harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k = - \infty}^{\infty}q^{k^{2}( 6 a + 1 2 b )}\left( q^{k ( a - 4 b )}+ q^{k ( 7 a + 8 b ) + 2 a + b}- q^{k ( a + 8 b ) + b}- q^{k ( 7 a + 2 0 b ) + 2 a + 8 b}\right) \, ,
h([\mu_{i_{01}}, \ldots, \mu_{i_{0p_0}}, w]) = \prod\limits_{k = 0}^n \left(\sum\limits_{j = 1}^{p_k}D_{i_{k1}}\cdots D_{i_{k(j-1)}}S_{i_{kj}}D_{i_{k(j+1)}}\cdots D_{i_{kp_k}}\right).
\begin{aligned}\left\Vert \int_{\sqrt{\epsilon}y}^\infty E_b(x,\theta)\mathrm{d}\theta\right\Vert_{L^2}\lesssim\epsilon^{-\frac{1}{4}}\Vert \langle Y\rangle^nE_b\Vert_{L^2}\lesssim\epsilon^{\frac{1}{4}},\end{aligned}
\Omega_{e^{-\rho\ln^2(t)}}\left(\frac{1+s}{2}\right)=\frac{1}{2}\sum_{i=0}^\infty\frac{(2i)!}{i!}(-\rho)^i\left({\left(\frac{4}{-1+s}\right)^{1+2i}-\left(\frac{4}{1+s}\right)^{1+2i}}\right)
I(g,{\bf A})=I_{WZNW}+ \frac{k}{2 \pi}\int d^2 z tr ({\bf A}\overline{\partial}g g^{-1}-\overline{{\bf A}}g^{-1}\partial g +{\bf A}\overline{{\bf A}}-g^{-1}{\bf A}g \overline{{\bf A}})
\frac{\partial\psi}{\partial t}=ie^{i\sigma_{z}\omega_{r}t/2}(\omega_{1}\sigma_{x}cos\omega_{r}t+\omega_{1}\sigma_{y}sin\omega_{r}t+(\omega_{0}+\frac{\omega_{r}}{2})\sigma_{z})e^{-i\sigma_{z}\omega_{r}t/2}\psi
\int_{0}^{T}|v(t)|^2dt = \int_{0}^{T}\left|-\sum_{m\in\mathbb{Z}^*}\frac{\hat{u}_{m}^{0}}{\hat{f}_m}e^{\frac{T}{2}\lambda_m}\zeta_m\left(t-\frac{T}{2}\right)\right|^2dt \leq C\sum_{m\in\mathbb{Z}^*}\frac{|\hat{u}_{m}^{0}|^2}{|\hat{f}_m|^2}e^{bm^6}\leq C ||u_0||_{\widetilde{\mathcal{H}}}^2,
M = \frac{i}{\sqrt{{\mathrm{\boldmath ~ \b e t a ~}}^{2}}}\left( \begin{matrix}{{E_{\mathrm{\boldmath ~ \b e t a ~}}}}&{{- E_{-{\mathrm{\boldmath ~ \m u ~}}}}}\\{{E_{\mathrm{\boldmath ~ \m u ~}}}}&{{\, \, \, E_{-{\mathrm{\boldmath ~ \b e t a ~}}}}}\\ \end{matrix}\right) \, .
\delta_3 \, = \, 2 \bigg ( |\alpha|^{-\frac{1+\gamma}{2}}\delta_1 + |\alpha|^{-\frac{\gamma}{2}}\big |\log |\alpha| \big | \, \| F \|_{L^\infty_{2+\gamma}(\Omega)}\, + d_\gamma [F] \bigg )\,,
\begin{aligned}F(\alpha, p) = \sum_{\substack{1\leq n\leq Z P^{+}(n)\leq y}}\frac{\psi_p(n)(1-\cos(2\pi n\alpha))}{n}+O\left(\frac{1}{\log y}\right) &= \sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\psi_p(n)(1-\cos(2\pi n\alpha))}{n}+O\left(\frac{1}{\log y}\right).\end{aligned}
x_t = 1 - t - \frac{t}{d'},y_t = \frac{t}{d'-5}\left(\frac{1}{3}- \frac{1}{d'}\right), z_t = \frac{t}{d'-5}\left( \frac{1}{3}- \frac{2}{d'}\right), t \leq \frac{d'}{d'+1}
\lim_{N \rightarrow \infty}\max_{l \leq N_k^{\delta}}\max_{\mu,\nu}\left | \mathbb{E}^V \theta(N \zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu))-\mathbb{E}^V \theta(\frac{N}{\pi}\int_I \tilde{G}_{\mu \nu}(z)q[\operatorname{Tr}(\mathcal{X}_E*\vartheta_{\tilde{\eta}})(Q_1)]dE) \right |=0,
\Delta S\propto \int d^{4}x\,n_{s}mv_{\theta}^{2}\propto \ell \Delta t\left( n_{s}/m\right) \int d^{2}x\,r^{-2}\sim \left( \ell \Delta t /\ell_{P}^{2}\right) \log \left( R_{0}/a_{0}\right) \sim \ell \Delta t /\ell_{P}^{2}
w_n'(t) = \begin{cases}w_{n-1}'(t) + q_{-}(t) t \in [ -T, x_n - R_n] \setminus \{x_i \}_{i =0}^{n-1}, m x_n -R_n < t < x_n - \tau_n, \tilde{w}_n'(t) x_n - \tau_n \leq t < x_n, \x_n < t \leq x_n + \tau_n, m x_n + \tau_n < t < x_n + R_n, w_{n-1}'(t) + q_{+}(t) t \in [x_n + R_n , T] \setminus \{x_i \}_{i =0}^{n-1}. \end{cases}
p(n_1, \ldots, n_V; \pmb{\rho}^{(N)}) \triangleq \prod_{v = 1}^V{N^{(N)}_v \choose n_v}\left(\rho^{(N)}_v p^{(N)}_v \right)^{n_v}&\! \left(1 - \rho^{(N)}_v p^{(N)}_v\right)^{N^{(N)}_v - n_v}.
\{T_{k}^{\mu}, T_{l}^{\nu}\}= 2 \, \delta_{k , l}^{\mu , \nu}\; , \hspace{1 . 3 c m}\{T_{0}^{\mu}, T_{k}^{\nu}\}= 0 \; . \hspace{2 c m}( k , l = 1 \ldots L - 1 ; \; \mu , \nu = 1 , 2 )
\Psi = \begin{pmatrix}\mathcal{F}^{-1}_1 [ f_+ (q_1) \hat \eta_+ (p_1) + f_- (q_1) \hat \eta_- (p_1)] \mathcal{F}^{-1}_2 [ f_+ (q_2) \hat \eta_+ (p_2) + f_- (q_2) \hat \eta_- (p_2)] \mathcal{F}^{-1}_3 [ f_+ (q_3) \hat \eta_+ (p_3) + f_- (q_3) \hat \eta_- (p_3)] \end{pmatrix}.
K_{J}:=\sum_{i=l}^{m}q^{i-l}(T_{i-1}^{-1}\cdots T_{l}^{-1})(T_{l}^{-1}\cdots T_{i-1}^{-1}){}\times \left\{X_{i}+(1-q)\sum_{j=i}^{m-1}(T_{i}^{-1}\cdots T_{j-1}^{-1})T_{j}^{-1}(T_{j-1}\cdots T_{i})X_{j+1}\right\}
|R^N(V,d\phi(e_j))\tau(\phi)|^2=K^2\big( |\langle d\phi,\tau(\phi)\rangle|^2|V|^2+|d\phi|^2|\langle V,\tau(\phi)\rangle|^2 &-2\langle d\phi(e_i),\tau(\phi)\rangle\langle V,\tau(\phi)\rangle\langle V,d\phi(e_i)\rangle\big).
L_{d}D1(p\|q)= \left[\frac{A^{a-1}\left(p,q\right)-A^{b-1}\left(p,q\right)}{a-b}\right] &-\left[\frac{(X+Y)^{a-1}\left(p,q\right)-(X+Y)^{b-1}\left(p,q\right)}{a-b}\right]
\phi^{( 0 )}\ = \ \left[ \begin{array}{c l}{\displaystyle 0 \ ,}&{r \le R_{g}}\\{\displaystyle v \left( 1 - \frac{K_{0}( m_{H}r )}{\ln ( 2 / m_{H}R_{g})}\right) ,}&{r \gg R_{g}.}\\ \end{array}\right.
\lim_{k \rightarrow \infty}\int_0^T \int_{\mathbb{R}}\rho_5(s,u) [ -(- \Delta)^{\gamma/2}G_k ] (s,u) du ds = -\frac{c_{\gamma}}{4}\iint_{\mathbb{R}^2}\frac{[ \int_0^T \rho_5(r,u) dr - \int_0^T \rho_5(s,v) ds ]^2}{|u-v|^{1 + \gamma}}du dv .
B^{(1+s)}\stackrel{def}{=}\widetilde{\Omega}_2 \, A^{(1+s)}= \widetilde{B}_2 - \left( \widetilde{\Omega}_1 + \widetilde{\Omega}_2 \, L_{21}\right)A_{11}\, L_{21}^T & = \widetilde{B}_2 - \widetilde{B}_1 \, L_{21}^T.
\dot{z}_{n}=\frac{\dot{\theta}_{n}}{\cos^{2}\theta_{n}}~,~~~\dot{z}_{n}~z_{m}=\frac{\dot{\theta}_{n}~\sin \theta_{m}}{\cos^{2}\theta_{n}~\cos \theta_{m}}~,~~~\dot{z}_{n}~\dot{z}_{m}=\frac{\dot{\theta}_{n}~\dot{\theta}_{m}}{\cos^{2}\theta_{n}~\cos^{2}\theta_{m}}~;
\frac{(q^{d+r-(d-2j-1)n};q^d)_{k-2}}{(q^{d-(d-2j)n};q^d)_k}=\frac{(q^{d-(d-2j)n+dk};q^d)_{(n+r)/d-2}}{(q^{d-(d-2j)n};q^d)_{(n+r)/d}}.
\left( N_{j}^{i}\right)_{i,j=1,2}=\begin{pmatrix}N_{1}^{1}N_{2}^{1}N_{1}^{2}N_{2}^{2}\end{pmatrix}=-\frac{1}{2}\begin{pmatrix}H_{21}H_{22}-H_{11}-H_{12}\end{pmatrix}=-\frac{1}{2}J_{H}\left( -H_{1},H_{2}\right) .
\frac{i}{2\pi}\delta^4 (p+q) \left[ \frac{f_+ ({\bf p})}{\omega ({\bf p})}\frac{p_0}{(p_0 + i\epsilon)^2 -{\bf p}^2 - m^2}+ f_- ({\bf p})\frac{1}{(p_0 + i\epsilon)^2 -{\bf p}^2 - m^2}\right] \ ,
\P\{m \}= \P\{Y_m = 0 \}=\prod\limits_{i=0}^{m-1}\P \{\ell^{(A)}_i \leq m-i \}= \prod\limits_{i=0}^{m-1}\left( 1 - \P \{\ell^{(A)}_i > m-i \}.\right)
m{d^2 x^\beta \over d\tau^2}(\delta^\alpha{}_\beta + \mu^2{\rm D}_{\beta \mu}g^{\mu \alpha})+{d x^\nu\over d\tau}{d x^\lambda\over d\tau}\left( \Gamma_{\nu \lambda}{}^\alpha + \frac{1}{2}\mu^2 g^{\alpha \gamma}(\partial_\gamma{\rm D}_{\nu \lambda}-\partial_\nu{\rm D}_{\gamma \lambda}- \partial_\lambda{\rm D}_{\gamma \nu})\right) = 0,
\|y_k-y_{k+1}\|\leq \frac{1+\theta}{1-\theta}\, \left[ \frac{1+\theta}{2}\frac{D^{-}f'(\lambda)}{|f'(\lambda)|}\|y_k-y_{k-1}\|+\theta\,\frac{2+f'(\lambda)}{|f'(\lambda)|}\right]\|y_k-y_{k-1}\| , k= 0,1,\ldots \,.
\begin{array}{lll}X \equiv X^{i=5}\quad X^i \sim \epsilon^{ijklm}\Omega_{jk}\Omega_{lm}\qquad 10 \times 10 \rightarrow 5^* Y \equiv Y_{i=5}\quad Y_i \sim \Phi^j \Omega_{ji}\qquad 5^* \times 10 \rightarrow 5 S S \sim{\rm tr}\left( W^{\dot\alpha}W_{\dot\alpha}\right) \qquad 24 \times 24 \rightarrow 1 \end{array}
\eta_t(x)\rightarrow i \begin{cases}1 \eta_t(x)=2 i=0, &\gamma \eta_t(x)=1 i=2, &1+\delta \eta_t(x)=1 i=0, &\lambda\sum_{y:y\sim x}1_{\{\eta_t(y)=2\}}\eta_t(x)=0 i=1, &0 \end{cases}
S_{n}\left( y\right)_{s,t}=1+\sum_{k=1}^{n}\sum_{l_{i}\in \left\{0,1,\dots,n-1\right\}}\left( \left( D^{l_{1}}p\right) \otimes \cdots \otimes \left(D^{l_{k}}p\right) \right) \left( x_{s}\right)\idotsint\limits_{s<u_{1}<\cdots <u_{k}<t}dx_{s,u_{1}}^{l_{1}+1}\otimes\cdots \otimes dx_{s,u_{k}}^{l_{k}+1}
S_{4 *}= \int d^{4}x \sqrt{- g}\left[ R + \frac{1}{4}\mathrm{T r}\left[ \nabla P \nabla P^{- 1}\right] - \frac{1}{2}\left( \nabla x \right)^{2}- \frac{1}{2}e^{\sqrt{3}x}\nabla \tau^{T}P \nabla \tau \right] .
\; = \; \frac{e^{2}}{( 2 \pi )^{2}}\; \mathrm{t r}( I ) \, \oint_{{\mathcal C}( \varepsilon )}\, d l \,{\hat{n}}_{\mu}\left[ \frac{A ( p ) p_{\mu}}{A^{2}( p ) p^{2}+ B^{2}}\right]
F^{\mu^{\prime}\nu^{\prime}}(x^{\prime})={\Lambda^{\mu^{\prime}}}_{\mu}{\Lambda^{\nu^{\prime}}}_{\nu}F^{\mu\nu}(\Lambda^{-1}x^{\prime})={\Lambda^{\mu^{\prime}}}_{\mu}{\Lambda^{\nu^{\prime}}}_{\nu}F^{\mu\nu}(x).
\left| \frac{\partial (\phi \circ F)}{\partial x_k}(x)\right|\leq C \, K \, \sqrt{\sum_{j=1}^n \left( \frac{\partial f_j}{\partial x_k}(x) \right)^2 + \left(\frac{\partial g_j}{\partial x_k}(x) \right)^2}\leq C \, K \, \left| \frac{\partial F}{\partial x_k}(x) \right|.
\tilde{S}_M=N\frac{1}{1-{\rm e}^{-4\tilde{\beta}}}\frac{1}{2}{\rm tr}_N\left[\tilde{M}_1^2+\tilde{M}_2^2-2{\rm e}^{-2\tilde{\beta}}\tilde{M}_1\tilde{M}_2\right]+N\sum_{a=1}^2{\rm tr}_N\ln (1-\tilde{g}\tilde{M}_a),
[V^B_A,\bar{\Omega}](-1)^{\varepsilon_B}+[\bar{V}^B_A,\Omega](-1)^{\varepsilon_B}+i\hbar\delta^B_A=i\hbar V^C_A\bar{V}^B_C+i\hbar\bar{V}^C_AV^B_C.
[ L_{- 1}, W_{0}^{3}] = ( \O_{- 4}^{2 3})_{- 1}= - 2 W_{- 1}^{3}\quad ; \quad [ L_{- 1}, W_{1}^{3}] = ( \O_{- 4}^{2 3})_{0}= - 3 W_{0}^{3}
\left( \tilde{\omega}^{(1, \delta)}(t) \right)_j :=\begin{cases}t \mbox{if}j = k(1), \omega_j \mbox{else},\end{cases}\quad\left( \tilde{\omega}^{(n, \delta)}(t) \right)_j :=\begin{cases}t \mbox{if}j = k(n), \left( \tilde{\omega}^{(n-1, \delta)}(\omega_j + \delta) \right)_j \mbox{else}.\end{cases}
\omega_- = \left( \begin{array}{cc}\Sigma^{(1)}_- 0 begin{align*}.7em]I_n \Sigma^{(2)}_-\end{array}\right), \qquad\omega_+ = \left( \begin{array}{cc}0 \Gamma^{(1)-1}\Gamma^{(2)}begin{align*}.7em]0 0\end{array}\right),
r^{\gamma^{\ast}}= \limsup_{T \rightarrow \infty}\,\inf_{\varphi(t) \in \Psi}\frac{1}{T}\biggl \{I_{T}^{\gamma^{\ast}}(\varphi(t)) \, \Bigl \vert \, \varphi(t) \in D \cup \partial D,\,\, t \in [0, T] \biggr\},
(\kappa_{a,\omega,n}^{-1})^*\circ \gamma^\dag_{p,(-\delta,\delta)}(w^u_{p}(\tau))=(0,\varphi(\tau),1),(\kappa_{a,\omega,n}^{-1})^*\circ \gamma^\dag_{p,(-\delta,\delta)}(w^s_{p}(\tau))=(\hat{\varphi}(\tau),0,1)
\Phi_{G n}^{\mathrm{o u t}}= \frac{e^{- i \omega_{n}^{\mathrm{o u t}}t}}{\sqrt{2 \pi \omega_{n}^{\mathrm{o u t}}}}e^{i k_{n}^{\mathrm{o u t}}z}e^{i m \phi}J_{\alpha | m |}( k_{n}^{\mathrm{o u t}}r ) ,
\left(\left(\left(i+j+\ell-kr\right)_{j=0}^{\phi-1}\right)_{i = 0}^{t-1}\right)_{\ell=kr+1}^{kr+r-1}= \left(\left(\left(i+j+\ell\right)_{j=0}^{\phi-1}\right)_{i = 0}^{t-1}\right)_{\ell=1}^{r-1}.
V ( \rho ) = \frac{1}{G}\rho^{2}- i \mathrm{t r}\int_{0}^{\rho}d s \S ( x , x ; s ) - 2 i \int_{0}^{\rho}s d s \ [ G_{1}( x , x ; s ) + G_{2}( x , x ; s ) ] +{\cal O}\left( \frac{1}{N}\right) ,
{\rm Hilb}({\rm gr}(JM(2) ,t)=(1,1),{\rm Hilb}({\rm gr}(JM(3) ,t)=(1,2,1),{\rm Hilb}({\rm gr}(JM(4) ,t)=(1,3,4,2),{\rm Hilb}({\rm gr}(JM(5) ,t)=(1,4,8,9,4),{\rm Hilb}({\rm gr}(JM(6) ,t)=(1,5,13,21,21,12,3),{\rm Hilb}({\rm gr}(JM(7) ,t)=(1,6,19,40,59,60,37,10).
\tilde{\chi}_{m,j,\gamma}(\lambda) = \gamma \int_{0}^{\infty}\left(1 - g_m\left(\frac{f(\gamma)}{\gamma}\lambda; x\right)\right)j(dx) - bf(\gamma)\lambda &= \int_{0}^{\infty}(1 - g_{m_\gamma}(\lambda; x)) j_\gamma(dx) - b_\gamma\lambda &= \chi_{m_{\gamma},j_{\gamma}}(\lambda) - b_\gamma \lambda
\lim_{K \to -\infty}\frac{\big(x q^{1-K}/\xi, q^{1-K}/(\beta_1 \xi ), q^{1-K}/(\beta_2 \xi ), q^{1-K}/(\beta_3 \xi ) ;q\big)_{\infty}}{\big(x q^{-\lambda -K}/ \xi, q^{1-K}/(\alpha_1 \xi ), q^{2-K}/(\alpha_2 \xi ), q^{1-K}/(\alpha_3 \xi ) ;q\big)_{\infty}}=1 .
g(r) := \int_0^r s_3^{-(2N-2)}\Big( \int_{0}^{s_3}s_2^{2N-2}\Big( \int_0^{s_2}s_1^{-(2N-2)}\Big(\int_{B(x, s_1)}u^{-(4N-1)}dy \Big)ds_1\Big) ds_2 \Big) ds_3 = r^{-(2N-2)}\int_{\partial B(x, r)}(-\Delta)^{N-2}u d\sigma + \omega_{2N-1}(-\Delta)^{N-2}u(x) +\frac{\omega_{2N-1}}{2 (2N-1)}(-\Delta)^{N-1}u(x) r^2.
\eta^{1/2}(\breve T_2-T_2)-(\breve \kappa_1-\kappa_1)\eta T_1^2 =O_p(n^{-1}), (\breve \kappa_1-\kappa_1)-{\textstyle{1\over 6}}(\breve \kappa_3-\kappa_3) =O(n^{-1}).
\tilde{\phi}_n^k (t, z) - \tilde{\phi}_n^k(t, y) = \psi_n^k (t)|z| - \psi_n^k(t) |y| \geq 0; \shortintertext{and so}|\tilde{\phi}_n^k (t,z) - \tilde{\phi}_n^k (t,y)| = \psi_n^k(t)(|z| - |y|) \leq \psi_n^k (t) |z-y|,
H_{a_1,b_1}(t_1,s_1) = \varphi((a_1(t_1)+h_{a_1}(t_1)1)(b_1(s_1)+h_{b_1}(s_1)1)) &= \varphi(a_1(t_1)b_1(s_1)) + h_{a_1}(t_1)h_{b_2}(s_2)
\langle{\cal W}(1){\cal W}(2) \rangle|_{\theta=0}=\langle \phi(1)\phi(2) \rangle = \frac{z_{1i}\delta^{ij}z_{2j}}{x^2_{12}}\equiv \frac{(12)}{x^2_{12}}= \frac{(21)}{x^2_{21}}\;.
M_{\mu}( x ) = \frac 1{\sqrt{2}}\left( \varphi_{\mu}( x ) - i \widetilde{\varphi}_{\mu}( x ) \right) , \hspace{0 . 3 i n}M_{\mu \nu}( x ) = \frac 1{\sqrt{2}}\left( \varphi_{\mu \nu}( x ) - i \widetilde{\varphi}_{\mu \nu}( x ) \right) ,
P_{match}=\sum_{i,j}p_{i,j}^{2}=\sum_{i,j}(i\cdot H_{M}\cdot j\cdot H_{N})^{-2}=(\sum_{i}^{M}i^{-2})\cdot(\sum_{j}^{N}j^{-2})\cdot(H_{M}\cdot H_{N})^{-2}.
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=\frac{\alpha\rho}{\tau}-\beta\tau\eta\leqslant \frac{2\alpha\rho^2}{1+3\rho^2}\leqslant\frac{2\alpha}{3},\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=\frac{\alpha\rho}{\tau}-\beta\rho\eta \leqslant \frac{2\alpha\rho^2}{1+3\rho^2}\leqslant\frac{2\alpha}{3},
(\varphi^2-\kappa\varphi^1-2\kappa)\varphi^1_\omega-(\kappa^2+1)\varphi^1_{\omega\omega}+2\varphi^2_\omega-(\varphi^1)^2-(\varphi^2)^2=0, (\varphi^2-\kappa\varphi^1-2\kappa)\varphi^2_\omega-(\kappa^2+1)\varphi^2_{\omega\omega}-2\varphi^1_\omega=0.
\sum_{k=1}^l \frac{(\tilde{C_2}t)^k}{k! (k-1)!}\frac{(l-1)!}{(l-k)!}\leq e^{\tilde{C}_2 t}\sum_{k=1}^l \frac{(l-1)!}{(k-1)! (l-k)!}= e^{\tilde{C}t}2^{l-1}\,.
\alpha = \sum_{i,j,k = 0}^2 A_{\sigma \nu \rho}^{ijk}\omega^\sigma_i \land \omega^\nu_j \land \omega^\rho_k \land dt, A_{\sigma \nu \rho}^{ijk}= - A_{\nu \sigma \rho}^{jik}= A_{\nu \rho \sigma}^{jki},
f(q,\mu) = \sum_{\rho\neq\sigma\in\Phi_{+},\,R_{\psi}= s_{\rho}s_{\sigma}}g_{|\rho|}\,g_{|\sigma|}\,{(\sigma^{\vee}\!\!\cdot\rho)\over{|\sigma^{\vee}\!\!\cdot\rho|}}\, x_{|\rho|}(\rho\cdot q,(\rho^{\vee}\!\!\cdot\mu)\xi)\,x_{|\sigma|}(\sigma\cdot q,(s_{\rho}(\sigma)^{\vee}\!\!\cdot\mu)\xi)
\mathbb{P}_{couple}\left(n^{-\frac{1}{\alpha}}\sum_{i=1}^{M_{1}\left(n\right)}Z>n^{-c'}\right) \leq n^{\left(c'-\frac{1}{\alpha}\right)\xi}nc'\log n\mathbb{E}\left(Z^{\xi}\right) +o\left(n^{-c}\right).
\left(V^{[n]}_{z_1z_1}-V^{[n]}\right)_{z_1}- U_{z_2}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0,\quad j=1\dots n-1, &\left(\Omega^{[1]}\,U\right)_{z_1}+\frac{r-1}{2n}\,U-r\,z_2\,U_{z_2}=0, &\Omega^{[j]}_{z_1}=U\,V^{[j]}_{z_1}.\quad j=1\dots n.
\mathcal{M}_{f_{1,\infty}}(N,\alpha+\delta,6\epsilon,\bigcup_{n=N}^{\infty}Z_{n,n^{-2}t},\psi)\leq\sum_{n=N}^{\infty}\mathcal{M}_{f_{1,\infty}}(N,\alpha+\delta,6\epsilon,Z_{n,n^{-2}t},\psi).
\left\langle x u_{\alpha}\,\bigg|\, \frac{(h_{\alpha}-e_{\alpha})^2}{(h_{\alpha}-e_{\alpha}+|k|)}\,\bigg|\, x u_{\alpha}\right\rangle = \alpha^2 \left\langle x u_{1}\,\bigg|\, \frac{(h_{1}-e_{1})^2}{(\alpha^2(h_{1}-e_{1})+|k|)}\,\bigg|\, x u_{1}\right\rangle,
S^2\mathbf R^{4k+2}=3\left( \bigoplus_{r=0}^{k}\; S^{2k-2r}_0\mathbf R^3 \right) \oplus\left( \bigoplus_{r=0}^{k-1}\;S^{(2k-1)-2r}_0\mathbf R^3 \right).
\sum_{i \in \mathcal{I}}{\!\! \pi_i(d_i',\boldsymbol{d}_{-i}' )}\! =\! \sum_{i \in \mathcal{I}}{\left\{d_i' \!-\! d_i'\gamma \sum_{j \in \mathcal{I}}d_j'\right\}}& = \frac{n}{\left( 1+ n\right)^2}\to 0\; \mathrm{as}\; n \to \infty.
H(f)=(x_{i+1}\frac{\partial f}{\partial x_{i+1}}-x_{i-1}\frac{\partial f}{\partial x_{i-1}})x_{i}\frac{\partial}{\partial x_{i}}+(y_{i-1}\frac{\partial f}{\partial y_{i-1}}-y_{i+1}\frac{\partial f}{\partial y_{i+1}})y_{i}\frac{\partial}{\partial y_{i}}
\chi \in C^\infty_0({\mathbb R}^d),0 \leqslant \chi({\mathbf x}) \leqslant 1; \chi({\mathbf x})=1,\{\mathbf x}\in{\mathcal O}' ; &{\rm supp}\,\chi \subset{\mathcal O}'';|{\mathbf D}^\alpha \chi({\mathbf x})| \leqslant \varsigma_s \delta^{-s}, \ |\alpha|=s, forall s\in{\mathbb N}.
(P\varphi_{\alpha,\beta}^{\lambda_{\alpha}})({\bf y}) = g_{\alpha,\beta}^{\lambda_{\alpha}}({\bf y})\,, g_{\alpha,\beta}^{\lambda_{\alpha}}({\bf y}) = -\big[\partial_{\bf y}^{\beta}V_{\alpha}^{\lambda_{\alpha}}({\bf y})\big] \psi_{\alpha}^{\lambda_{\alpha}}({\bf y})\,.
Q_j(x_4) =x_3 x_4^{2^{j-1}}+ x_{2}^{2^{j-1}}Q_{j-1}(x_4)+ x_{3}^{2^{j-1}}Q_{j-2}(x_4) &=x_3 \left( x_4^{2^{j-1}}+\alpha_{j-1,j-2}x_2^{2^{j-1}}x_4^{2^{j-2}}+ \sum_{k=0}^{j-3}(\alpha_{j-1,k}x_2^{2^{j-1}}+\alpha_{j-2,k}x_3^{2^{j-1}}) x_4^{2^k}\right).
\begin{array}{c c c}{{\left( \begin{array}{c}{{a}}\\{{b}}\\{{\varepsilon^{i}}}\\{{\rho^{i}}}\\{{\lambda}}\\{{\omega_{A B}}}\\{{T_{i}^{~ j}}}\\ \end{array}\right) ~}}&{{\longrightarrow}}&{{~ \left( \begin{array}{c}{{b}}\\{{a}}\\{{\rho^{i}}}\\{{\varepsilon^{i}}}\\{{- \lambda}}\\{{\omega_{A B}}}\\{{T_{i}^{~ j}}}\\ \end{array}\right)}}\\ \end{array}
\left( jO_{n}^{(3)}\right)^{2}-9\left( JO_{n}^{(3)}\right)^{2}=\frac{2^{n+1}}{7}(2\underline{\alpha}^{2}+3(\underline{\alpha}\cdot \widehat{\epsilon_{n}}+\widehat{\epsilon_{n}}\cdot \underline{\alpha})),
\left\lbrace \begin{array}{lll}^{C}D_{t}^{q}x(t)=Ax(t)+f(t, x(t), (Hx)(t))+Bu(t), \t\in (0, b]\setminus\lbrace t_1, t_2,\ldots, t_m\rbrace, x(0)+g(x)=x_{0}\in X,\triangle x(t_{i})=I_{i}(x(t_{i}^{-}))+Dv(t_{i}^{-}), i=1,2,\ldots,m,\end{array}\right.
\varphi=-\left( 1-\frac{\kappa\gamma^{2}}{1+\kappa}\right)\operatorname{div}\mathbf{Z}^{\left( e\right)}+\frac{\kappa\gamma^{2}}{\left( 1+\kappa\right) c}\mathbf{v}\cdot\left( \frac{\partial\mathbf{Z}^{\left( e\right)}}{c\partial t}+\operatorname{curl}\mathbf{Z}^{\left( m\right)}\right)
N \to \infty \ , \quad \mathrm{g_\mathrm{s}, \alpha ' , a n d s f i x e d \ ,}\quad R = ( 4 \pi \alpha^{2}g_{\mathrm{s}}N )^{1 / 4}\ , \quad \omega = \frac{1}{2}R s^{1 / 2}\ .
G ( s , \nu_{1}, \nu_{2}, \alpha ( \mu ) , N ) ={\frac{1}{s^{3}}}Z_{\sigma^{2}}^{- 3}( \alpha ( 1 / s ) , \alpha ( \mu ) ) G_{\mathrm{r e d}}( \nu_{1}, \nu_{2}, \alpha ( 1 / s ) , N )
{\cal{L}}^{p v}= q ( d e t g^{\eta \eta^{\prime}}) [ - \overline{{{\psi}}}\gamma_{\mu}\sigma_{\lambda \omega}\gamma_{5}\psi H_{\gamma \rho \delta}+ g_{\alpha \delta}\overline{{{\psi}}}\gamma^{\mu}\sigma_{\omega \nu}\gamma_{5}\psi H_{\mu \lambda \rho}- g_{\alpha \delta}\overline{{{\psi}}}\gamma_{\mu}\sigma_{\omega \nu}\gamma_{5}\psi H_{\beta \lambda \rho}]
\frac{1}{q_m q_{m + 1}}\lesssim_\times \alpha q^{-a}q_{n + 1}^{-A}\lesssim_\times \begin{cases}\alpha q_m^{-a}q_{n + 1}^{-A}a\geq 0 \alpha q_{m + 1}^{-a}q_{n + 1}^{-A}a\leq 0\end{cases}.
\Phi_{0}\cdot\widehat{\xi}_{0}=0,\,\,\,\,\Phi_{0}=\widehat{\varepsilon}^{*}_{0}(\Phi_{\underline{a}}), \,\,\widehat{\xi}_{0}=\widehat{\varepsilon}^{*}_{0}(\widehat{\xi}_{\underline{a}})\in J^{l}(b)\otimes\widehat{Q}_{1}.
S_{n,3}(\omega)=\mathrm{ord}(\omega)^4 a_{\frac{n}{\mathrm{ord}(\omega)},\frac{1}{2}y^2(x-y)^2 - \frac{(x-y)^2}{6}}+ \mathrm{ord}(\omega)^2 a_{\frac{n}{\mathrm{ord}(\omega)}, \frac{1}{6}(x-y)^2}.
{}_r\phi_s\left(\begin{array}{c}a_1,\ldots,a_r b_1,\ldots,b_s\end{array}\Bigg|q,z\right):=\sum_{k=0}^\infty\frac{(a_1,\ldots,a_r;q)_k}{(q,b_1,\ldots,b_s;q)_k}\left((-1)^kq^{k\choose2}\right)^{1+s-r}z^k.
\tfrac{1}{4}\mu(dp)^\Delta|V_i||W| = \tfrac{1}{8}\mu(dp)^\Delta|V_i||W|+\tfrac{1}{8}\mu(dp)^\Delta|V_i||W| > \frac{\mu d^\Delta}{16\rho}p^\Delta|Z'||W| +\frac{\mu d^\Delta}{8 \kappa r_1}p^\Delta n|W| > 7p^\Delta|Z'||W|+\rho p^\Delta n|W|/r_1\,,
P_{0}(z,w) =\frac{1}{|1-zw|^{2\lambda}}{}_2\!F_{1}\Big({\lambda,\lambda\atop 2\lambda+1};\frac{4({\rm Im}z)({\rm Im}w)}{|1-zw|^{2}}\Big) &=\frac{1}{|1-z\bar{w}|^{2\lambda}}{}_2\!F_{1}\Big({\lambda,\lambda+1\atop 2\lambda+1};-\frac{4({\rm Im}z)({\rm Im}w)}{|1-z\bar{w}|^{2}}\Big),
ds^2 \sim \, dx^2_{1,3}+ \alpha' g_s N\, \left[ d\rho^2 + \rho^2 \left( d\theta_1^2 + \sin^2\theta_1 \,d\phi_1^2 \right) + \sum_{a=1}^3\left(\sigma^a - A^a \right)^2 \right]
f^s L_1 \dots L_r f^{-s-1}= f^s \sum_{i=1}^m c_i \left( \sum_{\bar l_i}^{\bar k_i}f^{-s-1-|\bar l_i|}(-s-1)_{|\bar l_i|}f^{(\bar l_i)}\bar \partial^{(\bar l_i - \bar k_i)}\right) &= \sum_{i=1}^m c_i \left( \sum_{\bar l_i}^{\bar k_i}f^{|\bar l_i|}(-s-1)_{|\bar l_i|}f^{(\bar l_i)}\bar \partial^{(\bar l_i - \bar k_i)}\right) .
||U_{2m(n)}''(t)P_{m(n)}A(t)P_{m(n)}U_{2m(n)}''^*(t)-P_{m(n)}B(t)P_{m(n)}|| &\leq \frac{10}{n}+||P_{m(n)}U_{2m(n)}''(t)P_{m(n)}A(t)P_{m(n)}U_{2m(n)}''^*(t)P_{m(n)}-P_{m(n)}B(t)P_{m(n)}|| &=\frac{10}{n}+||U_{m(n)}'(t)A(t)U_{m(n)}'^*(t)-P_{m(n)}B(t)P_{m(n)}||\leq \frac{19}{n}
F =\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\left\{(x,y)\in [0,1]^2:\begin{aligned}|T_{\beta_1}^nx-f_1(x)|<\beta_1^{-n\tau_1(x)}|T_{\beta_2}^ny-f_2(y)|<\beta_2^{-n\tau_2(y)}\end{aligned}\right\}&=\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\bigcup_{w\in \Sigma_{\beta_1}^n, v\in \Sigma_{\beta_2}^n}J_{n,\beta_1}(w)\times J_{n,\beta_2}(v).
\left\lbrace\begin{array}{ll}z_{ii}= 0 i = 1, \ldots, K, z_{ij}= z_{ji}^* \approx{\mathbf{h}_i^{\mathrm{H}}\mathbf{h}_j \over M}\approx r_{ij}, j = 1, \ldots,K, \,\, j \neq i.\end{array}\right.
a_1=x_1-x_3+x_4-y_2=c_1-c_2+c_3-d_1, a_2=-x_1+x_3+x_4-y_2=-c_1+c_2+c_3-d_1, a_3=x_7-y_7=c_4-d_3, b_1=x_1+x_3-x_4+y_2=c_1+c_2-c_3+d_1, b_2=x_1-x_3+x_4+y_2=c_1-c_2+c_3+d_1, b_3=-x_1+x_3+x_4-y_2+2y_5=-c_1+c_2+c_3-d_1+2d_2, b_4=x_7+y_7=c_4+d_3,
u^e(S^e\otimes I)\Delta^em(a) =u^e(S^e\otimes I)(m\otimes m)\Delta(a) &=S^em(a_{(1)})m(a_{(2)})=mS(a_{(1)})m(a_{(2)}) &=m(S(a_{(1)})a_{(2)})=m(\eta\varepsilon(a))=\varepsilon(a)1_{A^e}
W(G) \leq W(G_1)+\sigma_G(v)+(G,G_1) &\leq W(G_1)+\sigma_G(v)+\frac{(n-1)^2}{18}&\leq \bigg(\frac{1}{12}(n-1)^3+\frac{7}{6}(n-1)-2\bigg)+\frac{1}{6}\bigg((n-1)^2+3(n-1)+2\bigg)+\frac{(n-1)^2}{18}&=\frac{n^3}{12}-\frac{n^2}{36}+\frac{53n}{36}-\frac{115}{36}
\big\|\sum_{i\in{I_n}}\Lambda_i^*g_i\big\|^2=\big\langle\sum_{i\in{I_n}}\Lambda_i^*g_i,\sum_{j\in{I_n}}\Lambda_j^*g_j\big\rangle =\sum_{i\in{I_n}}\sum_{j\in{I_n}}\langle\Lambda_i^*g_i,\Lambda_j^*g_j\rangle &=\sum_{i\in{I_n}}\langle g_i,g_i\rangle=\sum_{i\in{I_n}}\|g_i\|^2.
e^{-2\lambda}\widetilde{K}(X,Y,Z,W)= K(X,Y,Z,W)-|\nabla \l|^2\Big(h(X,Z)h(Y,W)-h(X,W)h(Y,Z)\Big) &-h(Y,W)\Big(D^2\lambda(X,Z)-X(\lambda)Z(\lambda)\Big)-h(X,Z)\Big(D^2\lambda(Y,W)-Y(\l)W(\l)\Big) &+h(X,W)\Big(D^2\l(Y,Z)-Y(\l)Z(\l)\Big)+h(Y,Z)\Big(D^2\l(X,W)-X(\l)W(\l)\Big),
l(y)=e^{-\frac{c}{a}}\Gamma \big(y; \frac{2a\theta}{\sigma^2}, \frac{\sigma^2}{2a}\big)+\int_{0}^{y}\Gamma \big(y-z; \frac{2a\theta}{\sigma^2}, \frac{\sigma^2}{2a}\big) d e^{-\frac{c}{a}-dz}\sqrt{\frac{c}{a d z}}\cdot I_1 \big(2\sqrt{\frac{cdz}{a}}\big)dz
\int_{\Omega}v \nabla_H \zeta \cdot \zeta = \frac{1}{2}\int_{\Omega}v_1 \partial_x \zeta^2 + v_2 \partial_y \zeta^2= -\frac{1}{2}\int_{\Omega}\left(\partial_x v_1 + \partial_y v_2\right) \zeta^2 = -\frac{1}{2}\int_{\Omega}(\div_H v) \zeta^2,
\nabla ( a \circ b ) = \nabla a \circ b + ( - 1 )^{\mathrm{d e g_{a}}a}a \circ \nabla b , \ \nabla_{L}( a \circ b ) = \nabla_{L}a \circ b + ( - 1 )^{\mathrm{d e g_{a}}a}a \circ \nabla_{L}b . \nonumber
P = - \frac{3}{8}g^{2}+ \rho T \left( 1 - \frac{2}{3}\frac{\rho T}{g^{2}}+ \frac{1 6}{9}\left( \frac{\rho T}{g^{2}}\right)^{2}- \frac{1 4 5}{2 7}\left( \frac{\rho T}{g^{2}}\right)^{3}+ \frac{1 4 7 2}{8 1}\left( \frac{\rho T}{g^{2}}\right)^{4}+ \ldots \right)
\int w(\lambda^{1/2}\alpha x_1+\beta,\zeta_0)\phi(x_1)dx_1=\lambda^{-1/2}\int w(\alpha x_1+\beta,\zeta_0)\phi(\lambda^{-1/2}x_1)dx_1 =\lambda^{-1/2}\Big[\sum_{k=0}^2\frac{\lambda^{-k/2}}{k!}\phi^{(k)}(0)\int w(\alpha x_1+\beta,\zeta_0)x_1^kdx_1+O(\lambda^{-3/2})\Big].
A^{\mu\nu}_L = -g^{\mu\nu}[{{7\zeta(3)}\over{4\pi^4}}\Lambda^2+ k^2 \sum_{n=0}{{(2^{2n-1}-1)n! B_{2n}\pi^{2n-2}{\sqrt\pi}}\over{2^{2n+4}(2n)!\Gamma(n +\frac{5}{2})}}\left({k^2 \over \Lambda^2}\right)^n].
Z_f(s, \chi, A_2) = q^{-\omega-1-rs}\int_{B_2}\chi(ac(\pi^{k+l}w^{k+r+l}u^p+w^{r+l}v^r\mathbb{H}_r^k(\pi vw, tw-\pi vw))) \times |\pi^{k+l}w^{k+r+l}u^p+w^{r+l}v^r\mathbb{H}_r^k(\pi vw, tw-\pi vw)|^s|w^{\frac{k+r+l}{p}+1}||dudvdw|, := q^{-(\omega+1)-rs}Z_{f_2}(s, \chi)\int_{\mathcal{O}_K^{\times}}\chi(ac(w^{k+r+l}))|dw|,
\mathcal{B}_{uu}^{i}=U_{uu}^{i}\begin{pmatrix}\alpha_{uu}^{1i}0 \dots 0 0 \alpha_{uu}^{2i}\dots 0 \vdots \vdots \ddots \vdots 0 0 \dots \alpha_{uu}^{Ni}\end{pmatrix}\left(U_{uu}^{i}\right)^{-1},\quad\mbox{for all $u,i=1,\dots,q$,}
\chi_{T}= \left( \frac{\partial J}{\partial \Omega}\right)_{T}= 2^{5 / 3}\pi^{1 / 3}\kappa^{- 4 / 3}N m r_{+}^{- 1}\frac{r_{+}^{4}+ 4 l^{2}r_{+}^{2}- l^{4}}{3 r_{+}^{4}+ 4 l^{2}r_{+}^{2}+ l^{4}}\frac{3 r_{+}^{2}+ l^{2}}{3 r_{+}^{2}- l^{2}}.
|\tilde \kappa_{\infty}(p,\tau)|^2\leq \begin{cases}\frac{\bar c_0}{2 (-\tau_0)}[\tau_2,\tau_0] \tau_2<\tau_0, \frac{\bar c_0}{2 (-\tau_2)}[\tau_0,\tau_2] \tau_0<\tau_2. \end{cases}
\small N_1 = \{E_{1, j}: 2 \leq j \leq p + 1 \}, \, N_2 = \{F_{1, j}: 2 \leq j \leq p + 1 \}, N_3 = \{E_{p+2, j}: p + 3 \leq j \leq n + 1 \}, \, N_4 = \{F_{p+2, j}: p + 3 \leq j \leq n + 1 \}.