sum _{r:x=k\atop k\le {\bar x} }\mu\big(r\big)
 },
 {
  "math_input": \hat{x} = (A^{T}A+ \Gamma^{T} \Gamma )^{-1}A^{T}\mathbf{b}
 },
 {
  "math_input": Ax = y.
 },
 {
  "math_input": P(k,k') = \frac {2 \pi} {\hbar} \mid \langle k' , q' | H_{el}| \ k , q \rangle \mid ^ {2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]
 },
 {
  "math_input": 
g(s) = \int_0^{\infty} (st)^{-k-1/2} \, e^{-st/2} \, W_{k+1/2,\,m}(st) \, f(t) \; dt,

 },
 {
  "math_input":  (R,W)= (Z_1 +Z_2, Z_1/(Z_1 + Z_2)) 
 },
 {
  "math_input": F \in [0,2]
 },
 {
  "math_input": {D}_{4}^{(3)}
 },
 {
  "math_input": \lambda_{in}
 },
 {
  "math_input": ~{\rm slog}_b(z)~
 },
 {
  "math_input": \Omega^8\operatorname{BSp}\simeq \mathbf Z\times \operatorname{BSp} ;\,
 },
 {
  "math_input": rpm_{motor}
 },
 {
  "math_input": \frac{\delta^3}{\delta J(x_1)\delta J(x_2) \delta J(x_3)}Z[J]
 },
 {
  "math_input": 1 + 2\;
 },
 {
  "math_input": \tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1}
 },
 {
  "math_input": a_{t+1} = (1 + r) (a_t - c_t), \; c_t \geq 0,
 },
 {
  "math_input": \Beta;\ G;\ \Upsilon
 },
 {
  "math_input": g\isin [(X\times Y)\to Z]
 },
 {
  "math_input": \underbrace{u_1(\mathbf{x},z_1)=v_1+\dot{u}_x}_{\text{By definition of }v_1}=\overbrace{-\frac{\partial V_x}{\partial \mathbf{x}}g_x(\mathbf{x})-k_1(\underbrace{z_1-u_x(\mathbf{x})}_{e_1})}^{v_1} \, + \, \overbrace{\frac{\partial u_x}{\partial \mathbf{x}}(\underbrace{f_x(\mathbf{x})+g_x(\mathbf{x})z_1}_{\dot{\mathbf{x}} \text{ (i.e., } \frac{\operatorname{d}\mathbf{x}}{\operatorname{d}t} \text{)}})}^{\dot{u}_x \text{ (i.e., } \frac{ \operatorname{d}u_x }{\operatorname{d}t} \text{)}}
 },
 {
  "math_input": \nabla_{\bold u}{\bold v}(P)
 },
 {
  "math_input":  \frac{dv}{dx} = \frac{dv\,k_{GT}}{d\,T_m(x)\,x^2}
 },
 {
  "math_input": a + b = 1 + (a + (b - 1)),\,\!
 },
 {
  "math_input":  P( | X - E( X ) | > t [ E( X - E( X ) )^{ 2k } ]^{ 1 / 2k } ) \le min[ 1, \frac{ 1 }{ t^{ 2k } } ] 
 },
 {
  "math_input": [ES] = \frac{[E]_0 [S]}{K_m + [S]}
 },
 {
  "math_input": \Delta_x \subset T_xM
 },
 {
  "math_input": E \left[ \hat{\sigma}^2\right]= \frac{n-1}{n} \sigma^2
 },
 {
  "math_input": v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \\ - \frac{1}{\sqrt{\mu}} \gamma^* & \alpha^* \end{matrix} \right).
 },
 {
  "math_input": \begin{align} \text{var} (a) &= \frac{3 \sigma^2}{2 \sqrt{\pi} \, \delta_x Q^2 c} \\ \text{var} (b) &= \frac{2 \sigma^2 c}{\delta_x \sqrt{\pi} \, Q^2 a^2} \\ \text{var} (c) &= \frac{2 \sigma^2 c}{\delta_x \sqrt{\pi} \, Q^2 a^2} \end{align}
 },
 {
  "math_input": i/k^2
 },
 {
  "math_input": X_1, X_2, Y_1, Y_2
 },
 {
  "math_input":  r = \cos^3 \theta + \sin^3 \theta 
 },
 {
  "math_input": \mathbb{H}P^2
 },
 {
  "math_input":  \gamma^\mu 
 },
 {
  "math_input": {V_{D}} = {V_{P}} + {V_{T}} \left(\frac{fu}{fu_{t}}\right)
 },
 {
  "math_input": a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\,
 },
 {
  "math_input": \begin{cases} y = t^5, \\ x = t^3. \end{cases}
 },
 {
  "math_input": \mathsf{fv}
 },
 {
  "math_input": \sum_x \sum_y I(x,y) \,\!
 }
]