harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\rm Vol}= \int_0^\pi d\theta \, \int_0^{2\pi}d\phi \, \int_0^{4\pi}d\psi \, \sqrt{\det g}= \int_0^\pi d\theta \, \int_0^{2\pi}d\phi \, \int_0^{4\pi}d\psi \left({\ell \over 2}\right)^3 \, \sin\theta = \ell^3 \, 2\pi^2 \ .
p(y_1^n|x^n) = \sum_{w^n}p(w^n) p(y_1^n | x^n, w^n) &= \prod_{i=1}^n \sum_{w_i}p_W(w_i)p_{Y_1|X,W}(y_{1i}|x_i,w_i) = \prod_{i=1}^n p_{Y_1|X}(y_{1i}|x_i).
H_{\rm sph}= \frac{1}{2}|\vec{L}_O|^2 +|\vec{r}|^2 V_{\rm Cal}\hphantom{H_{\rm sph}}{}= \frac{1}{2}\big ( (yp_z-zp_y)^2+(zp_x-xp_z)^2+(xp_y-yp_x)^2\big)+\big(x^2 +y^2+z^2\big) V_{\rm Cal}
\mathbb E_x^{(t)}(Z_t(Z_t-1)...(Z_t-n) s^{Z_t-n-1}, T_1^*\in dt_1, ..., T_n^*\in dt_n)/ dt_1...dt_n= & \frac{n! (n+1)!}{2^n}x\psi_t^{'}(s) \psi_t(s)^{x-1}\prod_{i=1}^n \psi_{t_i}^{'}(s) \big[ \sum_{k\geq 2}k(k-1)\mu (k) \psi_{t_i}(s)^{k-2}\big], s\in [0,1).
\sigma_k(B)= \sigma_k(B_1+B_2) = \sigma_k(B_1)+\sigma_k(B_1, B_1, \ldots, B_1, B_2) &+\cdots+\sigma_k(B_1, B_2,\ldots, B_2, B_2)+\sigma_k(B_2) = u_1^{4k}\sigma_k(b^1_{ij})+O(u_1^{4k-2}),
(E \otimes F_1) \otimes w_{p_1 - p_2}= (1 \otimes F_1)(E_{\Delta}- 1 \otimes \alpha(E)) \otimes w_{p_1 - p_2}& \in (p_1 - p_2 - 1)(1 \otimes F_1) \otimes w_{p_1 - p_2 - 1}- (1 \otimes F_2) \otimes w_{p_1 - p_2}+ Z.
{\rm Tutte}(\{q_{ij}\};x,y,z):= (-1)^{n-1}{\rm Coef\/f}_{[t_1 \cdots t_n]}{}\times \Biggl[\Biggl( \sum_{\ell_1,\ldots,\ell_n \atop \ell_i \in \{0,1 \}, \, \forall \,i}\prod_{1 \le i < j \le n}(z q_{ij}y+1)^{\ell_i \ell_j}y^{- (\sum_{j}\ell_{j})}t_1^{\ell_{1}}\cdots t_n^{\ell_{n}}\Biggr)^{x y}x^{-1}\Bigg] .
( \psi_{\omega}^{( \lambda )}, \psi_{\sigma}^{( \lambda )}) ={\frac{1}{( \omega - \sigma )}}\left[ ( \psi_{\sigma}^{( \lambda )}, H ( \lambda ) \psi_{\omega}^{( \lambda )})^{*}- ( \psi_{\omega}^{( \lambda )}, H ( \lambda ) \psi_{\sigma}^{( \lambda )}) \right]
\ln \left| \frac{Y_{+}^{+ \pm}}{Y_{-}^{+ \pm}}\right| \sim \ln \left| \frac{Y_{+}^{- \mp}}{Y_{-}^{- \mp}}\right| \sim \pm \ln 2 \pm \ln \left| \frac{k q}{\Omega \sqrt{k^{2}- q^{2}}}\right| +{\textstyle \frac{3}{2}}\ln \left| \frac{k + q}{k - q}\right| .
lim_{h\rightarrow0}g(a+h)-g(a)=lim_{h\rightarrow0}\frac{g(a+h)-g(a)}{h}\cdot h=lim_{h\rightarrow0}\frac{g(a+h)-g(a)}{h}\cdot lim_{h\rightarrow0}h=g^{\prime}(a)\cdot0=0
E_4 \left( \frac{z-1}{2z}\right) = E_4 \left(\frac{w+1}{2w+3}\right) = (2w+3)^4 E_4(w) &= z^4 E_4 \left(\frac{z-3}{2}\right) = z^4 E_4 \left(\frac{z}{2}+ \frac{1}{2}\right),
H_{y}=[jk_{x\epsilon}(Ae^{-jk_{x\epsilon}x}-Be^{jk_{x\epsilon}x})+\frac{k_{z}}{\omega\mu}\frac{m\pi}{a}(Ce^{-jk_{x\epsilon}x}+De^{jk_{x\epsilon}x})]sin(\frac{m\pi}{a}y)e^{-jk_{z}z}(37)
n_{x}=\begin{pmatrix}I_{m}-\frac{2x^{t}x}{1+|x|^2}-\frac{2x^{t}}{1+|x|^2}0 \frac{2x}{1+|x|^2}\frac{1-|x|^2}{1+|x|^2}0 0 0 1 \end{pmatrix}\exp(-\log(1+|x|^2)H_{0})\bar{n}_{\frac{x}{1+|x|^2}}.
ln2=7\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{15^{n}n}+3\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{80^{n}n}+5\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{24^{n}n}.
( r^{2}- r_{H}^{2}) \partial_{r}^{2}R_{( \gamma )}( r ) + \frac{1}{r}\left[ 2 r^{2}+ \gamma ( r^{2}- r_{H}^{2}) \right] \partial_{r}R_{( \gamma )}( r ) + \frac{\omega^{2}\ell^{4}}{( r^{2}- r_{H}^{2})}R_{( \gamma )}( r ) = 0 .
\sum_{i=1}^r\alpha_i^2+\alpha_1\alpha_2=\sum_{i=1}^r a_i^2+a_1a_2-\sum_{i=1}^r\frac{6a_i}{3r-2}+\frac{4\cdot 3+9(r-2)}{(3r-2)^2}=\sum_{i=1}^r a_i^2+a_1a_2-\frac{3}{3r-2}\,.
b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})=\epsilon_{1}^{2}-m_{1}^{2}=\epsilon_{2}^{2}-m_{2}^{2}=-\frac{1}{4P^{2}}(P^{4}+2P^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2}).
\sum_{n\in J}|\langle x,f_n\rangle|^2+\left\|\sum_{n\in \Bbb N\setminus J}\langle x,f_n\rangle f_n\right\|^2 =\sum_{n\in \Bbb N \setminus J}|\langle x,f_n\rangle|^2+\left\|\sum_{n\in J}\langle x,f_n\rangle f_n\right\|^2
h\varpi_p = \left(\begin{array}{cc}xp^{-1}y zp^{-1}t\end{array}\right) = \left(\begin{array}{cc}-z^{-1}\det(h) xp^{-1}0 zp^{-1}\end{array}\right) \left(\begin{array}{cc}0 1 1 z^{-1}tp\end{array}\right).
\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\mathbf{1}_{S_k^\mathbf{I}(\mathbf{x}')}(i) \leq \limsup_{n\to\infty}\frac{|{R'}_{k,N}^\mathbf{I}\cap[0,n-1]\cap G|}{|[0,n-1]\cap G|}\cdot 1+1\cdot (p_k^\mathbf{I}+\epsilon) \leq p_k^\mathbf{I}+\epsilon\left(1+\frac{1}{c}\right).
I_{(X_-,D_-)}(y,z)=ze^{t_-/z}\sum_{d\in\mathbb K_{-}}\tilde y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right)\textbf{1}_{[-d]}I_{D_-,d},
A_\mu=\pm\,[\,{i\over g}\,(\,H_2-H_1)\,{v_\mu^1\over\rho_1}+\kappa\,E^\pm_{1\,2}\,R_\mu^1]\pm\,[\,{i\over g}\,(H_4-H_3)\,{v_\mu^3\over\rho_3}+\kappa\,E^\pm_{3\,4}\,R_\mu^3\,].
0<Q^2<\frac{16}{3 \omega_4^2}\,\biggl[-\frac{\ell^2}{3}\left(\omega_4 M+ \frac{2\ell^2}{9}\right)+\frac{2\ell}{3\sqrt{3}}\left(\omega_4 M+ \frac{\ell^2}{3}\right)^{3/2}\biggr]\,,
\sup_{M \times [0, T_0]}\sum_a \big( | u^a_k - u^a_l| + | \nabla_H u^a_k - \nabla_H u^a_l | \big) \leq K^2 C_3 C_\beta e^{2 C_1}||d \phi||^2_{C^0}\sum_{i=k+1}^l \bigg( \frac{1}{2}\bigg)^{i-2}.
V_{p}([0,T];\Omega) := \Big\{u \in L^{2p}([0,T];L_{e}^{2}(\Omega)) \cap L^{2}([0,T];H_{e}^{\beta}(\Omega)) & g_{1-\alpha}*(u-u_{0}) \in C([0,T];L_{e}^{2}(\Omega)), (g_{1-\alpha}*(u-u_{0}))|_{t=0}= 0 \Big\},
{\frac{d M^{( 1 )}}{d m}}={\frac{d ( \delta M )}{d m}}+{\frac{1}{2}}\sum_{n}{\frac{d \omega_{n}^{B}}{d m}}-{\frac{1}{2}}\sum_{n}{\frac{d \omega_{n}^{F}}{d m}}- \left({\frac{1}{2}}\sum_{n}{\frac{d \tilde{\omega}_{n}^{B}}{d m}}-{\frac{1}{2}}\sum_{n}{\frac{d \tilde{\omega}_{n}^{F}}{d m}}\right)
\frac{\partial^{2}F}{( \partial v^{1})^{2}}= \phi_{1 1}^{m}+ \sum_{A = 2}^{m}[ 2 \phi_{1 A}^{m}f_{1}^{A}+ \phi_{A}^{m}f_{1 1}^{A}+ \sum_{B = 2}^{m}( \phi_{A B}^{m}f_{1}^{B}) f_{1}^{A}]
\lim_n \frac{\mu_{x}( Z( gz_n) )}{\mu_{x}(Z(z_n) )}=\lim_n \frac{\mu_{x}( Z(e f_{i_1-1}e f_{i_2-1}\ldots f_{i_n-1}e) )}{\mu_{x}( Z( e f_{i_1}e f_{i_2}\ldots f_{i_n}) )}=\lim_n \frac{T_{i_1-1,i_2-1}\cdots T_{i_{n-1}-1,i_n-1}}{T_{i_1,i_2}\cdots T_{i_{n-1},i_n}},
E(x,y;u):=\sum_{n=0}^{\infty}u^{n}\psi_{n}(x)\psi_{n}(y)=\frac{1}{\sqrt{\pi(1-u^{2})}}exp(-\frac{1-u}{1+u}\frac{(x+y)^{2}}{4}-\frac{1+u}{1-u}\frac{(x-y)^{2}}{4}),
f = \frac{\theta_{3}( u | \tau / 2 )}{\theta_{4}( u | \tau / 2 )}={\frac{\theta_{3}( 2 u | 2 \tau ) + \theta_{2}( 2 u | 2 \tau )}{\theta_{3}( 2 u | 2 \tau ) - \theta_{2}( 2 u | 2 \tau )}}~ .
\begin{array}{cc}\theta^{i}=\left(\begin{array}{c}\vartheta^{a}_{-}(\bar{\vartheta}^{-}_{a})^{t}\end{array}\right)~~~~~ ~~~~~\tilde{\theta}^{i}=\left(\begin{array}{c}\vartheta^{a}_{+}(\bar{\vartheta}^{+}_{a})^{t}\end{array}\right)\end{array}
W_{I_{1}\ldots I_{n}}\, \left( z_{1}, \bar{z}_{1}, \ldots , z_{n}, \bar{z}_{n}\right) \prod_{j = 1}^{n}( d \, z_{j})^{h_{I_{j}}}\, ( d \, \bar{z}_{j})^{\bar{h}_{I_{j}}}
\eta^{12}b_1= (x,y) + (xa+yc)e_1+(xb+yd)e_2 = (x,y) -u(xa+yc)(E_6,6E_4)+u(xb+yd)(E_4^2,6E_6) + (xa+yc)(\eta^{12},0)+(xb+yd)(0,6\eta^{12})
\begin{aligned}2\int_{r>r_{0},0<\theta<\pi}\eta|\nabla\omega|^{2}dxdy= \int_{r>r_{0},0<\theta<\pi}\omega^{2}(\Delta\eta+\boldsymbol{w}\cdot\nabla\eta)dxdy &+\int_{r>r_{0},0<\theta<\pi}\nabla\eta\cdot (0,b)\omega^{2}dxdy.\end{aligned}
D\left( \begin{array}{c}u^+(t)\frac{1}{\sqrt{z}}u^-(t)\frac{1}{\sqrt{\bar{z}}}\end{array}\right)=\left( \begin{array}{c}\partial_t u^+(t)\frac{1}{\sqrt{z}}\partial_t u^-(t)\frac{1}{\sqrt{\bar{z}}}\end{array}\right).
m_{k}\frac{dq^{k}}{dt}(t)=\hbar\nabla_{k}Imln\psi(q,t)=\hbar Im(\frac{\nabla_{k}\psi}{\psi})(q,t)=\frac{m_{k}j_{k}}{\psi^{*}\psi}=Re(\frac{\hat{P}_{k}\Psi}{\Psi}),
{\cal L}= \frac{1}{2}\, \partial \varphi \! \cdot \!{\overline{{\partial}}}\varphi - V ( \varphi ) \equiv \frac{1}{2}\, \partial \varphi \! \cdot \!{\overline{{\partial}}}\varphi - \sum_{r = 1}^{n}\; v_{r}( \varphi ) ,
\delta u_{\underline{{{m}}}}^{( 0 )}= u_{\underline{{{m}}}}^{i}i_{\delta}f^{i}, \qquad \delta u_{\underline{{{m}}}}^{i}= - u_{\underline{{{m}}}}^{j}i_{\delta}A^{j i}+ u_{\underline{{{m}}}}^{( 0 )}i_{\delta}f^{i},
\left(\begin{array}{c}\hat{{\bf b}}\hat{{\bf b}}^{\dagger}\end{array}\right)=\Omega\left(\begin{array}{c}\hat{{\bf a}}\hat{{\bf a}}^{\dagger}\end{array}\right)+\left(\begin{array}{c}{\bf d}{\bf d}^{*}\end{array}\right),\qquad\Omega =\left(\begin{array}{cc}\zeta \eta \eta^{*}\zeta^{*}\end{array}\right),
\sum_{j=m_0+1}^m F_j^*(\omega_{\mathbb{B}^{N_j}}^p) = \sum_{j=1}^{m_0}\Xi_j \wedge \left( \sum_{t=0}^{p-1}\omega^t_{\mathbb{B}^n}\wedge (F_j^*\omega_{\mathbb{B}^{N_j}})^{p-1-t}\right) \geq \sum_{j=1}^{m_0}\Xi_j \wedge \omega^{p-1}_{\mathbb{B}^n}.
\lim_{n\rightarrow \infty}\frac{\ell_R(R/I_n)}{n^2}=\lim_{n\rightarrow \infty}\frac{\ell_{\hat R}(\hat R/I_n\hat R)}{n^2}=\lim_{n\rightarrow\infty}\frac{\ell_{R_1}(R_1/I_nR_1)}{n^2}+\lim_{n\rightarrow\infty}\frac{\ell_{R_2}(R_2/I_nR_2)}{n^2}.
\lambda\|S(V|\Psi^3)\|^2_{2}=\lambda\iint_\Omega |\nabla V|^2\Psi^3\frac{dtdx}{|t|^{n-d-2}}\leq \iint_\Omega \mathcal{A}\nabla V \cdot \nabla V \Psi^3\frac{dtdx}{|t|^{n-d-2}}
\min\,\, \varphi_0(x(0),x(T)), &\dot{x}=F(x,u,v),{\rm a.e.}\{\rm on}\ [0,T], & \eta_j(x(0),x(T))=0,\mathrm{for}\j=1\hdots,d_{\eta}, & \varphi_i(x(0),x(T))\leq 0,\mathrm{for}\i=1,\hdots,d_{\varphi}, & u(t)\in U ,\,\, v(t)\in V,{\rm a.e.}\{\rm on}\ [0,T],
\int \eta_i(x) f(\sigma^x\xi, \sigma^x\omega) d\nu_{\widehat \theta}^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)}f(\xi, \omega) d\nu_{\widehat \theta}^N (\xi,\omega),
\mathfrak{L}_{1}(u):=\frac{1}{p}\sum_{j=1}^{\ell}\int\limits_{\mathbb{G}}|\mathcal{R}_{j}^{\frac{a_{j}}{\nu_{j}}}u(x)|^{p}dx-\frac{1}{q}\int\limits_{\mathbb{G}}|u(x)|^{q}dx
J_{\nu}(x)=\frac{2^{\nu+1}x^{-\nu}}{\sqrt{\pi}\Gamma\left(\frac{1}{2}-\nu\right)}\,\intop_{1}^{\infty}\frac{\sin(xt)}{(t^{2}-1)^{\nu+\frac{1}{2}}}\,dt,\,\,\,\,\,|(\nu)|<\frac{1}{2},\,x>0,
\frac{d\mu_{r,r}}{dx}\left(x\right)=\frac{2^{r+2}\sin(\varphi)^2 \sin(r\varphi)^{r+1}}{\pi \sin((r+1)\varphi)^r \left(4\sin(\varphi)^2\sin(r\varphi)^2+\sin((r-1)\varphi)^2\right)}
\Phi_{\mathfrak{p}}^{(i)}(x_i,\bar{\xi}_i, t) := \big( w^{(i)}_{\mathfrak{p}-1}+ u^{(i)}_{\mathfrak{p}-1}\big) \, \overline{V}^{(i)}\boldsymbol{\boldsymbol{\cdot}}\bar{\nu}_{\bar{\xi}_i},
u_x = D_x\, u = \frac{y_t\, u_s - y_s\, u_t}{x_s\, y_t - y_s\, x_t}u_y = D_y\, u = \frac{x_s\, u_t - x_t\, u_s}{x_s\, y_t - y_s\, x_t}.
\frac{P^{(n)}_{ij}\cdot P^{(m)}_{jk}}{P^{(n+m)}_{ik}}= \frac{Q^{(n)}_{\sigma(i) \sigma(j)}\cdot Q^{(m)}_{\sigma(j) \sigma(k)}}{Q^{(n+m)}_{\sigma(i) \sigma(k)}}
(\overline{C}_i + \delta_i)^{y_{r,i}}= \overline{C_i}^{y_{r,i}}+ \widetilde{R}_i\delta_i \widetilde{R}_i = y_{r,i}(\theta \overline{C_i}+ (1-\theta)\delta_i)^{y_{r,i}-1}\theta \in (0,1)
U_F=(v_F,y_F)=\left(\frac{-4\alpha^2 v_L^2+2\alpha \alpha_1 v_L(\sqrt{y_L^2-4v_L}-y_L)}{\alpha^2(\sqrt{y_L^2-4v_L}-y_L)^2-2\alpha_2 v_L(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_2v_F}{\alpha^2}-\frac{\alpha_1}{\alpha}\right);
csc^{2}\theta+sec^{2}\theta=\frac{h^{2}}{a^{2}}+\frac{h^{2}}{b^{2}}=\frac{a^{2}+b^{2}}{a^{2}}+\frac{a^{2}+b^{2}}{b^{2}}=2+\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{2}}
I_3 \le \frac{\lambda}{\lambda'}\, (1 - t)^p \int_{A_3}\frac{|w(x) - w(y)|^p}{|x - y|^{N+sp}}\, dx dy + t^p \int_{A_3}\frac{|v_0(x) - v_0(y)|^p}{|x - y|^{N+sp}}\, dx dy &+ C \int_{A_3}\frac{|w(x)|^{p-q}\, v_0(y)^q}{|x - y|^{N+sp}}\, dx dy =: \frac{\lambda}{\lambda'}\, (1 - t)^p\, I_4 + t^p\, I_5 + C J_q.
\mathbb{F}_2[x_{10},h_{10},h_{11},z,x_6,b_{20}] \mbox{\modulo relations\}x_{10}^4,x_{10}h_{11}=x_{10}^2h_{10}, h_{10}h_{11},x_{10}z = h_{10}x_6, h_{11}z,z^2, x_{10}x_6, zx_6,x_6^2,
\overline{{{\rho}}}= - 6 \left[{\overline{{{\Omega}}}}_{2}^{\phantom{2}}- \frac{1}{8 \pi \Im \tau}\right] \frac{\overline{{{\Omega}}}}{\overline{{{\eta}}}^{2 4}}- \frac{\overline{{{j}}}}{8}+ 1 2 6 \ .
1_{G}* ((\eta_{\hat{\nu}}\lambda )\!\uparrow^P_{P_{\nu}}) = (e - 1) \lambda\!\uparrow_{G_{\nu}}^{G}-\sum_{i = 2}^{e}\varepsilon_i (({{\chi}_i \downarrow_{G_{\nu}})\lambda})\!\uparrow_{G_{\nu}}^{G}+ \ \varepsilon ({{\chi}_{\nu}\lambda})\!\uparrow_{G_{\nu}}^{G}
P\left(T^*_n\leq q_n^Z(1-\alpha+\varepsilon_n)|\mathcal{F}^X\right)\geq P\left(\max_{\theta\in\mathcal{G}_n}|Z_n(\theta)|\leq q_n^Z(1-\alpha+\varepsilon_n)\right)-\varepsilon_n=1-\alpha,
\begin{array}{r c l c r c l}{\psi_{-}}{\rightarrow}{e^{- i \theta}g^{- 1}\psi_{-}}{\quad \quad}{\psi_{+}}{\rightarrow}{\psi_{+}}{\psi_{-}^{\dagger}}{\rightarrow}{\psi_{-}^{\dagger}g e^{i \theta}}{\quad \quad}{\psi_{+}^{\dagger}}{\rightarrow}{\psi_{+}^{\dagger}}\end{array}\quad ,
+(1-\mu^{2}) \stackrel{1}{J_{1}}(x){]}\; \delta(x-y)={[}\;\frac{P}{2(\lambda-\mu)}\;,\;\stackrel{1}{U^{\prime}}(x,\lambda) \; - \stackrel{1}{U^{\prime}}(x,\mu)\;{]}\; \delta(x-y) =
\dim C_k^r(\Delta)= \binom{k+3}{3}+12\binom{k+3-(r+1)}{3}&-\sum_{i=1}^{6}\biggl[2\binom{k+3-(r+1)}{3}-\binom{k+3-(2r+2)}{3}\biggr] &+\binom{k+3}{3}-\sum_{j=0}^k F(3,r+1,3)_j.
(\xi^{2}+1)\frac{d^{2}R_{mn}(-ic,i\xi)}{d\xi^{2}}+2\xi\frac{dR_{mn}(-ic,i\xi)}{d\xi}-(\lambda_{mn}(c)-c^{2}\xi^{2}-\frac{m^{2}}{\xi^{2}+1})R_{mn}(-ic,i\xi)=0
K ( X^{1}, X^{2}, X^{3}) = g^{2}e^{x^{A}K^{A B}y^{B}}, \K^{A B}= g^{- 1}\left( \begin{matrix}{{0}}&{{1}}&{{- 1}}\\{{- 1}}&{{0}}&{{1}}\\{{1}}&{{- 1}}&{{0}}\\ \end{matrix}\right)^{A B},
\begin{array}{l l}{{\{Q^{\alpha}, Q^{\beta}\}= - 2 \tilde{\Gamma}^{m \alpha \beta}P_{m},}}&{{[ P_{m}, Q^{\alpha}] = - i \tilde{\Gamma}_{m}{}^{\alpha \beta}\Sigma_{\beta},}}\\{{{}[ M_{m n}, Q^{\alpha}] = \displaystyle \frac i 4 Q^{\beta}( \Gamma_{m n})_{\beta}{}^{\alpha},}}&{{[ M_{m n}, \Sigma_{\alpha}] = - \displaystyle \frac i 4 ( \Gamma_{m n})_{\alpha}{}^{\beta}\Sigma_{\beta}}}\\ \end{array}
\|W(\xi_i \otimes \eta) - (\xi_i \otimes \eta)\|^2 = 2\|\eta\|^2 - 2 (W(\xi_i \otimes \eta)|\xi_i \otimes \eta) &= 2\|\eta\|^2- 2 (\omega_{\xi_i}\otimes \omega_\eta)W &\to 2\|\eta\|^2- 2 (\epsilon \otimes \omega_\eta)W &= 2\|\eta\|^2- 2 \omega_\eta(1) =0.
R = -\frac{2}{C^2}\left( \frac{A''}{A}+ 3 \frac{B''}{B}+ 3 \frac{{B'}^2}{B^2}+ 3 \frac{A'B'}{AB}-\frac{A'C'}{AC}- 3 \frac{B'C'}{BC}\right)=-h''(r)-6\frac{h'(r)}{r}-6 \frac{h(r)}{r^2}.
\sum_{k=1}^{r_1+r_2}\int_1^\infty \big\{g(x)+g^*(x)-\frac{M}{x^{M-a_k+ib_k}}g(1)\big\}\frac{x^{M-1-a_k+ib_k}}{x^M-1}dx &+\Big(\gamma+(\frac{2}{M})\log( \frac{2\pi}{M})\Big)g(1),
w(r)=\frac{r^2}{2n}+C_1r^{2-n}+C_2C_1=\frac{R_1^2-R_0^2}{2n(R_0^{2-n}-R_1^{2-n})}, C_2=\frac{R_0^n-R_1^n}{2n(R_1^{n-2}-R_0^{n-2})}.
\max \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{P}\mathbf{M}\mathbf{P}^\mathrm{H}\mathbf{W}^{\mathrm{RF}}\right] = \max \sum_{m=1}^M c_m \mu_m,
\begin{aligned}Q(p,q)\otimes Q(q,p) = \, \bigoplus_{n=0}^{p}\bigoplus_{m=0}^{q}\Bigg\{Q(p+q-n-m,p+q-n-m) \ \ \oplus \Bigg[\bigoplus_{k=1}^{\min(p-n,q-m)}\Bigg(Q(p+q-n-m-2k,p+q-n-m+k) \ \ \oplus Q(p+q-n-m+k,p+q-n-m-2k)\Bigg)\Bigg]\Bigg\}\ .\end{aligned}
\begin{cases}E=-\dfrac{i}{k}\gamma^{-1}\nabla\times(\mu^{-1}(\nabla\times B))-\gamma^{-1}(\nabla\times A), H=\dfrac{i}{k}\mu^{-1}\nabla\times(\gamma^{-1}(\nabla\times A))-\mu^{-1}(\nabla\times B).\end{cases}
T_{\phi\mu\nu}=B_{\mu\rho}B^\rho_\nu-\frac{1}{4}g_{\mu\nu}g^{\rho\sigma}g^{\lambda\tau}B_{\rho\lambda}B_{\sigma\tau}-2\frac{\partial V(\phi)}{\partial g^{\mu\nu}}+g_{\mu\nu}V(\phi).
\overline{{{P}}}= \varepsilon^{\mu , \mu}+ \varepsilon^{\widetilde{\mu}, \widetilde{\mu}}\hspace{0 . 5 i n}P = \varepsilon^{0 , 0}+ \frac 1 2 \varepsilon^{[ \mu \nu ] , [ \mu \nu ]}+ \varepsilon^{\widetilde{0}, \widetilde{0}}
W^{( 1 )}= \frac{B^{\prime}}{4}x_{0}^{4}+ \frac{b_{3}}{4}x_{3}^{4}+ \frac{b_{4}}{4}x_{4}^{4}+ \frac{b_{5}}{4}x_{5}^{4}- \psi_{0}x_{0}x_{3}x_{4}x_{5}\ .
U(q, N) :=\sum_{n=1}^{N}\frac{q^n (-q)_{n-1}}{1-q^n}, V(q, N) :=\frac{1-q^N}{2}\left(\sum_{n=1}^{\infty}nq^n (-q^{n+1})_{N-1}+ \sum_{n=1}^{\infty}nq^n (q^{n+1})_{N-1}\right).
\mathrm{d}\varepsilon = \frac{1}{48\pi^2}|J| \, \mathrm{d}^{(1/12,2/12)}+ \frac{1}{48\pi^2}|J| \, \mathrm{d}^{(1/12,4/12)}+ \frac{1}{384\pi^2}|J| \, \mathrm{d}_6 \times \mathrm{d}_6,
{H^{\dagger}}^{*}h= \begin{bmatrix}h_1 \frac{1}{2}h_2 \frac{1}{3}g_3 \frac{1}{T}h_T \frac{1}{2}h_2 \frac{1}{3}h_3 \frac{1}{T-1}h_{T+1}\frac{1}{3}h_3 \frac{1}{T-2}h_{T+2}\frac{1}{T}h_{T}\frac{1}{T-1}g_{T+1}\frac{1}{T-2}h_{T+2}h_{2T-1}\end{bmatrix}.
\int_{B_{d}(x,t^{H})}p_{l}^{\eta,\psi}(t,x,y)dy \geq\mathbb{P}\left(\|U_{t}\|_{\textsc{CC}}\leq\gamma t^{H}\right) =\mathbb{P}\left(\|\delta_{t^{H}}^{-1}U_{t}\|_{\textsc{CC}}\leq\gamma\right).
c_{\frac{n}{t_i}}\left((j,n)\right)=c_{\frac{n}{t_i}}\left(\left((j,n),\frac{n}{t_i}\right)\right)=c_{\frac{n}{t_i}}\left(\left(j,\left(n,\frac{n}{t_i}\right)\right)\right)=c_{\frac{n}{t_i}}\left(\left(j,\frac{n}{t_i}\right)\right)=c_{\frac{n}{t_i}}\left(j\right).
R_{ia}= \sum_{j,b}\mathcal{B}_{bi}^j \mathcal{B}_{aj}^b = \sum_{j,b}\mathcal{B}_{bij}\mathcal{B}_{ajb}(-\varepsilon_{j'})\varepsilon_b = \sum_{j,b}\mathcal{B}_{bij}\mathcal{B}_{jab}\varepsilon_{j'}\varepsilon_b
\eqref{eq:upper1}\le C_1e^{C_2\sqrt{i}}\left(\sum_{j=1}^D\binom{D}{j}j!\right) \le D! C_1e^{C_2\sqrt{i}}\left(\sum_{j=1}^D \binom{D}{j}\right) \le 2^D D! C_1e^{C_2\sqrt{i}}.
\vert x -y \vert^{n+2s+2}= R^{n+2s+2}\vert \tilde x - \tilde y \vert^{n+2s+2}\geqslant C R^{n+2s+2}\big ( 1 + R^{-n-2s-2}\vert y-a \vert^{n+2s+2}\big ).
\langle{\cal C}^{1 1}( x_{1}){\cal C}_{1 1}^{\dagger}( x_{3}){\cal K}_{{\bf 1}}( x_{2}) \rangle_{_{( 0 )}}={\frac{N^{2}- 1}{3 ( 2 \pi )^{6}( x_{1 3})^{2}( x_{1 2})^{2}( x_{2 3})^{2}}}.
e_G(V_1,V_2) \geq D|V_1| - e_G(U,W) - 2\binom{|V_1|}{2}\stackrel{(\ref{eWU})}{\geq}|V_1|(D-|V_1|+1)-D \geq \min \lbrace D^2/4-D/2, 2D-6 \rbrace \geq D+1.
N_2\leq C\|\partial_1\Delta_q\theta\|_{L^2}\|\Delta_q\omega\|_{L^2}\leq Cb_q2^{-2qs}\|\partial_1\theta\|_{H^s}\|\omega\|_{H^s}\leq Cb_q2^{-2qs}(\|\theta\|_{H^{1+s}}^2+\|\omega\|_{H^s}^2).
\begin{cases}c_t-\Delta c+c=g, \ \ (x,t)\in\Omega\times(0,T), \displaystyle{\frac{\partial c}{\partial\nu}}=0, \ \ (x,t)\in\partial\Omega\times(0,T), c(x,0)=c_0(x), \ \ (x,t)\in\Omega.\end{cases}
\widetilde{A}= \{(\lambda_1,\lambda_2,\eta_1,\eta_2)| \lambda_1^m=\zeta_m^a,\ \lambda_2=\zeta_n^x ,\ \eta_1=\zeta_m^y \zeta_{mn}^b,\ \eta_2^n=\zeta_n^c; \ \lambda_1 \ne 1, \ \eta_2\neq 1; (x,y)\in A\}
\varphi^{-1}: \mathcal{F}(L_{k}^{+}, (U^{\otimes k})_{k_{1}, \ldots , k_{r}}) \to F(\mathcal{S}_{k_{1}, \ldots , k_{r}}), (\varphi^{-1}\gamma)(\vec{x}, \vec{\nu})=\gamma_{\vec{\nu}}(\vec{x}).
\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-2}(q;q^d)_k^2q^{dk}}{(q^d;q^d)_k^d}&\equiv \frac{(1-q)^2(q^d;q^d)_{n-1-(n+1)/d}}{(q^d;q^d)_{(n+1)/d}^{d-1}}q^{(d(d+n)(n+1)-(n+1)^2)/(2d)-2}\pmod{\Phi_n(q)^2}.
(t^{p(p-1)i+p-1}, t^{p(p-1)j+p-1}) = \left( \sum_{\theta=1}^{d}(h_{p\theta+p-1}- g_{p\theta+p-1}a^{p\theta}) t^{p\theta+p-1}+ f_{h,g}(t) , \right. \, \, \left. \sum_{\theta=1}^{d}(g_{p\theta+p-1}- h_{p\theta+p-1}a^{p\theta}) t^{p\theta+p-1}+ f_{g,h}(t) \right).
\gamma(z) \sim 4 i \pi g^B_s \alpha^\prime \,\sum_{\epsilon=\pm1}\left(\sum_{p=1}^{N_1}\log\frac{(z+\epsilon z_{1p})}{\Lambda}- \sum_{q=1}^{N_2}\log\frac{(z+\epsilon z_{2q})}{\Lambda}\right)
h_n(\kappa_n) = \sum_{\substack{a \in \{p_1, \dots, p_{i-1}\}b\in \{q_1, \dots, q_{j-1}\}}}\{a, p_i\}\cdot \{b,q_j\}+ \{a, q_j\}\cdot \{b, p_i\}+ \{a,b\}\cdot \{p_i,q_j\}
\sup_{\overline{T}_k (\omega)\leq s \leq \underline{T}_{k+1}(\omega) \wedge t}|V_s (\omega)- V^{(n)}_s (\omega)| \leq \eta_{2k+1}, 0 \leq k \leq K_2(\omega), \sup_{\underline{T}_k (\omega)\leq s \leq \overline{T}_k (\omega) \wedge t}|V_s (\omega)- V^{(n)}_s (\omega)| \leq \eta_{2k}, 1 \leq k \leq K_1(\omega),
\sum_{i=1}^{\ell_1}a_i\in \Bigl[\frac{9}{25}+\frac{\epsilon}{2},\frac{17}{40}-\frac{\epsilon}{2}\Bigr]\cup\Bigl[\frac{23}{40}+\frac{\epsilon}{2},\frac{16}{25}-\frac{\epsilon}{2}\Bigr],
\frac{i}{\eta_{22}}\left(\Gamma_{22}^{1}\eta_{12}+\Gamma_{22}^{2}\eta_{22}\right) =\frac{i}{2}\frac{\eta_{22,2}}{\eta_{22}}-\frac{i}{\eta_{12}}\left(\Gamma_{12}^{1}\eta_{12}+\Gamma_{12}^{2}\eta_{22}\right) =-\frac{i}{2}\frac{\eta_{22,1}}{\eta_{12}},
\Delta E = - \frac{\beta_{0}^{4}}8 \left( \frac 1{m^{3}}+ \frac 1{M^{3}}\right) + \frac{\beta_{0}^{2}}{2 m M}\left< \phi_{0}\right| I \left| \phi_{0}\right> + \frac 1{2 \mu}\left( \frac{\mu^{2}}{m^{2}}+ \frac{\mu^{2}}{M^{2}}\right) \left< \phi_{0}\right| I^{2}\left| \phi_{0}\right>
S= \int_{0}^{L}dy\int d^{4}x\sqrt{-G}\left(-\frac{1}{2}G^{MN}\partial_{M}\Phi\partial_{N}\Phi+\frac{1}{\sqrt{-G}}\partial_{M}\left(\sqrt{-G}G^{MN}\Phi\partial_{N}\Phi\right)-\frac{1}{2}V(\Phi)\right)\ .
\psi(p) = \frac{1}{\sqrt{8\pi\sqrt{\beta}}\hbar}\int_{-\infty}^{+\infty}d{\xi}{\mbox{}}(1+\beta p^2)^{1/2}{\mbox{}}e^{-i{\xi}\frac{\tan^{-1}(\sqrt{\beta}p)}{\hbar \sqrt{\beta}}}{\mbox{}}\psi({\xi})
[\begin{matrix}|v|^{2}&-2v_{x}&-2v_{y}&-1\\ |A|^{2}&-2A_{x}&-2A_{y}&-1\\ |B|^{2}&-2B_{x}&-2B_{y}&-1\\ |C|^{2}&-2C_{x}&-2C_{y}&-1\end{matrix}]
( X )_{l l^{\prime}}= - \frac{\alpha}{n + \frac{\delta_{1}+ \delta_{2}}{2}}\sum_{n_{1}= 0}^{n - | m | - 1}\left( 2 n_{1}- n + | m | + \frac{\delta_{1}- \delta_{2}}{2}+ 1 \right) \widetilde W_{n l m}^{n_{1}}( \delta_{1}, \delta_{2}) \widetilde W_{n l^{\prime}m}^{n_{1}}( \delta_{1}, \delta_{2}) .
0=\iint_\Omega(\mathbf{x},y)\cdot\nabla\phi\Delta\phi~d\mathbf{x}dy=I_1+I_2+I_3+I_4-\frac12d\int_{\mathbb{R}^{n-1}}|\nabla\phi(\mathbf{x},-d)|^2~d\mathbf{x},
\lim_{M\to \infty}\frac{|\Lambda_{(\infty,m)}(\frac{M}{2},\Gamma)|}{|\Lambda(\frac{M}{2})|}= \lim_{M\to \infty}\frac{|\Lambda_{(\infty,m)}(\frac{M}{2})|}{|\Lambda(M)|}\frac{|\Lambda(M)|}{|\Lambda(\frac{M}{2})|}= \lim_{M\to \infty}\frac{|\Lambda_{(\infty,m)}(\frac{M}{2},\Gamma)|}{|\Lambda(M)|}\frac{M^2}{(\frac{M}{2})^2}=0
\det M^{( 1 )}= \frac{\det ( A^{( 1 )}- \Gamma^{( 1 )}B^{( 1 ) - 1}\Gamma^{( 1 ) t})}{\det B^{( 1 )}}= \frac{\det A^{( 1 )}}{\det B^{( 1 )}}\propto \lambda^{- 8}e^{- 3}.
-1 \leq \langle D^2 \varphi (x_0) \eta, \eta \rangle = \langle D^2 \psi (z_0) \zeta_0, \zeta_0 \rangle\,(\langle \nabla \delta (x_0), \eta \rangle )^2\leq \langle D^2 \psi (z_0) \zeta_0, \zeta_0 \rangle\, ,
f_n(\mu|\Psi_n) = \frac{1}{n}\sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i,\mu_j) + \lambda_n r(\mu) f_\infty(\mu) = \int \bigwedge_{j=1}^k d(x,\mu_j) P( x) + \lambda r(\mu).
-{\textstyle \frac{1}{4}}\omega^{a}{}_{b}\gamma_{a}{}^{b}= - \vartheta \Gamma_{s}( M ) -{\textstyle \frac{1}{8}}e^{c}f_{c b}{}^{a}\gamma_{a}{}^{b}\, , \hspace{1 c m}\Gamma_{s}( M ) \equiv{\textstyle \frac{1}{2}}( \cos{\alpha}\gamma^{0 1}- \sin{\alpha}\gamma^{2 3}) \, .
\hat{T}= \sum_{\{\Delta_{0}S_{\gamma}\}}\prod_{p \in \Delta_{0}S_{\gamma}}\hat{W}_{p}^{n_{p}}\exp{[ - \frac{1}{2 \beta_{0}}\sum_{\ell}\hat{E}_{\ell}^{2}- \frac{1}{2 \beta_{s}}\sum_{p_{s}\in ( S_{t + a_{0}}- S_{t})}n_{p}^{2}]},
F ( x ) ={\frac{E_{P}^{2}}{8 \pi}}\int_{- \infty}^{\infty}{\frac{d k}{2 \pi}}e^{- i k x}\ln \left( 1 +{\frac{8 \pi k^{2}}{E_{P}^{2}}}\right) = -{\frac{E_{P}^{3}}{8 \pi}}\left({\frac{L_{P}}{| x |}}\right) \exp \left( -{\frac{| x |}{\sqrt{8 \pi}L_{P}}}\right)
A_\lambda\partial_t(u_\lambda,v_\lambda) + B_\lambda(u_\lambda,v_\lambda)= (g_\lambda,g_{\Gamma\lambda}) - (T_\lambda\pi(f), T_\lambda\pi_\Gamma(f_\Gamma))\quad (0,T)\,, (u_\lambda, v_\lambda)(0) = (u_0, u_{0|\Gamma})\,.
\limsup_{n \rightarrow \infty}\limsup_{t \rightarrow \infty}P\big( \hat{W}^{n}(t) > x \big)\ \ =\ \ \limsup_{n \rightarrow \infty}P\big( \hat{W}^{n}(\infty) > x \big)\ \ =\ \P\big( W^{\infty}(\infty) > x \big)
< \Delta v_{-}, t_{1}^{- 1}\otimes t_{2}^{- 1}> = \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{q^{2}}}&{{0}}&{{0}}&{{0}}\\{{q}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{q^{3}}}&{{q^{2}}}&{{0}}\\ \end{array}\right) .
{\frac{1}{a + b z}}\left( \begin{array}{l}{{1}}\\{{z^{\prime}}}\\{{{z^{\prime}}^{2}}}\\{{\vdots}}\\ \end{array}\right) ={\frac{1}{a + b z}}\left( \begin{array}{l}{{1}}\\{{\left({\frac{c + d z}{a + b z}}\right)}}\\{{\left({\frac{c + d z}{a + b z}}\right)^{2}}}\\{{\vdots}}\\ \end{array}\right) .
H(U_{\bar{S}}\oplus Z^n) =H(U_{\bar{S},2}^n\oplus Z_{2}^n)+H(U_{\bar{S},1}\oplus Z_1|U_{\bar{S},2}^n\oplus Z_{2}^n) &=H(U_{S,2}^n\oplus Z_{2}^n)+H(U_{\bar{S},1}\oplus Z_1|U_{\bar{S},2}^n\oplus Z_{2}^n)
(M_1)_{ij}= a_{0ij}x_0 + a_{1ij}x_1 + a_{2ij}x_2 a_{3ij}x_3, (M_2)_{ij}= a_{i0j}y_0 + a_{i1j}y_1 + a_{i2j}y_2 a_{i3j}y_3.
\frac{d}{d \lambda}g(\lambda) = - 2 \sum_{i=1}^n \beta_i^2 [\lambda (\mu_i + \lambda)]^{-3}(\mu_i + 2\lambda), \frac{d^2}{d \lambda^2}g(\lambda) = 2 \sum_{i=1}^n \beta_i^2 [\lambda (\mu_i + \lambda)]^{-4}\bigl[10 \lambda^2 + 10 \mu_i \lambda + 3 \mu_i^2].
{\delta\over \delta y^\mu( \,\vec s \, )}= y^{\mu_2\dots \mu_{p+1}}(\, \vec s \, ){\delta\over \delta\sigma^{\mu\mu_2\dots \mu_{p+1}}(\, \vec s\, )}\ ,\qquad y^{\mu_2\dots \mu_{p+1}}(\, \vec s \, )\equiv \epsilon^{m_2\dots\mu_{p+1}}\, \partial_{m_2}\, y^{\mu_2}\dots \partial_{m_{p+1}}\, y^{\mu_{p+1}}\ .