harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \begin{array}{c}{0}\\{g^{\prime}\gamma_{d}^{\alpha}B_{\alpha}\psi_{l L}}\\ \end{array}\right) + \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{\psi_{1 R}}\\{\psi_{1 R}}\\ \end{array}\right) .
{m_2-k_1 \over n}=1-h_b(p_1*p_2*\alpha_1*\alpha_2)-\delta_1,{m_3-k_2 \over n}=1-h_b(p_2*P_3*\alpha_2)-\delta_2,{m_1-k_3 \over n}=1-h_b(p_1*P_3*\beta_1)-\delta_3,{m_2-k_4 \over n}=1-h_b(p_1*p_2*\beta_1*\beta_2)-\delta_4.
\mathbf{x}= \sqrt{P_0}\sum_{i \in \mathcal{K}_{\mathrm{0}}}{\sqrt{q_i}\mathbf{v}_i x_i}+ \sqrt{P^{(c)}}\mathbf{v}^{(c)}x^{(c)}& +\sum_{k \in \mathcal{K}_{\mathrm{\alpha}}}{\sqrt{P_k^{(p)}}\mathbf{v}_k^{(p)}x_k^{(p)}}
d\sum\limits_{i=0}^{q-2}\binom{q}{i}(lx+1)^{q-i}B_{i}\equiv d \bigg(\binom{q}{0}B_{0}+\sum\limits_{i=1}^{\frac{q-2}{2}}\binom{q}{2i}B_{2i}\bigg) \equiv0\pmod{4}.
\Delta(1/p,p) = 4 +{2}^{p+1}p + \frac{8}{p}+{2}^{p+1}- 4 \cdot 2^p - \frac{8 \cdot 2^p}{p}- p - 1 = p\, \left({2}^{p+1}-1 \right) + \left(3-{2}^{p+1}\right) +{\frac{8\,(1-{2}^{p})}{p}}.
\Delta^m H(B^e)\times \Big(\prod_{j=1}^h\big(2H^{-(y_j-(2y-1)/(ep))}\big(H/H(B^e)\big)^{1/(2h)}\big)^2\Big)\times \times\big(C_{12}H^{(2y-1)/(ep)}\big)^{pe-2h}V(pe-2h),
U_{i,j}=\begin{cases}\left\{\frac{1}{2}(2g-3j+12i+25),\frac{1}{2}(2g+3j-6i-11),g-3i-7\right\}_{g}\mbox{if $j\equiv 1\pmod{2}$,}\left\{\frac{1}{2}(2g-3j+12i+22),\frac{1}{2}(2g+3j-6i-14),g-3k-7\right\}_{g-1}\mbox{if $j\equiv 0\pmod{2}$,}\end{cases}
\|(d_j - \alpha 1)^{-1}\| = \frac{1}{\|\lambda_j - \alpha\|}\left\|\Big( 1 - \frac{d_j - \lambda_j 1}{\alpha - \lambda_j}\Big)^{-1}\right \|\leq \frac{2}{3}\sum_{l=0}^{\infty}(2/3)^l= 2,
\frac{1}{F_\mu(x')}\int_{F_\mu(x)}^{F_\mu(x')}F_\mu^{-1}(w)dw \ge\frac{1}{F_\mu(x')}\int_u^{F_\mu(x')}\pi_{F_{\mu}^{-1}(w)}((-\infty,F_{\mu}^{-1}(w))){F_\mu^{-1}(w)}dw.
D = t H - \frac{1}{2}\int_{- \infty}^{\infty}d x ( x - v t ){\cal P}( x - v t ) - \frac{v}{2}t \int_{\infty}^{\infty}d x{\cal P}( x - v t ) = t ( H - \frac{v}{2}P ) - \frac{1}{2}D_{0},
H\left(X_i|\tilde{X}_i,\{\hat{X}_i(u)\}_{u \in T_k(i)}, \{\tilde{Z}_j\}_{j \notin T_k(i)},\{X_w\}_{w \neq i}\right) = H\left(X_i|\tilde{X}_i,\{\hat{X}_i(u)\}_{u \in T_k(i)}\right)
(y^{(m)}_{r_1,\ldots,r_k})^2= (\log R)^2\frac{\varphi(VP_f)^2}{(VP_f)^2}\Bigl(\prod_{i=1}^k\frac{\varphi(r_i)^2}{r_i^2}\Bigr)(F_{r_1,\ldots,r_k}^{(m)})^2 &+O\left(\frac{(F_{max})^2\varphi(VP_f)^2(\log X)^2}{(VP_f)^2D_0}\right).
\langle \Psi_{1}^{\prime I_{1}}( z_{1}) \Psi_{2}^{\prime I_{2}}( z_{2}) \cdots \Psi_{n}^{\prime I_{n}}( z_{n}) \rangle = \langle \Psi_{1}^{I_{1}}( z_{1}) \Psi_{2}^{I_{2}}( z_{2}) \cdots \Psi_{n}^{I_{n}}( z_{n}) \rangle \, .
r(\rho) = \frac{(ga^3)^{1/2}}{2}\sqrt{\frac{3}{5}}\Big[ \frac{2}{3}\rho^{2/3}_{\: \quad 2}F_1 \big[ - \frac{9}{20}, \frac{1}{4}, \frac{11}{20}; \frac{l^{10/3}}{\rho^{10/3}}\big] + 3 l^{3/2}\frac{\Gamma\left(-\frac{9}{20}\right) \Gamma\left( \frac{3}{4}\right)}{10 \Gamma\left(\frac{3}{10}\right)}\Big],
E_{n,m}= t\left[\left(\sum_{i=1}^{n-1}l_i\right)\left(\sum_{i=1}^{m-1}k_i\right)\eta^2+\omega \eta\left(\sum_{i=1}^{m-1}k_i-\sum_{i=1}^{n-1}l_i\right)-\frac{\eta^2 N^2}{4}\right] = U\left(\sum_{i=1}^{n-1}l_i-\sum_{i=1}^{m-1}k_i \right)^2 +\mu \left(\sum_{i=1}^{n-1}l_i-\sum_{i=1}^{m-1}k_i\right).
\mathcal{A}_0 =A_{0}^{(1)}+A_{0}^{(2)}+A_0^{(3)}, \mathcal{A}_{\pm}=A_{\pm}^{(1)}P^{(2)}P^{(3)}+A_{\pm}^{(2)}P^{(3)}+A_{\pm}^{(3)}, \mathcal{P}=P^{(1)}P^{(2)}P^{(3)}.
\eqref{prop3}\leq -\frac1{N^d}\log \sum_{A\subset\Lambda_N \,:\, A \,is\, \Delta-sparse}\exp(J'{|A|})P_{A^c}(\Omega_{A^c}^+) &\leq -\frac1{N^{d}}\left( \left(\frac N\Delta\right)^d[(d\log\Delta+c_0)+J' -d\log\Delta+c_1\log\log\Delta]\right) &=-\frac{J'+c_0+c_1\log\log\Delta}{\Delta^d}<0 \mbox{for $\Delta=\Delta(J')$ large enough.}
\lim_{n \rightarrow \infty}\left\|\left(V_{j,n}(\cdot, 0) - U_{j,n}(\cdot, 0), \partial_t V_{j,n}(\cdot, 0) - \partial_t U_{j,n}(\cdot, 0)\right)\right\|_{\dot{H}^1 \times L^2}= 0.
\pi_{n+1}(x) - (x - \check S_{n^n(n+1)}^* \check S_{n^{n+1}}^{-1}+ \check S_{(n-1)^{n-1}n}^*\, \check S_{(n-1)^n}^{-1}) \pi_n(x)+\check S_{n^{n+1}}\,\check S_{(n-1)^n}^{-1}\, \pi_{n-1}(x) = 0
J_s=J_s^{(j)}=J_{s,m}^{(j)}(g(x))\equiv \frac{1}{2\pi}\int_Xg(x)e^{-\frac{\hat h_{j,+}(x,e^{-iu}\circ x)}{t}}\mathrm{Tr}\,\hat b^+_{j,s}(x,e^{-iu}\circ x)e^{-imu}dv_X(x)
\left[\pi_k(E_{\sqrt{c}}-F_{\sqrt{c}}) M(\cdot,x)\right](n) = -\sqrt{c}(2k+x) M(n,x+1) - \frac{x}{\sqrt{c}}M(n,x-1), \left[\pi_k(E_{\sqrt{c}}+F_{\sqrt{c}}) M(\cdot,x)\right](n) = \sqrt{c}(2k+x) M(n,x+1) - \frac{x}{\sqrt{c}}M(n,x-1),
\begin{array}{l}{{\omega_{1 \beta}= \displaystyle \frac 1{( \Lambda \Pi )^{2}}( \Lambda \Pi_{\sigma}) P^{+ \sigma \gamma}\Lambda_{n}\Gamma^{n}\tilde{\Gamma}^{m}\Pi_{m \gamma}k_{\beta},}}\\{{\omega_{2 \beta}= \displaystyle \frac 1{( \Lambda \Pi )^{2}}( \Lambda \Pi_{\sigma}) P^{+ \sigma \gamma}\Pi_{n \gamma}\Gamma^{n}\tilde{\Gamma}^{m}\Lambda_{m}s_{\beta}.}}\\ \end{array}
f ( x ) = c_{1}\epsilon ( x^{0}) \delta ( ( x^{0})^{2}- |{\bf x}|^{2}) + c_{2}\left( ( ( x^{0}+ i 0 )^{2}- |{\bf x}|^{2})^{- 1}+ ( ( x^{0}- i 0 )^{2}- |{\bf x}|^{2})^{- 1}\right) + c_{3}
\frac{1}{N}\sum_k^{(i)}\frac{\partial \mathbf{h}_i^*}{\partial g_{ik}}G\mathbf{e}_i \mathbf{e}_k^* X_i G\mathbf{e}_i= O_\prec(\frac{1}{N}), \frac{1}{N}\sum_k^{(i)}\mathbf{h}_i^* \frac{\partial G}{\partial g_{ik}}\mathbf{e}_i \mathbf{e}_k^* X_i G\mathbf{e}_i= O_\prec(\Pi_i^2).
\int_a^b \int_\Omega \chi^2(x)u(s,x)\overline{\phi(s,x)}\eta'(s)\,{\rm d}x\,{\rm d}s = \int_a^b \eta(s) \int_\Omega \Bigl(A(s,x)\nabla u(s,x)\cdot\nabla\bigl(\overline{\phi(s,x)}\chi^2(x)\bigr) -\nabla\bigl(u(s,x)\chi^2(x)\bigr)\cdot\overline{A(s,x)^{*}\nabla\phi(s,x)}\Bigr)\,{\rm d}x\,{\rm d}s.
\alpha_{\pm}\bigl(\mathcal{E}(n)\bigr) =\mathcal{E}(n\pm 1)-\mathcal{E}(n), R_{-1}\bigl(\mathcal{E}(n)\bigr)R_0\bigl(\mathcal{E}(n)\bigr)^{-1}=B_n=0, a^{(+)}\phi^{(\pm)}_n(x)=A_n\phi^{(\pm)}_{n+1}(x), a^{(-)}\phi^{(\pm)}_n(x)=C_n\phi^{(\pm)}_{n-1}(x).
(g_1,\varepsilon_1) \circ (g_2,\varepsilon_2) := (g_1g_2, \varepsilon ) ~~ , ~~ ~~ \varepsilon (z) = \varepsilon_1(g_2.z)\, \varepsilon_2(z) \ , (g,\varepsilon )^{-1}:= (g^{-1},\tilde{\varepsilon}) ~~ , ~~ ~~ \tilde{\varepsilon}(z) = \varepsilon (g^{-1}.z)^{-1}\ .
W_3= (I-MR(\lambda,A_{-1})B)^{-1}(MR(\lambda,A)Le_\lambda+Ke_\lambda)N_1R(\lambda,A)LR(\lambda,\mathfrak{A}) &+N_2(MR(\lambda,A)LR(\lambda,\mathfrak{A})+KR(\lambda,\mathfrak{A})),
II = |-2 r \int_{6r}^{\infty}dw \left(\lambda_{1}''(t+w)-\lambda_{2}''(t+w)\right)\left(\frac{w}{2(1+w^{2})}- \frac{w}{2(\lambda_{1}(t+w)^{2-2\alpha}+w^{2})}\right)| +E_{II}
\left\{\int_{\cal R}d^{n}p^{\prime}( \tilde{\omega}_{p^{\prime}}| e_{p^{\prime}p}| \eta_{p}) , \int_{\cal R^{\prime}}d^{n}q^{\prime}( \tilde{\omega}_{q^{\prime}}| f_{q^{\prime}q}| \eta_{q}) \right\}=
2\partial_{\bar z}(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})+A_1(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})-(\mathcal P_1\Theta^{-1}{\mathcal P_2^*})A_2=0\quad\mbox{in}\,\,\Omega \setminus \mathcal X.
(q_R)_{1,2}=\frac{2q_L-(u_L-u_R)(2u_R-1)}{2}\ \ \pm \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\sqrt{[-2q_L+(u_L-u_R)(2u_R-1)]^2+4\left[(u_L-u_R)\left((2u_L-1)q_L+\frac{u_L^2}{2}-\frac{u_R^2}{2}-\frac{2u_L^3}{3}+\frac{2u_R^3}{3}\right)-q_L^2\right]}}{2}.
te_{q}(xt)+\frac{t}{e_{q}(t)-1}e_{q}(tx)=\frac{te_{q}(xt)}{e_{q}(t)-1}e_{q}(t)=\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}(x,1)\frac{t^{n}}{\left[n\right]_{q}!},\left\vert t\right\vert <2\pi.
(\nabla h)(x) \cdot (\nabla q)(x) = 8{x_1}^4 + 8{x_1}^2{x_2}^2 - 8{x_1}^2 + 8{x_1}^2{x_2}^2 + 8{x_2}^4 - 4{x_2}^2 &= 8({x_1}^2 +{x_2}^2 )^2 - 8({x_1}^2 +{x_2}^2 ).
i D_{\mu \nu}( p ) = - i \frac{1}{p^{2}- \mu^{2}}\left( g_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}\right) + \mu \frac{1}{p^{2}- \mu^{2}}\frac{1}{p^{2}}\varepsilon_{\mu \nu \rho}p^{\rho}- i \alpha \frac{p_{\mu}p_{\nu}}{p^{4}}.
C_{h,u}(t_{j+1})u= -\frac{C}{2}\Bigg[ \int_{t_{j+1}-h}^{t_{j+1}}\left( \int_t^{t_{j+1}}\!\! A \int_0^s e^{A(s-r)}Bdu(r) \, ds + B\big(u(t)-u(t_{j+1})\big) \right) dt + \int_{t_{j+1}}^{t_{j+1}+h}\left( \int_{t_{j+1}}^t \!\! A \int_0^s e^{A(s-r)}Bdu(r) \, ds + B\big(u(t)-u(t_{j+1})\big) \right) dt \Bigg].
\Pr\left(Q_t=A|Q_{t-1}=A\right)= \Pr\left(X_t=A|X_{t-1}=A\right)=1- \alpha, &\Pr\left(Q_t=B|Q_{t-1}=A\right)= \Pr\left(X_t=B|X_{t-1}=A\right)= \alpha, &\Pr\left(Q_t=A|Q_{t-1}=B\right)= \Pr\left(X_t=A|X_{t-1}=B\right)= \beta, &\Pr\left(Q_t=B|Q_{t-1}=B\right)=\Pr\left(X_t=B|X_{t-1}=B\right)= 1- \beta.
n^2\exp\Bigl[-\Omega\Bigl(n^{1-2s}c^{\frac{1-2s}{1-s}}\Bigr)\Bigr]= n^2\exp\Bigl[-\Omega\Bigl(\log^{\frac{\alpha(1-2s)}{1-s}}n\Bigr)\Bigr]< \frac{1}{n^\beta},
\ddot{u}( k , t ) \, + \, \left[ \, k^{2}-{\frac{1}{2}}\left({\frac{\ddot{\kappa}}{\kappa}}\, - \,{\frac{1}{2}}{\frac{\dot{\kappa}^{2}}{\kappa^{2}}}\right) \right] \, u ( k , t ) \, = \, 0
x_i = (-1)^{i-1}\epsilon (1 \leq i \leq d), y_i = (-1)^{i-1}\epsilon (1 \leq i \leq d), z_i = \begin{cases}i (d-i+1- \epsilon \tau) , (d-i+1)(i + \epsilon \tau) \end{cases}(1 \leq i \leq d).
\Big(\sum_{t=1}^{T}\eta_t\Big)^{-1}= \frac{1}{\min\big(\frac{T}{16\kappa \bar{L}d},T\gamma\big)}= \max\Big(\frac{18\kappa \bar{L}d}{T},\frac{1}{T\gamma}\Big) \leq \frac{16\kappa \bar{L}d}{T}+\frac{1}{T\gamma}.
R ( \lambda , \mu ) = \left[ \begin{array}{c c c c}{f ( \mu , \lambda )}&{0}&{0}&{0}\\{0}&{1}&{g ( \mu , \lambda )}&{0}\\{0}&{g ( \mu , \lambda )}&{1}&{0}\\{0}&{0}&{0}&{f ( \mu , \lambda )}\\ \end{array}\right]
\sum_{m > 0}\int \frac{d^{2}\rho}{\pi}\e^{- | \rho |^{2}}( \rho | e^{\sqrt{\alpha^{\prime}/ 2}k_{\mu}\alpha_{- n}^{\mu}z^{n}/ n}\left[ z^{- m - 1}\alpha_{m}^{\mu}+ z^{m - 1}\alpha_{- m}^{\mu}\right] e^{- \sqrt{\alpha^{\prime}/ 2}k_{\mu}\alpha_{n}^{\mu}z^{- n}/ n}| q^{n}\rho )
\sum_{\substack{p \mid mn \Rightarrow T_{j-1}\leq p < T_j \Omega(n) = \Omega(n) = r}}\frac{r!^2 g(n) g(m)}{[n,m]}\leq r!^2 \sum_{j=0}^r \sum_{\substack{p \mid d \Rightarrow T_{j-1}\leq p < T_j \Omega(d) = j}}\frac{1}{d}\Bigg(\sum_{\substack{p \mid n \Rightarrow T_{j-1}\leq p < T_j \Omega(n) =r - j}}\frac{g(nd)}{n}\Bigg)^2.
P_{\rm{pro}}^{{\cal O},\rm{LB}}\left({{\gamma_{th}}}\right) ={F_{\Upsilon_{{T_1}}^{{\rm{UB}}}}}\left({{\gamma_{th}}}\right) +{F_{\Upsilon_{{T_2}}^{{\rm{UB}}}}}\left({{\gamma_{th}}}\right) -{F_{\Upsilon_{{T_1}}^{{\rm{UB}}}}}\left({{\gamma_{th}}}\right){F_{\Upsilon_{{T_2}}^{{\rm{UB}}}}}\left({{\gamma_{th}}}\right)
M_{\mathcal{R}}(f) = \begin{pmatrix}a_0 a_1 \dots a_{n-1}u \sigma(a_{n-1}) \sigma(a_0) \dots \sigma(a_{n-2}) \vdots \vdots \ddots \vdots u \sigma^{n-1}(a_1) u \sigma^{n-1}(a_2) \dots \sigma^{n-1}(a_0)\end{pmatrix}.
|U_2| \leq U_{21}+ U_{22}= O\biggl(\frac{k^{1/2}}{n}\max\biggl\{\frac{k^{\beta/d}}{n^{\beta/d}}\, , \, \frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
\begin{array}{l l}{{a^{b}= u^{a}\nabla_{a}u^{b}}}&{{={\frac{1}{V}}\, \xi^{a}\nabla_{a}\,{\frac{1}{V}}\, \xi^{b}={\frac{1}{V^{2}}}\, \xi^{a}\nabla_{a}\, \xi^{b}}}\\{{}}&{{= -{\frac{1}{V^{2}}}\, \xi^{a}\nabla^{b}\xi_{a}={\frac{1}{2}}\,{\frac{1}{V^{2}}}\, \nabla^{b}\left( - \xi^{a}\xi_{a}\right)}}\\{{}}&{{={\frac{1}{2}}\,{\frac{1}{V^{2}}}\, \nabla^{b}\, V^{2}= \nabla^{b}\ln V \; ,}}\\ \end{array}
{\cal L}_{\rm MCS}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\epsilon^{\mu\nu\rho}A_\mu \partial_\nu A_\rho +|D_\mu \phi |^2 +\frac{1}{2e^2}(\partial_\mu{\cal N})^2 -V(|\phi|,{\cal N})
\left\|\sum_{k=1}^{\infty}A_{\overline{\i}}^{k-1}x\right\|\leq \|x\|+\left\|\sum_{k=1}^{\infty}A_{\overline{\i}}^{k}x\right\|\leq \|x\|+\sum_{k=1}^{\infty}\lambda^k\|x\|\leq \|x\|\left(1+\frac{\lambda}{1-\lambda}\right)
{}_{2}F_{1}(\frac{3}{2}a,\frac{1}{2}(3a-1);a+\frac{1}{2};-\frac{z^{2}}{3})=(1+z)^{1-3a}{}_{2}F_{1}(a-\frac{1}{3},a;2a;2z(3+z^{2})(1+z)^{-3})
w(r_1) =\sum_{i=1}^k2^ix_i^{\prime}=\sum_{i=1}^{u}2^ix_i^{\prime}+\sum_{j=u+1}^{k}2^jx_j^{\prime}&\leq 2^u(1+x_u^{\prime})+\sum_{j=u+1}^{k}2^jx_j^{\prime}=2^u(1+x_u^{\prime})+\sum_{j=u+1}^{k}2^jx_j &\leq 2^ux_u +\sum_{j=u+1}^{k}2^jx_j\leq w(r_2).
(\tilde{F}_{p}v,\tilde{F}_{p}w) =(\tau_{f(p)}D_{p}f\tau_{p}^{-1}(v),\tau_{f(p)}D_{p}f\tau_{p}^{-1}(w))=\tau_{f(p)}D_{p}f\tau_{p}^{-1}(v,w) &=D_{0}(\tilde{f}_{p})(v,w)=(D_{0}(\tilde{f}_{p})_{1}(v,w),D_{0}(\tilde{f}_{p})_{2}(v,w)).
Q_1(t)=a_{0,\sup}(t)+\frac{\chi_1}{2d_3}\big(k{\bar r_1}+l{\bar r_2}\big)+\frac{k^2}{4\lambda d_3}\Big(\frac{\chi_1^2{\bar r_1}^2}{d_1}+\frac{\chi_2^2{\bar r_2}^2}{d_2}\Big)+\frac{a_{2,\sup}(t){\bar r_1}+b_{1,\sup}(t){\bar r_2}}{2},
(\begin{matrix}e_{+}\\e_{-}\\e_{0}\end{matrix})=U(\begin{matrix}e_{x}\\e_{y}\\e_{z}\end{matrix}),U=(\begin{matrix}-\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\ +\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{matrix}),
\int_{\mathbb{R}^{N}}f(\widetilde{\Psi}_{\epsilon_{n}}(y_{n}))\widetilde{\Psi}_{\epsilon_{n}}(y_{n})dx =\int_{\mathbb{R}^{N}}f(t_{\epsilon_{n}}\eta(|\epsilon_{n}z|)w(z))t_{\epsilon_{n}}\eta(|\epsilon_{n}z|)w(z)dx.
(1) \quad{\mbox{\bf{i}}}^2={\mbox{\bf{j}}}^2={\mbox{\bf{1}}}, \quad{\mbox{\bf{k}}}^2= -{\mbox{\bf{1}}}, \quad{\mbox{\bf{i}}}{\mbox{\bf{j}}}={\mbox{\bf{k}}}, \quad{\mbox{\bf{j}}}{\mbox{\bf{k}}}=-{\mbox{\bf{i}}}, \quad{\mbox{\bf{k}}}{\mbox{\bf{i}}}=-{\mbox{\bf{j}}},
\Sigma = \langle \Omega \vert T r ({\bar{\Psi}( 0 )}\Psi ( 0 ) ) \vert \Omega \rangle = -{\frac{s i n ( \theta )}{2 L \sqrt{2}}}\int_{- 1}^{0}\zeta ( Z_{R}) \zeta ( - Z_{R}- 1 ) d Z_{R}.
{\cal B}:=\Bigl\{(T_{W},V_{X|W},V_{Y|W}):\forall(w,x,y), \frac{(1-\epsilon)^{2}}{1+\epsilon'}\leq\frac{[T_{W}V_{X|W}V_{Y|W}](w,x,y)}{Q_{WXY}(w,x,y)}\leq\frac{(1+\epsilon)^{2}}{1-\epsilon'}\Bigr\}.
\frac{\delta^{2}\mathcal{A}}{\delta x_{\mu}( s ) \delta x_{\mu}( s^{\prime})}= \frac{\delta^{2}\mathcal{A}}{\delta x_{1}( s ) \delta x_{1}( s^{\prime})}+ \frac{\delta^{2}\mathcal{A}}{\delta \vec{\phi}( s ) \delta \vec{\phi}( s^{\prime})}.
\lim_{\tau' \to 0}\textbf{g}\left( S^\textnormal{coh}(\tau,z)(T_a),S^\textnormal{coh}(\tau,z)(T_b) \right) = \textbf{g}\left( e^{-\tau_2/z}T_a,e^{-\tau_2/z}T_b \right) = g(T_a,T_b)
\tilde{\beta}_{i}^{N}(t) =\frac{N\lambda+1}{N(\lambda+1)}\beta_{i}(t)+\frac{1}{N(\lambda+1)}\sum_{k:k\neq i}\beta_{k}(t) \widetilde{M}_{i}^{N}(t) =\sum_{k:k\neq i}(M_{ki}^{N}(t)-M_{ik}^{N}(t)).
[g] = \begin{pmatrix}m_{1,1}m_{1,2}0 \cdots 0 m_{2,1}m_{2,2}0 \cdots 0 m_{3,1}m_{3,2}m_{3,3}\cdots 0 \vdots \vdots \vdots \ddots \vdots m_{n,1}m_{n,2}m_{n,3}\cdots m_{n,n}\end{pmatrix}
x_+-y_+ = z_--y_++\alpha_\rho(y_--z_-)= z_--y_+-\alpha_\rho(z_++z_--y_+-y_-)+\alpha_\rho(z_+-y_+) &> z_--y_+ -\beta(z_++z_--y_+-y_-)+\alpha_\rho(z_+-y_+)=\alpha_\rho(z_+-y_+).
|\frac{1+w^{2}-R^{2}\lambda(t)^{2}- \sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}}{-4R^{2}\lambda(t)^{2}}| \leq C \frac{R^{2}\lambda(t)^{2}}{R^{2}\lambda(t)^{2}}\leq C, & R \lambda(t) > \frac{w}{2}, w \geq 1
V_{4}{}^{i j}\equiv \Phi^{i j}= -{\frac{1}{2}}e^{2 U}\epsilon_{i j m}\partial_{m}\left( \ln \chi^{( 1 )}\chi^{( 2 )}\right) \qquad V_{4}{}^{i 4}\equiv \Phi^{i 4}={\frac{1}{2}}e^{2 U}\partial_{i}\left( \ln{\frac{\chi^{( 1 )}}{\chi^{( 2 )}}}\right)
( - 1 )^{F_{L}}\left| B , \pm \right\rangle_{\mathrm{R R}}= \left| B , \mp \right\rangle_{\mathrm{R R}}, \qquad ( - 1 )^{F_{R}}\left| B , \pm \right\rangle_{\mathrm{R R}}= \left| B , \mp \right\rangle_{\mathrm{R R}}.
x_t = \frac{x_{yy}}{e^{2\phi}+ x_y^2}- \phi'(x) \left( 1 + \frac{x_y^2}{e^{2\phi}+ x_y^2}\right) = \frac{\partial}{\partial y}\left( e^{-\phi}\tan^{-1}(x_{y}e^{-\phi}) \right) + \phi'(x) \left( x_y e^{-\phi}\tan^{-1}(x_y )e^{-\phi}-1 \right),
\partial^{\mu}\widetilde{A}_{\mu}( x ) = - 4 \pi \int \delta \xi d s \, \epsilon_{\mu \nu \rho \sigma}\, \delta^{\mu}( s ) \left[ \Omega_{\xi}( s , 0 ) L^{\rho \sigma}[ \xi | s ] \Omega_{\xi}^{- 1}( s , 0 ) \right] \dot{\xi}^{\nu}( s ) \delta^{4}( x - \xi ( s ) ) .
\rho(x) - \frac{{\rm d}}{{\rm d}x}\left(\frac{\mathfrak{D}}{\sigma^*}\frac{{\rm d}\rho}{{\rm d}x}\right) = g(x)\mbox{for}x\in (-\ell, +\ell), \rho(x)=0\mbox{for}x=\pm\ell,
L_mw(\varepsilon) = q^{\kappa(m,\mu,\lambda)-(\varepsilon_1+ \cdots + \varepsilon_{m-1})}w(\varepsilon - \widehat{m}), R_mw(\varepsilon) = q^{\varepsilon_{m+1}+ \cdots + \varepsilon_N}w(\varepsilon + \widehat{m}),
G ( \{w \}) = \Bigg ( G_{i j}^{-}( \{w \}) \Bigg | \left. \begin{matrix}{{i}}&{{\downarrow}}&{{0 , 1 , \ldots , n}}\\{{j}}&{{\rightarrow}}&{{0 , 1 , \ldots , n}}\\ \end{matrix}\right. \Bigg ) ;
F_{\alpha}( a , t ) = \frac{\sinh ( b a_{\alpha}t ) \, \sinh ( b a_{\alpha}- 2 b Q_{\alpha}+ h ( 1 + b^{2}) ) t )}{\sinh t \, \sinh ( \alpha^{2}b^{2}t / 2 ) \, \sinh ( ( 1 + b^{2}) h t )}
(u^*_{pp})_{pp}+ y\hat{a}(x,y)(u^*_{pp})_{yy}+\hat{a}_x(x,y)yu^*_{yy}(u^*_{pp})_p + \left(\hat{a}_x(x,y)yu^*_{yyp}+\hat{a}_{xx}(x,y)yu^*_{yy}u^*_{pp}\right)(u^*_{pp}) = 0.
\vec{a}= (a_1,\cdots, a_N)^{T}\in \mathbb{R}^{N}, \, \vec{b}= (b_1,\cdots, b_N)^{T}\in \mathbb{C}^{N},\, \vec{c}= (c_1,\cdots, c_N)^{T}\in \mathbb{C}^{N}.
\lim_{(\theta,Q_{XYU})\to(0,Q_{XYU}')}\frac{\partial\Omega^{(\alpha,\theta)}(Q_{XYU})}{\partial\theta}=\sum_{x,y,u}Q_{XYU}'(x,y,u)\omega_{Q_{XYU}'}^{(\alpha)}(X,Y|U) =R^{(\alpha)}(Q_{XYU}').
{}_{N}\! \langle \psi_{n}^{\prime}, \bar{\psi}_{n}^{\prime}| \psi_{n}, \bar{\psi}_{n}\rangle_{N}= \prod_{\mu = 0}^{p}\delta \left( \mathrm{R e}\psi_{n \mu}^{\prime}- \mathrm{R e}\psi_{n \mu}\right) \delta \left( \mathrm{I m}\psi_{n \mu}^{\prime}- \mathrm{I m}\psi_{n \mu}\right) ~ .
f(P)Ag(P)A =\sum_{i=1}^n f(\lambda_i) E_iA \sum_{j=1}^n g(\lambda_j) E_jA=\sum_{i=1}^n f(\lambda_i)g(\lambda_i)E_iAE_iA +\sum_{i< j}\left(f(\lambda_i)g(\lambda_j)E_iAE_jA +f(\lambda_j)g(\lambda_i)E_jAE_iA\right).
G_{-2}=P_0, G_{-1}=0, G_{0}=(I-P_0)\bigl(G_{0,0}-G_{0,0}vM_0^\dagger v^*G_{0,0}\bigr)(1-P_0), G_1 =(I-P_0)\bigl(I-G_{0,0}vM_0^\dagger v^*\bigr)G_{0,1}&\times\bigl(I-vM_0^\dagger v^*G_{0,0}\bigr)(I-P_0) &-P_0G_{0,0}vM_0^\dagger v^*G_{0,1}vM_0^\dagger v^*G_{0,0}P_0,
{\rm{span}}\{Z, [Z_1, Z_2]\,:\, \mbox{with}Z, Z_1, Z_2 \in \mathcal{H}^0_\xi\}= &={\rm{span}}\left\{V_1, W_1,\ldots, V_{n-1}, W_{n-1}, \tau-\frac{k(\xi)|\mathcal{P}_H(\nu)|}{2}\,\eta\right\}.
\cos\theta = \frac{\dot{\chi}}{\sqrt{\dot{\chi}^2+\dot{\varphi}^2 \, e^{-\alpha\chi}}}\, , \,\,\,\sin\theta = \frac{\dot{\varphi}\, e^{-\frac{\alpha \chi}{2}}}{\sqrt{\dot{\chi}^2+\dot{\varphi}^2 \,e^{-\alpha\chi}}}\, .
\partial_t h =\sum_{j=0}^\infty S_j h_0 & \le h_0(t)\sum_{j=0}^\infty\frac{1}{(j)!}\left(\int_0^t s k^*(s)ds\right)^{j}&=h_0(t)\, \exp\left(\int_0^t s k^*(s)ds\right) &=\int_0^t k^*(s)ds \,\exp\left(\int_0^t s k^*(s)ds\right)
\begin{matrix}R_{CS}^{2}=1-(\frac{1}{L_{0}})^{\frac{2(R_{McF}^{2})}{n}}\\ [1.5em]R_{McF}^{2}=-\frac{n}{2}\cdot\frac{ln(1-R_{CS}^{2})}{lnL_{0}}\end{matrix}
\left[\begin{array}{ll}B^{(m)}_{n-m-1}B^{(m+1)}_{n-m-1}A^{(m)}_{n-m-1}A^{(m+1)}_{n-m-1}\end{array}\right]=\left[\begin{array}{cc}0 1 1 a_{n-m}\end{array}\right]\left[\begin{array}{ll}B^{(m-1)}_{n-m}B^{(m)}_{n-m}A^{(m-1)}_{n-m}A^{(m)}_{n-m}\end{array}\right]
\left| \nabla_{y'}^\alpha r'_\lambda(y',y_d,z_d)\right|+\left| \nabla_{y'}^\alpha r_{d,\lambda}(y',y_d,z_d)\right| & \leq \frac{Cy_d}{(y_d + z_d +|y'|)^{d-1+\alpha}}\frac{e^{-c|\lambda|^{\frac 12}z_d}}{\big (1+|\lambda|^{\frac 12}(y_d+z_d + |y'|)\big )\big (1+|\lambda|^{\frac 12}(y_d+z_d)\big )},
g_{5}\int_{0}^{\pi R}d y \left( \sqrt{\frac{2}{\pi R}}\cos \frac{y}{2 R}\right)^{2}\sqrt{\frac{2}{\pi R}}\cos \frac{y}{R}= g_{5}\sqrt{\frac{1}{2 \pi R}}= \frac{g_{4}}{\sqrt{2}}.
\begin{array}{c}{{\lambda_{1}\rightarrow e^{i ( \phi_{1}- \phi_{2}- \phi_{3}) / 2}\lambda_{1},}}\\{{\lambda_{2}\rightarrow e^{i ( - \phi_{1}+ \phi_{2}- \phi_{3}) / 2}\lambda_{2},}}\\{{\lambda_{3}\rightarrow e^{i ( - \phi_{1}- \phi_{2}+ \phi_{3}) / 2}\lambda_{3},}}\\{{\lambda_{4}\rightarrow e^{i ( \phi_{1}+ \phi_{2}+ \phi_{3}) / 2}\lambda_{4}.}}\\ \end{array}
\,L(z) = \sum_{\sigma \in \mathfrak{S}_N}(\sigma) \, q^{\ell(\sigma)}L^{+}_{1,\sigma(1)}(z) \, L^{+}_{2,\sigma(2)}(zq^{-2}) \dots L^{+}_{N,\sigma(N)}(zq^{2-2N}) \;,
|^k\mathbb{C}_N^{(f)}| = \sum_{\pi(N): N=\{N_1^{c_1};N_2^{c_2};\ldots N_f^{c_f}\}}\prod_{j=1}^f\binom{|^k\mathbb{C}_{N_j}^{(1)}|+c_j-1}{c_j}.
L = i \left( w_{\alpha}\nabla_{-}\varphi^{\alpha}- \tilde{w}_{\overline{{\alpha}}}\nabla_{-}\tilde{\varphi}^{\overline{{\alpha}}}\right) ( \varphi , \tilde{\varphi}) - q A_{-}^{a}{W}_{a}( \varphi , \tilde{\varphi})
\sum(-1)^{f_\sigma}\cdots E_{ij}E_{jk}\cdots-\sum (-1)^{f_\sigma+(\bar i +\bar j)(\bar j +\bar k)}\cdots E_{jk}E_{ij}\cdots = \sum \delta_{jj}(-1)^{f_\sigma}\cdots E_{ik}\cdots-\sum \delta_{ik}(-1)^{f_\sigma+(\bar i +\bar j)(\bar j +\bar k)}\cdots E_{jj}\cdots
\phi_{R}\rightarrow B^{\prime}e^{- \gamma a x}e^{\gamma a L}\frac{\Gamma ( 2 \gamma ) \Gamma ( 1 + 2 \sigma )}{\Gamma ( 1 + \gamma + \sigma + \lambda ) \Gamma ( \gamma + \sigma - \lambda )}+ B^{\prime}e^{\gamma a x}e^{- \gamma a L}\frac{\Gamma ( - 2 \gamma ) \Gamma ( 1 + 2 \sigma )}{\Gamma ( 1 - \gamma + \sigma - \lambda ) \Gamma ( - \gamma + \sigma - \lambda )}
\Lambda^{s}f(x)=c_{s}\; \int_{\mathbb{T}}\frac{f(x)-f(y)}{|x-y|^{1+s}}dy+c_{s}\sum_{k\in\mathbb{Z}\setminus\{0\}}\int_{\mathbb{T}}\frac{f(x)-f(y)}{|x-y+2k\pi|^{1+s}}dy,
\begin{aligned}\widetilde L := L + \sum_{i=1}^{n_0}\left(2+2x_i\sum_{j=1}^{n_0}a_{ij}(z)\right)\partial_{x_i}+ \sum_{j=n_0+1}^n 2x_j\sum_{i=1}^{n_0}a_{ij}(z) \partial_{x_j}&+ \sum_{l=1}^m\sum_{i=1}^{n_0}c_{il}(z)\partial_{y_l}+\left(\sum_{i,j=1}^{n_0}a_{ij}(z)+\sum_{i=1}^{n_0}\frac{b_i(z)}{x_i}\right).\end{aligned}
\hat{H}= \frac{1}{2}\hat{p}_{z}^{2}+ \hbar \omega \left[ a_{+}^{\dagger}a_{+}+ a_{-}^{\dagger}a_{-}\right] \ \ \ , \ \ \ \hat{\phi}= \hat{p}_{z}+ \hbar g \left[ a_{+}^{\dagger}a_{+}- a_{-}^{\dagger}a_{-}\right] \ \ \ .
\mathfrak{R}_{1}= \mathfrak{Q}_{1}; \mathfrak{R}_{2}= \mathfrak{Q}_{2}; \mathfrak{R}_{3}= [\mathfrak{Q}_{3}, \mathfrak{Q}_{4}] = (x^{4}, x^{3}y, x^{2}y^{2}, xy^{3}, xz, yz, z^{2}),
v_{4}(t,r) = \frac{-1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{2 \pi}d\theta \frac{v_{4,c}(s,\sqrt{r^{2}+2 r \rho \cos(\theta) + \rho^{2}})}{\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}}(r+\rho \cos(\theta))
\binom{s}{2}g''h-\binom{r}{2}h''g+(s-1)g'h_1-rh_1'g+sg_1'h-(r-1)h'g_1= = (r+s-1)\left( sg''h+sg'h'-rh'g'-rh''g\right)+2r gh''+ + (s-1)g'h_{11}-r h_{11}'g+s g_{10}'h-(r-1)h'g_{10}.
\left(\begin{array}{c}x y\end{array}\right)= F_K(\xi,\eta)=\left(\begin{array}{c}a_0^K+a_1^K\xi+a_2^K\eta+a_{12}^K\xi\eta b_0^K+b_1^K\xi+b_2^K\eta+b_{12}^K\xi\eta\end{array}\right),
\mathbf{B}=\frac{1}{N}\begin{bmatrix}1 1 \cdots 1 j2\pi \frac{0}{N}j2\pi \frac{1}{N}\cdots j2\pi \frac{N-1}{N}\vdots \vdots{}\vdots \left( j2\pi \frac{0}{N}\right)^V \left( j2\pi \frac{1}{N}\right)^V \cdots \left( j2\pi \frac{N-1}{N}\right)^V \end{bmatrix},
| G(\xi_2') | \leq 2^{70}A^{-1}N_1^3 \textnormal{and}| G'(\xi_2')| =\Bigl| \Bigl( \frac{dG}{d\xi_2'}\Bigr) (\xi_2') \Bigr| \leq 2^{-100}N_1^2.
p = \frac{S^2}{\vert g\vert}= \frac{2MS^2}{\vert \alpha \vert \vert M^2 -(m_1 - \kappa m_2)^2\vert},\ \e = \frac{R}{\vert g\vert}= \sqrt{1 + \frac{S^2}{\alpha^2}\hspace*{0.06cm}\frac{M^2 - (m_1 + \kappa m_2)^2}{M^2 - (m_1 - \kappa m_2)^2}}.
\textstyle{\frac{P_T}{\sigma^2}}\left(\prod\limits_{k=1}^{r}\frac{\beta\,[\boldsymbol{\Lambda}]_{k,k}^2}{\beta+[\boldsymbol{\Lambda}]_{1,1}^2-[\boldsymbol{\Lambda}]_{k,k}^2}\right)^{\frac{1}{r}}\,=2^{\frac{R}{r}}\,{\textstyle\sum\limits_{k=1}^{r}\left(\frac{\beta}{\beta+[\boldsymbol{\Lambda}]_{1,1}^2-[\boldsymbol{\Lambda}]_{k,k}^2}\right)}.
\mathcal{R}(\mu,m,K,\Delta;s) =\left[{{\left( m+\mu K \right)}^{2}}-{{\left( \mu K\Delta \right)}^{2}}\right]{{\overline{\gamma}}^{2}}{{s}^{2}}-2m\mu \times \left( 1+K \right) \left( m+\mu K \right)\overline{\gamma}s+{{\left[ \left( 1+K \right)m\mu \right]}^{2}}.
H = \frac{1}{2}N^{2}\sum_{- \infty}^{\infty}( \alpha_{- n}\cdot \alpha_{n}+ \tilde{\alpha}_{- n}\cdot \tilde{\alpha}_{n}) + \frac{N^{2}}{4 x}\sum_{- \infty}^{\infty}\gamma_{- n}\cdot \gamma_{n}.
u(x)\geqslant \,u(\xi_k(-t))-\int^{-t}_{-T_k}L(\xi_k,\dot{\xi}_k)\ds+\int^0_{-T_k}L(\xi_k,\dot{\xi}_k)\ds-\frac 1k = \,u(\xi_k(-t))+\int^0_{-t}L(\xi_k,\dot{\xi}_k)\ds-\frac 1k.
\frac{d\lambda_\pm}{d\theta}=2\sin\tau \cdot\left(\pm \frac{1}{2}\lambda_{0,1}h^{-1/2}+ \lambda_{0,2}\pm \frac{3}{2}\lambda_{0,3}h^{1/2}+ 2\lambda_{0,4}h\pm \frac{5}{2}\lambda_{0,5}h^{3/2}+ O(h^2)\right).
\| s' \|_{(f^{\mathrm{an}})^*(h^n), \hat{\kappa}(\zeta)}= \|f^*_{\hat{\kappa}(f^{\mathrm{an}}(\zeta))}(s) \|_{(f^{\mathrm{an}})^*(h^n), \hat{\kappa}(f^{\mathrm{an}}(\zeta))}& \leqslant \| s \|_{h^n, \hat{\kappa}(f^{\mathrm{an}}(\zeta))}\leqslant e^{n\epsilon}|s|_{h^n}(f^{\mathrm{an}}(\zeta)) & = e^{n\epsilon}|s'|_{(f^{\mathrm{an}})^*(h^n)}(\zeta),