harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|I_{n,2}^{\pi}(\theta)\right| \lesssim_{u}\int_{A}|x_{1}|^{-\psi_{p}(\theta)_{+}-1}|x_{2J}|\left|D_{n}(\overline{x}_{2p}-x_{1})^{\ast}\prod_{j=2}^{2p}D_{n}(x_{j})\right|\,\mathrm{d}x_{1}\cdots\,\mathrm{d}x_{2p}&\lesssim_{u}(\log{n})^{2p-2}\int_{A_{J}}|x_{1}|^{-\psi_{p}(\theta)_{+}-1}|x_{2J}|L_{n}(x_{2J})^{2}\,\mathrm{d}x_{1}\,\mathrm{d}x_{2J}
0=\sum_{i=0}^{n-1}\dfrac{1}{\binom{n-1+1}{i}}\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot \tilde{f}_{(n-1)+1-i}\left(\prod_{k\in \left\{1, \ldots, n\right\}\setminus I}x_{k}\right),
k_{n_{x},n_{y},n_{z}}=k_{n_{x}}\hat{x}+k_{n_{y}}\hat{y}+k_{n_{z}}\hat{z}=\frac{n_{x}\pi}{L_{x}}\hat{x}+\frac{n_{y}\pi}{L_{y}}\hat{y}+\frac{n_{z}\pi}{L_{z}}\hat{z}.
\left({T^{*_{op}}}\right)^{\dagger_{op}}= \left({T^{\dagger_{op}}}\right)^{*_{op}}=T{^{\widetilde{}_{op}}}, \left({T^{*_{op}}}\right)^{\widetilde{}_{op}}=\left({T{^{\widetilde{}_{op}}}}\right)^{*_{op}}= T{^{\dagger_{op}}}, &\left({T^{\dagger_{op}}}\right)^{\widetilde{}_{op}}=\left({T{^{\widetilde{}_{op}}}}\right)^{\dagger_{op}}= T^{*_{op}}.
\int_{R}^{2R}\big|J_{\nu(k)}( r\rho)\big|^2 \frac{\mathrm{d}r}{(1+\frac r M)^\beta}\lesssim\int_{R}^{2R}\big|J_{\nu(k)}( r\rho)\big|^2 \mathrm{d}r\lesssim 1.
V^{n}_{l,m}(f):=({\rm ev}_{\overline{a}^{l}}\otimes 1_{a^{n-(m+l)}}\otimes{\rm ev}_{\overline{a}^{m}})\circ( 1_{\overline{a}^{l}}\otimes f\otimes 1_{\overline{a}^{m}})\circ (1_{\overline{a}^{l}}\otimes c_{\overline{a}^{m},b}).
\mbox{$a(x,y,t)\sim\sum^{+\infty}_{j=0}a_j(x,y)t^{n-1-\frac{d}{2}-j}$ in $S^{n-1-\frac{d}{2}}_{1,0}(D_0\times D_0\times\mathbb R_+)$}, &\mbox{$a_j(x,y)\in C^\infty(D_0\times D_0)$, $j=0,1,\ldots$}, &\mbox{$a_0(x,x)$ is given by~\cite[Theorem 1.6]{HH}}.
\mathrm{Tr}\left(U\int_{-\infty}^{0}\Lambda e^{s\Lambda}s^{k}\,\mathrm{d}s\,U^{\dagger}\right) = \mathrm{Tr}\left(\int_{-\infty}^{0}\Lambda e^{s\Lambda}s^{k}\,\mathrm{d}s\,\right)\ ,
\mathcal{U}= \left \{\alpha_{k_{0}}\times \alpha_{k_{1}}\times \ldots \times \alpha_{k_{(M+1)N}}\times I^{\infty}: \{k_{j}\}_{j=0}^{(M+1)N}\right. & \{1, \ldots, n \}(M+1)N+1 \Big \}.
C_{0,k}\subseteq (A_{3k,k}+ \{3a, 2b, 2 c \}), C_{1,k}= C_{0,k}+ \{a \}, D_{0,k}\subseteq (B_{3k+2,k}+ \{a \}) \cup (B_{3k+1,k}+ \{c \}), D_{1,k}= D_{0,k}+ \{a \}.
\varphi_{n-1}(st) =[1,cg_n(xz\i)]_M &=[bp_n(x),cg_n(x)]_M &=[b_sp_n(x_s),c_sg_n(x_s)]_M[b_tp_n(x_t),c_tg_n(x_t)]_M &=[1,c_sg_n(x_sz_s\i)]_M[1,c_tg_n(x_tz_t\i)]_M &=\varphi_{n-1}(s)\varphi_{n-1}(t)
I = ( \frac{6}{\pi^{2}})^{4}[ 1 6 x_{2 4}^{2}( \frac{1}{2 s}- 1 ) D_{4 4 5 5}+ \frac{6 4}{9}\frac{x_{2 4}^{2}}{x_{1 3}^{2}}\frac{1}{s}D_{3 3 5 5}+ \frac{1 6}{3}\frac{x_{2 4}^{2}}{x_{1 3}^{4}}\frac{1}{s}D_{2 2 5 5}
\int_{\Sigma}\tilde{f}Bf = \int_{\Sigma}fB\tilde{f}\:\: , \:\:\: \int_{\Sigma}\tilde{f}Df = \int_{\Sigma}fD \tilde{f}\:\: , \:\:\: \int_{\Sigma}\tilde{f}Af = -\int_{\Sigma}fC \tilde{f}
h_b(x):=\left\lbrace\begin{aligned}\mathbf{E}_x(T_b),x\in [l,b]\cap E, &\frac{h_{b}(b_k)-h_{b}(a_k)}{b_k-a_k}\cdot (x-a_k)+h_{b}(a_k),x\in (a_k,b_k)\subset (l,b), k\geq 1. \end{aligned}\right.
2^{s_tq}\|\Delta_q\rho(t)\|_{L^2}\leq d_q \|\rho_0\|_{B^{s_t}_{2,r}}+d_q\int_0^t\|f(\tau)\|_{B^{s_t}_{2,1}}~d\tau &\quad+Cd_q\int_0^t2^{\big(-\eta\int_{\tau}^{t}V(s)ds\big) q}(\sqrt{q+2}V(\tau)+1)\|\rho\|_{B^{s_\tau}_{2,r}}~d\tau.
((\eth_k + \beta_k)(a + u)) \cdot (b + v) = \eth_k(a) \cdot b + r(b)\beta_k(u) + l(\eth_k(a)) v, (a + u) \cdot ((\partial_k + \alpha_k)(b + v)) = a \cdot \partial_k(b) + l(a)\alpha_k(v) + r(\partial_k(b)) u, (\eth_k + \beta_k)((a + u)\cdot (b + v)) = \eth_k(a \cdot b) + \beta_k(l(a)v) + \beta_k(r(b)u).
\begin{array}{r c l}{{N = \sum_{a = 1}^{3}{\alpha^{a}}^{\dagger}\alpha^{a}\ \ \}}&{{,}}&{{\ \ \N^{\dagger}= N \ \ \ ,}}\\{{B^{\dagger}= \sum_{a = 1}^{3}{\alpha^{a}}^{\dagger}{\alpha^{a}}^{\dagger}\ \ \}}&{{,}}&{{\ \ \B = \sum_{a = 1}^{3}\alpha^{a}\alpha^{a}\ \ \ ,}}\\ \end{array}
\int^\infty_0 |u (x, t) |^2 dx \leq \, \frac43 \big( c_1 +2C_1 (t) \big(c_1 +C_1(t) )^{1/2}\big) + \left ( 2C_1 (t) \right )^{4/3}\left (\int^\infty_0 | u_x( x , t)|^2 dx \right )^{1/3}= \, D_1 + \left ( 2C_1 (t) \right )^{4/3}\left ( \int^\infty_0 |u_x( x , t)|^2 dx \right )^{1/3}
D_{1}(\sum_{k=1}^{n}|\alpha_{k}|^{p})^{\frac{1}{p}}\leq \|\sum_{k=1}^{n}\alpha_{k}(x_{2k-1}-x_{2k})\|\leq D_{2}(\sum_{k=1}^{n}|\alpha_{k}|^{p})^{\frac{1}{p}}.
\left. \begin{array}{r}{{h ( a , c ) A + l ( a , b , c ) \bar{A}\bar{B}}}\\{{h ( b , c ) \bar{B}+ l ( a , b , c ) B A}}\\{{e^{c}+ j^{2}. f ( c , a ) A \bar{C}+ f ( c , b ) C \bar{B}+ j^{2}. g ( c , a , b ) ( \bar{A}\bar{B}\bar{C}+ j . C B A )}}\\ \end{array}\right)
a_{m}(x)+\sum_{k=0}^{n-1}\frac{(-1)^{k+m+1}}{2\pi}a_{k}(x)\int_{-\infty}^{\infty}\frac{\varphi\left( \tau\right) z^{k+m+1}(\tau)d\tau}{\frac{1}{4}+\tau^{2}}& =\frac{(-1)^{m}}{2\pi}\int_{-\infty}^{\infty}\frac{\varphi\left(\tau\right) z^{m+1}(\tau)d\tau}{\frac{1}{2}+i\tau},m=0,1,\ldots,n-1.
= \int_{0}^{\frac 1 4}\d s \ \frac{1}{( 1 - 4 s )^{\frac 1 2}s^{\frac 1 2}}\ \exp \left( - t k^{2}s \right) = \frac{\pi}{2}\exp \left( - t \frac{k^{2}}{8}\right) I_{0}( t \frac{k^{2}}{8}) \, ,
\Delta_{v}={\frac{1}{2}}+{\frac{1}{2}}\pi \beta_{n = 1}q^{2}={\frac{1}{2}}+{\frac{q^{2}}{|{\cal D}- 9 |}}, \; \; \, \Delta_{m}={\frac{1}{2}}+{\frac{1}{8 \pi \beta_{n = 1}}}e^{2}={\frac{1}{2}}+{\frac{e^{2}|{\cal D}- 9 |}{1 6}}
F_j(x,z):= \frac{F(x_j+r_jx,\xi_j+\lambda_jz)-F(x_j+r_jx,\xi_j)-\lambda_jF_{z}(x_j+r_jx,\xi_j)[z]}{\lambda_j^2}= \int_0^1(1-t)F_{zz}(x_j+r_jx,\xi_j+t\lambda_jz)[z,z];
\overline{v}^{s,\pm}(\mathbf{p})v^{r,\mp}(\mathbf{p})=\varepsilon_s\delta_{sr}\frac m{p_0},\hspace{0.3in}\left( \overline{v}^{s,\pm}(\mathbf{p})=v^{*s,\pm}(\mathbf{p})\Gamma_4\overline{\Gamma}_4\right) ,
\begin{array}{r l | r l}{{L_{2}L_{- 2}\Psi}}&{{= \ ( 4 h + c / 2 ) \Psi}}&{{L_{2}L_{- 1}{}^{2}\Psi}}&{{= \ 6 h \Psi}}\\{{L_{1}{}^{2}L_{- 2}\Psi}}&{{= \ 6 h \Psi}}&{{L_{1}{}^{2}L_{- 1}{}^{2}\Psi}}&{{= \ 4 h ( 2 h + 1 ) \Psi \, ,}}\\ \end{array}
\Phi_K(t,z)= \frac{\mathbb{E}[D_tM(T,z)|\mathcal{F}_t]}{M(t,z)}=\frac{\mathbb{E}[D_t[K(z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]] | \mathcal{F}_t]}{\mathbb{E}[K(z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T] | \mathcal{F}_t]}.
{\cal T}_o\,=\,\quad\left[\begin{array}{cccc}1 \qquad 0 \qquad 0 \qquad 0 & 1 \qquad 1 \qquad 0 \qquad 0 & -2 \qquad 0 \qquad 1 \qquad 0 & 0 \qquad 0 \qquad 0 \qquad q\end{array}\right]\hphantom{xxxxxxxxxxxxxxxxxx}
d s_{( , 1 )}^{2}= - \int_{s_{2}}^{s_{3}}\sum_{a , b = 1}^{8}\sum_{\mu}{y}_{\mu}^{a}{y}_{\mu}^{b}d s \rho_{a}\rho_{b}= - \int_{s_{2}}^{s_{3}}\sum_{\mu}d{t}_{\mu}d{t}_{\mu}.
\tau(x)\cdot\dot{\rho}\tau^{-1}(x)\dot{\rho}^{-1}={\rm diag}(1,xE_{d-2},x^{-1},1,x^{-1}E_{d-2},x){}=(\frac{d-1}{2}\alpha^{\vee}_{d-1}+\frac{d-3}{2}\alpha^{\vee}_{d}+\sum_{i=2}^{d-2}(i-1)\alpha^{\vee}_i)(x),
\sum_{u=1}^3N^{(1)}_{F_u^{ti}h_t}(r)\leqslant N^{(1)}_{(f,H_i),\leqslant k_i}(r)+N^{(1)}_{(f,H_t),\leqslant k_t}(r)+N(r,S_i)+N(r,S_t).
(x_1^2 - q^{2i-1}) z_i - q^2 (x_1^2 - q^{2i-5}) z_{i-1}\qquad - q^d (x_1^2 + q^{2i-3}) (q^2-1) + \tau x_1 q^{i-1}(q-1)(q^{d+1}+1) =0.
\frac{1}{\lambda_n}\asymp\max\left\{\max_{j:j<M_n}\sum_{k=j}^{M_n-1}\frac{\pi_n([0,j])}{\pi_n(k)p_{n,k}}, \max_{j:j>M_n}\sum_{k=M_n+1}^j\frac{\pi_n([j,n])}{\pi_n(k)q_{n,k}}\right\},
\left(v^{lk'}_{i k}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\overline{\left(\left(v^{l'k'}_{jk}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\right)^{t}}=a_{ij}^{ll'}I_{N},\quad\mbox{for all $i,j,l,l'=1,2$.}
T_{\mu \nu}^{m a t t e r}= \frac{1}{4}\left( H_{\mu \nu}^{2}- \frac{1}{6}\tilde{G}_{\mu \nu}H^{2}\right) e^{- \frac{8}{\sqrt{2 | D - 2 |}}\tilde{\phi}}+ \left( \partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}- \frac{1}{2}( \partial \tilde{\phi})^{2}\tilde{G}_{\mu \nu}\right) .
\partial_x^{(i)}p_t(x,y)= \sum_{}^{}\frac{i!}{k_1! \cdots k_i!}\partial^{(k)}_{f(x)}q_t(f(x),f(y))\prod_{j=1}^{i}\left(\frac{\partial_x^{(j)}f(x)}{j!}\right)^{k_j}\partial_yf(y),
f'(t) = \left( \log \Big( e+\frac{c_1 t}{a}\Big) \right)^{-\lambda}\left( \frac{-\lambda c_1^2 t}{a^2 \log (e+ c_1 t/a)(e+c_1 t/a)}+\frac{c_1}{a}\right) - \frac{C(n-1)}{t}= \frac{\frac{c_1}{a}\left( 1- \frac{\lambda}{\log(e+ c_1 t/a)((ea)/(c_1 t)+1)}\right)}{\left(\log\big(e+\frac{c_1 t}{a}\big) \right)^{\lambda}}- \frac{C(n-1)}{t},
J_{s_2}= \int\limits_{\substack{|\vec{w}|>1 |\vec{w}|\le\lambda |\vec{v_0}|^B}}|\vec{w}|^s e^{-f(\vec{w})}\, d\vec{w}+ \int\limits_{\substack{|\vec{w}|>1 |\vec{w}|>\lambda |\vec{v_0}|^B}}|\vec{w}|^s e^{-f(\vec{w})}\, d\vec{w},
{\rm Cov}_{\rho,\Phi}^{f,g}(A,B) := \Phi\left(f(\rho) A^*B \right)-\Phi\left(f(\rho)g(\rho)A^*\right) \Phi\left(f(\rho)g(\rho)^2\right)^{-1}\Phi\left(f(\rho) g(\rho) B\right){\rm Var}_{\rho,\Phi}^{f,g}(A) :={\rm Cov}_{\rho,\Phi}^{f,g}(A,A),
X_1^\prime(\theta) = p-2-\gamma-\frac{(p-2) R_\theta-\gamma P_\theta}{\sqrt{P_\theta R_\theta}}= \frac{(\sqrt{P_\theta}-\sqrt{R_\theta})\big((p-2)\sqrt{R_\theta}+\gamma\sqrt{P_\theta}\big)}{\sqrt{P_\theta R_\theta}}.
\varrho_4 = \prod_{i=1}^{m-1}[E_{\alpha_{mk}}, E_{\beta_{ir}}^*]\prod_{i=1}^m [E_{\beta_{ir}}^*, E_{\gamma_{mk}}^*] \prod_{i=1}^{m}[E_{\alpha_{ir}}, E_{\delta_{mk}}] E_{\zeta_{mj}}
\frac{d}{d \tau}( g_{\mu \nu}{\dot{x}}^{\nu}) - \frac{1}{2}\partial_{\mu}g_{\nu \lambda}{\dot{x}}^{\nu}{\dot{x}}^{\lambda}= ( \frac{k d_{\tau}}{R}F_{\mu \nu}+ \frac{m k R}{\sqrt{\alpha^{\prime}}}\tilde{F}_{\mu \nu}){\dot{x}}^{\nu},
\begin{array}{rcl}\{\Gamma^A_0,\Gamma^B_0\}= 0 \cr \{\Gamma^A_0,\Gamma^B_1\}= E^{AB}(\Gamma_0) \cr\{\Gamma^A_1,\Gamma^B_1\}= \Gamma^C_1\partial_C E^{AB}(\Gamma_0) \,.\end{array}
\frac{p_{\beta_{1},\lambda}}{p_{0}}\frac{p_{\beta_{2},\lambda}}{p_{0}}=(1+\lambda^{2})\exp^{-\frac{y^{2}+(1+\lambda^{2})(y-\lambda\beta_{1}^{\tau}x)^{2}}{2(1+\lambda^{2})}-\frac{y^{2}+(1+\lambda^{2})(y-\lambda\beta_{2}^{\tau}x)^{2}}{2(1+\lambda^{2})}}
\sum_{e=n+1}^{n+m}\sum_{f=n+1}^{n+m}\langle \gamma_b, \gamma_e, g^{ef}\gamma_f,\gamma_c\rangle_{0,1}=\sum_{e=1}^{n+3}\langle \epsilon_b, \epsilon_c,\epsilon_e,\epsilon_e\rangle_{0,1}.
H(a,x) = \,\frac{x(\sum_{n=0}^\infty (1-a)^n x^n/n!) - \sum_{n=1}^\infty x^n/n!}{x(\sum_{n=1}^\infty x^n/n!)}= \,\frac{(1/2-a)x^2 + ((1-a)^2/2!-1/3!)x^3+ \cdots}{x^2+ x^3/2! + \cdots}.
\frac{1}{\ell}\sum_{i=1}^{\ell}|b_i| = \frac{1}{\ell}\sum_{s\in\mathbb{S}}\sum_{s\in\mathbf{s}}|b_s| = \sum_{s\in\mathbb{S}}\frac{\sum_{s\in\mathbf{s}}|b_s|}{\ell}.
\langle W \rangle = \exp - \{\frac{2 \rho A}{N - 1}[ 1 - \cos ( \frac{2 \pi}{N}) + 1 - \cos ( \frac{4 \pi}{N}) + \dots + 1 - \cos ( \frac{2 \pi}{N}( \frac{N - 1}{2}) \}.
\sup_{\gamma \in \mathcal{P}(\mathcal{S})}\left\{\int_{\mathcal{S}}g \, d \gamma - \lambda_1 \boldsymbol{K}_1(\mu_{13}, \gamma_{13}) - \lambda_2 \boldsymbol{K}_2(\mu_{23}, \gamma_{23}): \boldsymbol{K}_\ell(\mu_{\ell3}, \gamma_{\ell3}) < \infty \ell = 1, 2\right\},
F^\beta_i{\frac{\partial F^\alpha_j}{\partial u^\beta}}-F^\beta_j{\frac{\partial F^\alpha_i}{\partial u^\beta}}- C^l_{ij}\,F^\alpha_l={\frac{\partial F^\alpha_i}{\partial v^\rho}}G^\rho_j -{\frac{\partial F^\alpha_j}{\partial v^\rho}}G^\rho_i\,,
( 2 \pi )^{3}{\hat{G}}({\hat{\xi}}{\hat{\eta}};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime}) = \sum_{1 2 3}D^{- 1}({\hat{q}}_{1 2}) \int d{{\hat{q}}_{1 2}}^{\prime \prime}K ({\hat{q}}_{1 2},{{\hat{q}}_{1 2}}^{\prime \prime}){\hat{G}}({\hat{\xi}}^{\prime \prime}{\hat{\eta}}^{\prime \prime};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime})
\begin{aligned}G_\beta(r, p, X, A) &={1\over 1-\beta}r^\beta F\left(r^{1-\beta}, (1-\beta) r^{-\beta}p, (1-\beta) r^{-\beta}X+(\beta^2-\beta)r^{-\beta-1}p\otimes p, A\right) \end{aligned}
\begin{aligned}\sup_{0\leq t\leq T}\big(\|V_\varepsilon\|_2^2+\varepsilon^2\|W_\varepsilon\|_2^2\big)(t) &\qquad+\int_0^{T}\Big(\|\nabla_HV_\varepsilon\|_2^2+\varepsilon^2\|\nabla_HW_\varepsilon\|_2^2+\varepsilon^{\alpha-2}\|\partial_zV_\varepsilon\|_2^2+\varepsilon^\alpha\|\partial_zW_\varepsilon\|_2^2\Big)dt &\leq K(T)\varepsilon^\beta,\end{aligned}
{\cal L}= - \frac{i}{2}\epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}- i g a_{\nu}) B_{\lambda}+ \frac{M}{2}B_{\mu}^{*}B_{\mu}- \frac{i}{2}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}+ ( \partial_{\mu}\bar{c}) ( \partial_{\mu}c ) + \frac{1}{2 \rho}( \partial_{\mu}a_{\mu})^{2}\; .
b_t^n(x)=\frac{\beta_t(\xi_{\delta_1(t)}, \ldots, \xi_{\delta_k(t)}, x)}{1+n^{-\theta}|\beta_t(\xi_{\delta_1(t)}, \ldots, \xi_{\delta_k(t)}, x)|}\rightarrow \beta_t(\xi_{\delta_1(t)}, \ldots, \xi_{\delta_k(t)}, x)=b_t(x)
\sigma_{I I}={\frac{2}{\kappa^{2}[ B_{0}( y_{0}+ y_{*}) ]^{1 / 3}( y_{0}+ y_{*})}}={\frac{2}{\kappa^{2}( B_{0}y_{0})^{1 / 3}y_{0}e^{4 n}}}> 0 \, ,
u \star v = \operatorname{e x p}\Big ({\frac{\kappa}{2}}\, \omega^{i j}{\frac{\partial \phantom{x^{i}}}{\partial x^{i}}}{\frac{\partial \phantom{x^{j}}}{\partial{\tilde{x}}^{j}}}\Big ) u ({\bf x}) v ({\bf{\tilde{x}}}) \Big \vert_{{\bf x}={\bf{\tilde{x}}}}
u_t = \Delta u x\in \Omega, \quad 0<t<T v_t =\Delta_{M}v-u^2 v^2 x\in M,\quad 0<t<T \frac{\partial u}{\partial \eta}=u^2 v^2 x\in M, \quad 0<t<T u =u_0 x\in\Omega ,\quad t=0 v =v_0 x\in M , t=0
Y_{l_{1}}^{m_{1}}\star_{1}^{n}Y_{l_{2}}^{m_{2}}=(-1)^{n+m}\sqrt{n+1}\sum_{l=0}^{n}\left[\begin{array}[c]{ccc}l_{1}l_{2}l m_{1}m_{2}-m\end{array}\right] \!\!{[j]}\Y_{l}^{m}\ .
\left(\begin{aligned}\mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_1}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})|\mathcal{F}_t^1]=0 \mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_2}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})|\mathcal{F}_t^2]=0\end{aligned}\right),
\left| \left< \Phi, \psi^{(x)}\right> \right| \geq 1 - K \tau_n^2 (2 \sqrt K \tau_n)^{-2}= \frac 34 \textnormal{and}\left| \left< \Phi, \psi^{(y)}\right> \right| \geq \frac 34 \, .
{\displaystyle(2\pi)^4\delta^{(4)}(q+\sum p)(\frac{2i\beta\lambda}{4!}(p_{20}+p_{30})\delta_{ii_1}\delta_{i_2i_3}-\frac{i\lambda\beta}{3!}p_{10}\delta_{ii_1}\delta_{i_2i_3}+\mbox{\rm perm})}
b(\vec\lambda,m,l,w)=\sum_{\vec t,\; \sum t_i=0}\,\det_{\frac{w}{2}\times \frac{w}{2}}\,\left| \Biggl({m \atop (m/2)+\frac{l+w}{2}\, t_i+\lambda_j-\lambda_i}\Biggr) \right|,
t \,\frac{\partial w}{\partial t}= t^{-d}a(t,x) + (\lambda(t,x)-d)w+ b(t,x) \frac{\partial w}{\partial x}\qquad+ \sum_{j+\alpha \geq 2}a_{j,\alpha}(t,x) t^{d(j+\alpha-1)}w^j\Bigl(\frac{\partial w}{\partial x}\Bigr)^{\alpha}.
{\Pr}\bigg({{x}_R}= 1 ~\bigg|~{x_T}={s}\bigg)={\Pr}\bigg(g_{T,R}^\theta \ge{\tau_R}~\bigg|~{x_T}={s}\bigg)= \frac{1}{2}\left(1 - \left(\frac{{{\tau_R}-{{\mu}_{{{s}_{T,R}}}}}}{{\sqrt{2{{\sigma}_{{{s}_{T,R}}}}^2}}}\right)\right),
f(z) = \frac{8}{729}z f_1^3(z) & = \frac{8}{729}\times \begin{cases}\left(\omega^2 \varphi_1(z) - \varphi_2(z) \right)^3, \Im z > 0, begin{align*}5pt]\left(\omega \varphi_1(z) - \varphi_2(z) \right)^3, \Im z < 0. \end{cases}
x_{nt}^-(1) = x_{\alpha_{nt}}^-(1)=[\,x_t^-(0),\, x_{n(t-1)}^-(1)\,]_{\langle t, t+1\rangle\cdots \langle t, n\rangle}, x_{nt}^+(-1) = x_{\alpha_{nt}}^+(-1)=[\,x_{n(t-1)}^+(-1),\, x_{t}^+(0)\,]_{\langle t+1, t\rangle^{-1}\cdots \langle n, t\rangle^{-1}}.
\frac{\delta S}{\delta \omega}= \Pi^{n-2}\left((-1)^{n-1}d\theta^{n-2}+ \theta^{n-2}\omega+(-1)^{n-1}\omega\theta^{n-2}\right)= = \Pi^{n-2}(\theta^{n-3}(d\theta+\omega\theta+\theta\omega)).
\det{\mathbf{D}}_{Q,n-k+1}= \prod_{k+1 \le p\le n}\det({\mathbf{I}}-({\mathbf{T}}_{k})^{2}\,{\mathbf{T}}_{p,k+1}) = \prod_{k+1 \le p\le n}\, \prod_{T\in{Q \choose p-k+1}}(1-\sigma_T)^{(p-k-1)!\cdot (p-k+1)\cdot (n-p+k-1)!}&= \prod_{2 \le m\le n-k+1}\, \prod_{T\in{Q \choose m}}(1-\sigma_T)^{(m-2)!\cdot m\cdot (n-m)!}
\widetilde{d}_{r}\omega_{h_{k}....h_{1},j}^{i}\overset{def}{=}\frac{\partial \omega_{h_{k}....h_{1},j}^{i}}{\partial x^{r}}-\Gamma_{ra}^{i}\omega_{h_{k}....h_{1},j}^{a}+\Gamma_{rj}^{a}\omega_{h_{k}....h_{1},a}^{i}
-(-1)^{\varepsilon_\alpha}\frac{\delta\Gamma}{\delta\psi^*_\alpha}\bigg( \frac{\delta_l}{\delta\psi^\alpha}-(-1)^{\varepsilon_i\;\varepsilon_\alpha}\;\cdot\;\;(\Gamma^{''-1})^{ij}\;\; \bigg(\frac{\delta_l}{\delta\varphi^j}\frac{\delta_l\Gamma}{\delta\psi^\alpha}\bigg) \frac{\delta_l}{\delta\varphi^i}\bigg). \rule{1.5cm}{0cm}
f\big(\Re\{\mathbf{y}\};\mathcal{H}_k\big)=\!\int\! f\big(\Re\{\mathbf{y}\}|h';\mathcal{H}_k\big) f_{H'}(h')\mathrm{d}h',\ \f\big(\Im\{\mathbf{y}\};\mathcal{H}_k\big)=\!\int\! f\big(\Im\{\mathbf{y}\}|h';\mathcal{H}_k\big) f_{H'}(h')\mathrm{d}h'
\int_{0}^{1}\sqrt[m]{1-x^{n}}dx=\int_{0}^{1}\sqrt[n]{1-x^{m}}dx=\frac{\frac{1}{m}!\frac{1}{n}!}{(\frac{1}{m}+\frac{1}{n})!}=\frac{\int_{0}^{\infty}e^{-x^{n}}dx\cdot\int_{0}^{\infty}e^{-x^{m}}dx}{\int_{0}^{\infty}e^{-x^{\frac{mn}{m+n}}}dx}
\begin{aligned}T_{f}\left(A^{{'E}_{1}}_{1}, h^{{'E}_{1}}_{L^{2}}\right)=\sum_{q=0}^{m}(-1)^{q}T_{f}({'E}^{*, q}_{1})=T_{f}(A_{0})+\sum_{q=1}^{m}(-1)^{q-1}T_{f}(A_{q}), T_{f}\left(A^{{'E}_{2}}_{2},h^{{'E}_{2}}_{L^{2}}\right)=\sum_{q=1}^{m}(-1)^{q}T_{f}(B_{q})=\sum_{q=1}^{m}(-1)^{q-1}T_{f}(B^{-1}_{q}).\end{aligned}
R =\sum_{i}W_{i i}\otimes I_{k}-\sum_{i,j}W_{i j}\otimes e_{i}e_{j}^{*}= \sum_{i}W_{i i}\otimes \left( \sum_{j}e_{j}e_{j}^{*}\right)-\sum_{i,j}W_{i j}\otimes e_{i}e_{j}^{*}&= \sum_{i\neq j}W_{i i}\otimes e_{j}e_{j}^{*}-\sum_{i \neq j}W_{i j}\otimes e_{i}e_{j}^{*}= \sum_{i \neq j}W_{jj}^{ii}- \sum_{i \neq j}W_{ij}^{ij}.
\varphi (g_2 g_3) \varphi (g_1 g^{-1}_3) \overset{{\bf A6}}{=}\varphi (g_2) \varphi (g_3) \varphi (g^{-1}_3) \varphi (g_1) \overset{{\bf A2}}{=}\varphi (g_2) \varphi (g_1) \overset{(\ref{eq:anti3})}{=}\varphi (g_2 g_1) \,,
C_{N-1}^{k_{1}-1}\left[ \frac{r^{N-k_{1}}}{k_{1}}e^{\int_{0}^{r}\frac{1}{C_{0}}t^{k_{1}-1}b_{1}\left( t\right) dt}\left( u_{1}^{^{\prime}}\left(r\right) \right)^{k_{1}}\right]^{\prime}=r^{N-1}e^{\int_{0}^{r}\frac{1}{C_{0}}t^{k_{1}-1}b_{1}\left( t\right) dt}p_{1}\left( r\right) f_{1}\left(u_{1}\left( r\right) ,u_{2}\left( r\right) \right) ,
y_2y_1 = py_1 y_2, x_2x_1 = px_1x_2, y_1x_1 = -x_1 y_1 + p^2 x_2y_1 + x_1y_2 - p x_2y_2, y_1x_2 = -px_1y_1 + x_2y_1 + x_1y_2 - px_2y_2, y_2x_1 = -px_1y_1 - 2p^2x_2y_1 + p x_1 y_2 - p x_2 y_2, y_2x_2 = -px_1y_1 + p^2x_2y_1 + x_1y_2 - x_2y_2,
[\begin{matrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}&c_{2}c_{3}s_{1}+c_{1}s_{3}\\c_{1}s_{2}&c_{2}&s_{1}s_{2}\\ -c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{matrix}]
\mathbb{R}_{12}(u-v|g_1,g_2) \,\mathbb{R}_{13}(u|g_1,g_3) \,\mathbb{R}_{23}(v|g_2,g_3) =\mathbb{R}_{23}(v|g_2,g_3) \,\mathbb{R}_{13}(u|g_1,g_3)\, \mathbb{R}_{12}(u-v|g_1,g_2).
s_i:=|\{j\colon 1\leq j\leq r,\i_j=i\}|\i=1,2; G^{(i_1,\dots,i_r)}:=\bigl[G^{(i_1)}, \bigl[ G^{(i_2)},\dots,\bigl[G^{(i_{r-1})},G^{(i_r)}\bigr]\dots\bigr]\bigr]\subset G_r.
\left\{\mu \frac{\partial}{\partial \mu}+ \left( \beta_{p}- d_{p}p \right) \frac{\partial}{\partial p}+ \gamma_{i}\int d^{4}x \sqrt{- g}\Phi_{i}\frac{\delta}{\delta \Phi_{i}}\right\}\Gamma \left[ g_{\alpha \beta}, p , \Psi_{i}, \mu \right] = 0 .
\|\phi^{(\nu)}(t,\nu x)-\phi^{(0)}(t,\nu x)\|_{L_x^\infty}=\|\phi^{(\nu)}(t, x)-\phi^{(0)}(t,x)\|_{L_x^\infty}&\lesssim\|\phi^{(\nu)}(t)-\phi^{(0)}(t)\|_{H^k}\lesssim\nu^{2\sigma}.
\min\Big(\alpha, \,\frac{d}{\alpha}T^{-\frac{\beta-1}{\beta}}\Big)=\min\Big(\max(\alpha, T^{-\frac{\beta+2}{2\beta}}), \frac{d}{\sqrt{T}}, \,\frac{d}{\alpha}T^{-\frac{\beta-1}{\beta}}\Big)\enspace.
\frac{1}{N}\sum_{i\in B}w_i \epsilon_i = \sum_t \frac{w_t}{N}\sum_{i\in U_t}\delta_i \epsilon_i = \sum_t w_t \frac{N_t}{N}\big( Cov_{N_t}(\delta_i, \epsilon_i) + (\frac{1}{N_t}\sum_{i\in U_t}\delta_i) (\frac{1}{N_t}\sum_{i\in U_t}\epsilon_i) \big)\rightarrow 0
v_p([z^{dp^{i}n}]\log F_i(z)) \geq c_{\lambda_i}-i+v_p\left([z^{dp^{i}n}]\left(h(q^{dp^i},z^{dp^i})-\frac{1}{p}h(q^{dp^{i+1}},z^{dp^{i+1}})\right)\right) &=c_{\lambda_i}-i+v_p\left([z^n]\left(h(q^{dp^{i}},z)-\frac{1}{p}h(q^{dp^{i+1}},z^p)\right)\right) &\geq c_{\lambda_i}-i-v_p(q^{p^{i}}-1) &=c_{\lambda_i}-\lambda_i-i &=b_i.
\int_{(G\setminus B(r,x))\cap\Omega}v\, \left( \frac{\partial u}{\partial n_{+}}+ \frac{\partial u}{\partial n_{-}}\right) d\sigma = \int_{\partial G}\left( u \frac{\partial v}{\partial n}- v \frac{\partial u}{\partial n}\right) d\sigma + \int_{\partial B(r,x)}\left( u \frac{\partial v}{\partial n}- v \frac{\partial u}{\partial n}\right) d\sigma.
\Lambda_{q}^{2}\mathfrak{u}_{+}= \mathrm{span}\{e_{i}\otimes e_{j}- q e_{j}\otimes e_{i}: 1 \leq i < j \leq N\}, S_{q}^{2}\mathfrak{u}_{+}= \mathrm{span}\{e_{i}\otimes e_{i}\}_{i = 1}^{N}\cup \mathrm{span}\{e_{i}\otimes e_{j}+ q^{-1}e_{j}\otimes e_{i}: 1 \leq i < j \leq N\}.
R_{1}=\rho_{0}^{3}\frac{v_{0}v_{1}}{2}=\rho_{0}^{3}\left[ \varphi_{1}\varphi_{2}+\frac{1}{4}f_{0}\varphi_{1}\theta_{1}^{2}+g_{0}\theta_{1}\theta_{2}\right] & =\varphi_{1}(\rho_{0}^{3}\varphi_{2})+\theta_{1}(g_{0}\rho_{0}^{3}\theta_{2})+\frac{1}{4}f_{0}\rho_{0}^{3}\varphi_{1}\theta_{1}^{2}
i\hbar{\partial\Phi \over \partial t}(1, \cdots, N)={1 \over 2m}\sum^N_{p=1}\left\{i\hbar \partial_i^{(p)}-{e \over c}\sum^N_{r=1(\ne p)}{\cal A}_i (p,r) \right\}^2\Phi(1, \cdots, N)\,.
\left( \mathbf{I}+K(\lambda,0)\right) e^{ik\frac{2\pi}{P_{0}}x}=\{\mathbf{I}+\partial_{x}^{-2}\rho(\lambda,0)\}e^{ik\frac{2\pi}{P_{0}}x}& =\left\{1-\frac{4\pi^{2}}{P_{0}^{2}k^{2}}\int\frac{v[\mu_{+}^{\prime}+\mu_{-}^{\prime}]}{v+\frac{\lambda}{ik}}dv\right\}e^{ik\frac{2\pi}{P_{0}}x},k\neq0.
\displaystyle\begin{array}{lcl}AC+BD = \alpha\omega \tau_\alpha\left[\sin(\omega \tau_\alpha)+\omega\cos(\omega \tau_\alpha)\right]+\frac{\alpha\omega^2}{a}(1-\tau_\alpha)\cos(\omega \tau_\alpha) & +\frac{\alpha\omega}{a}\left(\omega^2 \tau_\alpha+a\right)\sin(\omega \tau_\alpha) +\frac{\omega^2}{a}\left(a-1+\alpha^2 \tau_\alpha\right)\;.\end{array}
\Gamma_{\alpha_0}(x_k) \leq\left(\frac{1}{1+\alpha_0\mu}\right)^{k}\Gamma_{\alpha_0}(x_0) &= \left(1-\frac{1}{1+\ell(\tau+1)Q}\right)^{\frac{k}{\tau+1}}\Gamma_{\alpha_0}(x_0) &\leq \left(1-\frac{1}{[1+\ell(\tau+1)Q](\tau+1)}\right)^k \Gamma_{\alpha_0}(x_0),
\widehat{R}_{j , j + 1}( \zeta ) \widehat{R}_{j + 1 , j + 2}( \xi \zeta ) \widehat{R}_{j , j + 1}( \xi ) = \widehat{R}_{j + 1 , j + 2}( \xi ) \widehat{R}_{j , j + 1}( \xi \zeta ) \widehat{R}_{j + 1 , j + 2}( \zeta ) .
Cd\left( z_{k}\right) \leq d\left( z_{j},z_{k}\right) \leq d\left(z_{j},\xi_{k}\right) +d\left( \xi_{k},z_{k}\right) & =d\left( z_{j}\right) -d\left( \xi_{k}\right) +d\left( z_{k}\right)-d\left( \xi_{k}\right) & \leq2d\left( z_{k}\right) -2d\left( \xi_{k}\right) =2d\left( z_{k},\xi_{k}\right) .
|F(\xi_1,\eta_1,\xi_2,\eta_2)| =|\xi_1 \eta_2+ \xi_2 \eta_1 + 2(\xi_1 \eta_1 + \xi_2 \eta_2)| & \geq 2| \xi_2 \eta_1 + \xi_1 \eta_1 + \xi_2 \eta_2| - |\xi_1 \eta_2- \xi_2 \eta_1| & \geq 2|\xi_2 \eta_2| - 2|\xi_1 + \xi_2| |\eta_2| - 2^7 A^{-1}N_1^2 & \geq K^{-1}N_1^2 -2^8 A^{-1}N_1^2 & \geq 2^{-1}K^{-1}N_1^2.
{\bf n}\sp{T}T{\bf n}={\bf n}\sp{T}M\sp{-1}\, M(\tau P+\tilde T)M\sp{T}M\sp{T\,-1}{\bf n}=\frac{\tau}{N}k\sp2+{\bf n}\sp{T}M\sp{-1}\, M\tilde TM\sp{T}M\sp{T\,-1}{\bf n}.
E_2 \ll_{\delta}\begin{cases}x\exp \left( -4 \exp\left( \kappa (c_k \alpha x^\delta)^{\frac{1}{4}}\right)^{-1}\right) \k \ , x \exp \left( -2 \exp\left( \kappa ( \alpha x^\delta)^{\frac{1}{2}}\right)^{-1}\right) \k \ .\end{cases}
{\mathbb E}\|\hat \Sigma-\Sigma\|={\mathbb E}\sup_{\|u\|\leq 1}\biggl\|n^{-1}\sum_{j=1}^n \langle X_j,u\rangle X_j-{\mathbb E}\langle X,u\rangle X\biggr\| &\geq \sup_{\|u\|\leq 1}{\mathbb E}\biggl\|n^{-1}\sum_{j=1}^n \langle X_j,u\rangle X_j-{\mathbb E}\langle X,u\rangle X\biggr\|.
C_{\tau_{\sigma}}= -\sum_{n=1}^{+\infty}\log|1-q^{n}|\leq-\sum_{n=1}^{+\infty}\log(1-e^{-2\pi\mathrm{Im}\tau_{\sigma}n})\leq -\sum_{n=1}^{+\infty}\log(1-e^{-\pi\sqrt{3}n}),
F_q(J_1,J_2,\gamma_1) := \sum_{n_1,n_2 =0}^\infty \frac{J_1^{n_1}J_2^{n_2}e^{i\gamma_1 q^{2 [n_1]_q}}}{[n_1]_q![n_2]_q!}; F_q(J_1,J_2,\gamma_2) := \sum_{n_1,n_2 =0}^\infty \frac{J_1^{n_1}J_2^{n_2}e^{i\gamma_2 q^{2 [n_2]_q}}}{[n_1]_q![n_2]_q!}
x_r(z,\tau):=(-1)^r q^{\frac{(2r-n)^2}{8n}}u^{-r}\prod_{m \geq 1}(1-q^{nm-r}u^n)\prod_{m \geq 0}(1-q^{nm+r}u^{-n})\prod_{m \geq 1}(1-q^{nm}).
D_{\phi}= D_{\phi}^{( 2 )}+ \frac{\lambda^{1}}{N^{2}}\sum_{n = 3}^{\infty}\frac{1}{n 2^{n}}\frac{V_{n}}{N^{n}}T r_{p}( 6 \Sigma_{\phi}^{( 1 )}( p ) N^{2 / 3})^{n}.
\mathcal{P}_0 = \big\{\mathbf{x}\in \mathcal{P}\,:\, x^{(j)}\le \kappa j \in [d] \big\}, \mathcal{P}_1 = \big\{\mathbf{x}\in \mathcal{P}\,:\, x^{(j)}> \kappa j \in [d] \big\}, \mathcal{P}_2 = \big\{\mathbf{x}\in \mathcal{P}\,:\, x^{(j)}> \kappa j \in [d] \big\}.
p_{1}=p_{1}^{\nu}=\lim_{\varepsilon \rightarrow 0}\frac{\log r_{1}\left(\varepsilon \right)}{\log \varepsilon},q_{1}=q_{1}^{\nu}=\lim_{\varepsilon \rightarrow \infty}\frac{\log r_{1}\left( \varepsilon\right)}{\log \varepsilon},