harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar{\check{y}}_{\mathrm{d},i,k}[n] &=\sum_{k'\in\mathcal{K}}\sum_{\ell'\in\mathcal{L}_{m_i,k'}}h_{i,k',\ell'}x_{\mathrm{d},k}\big[n-\big(d_{m_i,k',\ell'}-d_{m_i,k,\ell^\star_{i,k}}\big)\big] &\quad+z_{\mathrm{d},i}\big[n+d_{m_i,k,\ell^\star_{i,k}}\big]+e_{\mathrm{d},i}\big[n+d_{m_i,k,\ell^\star_{i,k}}\big].
G_t(x,y) = \frac{1}{(2\pi \sinh 2t)^{1/2}}\exp\bigg( -\frac{1}4 \big[ \tanh(t)\, (x+y)^2 + \coth(t)\,(x-y)^2\big]\bigg) & = \frac{1}{(2\pi \sinh 2t)^{1/2}}\exp\bigg( -\frac{1}2 \coth(2t) \,\big(x^2+y^2\big) + \frac{xy}{\sinh 2t}\bigg)
S_{0}[ B_{\mu \nu}( x ) , A_{\mu}( x ) ] = \int d^{4}x \; \Bigg \{\frac{1}{4}\epsilon^{\mu \nu \rho \sigma}B_{\mu \nu}( x ) F_{\rho \sigma}( x ) + \frac{1}{2}A_{\mu}( x ) A^{\mu}( x ) \Bigg \},
ds^2 = -2 dx^+ dx^-- \beta^2 \sum_{k=1}^n a_k^2 \left[ (\tilde{x}^{2k-1})^2 +(\tilde{x}^{2k})^2 \right]+ \sum_{i=1}^{2n}(d\tilde{x}^i)^2 + \sum_{i=2n+1}^9 (dx^i)^2,
M_a^{-T}= M_b^{-T}\left(\begin{array}{ccccc}1 O(\delta) O(\delta^2) \dots O(\delta^{k-1}) 0 1 O(\delta) \dots O(\delta^{k-2}) 0 0 1 \dots O(\delta^{k-3}) \vdots \ddots 0 0 0 \dots 1\end{array}\right),
G \, \bar{\psi}\gamma^{\mu}\psi \, \bar{\psi}\gamma_{\mu}\psi ={\frac{\pi ( \pi - g_{\mathrm{e f f}}^{2})}{g_{\mathrm{e f f}}^{2}}}\, \Bigl ( \bigl ( j^{0}\bigr )^{2}-{\frac{g_{\mathrm{e f f}}^{2}}{\pi}}\, \bigl ( j^{1}\bigr )^{2}\Bigr ) .
\partial_{+}\left( \Gamma^{( 1 ) - 1}\partial_{-}\Gamma^{( 1 )}\right) = - \Gamma^{( 1 ) - 1}\Gamma^{( 2 )}, \quad \partial_{+}\left( \Gamma^{( 2 ) - 1}\partial_{-}\Gamma^{( 2 )}\right) = \Gamma^{( 1 ) - 1}\Gamma^{( 2 )}.
\frac{\partial}{\partial t}<u_1(t)>^{\widehat{}}(k)+\nu (\frac{\pi k}{h})^2<u_1(t)>^{\widehat{}}(k) =\int_0^h -\frac{p_1(t)}{\Pi_1}\sqrt{\frac{2}{h}}\sin \frac{\pi k x_3}{h}dx_3 &=-\sqrt{\frac{2}{h}}\frac{p_1(t)}{\Pi_1}\frac{h}{\pi k}\bigg(1-(-1)^k\bigg),
\langle D_w f ,f \rangle_{L^2}= \sum_{I \in \mathcal{D}}\langle w \rangle_I \left | \widehat{f}(I) \right |^2 \le [w]_{A_2}\sum_{I \in \mathcal{D}}\langle w^{-1}\rangle_I^{-1}\left | \widehat{f}(I) \right |^2 = [w]_{A_2}\langle \left( D_{w^{-1}}\right)^{-1}f ,f \rangle_{L^2}.
u =(c_1\cos (n\theta)+c_2\sin(n\theta))\Big((n-\tanh{t})e^{nt}+(n+\tanh{t})e^{-nt}\Big), \delta = T\frac{\Big(n(n-\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{nT}-\Big(n(n+\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{-nT}}{(n-\tanh{T})e^{nT}+(n+\tanh{T})e^{-nT}},
S_{E}= M_{0}^{2}\int_{0}^{T}\! \! d t \int_{0}^{R}\! \! d r \left[ 1 + \frac{1}{2}{\bf u}\partial^{2}{\bf u}\right] + \sum_{a = 1}^{2}\frac{m_{a}}{2}\int_{t_{1}}^{t_{2}}\! \! d t{\dot{\bf u}}^{2}( t , r_{a}){,}
\Psi_{E}( r , \theta ) = \sum_{n = - \infty}^{\infty}e^{i n \theta}\left( \begin{array}{c}{{A_{n}\, J_{n - \kappa}\left( | E | r \right) + B_{n}\, J_{\kappa - n}\left( | E | r \right)}}\\{{- i \frac{| E |}{E}\left( A_{n}\, J_{n + 1 - \kappa}\left( | E | r \right) - B_{n}\, J_{\kappa - n - 1}\left( | E | r \right) \right) e^{i \theta}}}\\ \end{array}\right)
\hat{s}_{g}\left( q, r, \tau_t \right) = \frac{q \exp \left( - \frac{\left( r - d_{g}\right)^2}{2 \tau_t^2}\right)}{q \exp \left( - \frac{\left( r - d_{g}\right)^2}{2 \tau_t^2}\right) + (1-q) \exp \left( -\frac{r^2}{2 \tau_t^2}\right)},
a_{1}^{\prime}=a_{1}^{*}+\frac{a_{1}^{*}}{2}(1-\sqrt{\frac{\overline{C}^{7}}{\overline{C}^{7}+25^{7}}})a_{2}^{\prime}=a_{2}^{*}+\frac{a_{2}^{*}}{2}(1-\sqrt{\frac{\overline{C}^{7}}{\overline{C}^{7}+25^{7}}})
S={1\over{4\pi}}\int^{2\pi}_0d\sigma\int d\tau\left[\sqrt{g}g^{\alpha\beta}G_{ij}\partial_{\alpha}X^i\partial_{\beta}X^j+\epsilon^{\alpha\beta}B_{ij}\partial_{\alpha}X^i\partial_{\beta}X^j-{1\over 2}\sqrt{g}\phi{\cal R}^{(2)}\right],
|\tilde{f}(y_2)-\tilde{f}(\theta_2)-\tilde{f}'(\theta_2)(y_2-\theta_2)|\lesssim |y_2-\theta_2|^2, \tilde{f}(y_2)=y_2^{-1}\left[\chi\left(\sqrt{2}y_2\right)-\chi\left(8\sqrt{2}y_2\right)\right],
\displaystyle \mathcal F^{+}(h,h)= \int_{\mathbb R^{d}\times S^{d-1}}\, \phi(|v-v_{\ast}|)\, b(\cos\theta)M_{\ast}\, h_{\ast}^{\prime}\, h^{\prime}\, dv_{\ast}d\sigma\,, begin{align*}2pt] \displaystyle \mathcal F^{-}(h,h) = -\int_{\mathbb R^{d}\times S^{d-1}}\, \phi(|v-v_{\ast}|)\, b(\cos\theta)M_{\ast}\, h_{\ast}\, h\, dv_{\ast}d\sigma\,.
du(t,x,z)=-x^2u(t,x,z)dt+k(t,x,z)[dF(t,x,z)-xy(t,x,z)q(t,x,z)dt] &=-x^2u(t,x,z)dt+k(t,x,z)y(t,x,z)q(t,x,z)dG(t) &-k(t,x,z)xy(t,x,z)q(t,x,z)dt &=f(t,x,z)dt+k(t,x,z)y(t,x,z)q(t,x,z)dG(t)
\sum_{a=2}^{n-3}\left(2^{n-2-a}-1-\sum_{i=0}^{n-3-a}\sum_{j=0}^{n-3-a-i}1\right) = 2^{n-3}-(n-2)-\sum_{a=2}^{n-3}\binom{n-1-a}{2}=2^{n-3}-(n-2)-\binom{n-2}{3}
2p \sum_{2p-s \geq l \geq s}{\binom{2p-l}{s-1}\cdot \binom{l-1}{s-1}\cdot s!}\leq 2p \sum_{2p-s \geq l \geq s}{\binom{2p-l}{s-1}\cdot l^s}\leq (2p)^2(2p-s)^{s-1}(2p-s)^{s}\leq (2p)^{2s+1},
{\tilde \Delta}_{n,k,2}^{(1,3)}(i) := \sum_{j=i+1}^{k-1}\big \{\varphi^{(3)}_{n-k}(\tilde S_{k-j -1}) ( Z_{k-j}( Z_{k-i}Z_k )^{(0)})^{(0)}\big \}\, .
\xi_r^{{(s)}}:= \sum_{k=\tau_{r-1}^{{(s)}}+1}^{\tau_r^{{(s)}}}g(S_k^{(s)}(\Delta)),\;\eta_r^{{(s)}}:= \tau_r^{{(s)}}-\tau_{r-1}^{{(s)}},\;\omega_r^{{(s)}}:=\xi_r^{{(s)}}-\lambda\eta_r^{{(s)}},
2(\ln (12C_1|u^0|) )^+ -T_0-1< 2(\ln (12C_1|u^0|) )^+-(\ln(4C_{1/2}|u^0|))^+ -1 &< 2(\ln (12C_1|u^0|) )^+-(\ln(12C_1|u^0|))^+ = (\ln (12C_1|u^0|) )^+ .
( \bar{M}_{m})_{j k}= i g{\frac{( 1 - \delta_{j k})}{\sin^{2}[ ( j - k ) \pi / ( r + 1 ) ]}}- i g \delta_{j k}\sum_{l \ne j}{\frac{\cos [ ( j - l ) \pi / ( r + 1 ) ]}{\sin^{2}[ ( j - l ) \pi / ( r + 1 ) ]}}.
G_{\omega}\simeq c_{\lambda}{\frac{e^{- ( \lambda_{\omega}+ d ) z^{\prime}/ \beta_{H}}}{( \operatorname{c o s h}{\frac{z}{\beta_{H}}}- \operatorname{s i n h}{\frac{z}{\beta_{H}}}\operatorname{c o s}\gamma )^{( \lambda_{\omega}+ d )}}}~ ~ ,
\|\tilde a\|_{l_1^d}=\|\Delta_{l_1^d}(\Phi(g'(0)a+u))\|_{l_1^d}&\geq \|g'(0)a+u\|_{l_1^d}-\|\Delta_{l_1^d}(\Phi(g'(0)a+u))-g'(0)a-u\|_{l_1^d}&\geq g'(0)\|a\|_{l_1^d}-\|u\|_{l_1^d}- \sigma_s^d(g'(0)a+u)_{1}&\geq g'(0)-2\|u\|_{l_1^d}-g'(0)\sigma_s^d(a)_{1}&\geq g'(0)-2\sqrt{2}c_1h-g'(0)\sigma_s^d(a)_{1}.
p^1(\mu,r) =\int_0^{2r}g_r(s)(1-e^{-\mu s})\,ds &=e^{-4r}(1-e^{-2r\mu})+\int_0^{2r}(2+2r-s)e^{-(2r+s)}(1-e^{-\mu s})\,ds &=\frac{\mu e^{-2r}(\mu+2r+2r\mu)-\mu^2e^{-2r(2+\mu)}}{(1+\mu)^2}.
\delta_{t}(\tilde{U}_C)^{k}_{ij}-\mu\Delta_{h}(\tilde{U}_C)^{k}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(\tilde{U}_C)^{k}_{ij}=b^{k}_{ij}+\frac{g((\tilde{U}_C)^k_{ij})+g((\tilde{U}_C)^{0}_{ij})}{2}, (x_i,y_j)\in\Omega_{h},
\pi'_0(\nabla(e_0)) = \pi'_0\Big(e_1\otimes \frac{dz}{2z^2}\Big) =(w+w^2)\frac{dw}{w^3}=\nu'(w)\pi'_0(e_0) \pi'_0(\nabla(e_1)) = \pi'_0\Big( ze_0\otimes \frac{dz}{2z^2}+e_1\otimes\frac{dz}{2z}\Big) =w^2\frac{dw}{w^3}+w\frac{dw}{w}=\nu'(w)(w+w^2)=\nu'(w)\pi'_0(e_1).
\rho^D_{m_0\rightarrow m_1+m_2}=\frac{1}{4\pi R^2}\sum_j(2j+1) \rho^{D-2}_{m_0\rightarrow m_{1j}+m_{2j}};\qquad m_{ij}^2=m_i^2+\left(\frac{j}{R}\right)^2.
\mathfrak{F}_n^{(2)}=\left[\displaystyle \frac{(\lambda_1 - 1)t_0 + t_1}{\lambda_1 - \lambda_2}\right]\lambda_1^n +\left[\displaystyle \frac{(\lambda_2 - 1)t_0 + t_1}{\lambda_2 - \lambda_1}\right]\lambda_2^n
[\varphi_\lambda([x_\mu y_1], y_2)_{\lambda+\mu}[w_\gamma v ]]+ (-1)^{xy_1}[\varphi_\lambda(y_1, [ x_\mu y_2])_{\lambda+\mu}[w_\gamma v ]] = [(\varphi_\lambda(x, [{y_1}_\eta y_2]))_{\lambda+\mu}[w_\gamma v]].
R_{ach}= \log_{2}\bigg(1+\frac{\mathbf{h}_{s,k}^{H}\mathbf{Q}^{*}_{s}\mathbf{h}_{s,k}}{\mathbf{h}_{s,k}^{H}\mathbf{V}^{*}\mathbf{h}_{s,k}+\sigma_{sa,k}^{2}+\frac{\sigma_{sp,k}^{2}}{\rho^{*}_{s,k}}}\bigg) -\log_{2}\bigg|\mathbf{I}\!+\!(\mathbf{H}_{e,l}^{H}\mathbf{V}^{*}\mathbf{H}_{e,l})^{-1}\mathbf{H}_{e,l}^{H}\mathbf{Q}^{*}_{s}\mathbf{H}_{e,l}\bigg|
T = \left( \begin{array}{c c c c}{{0}}&{{x_{3}}}&{{- x_{2}}}&{{- x_{1}}}\\{{- x_{3}}}&{{0}}&{{x_{1}}}&{{- x_{2}}}\\{{x_{2}}}&{{- x_{1}}}&{{0}}&{{- x_{3}}}\\{{x_{1}}}&{{x_{2}}}&{{x_{3}}}&{{0}}\\ \end{array}\right)
\| N^{n,5}_\tau\|_2 \le \alpha^2 \sup_{0\le s\le \tau}\| (\sigma'\sigma)^2(X_s) \|_2 \frac 1n \sum_{j=0}^{\lfloor n\tau \rfloor}\int_{0}^{1\wedge(n \tau -j )}\gamma_{\frac{x+j}n}dx \sum_{k=j+1}^{\infty}k^{2\alpha -2}.
\begin{aligned}p(t,v,g):=\left(\frac{1}{V_F}\sum_{k=-\infty}^{+\infty}c_k\exp\left(ik\frac{2\pi}{V_F}(v-\int_0^tg_{}(s)ds)\right)\right)\frac{1}{\sqrt{2\pi a(t)}}\exp\left(-\frac{(g-g_{}(t))^2}{2a(t)}\right), \end{aligned}
E(t) + \int_0^t ||u'(\tau)||_{1,2}^2\,d\tau \leq \liminf_{N\to\infty}\left( E_N(t) + \int_0^t ||u_N'(\tau)||_{1,2}^2 \,d\tau \right) &= \liminf_{N\to\infty}E_N(0) &=E(0)
\begin{array}{l}A_i=A_i^0+\sum\limits_{j=1}^dA_{i}^{j}D_{x^{j}},B_i=B_i^0+\sum\limits_{j=1}^d B_{i}^{j}D_{x^{j}},i=1,2, L=L^0+\sum\limits_{k=1}^d L^{k}D_{x^{k}},M=M^0+\sum\limits_{k=1}^d M^{k}D_{x^{k}}.\end{array}
J^{i}( z e^{i 2 \pi n}) = e^{- 2 i \pi n \alpha_{0}^{-}\left( J_{0}^{0}+ J_{0}^{1}\right)}J^{i}( z ) e^{2 i \pi n \alpha_{0}^{-}\left( J_{0}^{0}+ J_{0}^{1}\right)}
\widehat{\Phi}_{\mathrm{HG}}(z) = \begin{pmatrix}\Gamma(1 -\beta)G(\beta; z) -\frac{\Gamma(1 -\beta)}{\Gamma(\beta)}H(1-\beta;ze^{-i\pi}) \Gamma(1 +\beta)G(1+\beta;z) H(-\beta;ze^{-i\pi})\end{pmatrix},
\int_{1}^{\infty}e^{-r}r^{1-N}\int_{0}^{r}e^{t}t^{N-1}p_{1}\left( t\right)dtdr=\int_{1}^{\infty}e^{-r}r^{1-N}\int_{0}^{r}e^{t}t^{N-1}p_{2}\left(t\right) dtdr=\infty ,
\begin{aligned}\|v\|_{C^{(1+\alpha)/2,1+\alpha}}\leq K_2 P_1(a_{\alpha/2,\alpha})\big(\|v_0\|_{C^{1+\alpha}(\overline D)}+\|g\|_{C^{\alpha/2,\alpha}}+\|f\|_{L^p(D_T)}\big),\end{aligned}
\alpha' = \gamma\alpha^{(1)}+ (1-\gamma)\alpha^{(1)}, \tilde{p}' = \frac{\gamma\alpha^{(1)}}{{\alpha'}}\tilde{p}^{(1)}+\frac{(1 - \gamma) \alpha^{(2)}}{{\alpha'}}\tilde{p}^{(2)}, p_i' = \frac{\gamma(1-\alpha^{(1)})}{{1-\alpha'}}{p}_i^{(1)}+\frac{(1 - \gamma) (1-\alpha^{(2)})}{{1-\alpha'}}{p}_i^{(2)}, i = 1, 2,
\partial_{x_j}^p\frac{x_j^{2r}}{(1+\epsilon x^{2m})^2}\leq\sum_{\ell=0}^p \binom{p}{\ell}(2r)\cdots(2r-\ell+1)x_j^{2r-\ell}\sum_{(m_1,\dots,m_{p-\ell})}C_{p-\ell,m_i}\frac{x_j^{\ell-p}}{(1+\epsilon x^{2m})^2}&= \sum_{\ell=0}^p c_{p,\ell,r}\frac{x_j^{2r-p}}{(1+\epsilon x^{2m})^2}.
\mathrm{D}_{z}^{( s )}( E_{i}) = z^{s_{i}}E_{i}\, , \mathrm{D}_{z}^{( s )}( F_{i}) = z^{- s_{i}}F_{i}\, , \mathrm{D}_{z}^{( s )}( h_{i}) = h_{i}\, , i = 0 , \cdots , r
[x+a+u,y+b+v]_{\mathfrak{g}}= [x,y]_{\mathfrak{g}_1}+[a,b]_{\mathfrak{g}_2}+\omega(x+a,y+b) = [x,y]_{\mathfrak{g}_1}+[a,b]_{\mathfrak{g}_2}+\omega_{2,0}(x,y)+\omega_{1,1}(x,b)-\omega_{1,1}(y,a)+\omega_{0,2}(a,b)
u_{\pm}= \frac{1}{4 \sqrt{2}}\Big ( ( F_{\mu \nu}F_{\mu \nu}+ F_{\mu \nu}^{*}F_{\mu \nu})^{\frac{1}{2}}\pm ( F_{\mu \nu}F_{\mu \nu}- F_{\mu \nu}^{*}F_{\mu \nu})^{\frac{1}{2}}\Big )
\left(1 - \frac{Rt}{p_n}\right) \left(1 - \frac{Rt}{p_{-n}}\right) = \left(1 - \frac{Rt}{p_n}\right) \left(1 + \frac{Rt}{p_n}- Rt \left(\frac{1}{p_n}+ \frac{1}{p_{-n}}\right) \right) & = \left(1 - \frac{R^2t^2}{p_n^2}\right)\left(1 + \frac{Rt}{1+ \frac{Rt}{p_n}}\left(-\frac{1}{p_n}- \frac{1}{p_{-n}}\right)\right).
\ell(t_\lambda) = (n-1)\lambda_1+(n-2)\lambda_2+\cdots+(n-n)\lambda_n- \big( (n-1)\lambda_n + (n-2)\lambda_{n-1}+ \cdots + (n-n)\lambda_1\big) &= (n-1)\lambda_1+(n-3)\lambda_2+(n-5)\lambda_3+\cdots -(n-3)\lambda_{n-1}-(n-1)\lambda_n = \langle \lambda, 2\rho^\vee\rangle,
\chi_{\rm out}(x)=U^{\dag}\left( \eta \right) \left( \begin{array}{c}G\left( x\right) F\left( x\right) \end{array}\right)\equiv \left( \begin{array}{c}h_{\rm out}\left( x\right) g_{\rm out}\left( x\right) \end{array}\right) \;.
\begin{aligned}0 \geq S^{ij}_k (u_1 + B |x|^2)_{ij}= f_1 + 2 B \sum_i S^{ii}_k \geq - A f^{1 - 1/(k-1)}+ 2 c_0 B S_1^{1/(k-1)}S_k^{1 - 1/(k-1)}= - A f^{1 - 1/(k-1)}+ 2 c_0 B (\Delta u)^{1/(k-1)}f^{1 - 1/(k-1)}.\end{aligned}
\frac d{dt}E_{L,M}(q_t,\dot q_t) = \dot q_t\cdot \left( \frac{d}{dt}\frac{\partial L}{\partial \dot q}- \frac{\partial L}{\partial q}\right) (q_t,\dot q_t) = \dot q_t\cdot R_{L,M}(q_t,\dot q_t) \,.
\gamma^*\beta_{\alpha}(U)s =\beta_{\alpha}(U)(\gamma(U)s) &=\alpha(U)[U\rightarrow U, \overline{\gamma(U)s}], \beta_{(\xi_H \gamma)^*\alpha}(U)s =\beta_{\alpha\xi_H\gamma}(U)s &=(\alpha\xi_H\gamma)(U)[U\rightarrow U, \bar s] &=\alpha(U)(\xi_H\gamma)(U)[U\rightarrow U,\bar s] &=\alpha(U)[U \rightarrow U,\overline{\gamma(U)s}].
\begin{aligned}\hat{g}U_b= -\sigma_y\bar{y}Y+\sigma_x\bar{y}X-\hat d_bD_b s\bar{y}X= \sigma_y\bar{y}Y-\sigma_x\bar{y}X-a_{xx}\bar{y}X+\hat d_x D_x+\hat d_bD_b \end{aligned}
I=\,\sum^N_{k=1}\int_{T^*_{(x,y)}TM^k_4}\,\frac{1}{\sqrt{-\det \eta^k_4}}\,d^4p_{xk\mu}\,\wedge \frac{1}{\sqrt{-\det \eta^k_4}}\, d^4p_{zk\mu}\,|p_{xk\mu},p_{zk\mu}\rangle\langle p_{xk\mu},p_{zk\mu}|.
\rho_{x_i}\rho_{x_j}= q_{ji}\rho_{x_j}\rho_{x_i}, \delta_i \delta_j = q_{ji}\delta_j \delta_i, \delta_i \rho_{x_j}=q_{ij}\rho_{x_j}\delta_i\mbox{if $i\neq j$}, \delta_i \rho_{x_i}-\rho_{x_i}\delta_i =1.
\Gamma_{S}^{2 2}( p ) + e^{- \eta ( p^{0}) \bar{\Theta}}\Gamma_{S}^{1 2}( p ) = - \left[ \Gamma_{S}^{1 1}( p ) + e^{\eta ( p^{0}) \bar{\Theta}}\Gamma_{S}^{1 2}( p ) \right]
\frac{\eta}{\sqrt{\eta^2+2m+2}}=(\eta)\left(1-\frac{2m+2}{\eta^2}+o_{|\eta|\rightarrow+\infty}\left(\frac{1}{\eta^2}\right)\right), \frac{\eta-\xi}{\sqrt{(\eta-\xi)^2+2n+2}}=(\eta-\xi)\left(1-\frac{2n+2}{(\eta-\xi)^2}+o_{|\eta-\xi|\rightarrow+\infty}\left(\left(\frac{1}{\eta-\xi}\right)^2\right)\right).
\xi_t(i) &\begin{cases}\phi\left(\frac{i}{N}\right) k=2l=3, \psi\left(\frac{i}{N}\right) k=1l=2, \frac{1}{N}\sum_{j\neq i}\lambda\left(\frac{i}{N}, \frac{j}{N}\right)1_{\{\xi_t(j)=2\}}k=0l=1, 0 \end{cases}
f_{{\bf{q}}}= (\frac{\omega_{{\bf{q}}}+ \epsilon_{{\bf{q}}}+ \rho_{0}v_{{\bf{q}}}}{2 \mbox{}\omega_{{\bf{q}}}})^{\frac{1}{2}}d_{{\bf{q}}/2}({\bf{q}}) + (\frac{-\omega_{{\bf{q}}}+ \epsilon_{{\bf{q}}}+ \rho_{0}v_{{\bf{q}}}}{2 \mbox{}\omega_{{\bf{q}}}})^{\frac{1}{2}}d^{\dagger}_{-{\bf{q}}/2}(-{\bf{q}})
\mathfrak{F}_{l+1}^{(k)}=\sum_{m=1}^{k}\left[\displaystyle \frac{\displaystyle\sum_{p=1}^{k}\left(\lambda_m^{k-p}-\displaystyle\sum_{i=1}^{k-p}\lambda_m^{(k-p)-i}\right)t_{p-1}}{\displaystyle \prod_{j=1, j\neq m}^{k}(\lambda_m - \lambda_j)}\right] \lambda_m^{l+1}
d s^{2}= \frac{1}{1 6}( d \psi - \sum_{i = 1}^{3}\cos{\theta_{i}}d \phi_{i})^{2}+ \frac{1}{8}\sum_{i = 1}^{3}( d \theta_{i}^{2}+ \sin^{2}{\theta_{i}}d \phi_{i}^{2}) \ ,
\exp({\lambda_{-k}{\cal L}_{-k}})\exp({\lambda_k{\cal L}_k})=\exp\bigg[{\lambda_+\over \sinh (k\lambda_+)}(k\lambda_{-k}{\cal L}_{-k}+k^2\lambda_{-k}\lambda_k{\cal L}_0+k\lambda_k{\cal L}_k+c_kI)\bigg]\ ,
(\mu+\partial_t-\Delta + b \cdot \nabla)^{-1}f :=(\mu+\partial_t-\Delta )^{-1}&- (\mu +\partial_t - \Delta)^{-\frac{1}{2}-\frac{1}{q}}Q_{p}(1 + T_p)^{-1}G_{p}(\mu +\partial_t - \Delta)^{-\frac{1}{2}+\frac{1}{r}}f,
H=\frac{b_{11}g_{22}+b_{22}g_{11}-b_{12}g_{21}-b_{21}g_{12}}{2(g_{11}g_{22}-g_{12}g_{21})}=\frac{b_{11}g_{22}+b_{22}g_{11}}{2(g_{11}g_{22}-(g_{12})^2)},
M_{1,\alpha}(V= \square) = - \frac{q^{2g+1}}{10! \zeta_q(2)}\Bigg( g^{10}R_0(\alpha) + g^9 R_1(\alpha) +g^8 R_2(\alpha) \Bigg) + O\Big(q^{2g+1}g^{7+\epsilon}\Big),
\gamma^{(1)}= \gamma(\begin{bmatrix}\varpi &\varpi & 1 & 1\end{bmatrix}), \gamma^{(2)}= \gamma(\begin{bmatrix}\varpi &1 & 1 & \varpi\end{bmatrix}), &\gamma^{(3)}= \gamma(\begin{bmatrix}1 &1 & \varpi & \varpi\end{bmatrix}), \gamma^{(4)}= \gamma(\begin{bmatrix}1 &\varpi & \varpi & 1\end{bmatrix}),
\exp \left\{i N T [ \sigma ] \right\}\equiv \int \left[ \prod_{n , x}d \psi_{r}^{\prime}( x ) \right] \exp \left\{i \int d^{d}x \; \left[ - \overline{{{\psi}}}_{r}^{\prime}\gamma^{\mu}\partial_{\mu}\psi_{r}^{\prime}+ \sigma_{\ell}( x ) \, j_{\ell}^{\prime}( x ) \right] \right\}\; .
R_{1 2}= R_{j_{1}j_{2}}^{i_{1}i_{2}}= \delta_{j_{1}}^{i_{1}}\delta_{j_{2}}^{i_{2}}\left( 1 + ( q - 1 ) \delta^{i_{1}i_{2}}\right) + ( q - q^{- 1}) \delta_{j_{2}}^{i_{1}}\, \delta_{j_{1}}^{i_{2}}\, \theta ( i_{1}- i_{2})
\langle f|g\rangle_{{\cal L}^{1|2}}=\langle f_0|g_0\rangle_{\cal L}^s+\langle f_1|g_1\rangle_{\cal L}^{s+1/2}+\langle f_2|g_2\rangle_{\cal L}^{s+1/2}+\langle f_3|g_3\rangle_{\cal L}^{s+1}\,,
\|\Psi\|_{L^r(\mathbb{R}^{d-1})}= \mu^{\frac{d-1}{p}-\frac{d-1}{r}}\|\Phi\|_{L^r(\mathbb{R}^{d-1})}= \mu^{\frac{d-1}{p}-\frac{d-1}{r}}\|\Phi\|_{L^r([0,1]^{d-1})}\leq M_1 \mu^{\frac{d-1}{p}-\frac{d-1}{r}}.
m_n^{(\alpha,\beta,\gamma,\delta,q)}=\frac{(abcd;q)_\infty}{(q, ab, ac, ad, bc, bd, cd ; q)_\infty}\sum_{\ell=0}^n(-1)^{n-\ell}{n\choose \ell}\left(\frac{1-q}{2}\right)^\ell\frac{Z_\ell}{\Pi_{i=0}^{\ell-1}(\alpha\beta-\gamma\delta q^i)}.
\mathbb{P}(Y_j \not\in U \ \big| \Y_1 = y_1, \dots ,Y_{j-1}= y_{j-1}) = \sum_{y \in V-U}\mathbb{P}(Y_j = y \ \big| \Y_1 = y_1, \dots ,Y_{j-1}= y_{j-1}) \ge |V-U|\alpha \beta= (q^n - q^m)\alpha \beta.
K(T,r)=\frac{1}{3}\underset{t>0}\int \underset{i \in \phi}\prod\frac{1}{1+\max\{e^t-1,T\}r^\alpha d_i^{-\alpha}}\mathrm{d}t\approx \frac{1}{3}\underset{t>0}\int \underset{i \in \phi}\prod\frac{1}{1+(e^t-1)r^\alpha d_i^{-\alpha}}\mathrm{d}t=K(r)
Z [ Y , \psi , \eta ] = \exp \left\{-{S_{c}}[{X_{c}}, \tilde{g}] \right\}\int{\mathcal{D}_{\tilde{g}}}\varphi{{\left( D e t^{\prime}\hat{\Delta}\right)}^{- d / 2}}{\frac{\sqrt{D e t^{\prime}{\hat{P}^{\dagger}}\hat{P}}}{V o l ( C K V )}}\exp \left\{-{S_{L}}[ \varphi , \hat{g}] \right\},
\liminf_{x\to \infty}\frac{\vert \mathcal{T}_2(x)\vert}{\pi(x)}=\liminf_{x\to \infty}\frac{\vert \mathcal{T}'_2(x)\vert}{\pi(x)}\ \mbox{and}\ \limsup_{x\to \infty}\frac{\vert \mathcal{T}_2(x)\vert}{\pi(x)}=\limsup_{x\to \infty}\frac{\vert \mathcal{T}'_2(x)\vert}{\pi(x)}.
\mathbb{B}''(({\mathbf V},\psi),(\tilde{{\mathbf V}},\tilde{\psi})):=\frac{d}{d\varepsilon}\mathbb{B}'(\hat{\mathbf U}+\varepsilon \tilde{{\mathbf V}},\hat{\varphi}+\varepsilon\tilde{\psi})({\mathbf V},\psi)\Big|_{\varepsilon=0},
A_{\mu}^{( c l )}= \left( \begin{array}{c c c}{{X_{\mu}^{( 1 )}}}&{{}}&{{}}\\{{}}&{{X_{\mu}^{( 2 )}}}&{{}}\\{{}}&{{}}&{{X_{\mu}^{( 3 )}}}\\ \end{array}\right) = \sum_{a = 0}^{2}X_{\mu}^{a}\otimes U^{a}.
\left( \begin{matrix}{{0}}&{{0}}&{{s 2}}&{{s 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{s 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{s 2}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{s 3}}&{{0}}&{{s 2}}\\{{0}}&{{s 3}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{matrix}\right)
\sum_{n_1,n_2}(C(n_1,n_2) \bar{\chi}\partial_{\mu}\chi A_{\mu}(n_1,n_2)+c.c.) +\sum_{(n_1,n_2)\neq (0,0)}(m C(n_1,n_2) \chi \bar{\chi}\phi(n_1,n_2) + c.c.)
{\cal L}= \frac{1}{g^{2}}\left[ ( D_{\mu}z )^{\dagger}( D^{\mu}z ) + i b \epsilon^{\mu \nu}( \partial_{\mu}z )^{\dagger}( \partial_{\nu}z ) - \frac{M^{2}}{2}b^{2}- \lambda ( z^{\dagger}z - 1 ) \right] ,
f^{\prime}_x(s)=-\frac{{(\psi^{-1})}^{\prime}\left(\sqrt{\frac{s_0-s}{\beta(G(s_0)-s_0)}}\right)}{2\sqrt{\beta(s_0-s)(G(s_0)-s_0)}}Q^{\prime}\left(\psi^{-1}\left(\sqrt{\frac{s_0-s}{\beta(G(s_0)-s_0)}}\right)+x\right),
\eta_{w}^{\ast}([\varphi_{T_{k}}\varphi_{T_{k}^{\ast}}])=\eta_{w}^{\ast}([\varphi_{T_{+}}]+[\varphi_{T_{-}}])=\frac{[\varphi_{\Omega_{w}^{-1}(T_{+})}]}{[\varphi_{I(T_{+})}]}+\frac{[\varphi_{\Omega_{w}^{-1}(T_{-})}]}{[\varphi_{I(T_{-})}]},
\hat{B}_N = \left(\begin{array}{ccc}\hat{b}_{0}\ddots \hat{b}_{N+1}\end{array}\right), R_N = \left(\begin{array}{ccccc}\xi_0 \eta_0 \ddots \theta_0 \ddots \ddots \ddots \ddots \xi_{N-1}\ddots \eta_{N-1}\theta_{N-1}\end{array}\right),
\{D_{\alpha \beta},{\bar{D}}_{\dot{\alpha}\dot{\beta}}\}T | ={\cal A}D_{\alpha \beta}{\bar{D}}_{\dot{\alpha}\dot{\beta}}{\bar{\cal A}}| + c . c . = 1 6{\cal A}\partial_{( \alpha ( \dot{\alpha}}\partial_{\beta ) \dot{\beta})}{\bar{\cal A}}| + c . c .
\big\{Z^{(\alpha)}_{j,m,n}\big\}_{j=1}^{n-1}=\big\{{-}x_{j,m}^{(-\alpha-1)}\big\}_{j=1}^m \cup\big\{x_{j-m,n-m-1}^{(\alpha+1)}\big\}_{j=m+1}^{n-1}.
\sigma_{e_\beta}(z_k)=-\frac{2\langle \beta,\alpha_k \rangle}{\langle \beta,\beta \rangle}\left(\sum_{1\leq i\leq k-1,\,\alpha_i=\beta}z_{i}\right)z_k+\begin{cases}0, \textrm{if}\alpha_k\neq\beta, -z_k^2, \textrm{if}\alpha_k=\beta,\end{cases}1 \leq k \leq n.
B = \left( \begin{array}{c c c c c c c c}{{0}}&{{x}}&{{z}}&{{y}}&{{0}}&{{- y}}\\{{- x}}&{{0}}&{{x}}&{{z}}&{{y}}&{{0}}\\{{- z}}&{{- x}}&{{0}}&{{x}}&{{z}}&{{y}}\\{{- y}}&{{- z}}&{{- x}}&{{0}}&{{x}}&{{z}}\\{{0}}&{{- y}}&{{- z}}&{{- x}}&{{0}}&{{x}}\\{{y}}&{{0}}&{{- y}}&{{- z}}&{{- x}}&{{0}}\\ \end{array}\right) ,
\int_{-\infty}^{+\infty}e^{xz}\operatorname{Ai}(z+a) \operatorname{Ai}(z+b) dz =\frac{1}{2\sqrt{\pi x}}\exp\left( \frac{x^3}{12}- \frac{a+b}{2}x - \frac{(a-b)^2}{4x}\right), x>0.
(-\mathcal L_{\beta,\Omega}^{})^{-1}(\mathcal L_\Omega \phi_0)(t,x) = \mathbf E\left[\int_0^{\tau_{t,x}}\mathcal L_{\beta,\Omega}\phi_0\left(X^{x,\alpha}(s)\right) ds\right]=\mathbf E\left[\phi_0(X^{x,\alpha}(\tau_{t,x})) \right]-\phi_0(x).
[\nabla u]^p_{W^{\tau,p}(B_{R/4})}\le \frac{(2-\Gamma)^{-p}\,(\Gamma-1)^{-p}}{(\Gamma-1-\tau)\,\tau\,}\,\frac{\mathcal{C}_3}{R^{p\,(1+\tau)}}\,\mathcal{A}_R(u,f).
N^{-7/4}(x-N)^3= \;\alpha\,N^{-3/8}\,(x-N)\,\frac{N^{-11/8}(x-N)^2}{\alpha}\le \; \frac12\Big(\alpha^2N^{-3/4}(x-N)^2+\frac{N^{-11/4}(x-N)^4}{\alpha^2}\Big)
(l+1)N\gamma_cw^cf^{a_1...a_l}=\sum_{i=1}^{l}w^{a_i}\left[(\gamma_cw^c-N)\gamma_bf^{a_1...{\not a_i}...a_lb}\right]+d\left[\left(\frac{l}{l+2}\gamma_cw^c+\frac{2}{l+2}N\right) \gamma_bj^{a_1...a_lb}\right],
\left\langle \left[ \sigma_{+}[ Q^{W}] \right]^{q}\left[ \sigma_{-}[ Q^{W}] \right]^{n - q}\right\rangle_{Q^{\mu}}= \left\langle \left[ \sigma_{+}[ Q^{W}] \right]^{n - q}\left[ \sigma_{-}[ Q^{W}] \right]^{q}\right\rangle_{Q^{\mu}}.
\begin{aligned}\int_{B(x, 2R)\backslash E}|H(z, y, 1)\lambda(y)|dy \leq C(n)\int_{B(x, 2R)\backslash E}|\lambda(y)|dy & \leq C(n)(\int_{B(x, 2R)\backslash E}|\lambda^2(y)|dy)^{\frac{1}{2}}(vol(B(x, 2R)\backslash E))^{\frac{1}{2}}&\leq \Phi(\epsilon|n, R)\end{aligned}
\sum_{j> m_1}p_j^{(n)}=\sum_{j\geq 1}p_j^{(n)}-\sum_{j=1}^{m_1}p_j^{(n)}&\leq |\sum_{j\geq 1}p_j^{(n)}-\sum_{j\geq 1}p_j^{(0)}|+|\sum_{j\geq 1}p_j^{(0)}-\sum_{j=1}^{m_1}p_j^{(0)}|+|\sum_{j=1}^{m_1}p_j^{(0)}-\sum_{j=1}^{m_1}p_j^{(n)}| &<\varepsilon.
\nabla = \beta_{g}g \frac{\partial}{\partial g}+ \beta_{\xi}\xi \frac{\partial}{\partial \xi}+ \beta_{\sigma}\sigma \frac{\partial}{\partial \sigma}+ \beta_{\rho}\rho \frac{\partial}{\partial \rho}+ \gamma_{Q}\mathbf{N}_{Q}+ \gamma_{\bar{Q}}\mathbf{N}_{\bar{Q}}+ \gamma_{B}\mathbf{N}_{B}.
\begin{aligned}1+ 1 \cdot(-3x_2) + (-3x_1) \cdot (-3x_2 + 1) + (-3x_1 + 1)\cdot (-x_2^3 + 3x_2^2) + (-x_2^3 + 3x_2^2) \geq &0.5(-0.75x_1 + 0.5) + 0.5(1-x_1)^3 + 0.5 (-0.75x_2 + 0.5) + 0.5(1-x_2)^3 - 0.75.\end{aligned}
\sigma_{3}= \left[ \begin{array}{c c}{{- q}}&{{0}}\\{{0}}&{{q^{- 1}}}\\ \end{array}\right] \; , \qquad \sigma_{-}= \left[ 2 \right]^{1 / 2}\left[ \begin{array}{c c}{{0}}&{{0}}\\{{- q^{1 / 2}}}&{{0}}\\ \end{array}\right] \; ,
\widetilde \Omega := \Big\{\big|\widehat f_{3,n}(x) -f_0(x)\big|\leq C_3 L_n^{\frac{\beta}{\beta+1}}+C_3 \big(f_0(x)L_n\big)^{\frac{\beta}{2\beta+1}}, \ \x\in [0,1]\Big\}.
\underset{\left\{\pmb{F}_k\right\}_{k=1}^K,\left\{\pmb{W}_k\right\}_{k=1}^{K}}{}\,\,\,\,\,\sum\limits_{k=1}^K||\pmb{\Xi}_k^{(\kappa)}\pmb{J}_k||_{*}+||\pmb{S}_e\pmb{\Phi}_e^{(\kappa)}||_{*}\,\,\,\,\, \qquad\pmb{S}_k \succeq \pmb{0},\,\,\, \sigma_{min}(\pmb{S}_k) \geq \epsilon,
\begin{array}{c c c}{\psi^{a +}}{= - \frac 1{\sqrt{2}}\left( \chi^{a}+ \overline{{\chi}}^{a}\right)}{}{\psi^{a -}}{= - \frac i{\sqrt{2}}\left( \chi^{a}- \overline{{\chi}}^{a}\right)}{}\end{array}
\{\mathbf{S}_{Q}, \mathbf{S}_{Q}\}= - 2 \mathbf{Z}, \qquad \{\mathbf{S}_{Q}, \mathbf{\bar{Q}}_{\mu}\}= - i \mathbf{P}_{\mu}, \qquad \{\mathbf{\bar{Q}}_{\mu}, \mathbf{\bar{Q}}_{\nu}\}= - 2 \delta_{\mu \nu}\mathbf{\bar{Z}}.
D^\alpha{}_\beta=\begin{pmatrix}1 0 -a\sin(x^3+x^4) -a\sin(x^3+x^4) 0 1 \pm a\cos(x^3+x^4) \pm a\cos(x^3+x^4) 0 0 1 0 0 0 0 1\end{pmatrix},
(D_{-\frac{\nu}{2},k})_nK_{-\frac{1}{2}\nu}(n'^{-1}n) = (D_{-\frac{\nu}{2},k}K_{-\frac{1}{2}\nu})(n'^{-1}n) = |(n'^{-1}n)_2|^{2k}K_{-\frac{1}{2}\nu-k}(n'^{-1}n),
B=\begin{bmatrix}b_{11}b_{12}q^mb_{13}b_{14}b_{21}b_{22}(q^{2m-1}-1)b_{23}-q^mb_{13}q^{m-1}b_{31}b_{32}b_{33}b_{34}b_{41}b_{42}q^{m-1}(q^{2m-1}-1)b_{43}(q-2)(q^{2m}-1)b_{44}\end{bmatrix},
\Phi(v_1,v_2,v_3) = \bar G (v_1, v_2,v_3,_2v_3,v_1v_3,v_1v_2,v_1v_2v_3) &= \varphi(v_1) + \varphi(v_2) + \varphi(v_3)+g(v_2v_3) + g(v_1v_3) + g(v_1v_2) + h(v_1v_2v_3).
\mathbb{P}\left\{f^{(i)}_t V^{(i)}_t \leq \bar{t}\right\}= \mathbb{P}\left\{V^{(i)}_t \leq \frac{4\bar{t}}{\sigma^2_{w,i}\sqrt{n_x}\left((1+2\alpha)^2\rho^{(i)}\frac{\| \Sigma^{(i)}_t\|}{\sqrt{n_x}}+ 4\right)}\right\},
\prod \limits_{i=t_r}^{n - 1}|c(\theta_i)p(x_i)| \leq 4^{2K_{p_r}+ 20}\cdot 4^{K_{p_r}+ 10}\cdot (3/5)^{\left(1 - \frac1{M_0}\right)(n - t_r)/2}\leq &\leq 4^{4K_{p_r}}\cdot (3/5)^{\left(1 - \frac1{M_0}\right)(n - t_r)/2},