harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\fbox{\displaystyle \partial_{z}\partial_{\bar{z}}\chi_{n}+ \frac{1}{2}\left[ \partial_{\varphi}^{2}V ( \varphi_{0}) \right] \chi_{n}= \widetilde{\mathrm{I n h}}\left( \chi_{0}, \ldots , \chi_{n - 1}\right) ,}
D =\mathbb{Z}_{(2)}[ x_2^2, x_3^2, x_{41}^2, \dots, x_{4n}^2, v_1, \dots, v_m]/(v_0 x_3^2), C =\mathbb{Z}_{(2)}[ x_2^2, x_{41}^2, \dots, x_{4n}^2, v_1, \dots, v_m].
\delta A_{( 3 )}\; = \; d \Delta^{( 2 )}+{\textstyle \frac{1}{2}}\vec{\Sigma}_{( 1 )}^{\T}\eta \vec{F}_{( 2 )}-{\textstyle \frac{1}{2}}\vec{\Sigma}_{( 0 )}{}^{T}\eta \left[ \vec{F}_{( 3 )}- A_{( 1 )}\vec{F}_{( 2 )}\right] \; .
\limsup_{\varepsilon\,\to\,0^+}\,\frac{\log \log\mathcal E_{\mathrm{top}}(f)(\varepsilon)}{-\log\varepsilon}\,=\, \overline{\mathrm{mo}}\,(\mathcal M_f^{\mathrm{erg}}(X),\mathrm W_p) \,\leqslant\, \overline{\mathrm{mo}}\,(\mathcal M_1(X),\mathrm W_p) \,\leqslant \, \mathrm{\overline{dim}_B}\, X.
\alpha(\lambda) = 1-\frac{e^{-2\sigma_1(T-t_0)}}{e^{-2\sigma_1(T-t_0)}+e^{2\sigma_1(T-t_0)}}\left(\frac{2j\sigma_1}{\lambda+j\sigma_1}\right) \beta(\lambda) =e^{j\phi+jN\pi}\left(\frac{-2j\sigma_1}{\lambda+j\sigma_1}\right)\frac{e^{2j\lambda T}}{e^{-2\sigma_1(T-t_0)}+e^{2\sigma_1(T-t_0)}}
L_1(X, Y, Z) := u^2 a_{011}X + u^2 a_{111}Y +u (a_{111}t + a_{112}u) Z, L_2(X, Y, Z) :=u(a_{011}t + a_{012}u) X+ (a_{111}t^2 + a_{112}tu + a_{122}u^2)Z, Q(Y, Z) := a_{011}Y^2 + a_{012}YZ + a_{022}Z^2.
0 = \nabla^2(\hat yc)(\hat x)[x_2-x_1] + \nabla c(\hat x)^\top (y_2-y_1) y_2-y_1 = Q_{k_j}\nabla c(\hat x)[x_2-x_1] + AP_j(\mu_{j_2}-\mu_{j_1}) 0 = A^\top \nabla c(\hat x)[x_2-x_1].
(x\ast y) \ast z-x\ast (y\ast z)= \alpha(\alpha(xy)z)-\alpha(x\alpha(yz)) = (xy)\alpha(z)-\alpha(x)(yz) = (-1)^{|x||y|}((yx)\alpha(z)-\alpha(y)(xz)) = (-1)^{|x||y|}((y\ast x)\ast z-y\ast(x\ast z)),
1+(r\!+\!1)(n\!+\!1)f^\sigma + \sum_{i=1}^{r+1}(r\!+\!1\!-\!i)\!\cdot\! f^{{}^i\sigma}- \sum_{i=1}^{r+1}(r\!+2\!-\!i)\!\cdot\! f^{\sigma^i}.
\frac{\left| u'_{\ell,a_1 \ldots a_n}(t) \right|}{\left| u'_{\ell,a_1 \ldots a_n}(r) \right|}=\left( \frac{q_{\ell,n}+r(\ell-1)\ell^{a_n}q_{\ell,n-1}}{q_{\ell,n}+t(\ell-1)\ell^{a_n}q_{\ell,n-1}}\right)^2 \leq\left( \frac{q_{\ell,n}+(\ell-1)\ell^{a_n}q_{\ell,n-1}}{q_{\ell,n}}\right)^2 \leq \ell^2.
\begin{cases}\E^\prime(n) \ = \ \min\{E(i) : 0 \leq i \leq n\}begin{align*}8pt]\k(n) \ = \ \min\{i : 0 \leq i \leq n, \E(i) = E^\prime(n)\}begin{align*}8pt]\A^\prime(n) \ = \A(k(n))\end{cases}.
\int_\zeta^\pi \frac{Q(\theta) \, \cos^{d-3}(\theta/2)\, \sin\theta \, d\theta}{\sqrt{\cos\zeta - \cos\theta}}= q\, 2^{-(d-3)/2}\, \int_\zeta^\pi \frac{(1-\cos\theta)^{-(d-2)/2}\, (1+\cos\theta)^{(d-3)/2}\, \sin\theta \, d\theta}{\sqrt{\cos\zeta - \cos\theta}}
T = Q(z + f) = \sum_{t=0}^{\ell}(z + f)^t Q^{(t)}= \sum_{t=0}^{\ell}\sum_{u=0}^{t}\binom{t}{u}z^u f^{t-u}Q^{(t)}= \sum_{u=0}^{\ell}z^u T_u \ ,
\begin{array}{c c}{\left\{\, b ( \sigma , \tau ) , c ({\sigma}^{\prime}, \tau ) \right\}\, = 2 \pi \delta ( \sigma -{\sigma}^{\prime})}{}{\left[ \beta ( \sigma , \tau ) , \gamma ({\sigma}^{\prime}, \tau ) \right] = 2 \pi \delta ( \sigma -{\sigma}^{\prime})}{}\end{array}
\tilde{\delta}(x+y)= t(\mathcal M)-{\lim_{n \to \infty}\delta((x+y)z_nz_n')}= (t(\mathcal M)-{\lim_{n \to \infty}\delta(xz_nz_n')})+(t(\mathcal M)-{\lim_{n \to \infty}\delta(yz_nz_n')}) = \tilde{\delta}(x)+\tilde{\delta}(y).
\Phi \sim B \left[ \frac{( \sqrt{2 \mu k}u )^{\nu + 1 / 2}}{2^{\nu}}\frac{( \sqrt{E_{b}})^{\nu + \frac{1}{2}}e^{i \frac{\nu \pi}{2}}}{\Gamma ( 1 + \nu )}- \frac{( \sqrt{2 \mu k}u )^{- \nu + 1 / 2}}{2^{- \nu}}\frac{( \sqrt{E_{b}})^{- \nu + \frac{1}{2}}e^{i \frac{\nu \pi}{2}}}{\Gamma ( 1 - \nu )}\right] .
\begin{aligned}|\Omega_{\rho_{t}}-\Omega_{\infty}| \le\frac{2|J_{\rho_t}- J_{\infty}|}{|J_{\infty}|}\le \frac{4}{m}|J_{\rho_t}- J_{\infty}| &\le C\frac{4}{m}H(\rho_{0}|M_{\Omega_{\rho_{0}}})\int_{t}^{\infty}e^{-\int_{0}^{r}B(s)ds}dr.\end{aligned}
L_{9}(q) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{6n^2 - 4j^2 -3j}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 6n + 2- 4j^2 - j}
\frac{\partial\overline{\rho}\tilde{u_{i}}}{\partial t}+\frac{\partial\overline{\rho}\tilde{u_{i}}\tilde{u_{j}}}{\partial x_{j}}+\frac{\partial\overline{p}}{\partial x_{i}}-\frac{\partial\tilde{\sigma}_{ij}}{\partial x_{j}}=-\frac{\partial\overline{\rho}\tau_{ij}^{r}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}(\overline{\sigma}_{ij}-\tilde{\sigma}_{ij})
\check{\mathbf{u}}^{(t+1)}_N = \exp\left[j\arg\left(\left(\|\mathbf{y}\|^2_{\infty}N_s\mathbf{I}_N - \mathbf{T}_N^H\tilde{\mathbf{V}}\mathbf{T}_N\right)\check{\mathbf{u}}^{(t)}_N\right)\right].
\partial_{i}G_{j k}= \hat{D}_{i}G_{j k}= \hat{D}_{i}( e_{j}^{a}e_{k}^{a}) = M_{i j}^{l}e_{l}^{a}e_{k}^{a}+ M_{i k}^{l}e_{l}^{a}e_{j}^{a}= M_{i j}^{l}G_{k l}+ M_{i k}^{l}G_{j l}.
\sum_{1\leq k\leq n}w_kw_{ki}= -\frac{h'}{2}|Dw|^2\log|Dw|^2u_i-\frac{g'\gamma^i}{2}|Dw|^2\log|Dw|^2 = -\frac{h'}{2}Au_i-\frac{g'\gamma^i}{2}A,i=1,2,\ldots,n.
G_{l + \nu}( r^{\prime \prime}, r^{\prime}; T ) = - \frac{2 a}{\hbar}\sqrt{r^{\prime}r^{\prime \prime}}\int_{0}^{\infty}\frac{d T}{T}\, I_{l + \nu}\left( \frac{2 a \, r^{\prime \prime}r^{\prime}}{i T}\right) \, \exp \left\{i \left[ \frac{( E + i 0^{+})}{\hbar}T + a \left( r^{2}+ r^{2}\right) \frac{1}{T}\right] \right\}\; ,
\frac{1}{n}\mathbb{E}Z_n(\epsilon_0) = \frac{1}{n}\sum_{i=1}^{n}\mathbb{P}(1 \leq \#{\cal E}_i \leq \epsilon_0 n) \leq \sum_{r=1}^{\epsilon_0 n}\frac{1}{C}\frac{T_re^{-r}}{(r-1)!}e^{-\delta_0 r}\leq q_0(C,\epsilon_0,\omega_0).
\int_{-\sqrt{1-y^{2}-x^{2}}}^{+\sqrt{1-y^{2}-x^{2}}}\int_{-\sqrt{1-x^{2}}}^{+\sqrt{1-x^{2}}}\int_{0}^{1}f_{X,Y,Z}(x,y,z)\sqrt{1-x^{2}-y^{2}-z^{2}}^{(n)}dxdydz=1;
{\bf V}_{\rm vec}(\xi^{1},\theta) \equiv V^{(-1)}_{\rm vec}(\xi^{1})+\theta V^{(0)}_{\rm vec}(\xi^{1})\equiv \frac{i}{2}\sum_{\mu=0}^{p^{\prime}}: \zeta_{\mu}(k){\bf \dot{X}}^{\mu}(\xi^{1},\theta) \,\exp\left( i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{\rho=0}^{p^{\prime}}k_{\rho}{\bf X}^{\rho}(\xi^{1},\theta)\right):~,
\sum_{j=1}^3\sum_k\int_{\mathbb{R}}(\mathbf{1}_{|k-k_*|\geq D\varepsilon_0^{-1}k_0}+\mathbf{1}_{|\xi-\eta_*|\geq D\varepsilon_0^{-1/20}k_0})e^{|k-k_*|+|\xi-\eta_*|}\big|\widehat{g_j'}(t,k,\xi)\big|^2\,\mathrm{d}\xi\lesssim (Z(t))^2 e^{-D\varepsilon_0^{-1/20}k_0/2}.
|c_n| =|h_1(id)-Hw_n'(id)|=|H(v-w_n')(id)|\leq \int_{\partial\Gamma}|v(\xi)-w_n'(\xi)|d\nu(\xi) \leq \|v-w_n'\|_{L^1(\nu)}\leq \|v-w_n'\|_{L^2(\nu)}.
{\scriptstyle \begin{pmatrix}1 0 0 1 1 1 1 0 1 0 1 -c c\cdot \xi c\cdot(\xi-1) 0 0 1 1 -c\cdot(\xi+1) -c\cdot(1+\xi^{-1}) -c\end{pmatrix}}^{\mathtt{T}},c\in\mathbb{C}.
|\kappa_{N,s}^{-1}F_{N,s}\ast f(x)|\leq \int_{\{|z|< 1\}}|\ln|z|\f(y)|\dy + \int_{\{|z|\geq 1\}}|\ln (|z|)|z|^{2s-N}f(y)|\dy=:f_1(x)+f_2(x).
U^+\cap T_i(U^{\geqq0})=U^+\cap T_i(U^+),U^-\cap T_i(U^{\leqq0})=U^-\cap T_i(U^-), &U^+\cap T_i^{-1}(U^{\geqq0})=U^+\cap T_i^{-1}(U^+),U^-\cap T_i^{-1}(U^{\leqq0})=U^-\cap T_i^{-1}(U^-).
\div(T^\varphi(\nabla w,\,\cdot\,)^\sharp) = (T^\varphi_{ik}w_k)_i & = T^\varphi_{ik,i}w_k + T^\varphi_{ik}w_{ki}& = \left(R^\varphi_{ik,i}- \frac{S^\varphi_i}{m}\delta_{ik}\right) w_k + T^\varphi_{ik}w_{ki}\, .
-\frac{\partial L_{d}DAGI(p\|q)}{\partial q_{j}}=\sum_{j}p_{j}\left[\underbrace{\frac{a-1}{a-b}\left(\overline{MG}\right)^{a-2}-\frac{b-1}{a-b}\left(\overline{MG}\right)^{b-2}}_{Z}\right]\frac{\partial \left(\overline{MG}\right)}{\partial q_{j}}
\|u_n\|^p_{L^p(B_1)}\le C\,[u_n]^p_{W^{s,p}(B_1)}+C\,\|u_n\|^p_{L^{p-1}(B_1)}&\le C\,[u]^p_{W^{s,p}(B_1)}+C\,\|u_n\|^p_{L^{p-1}(B_1)}.
1 = \left( \prod_{\sigma \in S_n}x_{\sigma(1)}^{c_1}x_{\sigma(2)}^{c_2}\cdots x_{\sigma(n)}^{c_n}\right)^{1/n!}< \frac{1}{n!}\sum_{\sigma \in S_n}x_{\sigma(1)}^{c_1}x_{\sigma(2)}^{c_2}\cdots x_{\sigma(n)}^{c_n}
\beta_1(t)-\beta_2(t)= =[x(u_k,t+\widetilde{\sigma}_1)-x(u_k,\widetilde{\sigma}_1)]- [x(u_j,t+\widetilde{\sigma}_1)-x(u_j,\widetilde{\sigma}_1)]= =[x(u_k,t+\widetilde{\sigma}_1)-x(u_j,t+\widetilde{\sigma}_1)]-(k-j) \cdot \varepsilon\geqslant -(k-j) \cdot \varepsilon.
F^{( 0 )}( x , y ) = \left( \begin{array}{c c c}{{O}}&{{- I}}&{{\vec{0}}}\\{{I}}&{{O}}&{{\vec{0}}}\\{{\vec{0}^{T}}}&{{\vec{0}^{T}}}&{{0}}\\ \end{array}\right) \delta^{2}( x - y ) ,
G_1 (x,y,q): =z \frac{j(qyz;q^2)}{j(q^2y^2;q^4)j(z^2;q^4)}\frac{J_4^4}{J_2^2}\sum_{k}\frac{q^{k^2}(yz)^k}{1-q^{2k}x}& + z \frac{j(qxz;q^2)}{j(q^2x^2;q^4)j(z^2;q^4)}\frac{J_4^4}{J_2^2}\sum_{k}\frac{q^{k^2}(xz)^k}{1-q^{2k}y}- \frac{q}{xy}\frac{j(xy;q^2)}{j(q^2x^2;q^4)j(q^2y^2;q^4)}\frac{J_4^4}{J_2^2}\sum_{k}\frac{q^{k^2-k}(xy)^k}{1-q^{2k-1}z}
\tilde{\pi}\circ \lambda(g) = \tilde{\pi}\left ( \sum_{A \ni g^{-1}}(A,g) \right ) = \sum_{A \ni g^{-1}}\tilde{\pi}(A,g) = \pi(g) \cdot \sum_{A \ni g^{-1}}\sum_{h \in A}\varepsilon(h) \cdot \prod_{h \in Y_g \setminus A}(\pi(d(g)) - \varepsilon(h))
\sup_{y\mathbb{\in R}_{\geqslant0}}\left\vert (\mathcal{D}V)(y)\right\vert =\sup_{y\mathbb{\in R}_{\geqslant0}}\left\vert (a-by)(1+y)^{-1}-\frac{\sigma^{2}}{2}y(1+y)^{-2}\right\vert <\infty.
\frac{\partial}{\partial x}\log(C(x,u^2/x)) =\frac{x^{-1/\tilde{\gamma}_1}+((\gamma_1 / \tilde{\gamma}_1)-1)(u^2/x)^{-1/{\gamma}_0}-(\gamma_1 / \tilde{\gamma}_1)}{x(x^{-1/\tilde{\gamma}_1}+(u^2/x)^{-1/{\gamma}_0}-1)}.
\sum_{k=s}^{\ell}C_k = \sum_{n=0}^{s-2}\left(p_{n+1}^{(+)}-p_{n}^{(+)}\right)\prod_{i=1}^{n}\left(q- p_i \right)\prod_{i=n+1}^{s-2}\left(q- p_i+1 \right).
\phi_k(x) \leq \Big(1 -\frac{A_0}{A_k}\Big) f(x) + \frac{A_0}{A_k}\phi_0(x) + \frac{\tilde \varepsilon_{k}}{A_k}=\Big(1 - \frac{1}{A_k}\Big)f(x) + \frac{1}{A_k}\phi_0(x)+ \frac{\tilde \varepsilon_{k}}{A_k}.
t_{8}^{r_{1}\cdots r_{8}}M_{r_{1}r_{2}}N_{r_{3}r_{4}}\cdots N_{r_{7}r_{8}}= - 6 \, M_{t_{1}t_{2}}N_{t_{2}t_{1}}N_{m n}N_{n m}+ 2 4 \, M_{t_{1}t_{2}}N_{t_{2}m}N_{m n}N_{n t_{1}}\, .
\begin{array}{c c c c c c c}{\nabla Z_{A B}}&{=}&{P_{A B C D}\overline{{Z}}^{C D}}&{\longrightarrow}&{\nabla \widehat{Z}_{A B}}&{=}&{{\cal P}_{A}^{A^{\prime}}{\cal P}_{B}^{B^{\prime}}P_{A^{\prime}B^{\prime}C D}\overline{{Z}}^{C D}\ ,}\\ \end{array}
f ( x ) = - 2 \pi c_{2}^{\prime}\left( ( ( x^{0}+ i 0 )^{2}- |{\bf x}|^{2})^{- 1}+ ( ( x^{0}- i 0 )^{2}- |{\bf x}|^{2})^{- 1}\right) + ( 2 \pi )^{- 4}a_{0}.
\phi(D):=\phi(P_1) \dots \phi(P_g) = E\prod_{i=1}^m \dfrac{\theta(\sum_{j=1}^g \int_P^{P_j}\omega - \int_P^{A_i}\omega - \Delta, \Omega )}{\theta(\sum_{j=1}^g \int_P^{P_j}\omega - \int_P^{B_i}\omega - \Delta, \Omega )},
\partial_t f+\hat{p}\cdot\nabla_x f =\frac{U_\mu p^\mu}{c\tau p^0}\frac{1}{\sqrt{F_E^0}}\left\{\left(1-p^\mu q_\mu\frac{1+\frac{\mathcal{I}}{mc^2}}{bmc^2}\right)F_E-F_E^0-f\sqrt{F_E^0}\right\}.
\overline{{\mathrm{\Large v}}}_{3}( x ) ={\frac{2 \pi^{3 / 2}}{g^{2}\Lambda}}x +{\frac{1}{3}}\left( 4 x -{\frac{{\cal E}^{2}}{x}}\right) x^{1 / 2}{\mit \Gamma}( 1 / 2 , x ) +{\frac{2}{3}}( 1 - 2 x ){\mathrm e}^{- x}-{\frac{2}{3}}\ ,
D_{i;I}=\partial_{x^i}+\frac{\partial u_{\bold{0};I}^{\alpha}}{\partial x^i}\partial_{u_{\bold{0};I}^{\alpha}}+\cdots+\sum_{\alpha,J_1}u^{\alpha}_{J_1+\bold{1}_i;I}\partial_{u^{\alpha}_{J_1;I}}.
\sum_{j = 1}^{M}\frac{\partial}{\partial x_{j}}f ( x_{1}, \dots , x_{M}) = \biggl ( \, \sum_{j = 1}^{p - 1}+ \sum_{j = p}^{M}\, \biggr ) \frac{\partial}{\partial x_{j}}f ( x_{1}, \dots , x_{M}) .
\partial_{t}(\rho_{n}W_{n}) + \partial_{x}(\rho_{n}W_{n}u_{n}) = \rho_{n}( \partial_{t}W_{n}+ u_{n}\partial_{x}W_{n}) + W_{n}( \partial_{t}\rho_{n}+ \partial_{x}(\rho_{n}u_{n})) = 0,
\theta(\theta(\rho_{i}, \rho_{j}), \theta(\rho_{i}, \rho_{\ell})) =\theta\left(\rho_n^{qi+B(1,n-i+1)}\rho_{{j-i}},\rho_n^{qi+B(1,n-i+1)}\rho_{\ell-i}\right)=\theta(\rho_{j-i}, \rho_{{\ell-i}}) &=\rho_n^{q(\ell-i)+B(j-\ell, n-\ell+i)},
\left\{\prod\limits_{j=1}^{N-1}(1-R\kappa_j(x))\right\}^{\frac 12}+ \left\{\prod\limits_{j=1}^{N-1}(1+R\kappa_j(x))\right\}^{\frac 12}= c\ \mbox{for every}x \in \partial\Omega,
L(x_{\rm th})=\sqrt{\frac{\pi}{2}}\sigma f_D begin{align*}-0.2em] \times \sum_{i=1}^N \underbrace{\int_0^{x_{\rm th}}\!\!\cdots\int_0^{x_{\rm th}}}_{(N-1)-{\rm fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_1,\dots,x_i=x_{\rm th},\cdots,x_N) begin{align*}-0.2em] \times \underbrace{dx_1 \cdots dx_k\cdots dx_N}_{\substack{(N-1)-{\rm fold}k \neq i}}.
\begin{array}{l}- \lambda_1(0,1,1) = - \lambda_1(1,1,1) = - \lambda_2(0,0,1) = - \lambda_2(1,0,1) = - \lambda_3(1,0,0) = - \lambda_3(1,1,0) = - \lambda_4(0,0,0) = - \lambda_4(0,1,0) = \frac{1}{2}.\end{array}
\sigma_{k}(\overbrace{B,\dotsc,B}^{l},C,\dotsc,C) :=\sigma_{k}(\overbrace{B,\dotsc,B}^{l},\overbrace{C,\dotsc,C}^{k-l}), T_{k}(\overbrace{B,\dotsc,B}^{l},C,\dotsc,C)_{ij}:=T_{k}(\overbrace{B,\dotsc,B}^{l},\overbrace{C,\dotsc,C}^{k-l})_{ij}.
\delta \bigg ( \! \operatorname{e x p}\bigg \{\frac{i}{\hbar}S_{\mathrm{e x t}}\bigg \}\bigg ) = - i \hbar \hat{T}( \delta Y ) \operatorname{e x p}\bigg \{\frac{i}{\hbar}S_{\mathrm{e x t}}\bigg \},
\big|\big\{u\in M_{n,\ell,{\bf t}}:\max(u)=k+\sum_{j=1}^{\ell-1}t_j+1\big\}\big| = =\binom{k+\sum_{j=1}^{\ell-1}t_j+1+(\ell-1)-\sum_{j=1}^{\ell-1}t_j-1}{\ell -1}= \binom{k+\ell-1}{\ell-1}.
M_{1+21-2}(\mathbf{z}) = \begin{pmatrix}1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 -1 0 0 1 0 0 z_{5,1}z_{5,2}0 0 0 1 z_{6,1}z_{6,2}-1 -z_{6,1}1 0 \end{pmatrix}.
\hat{w}\left(z,\,\tau\right)=\left.\frac{1}{\left(-t\right)^{\frac{1}{2}+\sigma}}\,\hat{u}\left(\left(-t\right)^{\frac{1}{2}+\sigma}z,\,t\right)\right|_{t=-\left(2\sigma\tau\right)^{\frac{-1}{2\sigma}}}
[{\cal L}_\alpha,{\cal P}_\beta]=-i(1+\nu R)(P\gamma)_{\alpha\beta},\quad\{{\cal L}_\alpha,{\cal P}_\beta\}=4i(PJ)\epsilon_{\alpha\beta}-4\Gamma_{\alpha\beta},\quad\Gamma_{\alpha\beta}\equiv\epsilon_{\mu\nu\lambda}P^\mu J^\nu \gamma^\lambda_{\alpha\beta},
\mathbb{E}_f(\hat{H}_n) - H = O\biggl(\max\biggl\{\frac{k^{\tilde{\beta}/d}}{n^{\tilde{\beta}/d}}\int_{\mathcal{X}_n}\frac{C_{n,\tilde{\beta}}(x)}{f(x)^{\tilde{\beta}/d}}\, dx \, , \, q_n^{1-\epsilon}\, , \, q_n\log n \, , \, \frac{1}{n}\biggr\}\biggr),
\prod_{r=1}^{2^q}(s_{r,e}2^{-q-1})^{s_{r,e}}((q-s_{r,e})2^{-q-1})^{q-s_{r,e}}\leq \prod_{r=1}^{2^q}(s_{r,e}2^{-q-1})^{s_{r,f}}((q-s_{r,e})2^{-q-1})^{q-s_{r,f}}&< \prod_{r=1}^{2^q}(s_{r,f}2^{-q-1})^{s_{r,f}}((q-s_{r,f})2^{-q-1})^{q-s_{r,f}}
D_{\Phi(y)_1}\log|{\bf k}+\Phi(x)-\Phi(y)||D_{y_1}\Phi_1(y)|=|\frac{\Phi(y)_1-\Phi(x)_1-k}{|{\bf{k}}+\Phi(x)-\Phi(y)|^2}||D_{y_1}\Phi_1(y)| \le 2\frac{1}{|{\bf{k}}+\Phi(x)-\Phi(y)||y|^2},
T_0 T_1 T_0 (E_{2,1}(1)) = T_0 T_1 T_0 \big(T_2 T_3 \cdots T_{N-1}\big(x_0^+\big)\big) = T_0 T_1 T_2 T_3 \cdots T_{N-2}\big(x_{N-1}^+\big) = -E_{N,1}(2).
I_{2,N}:=\int_{r\in N,|t|/2\leq|r|\leq 2|t|}\frac{|F_s(r) - F_s(t)|}{|r-t|^{2-\frac1{2s}}}\,dr\lesssim \frac1{|t|^{2-\frac1{2s}}}\int_0^{2|t|}\frac{r}{(1+r^{\frac1s})^{\frac{N+2s}2}}\,dr+ \frac{|F_s(t)|}{|t|^{1-\frac1{2s}}}.
G_{m,k,r}^{(n)}\big(\theta_1^{(n)},\ldots,\theta_n^{(n)}, \{p_{ij}\}\big)=0 \ \m >k, G_{0,0,r}^{(n)}\big(\theta_1^{(n)},\ldots,\theta_n^{(n)},\{p_{ij}\}\big)=e_{r}(d_2, \ldots,d_n),
zn(x;k)=\sum_{n=1}^{\infty}\frac{2\pi K(k)^{-1}sin[\pi K(k)^{-1}x]q(k)^{2n-1}}{1-2cos[\pi K(k)^{-1}x]q(k)^{2n-1}+q(k)^{4n-2}}
\frac{\mathbb{P}( M_f^+, \alpha; \tau)}{(c \tau + d)^{k-2}}= \sum_{n=0}^{k-2}\frac{\overline{L(f, e^{- 2 \pi i d/c}, n+1)}}{(k-2-n!)}(- 2 \pi i )^{k-2-n}\left( \frac{1}{c^{k-2-n}(c \tau + d)^n}\right).
a =\langle e,e,e,e,e,e \rangle (1,2)(3,4)(5,6) b = \langle e,e,e,e,e,b \rangle (4,5) c = \langle a,e,e,e,e,c \rangle (2,3) d = \langle a,e,e,e,e,d \rangle (2,3)(4,5)
s_{m_\textbf{s}}(t) =\sum\limits_{n_\textbf{s}\in \mathbb{Z}}(-1)^{m_\textbf{s}n_\textbf{s}}\mathbf{d}_{m_\textbf{s}}[n_\textbf{s}]\mathbf{\theta}_{m_\textbf{s}}[n_\textbf{s}]f_{,\textbf{s}}\left(t-n_\textbf{s}\frac{T}{2}\right)e^{j2\pi m_\textbf{s}\frac{t}{T}}
\begin{aligned}{\rm Bias}(\hat D_{\lambda}) =\mathbb{E}\bigg \{\sum_{l\in \bar l}\lambda_l \hat D_l -D \bigg \}= \sum_{l\in \bar l}\lambda_l{\rm Bias}(\hat D_l) = \sum_{l\in \bar l}\sum_{j\in J}\gamma_j \lambda_l \varphi_j(l) \varGamma^{-j/2d}+\rho_{\lambda bias}(\varGamma^{-1/2}).\end{aligned}
\zeta_{\widetilde{L}_p}^\mathrm{iso}(s) = \prod_{\mathfrak{p}\mid p}\frac{(1 - q_\mathfrak{p}^{\, 24d + 5 -10s}) \, V_{\mathfrak{p}}(s)}{(1 - q_\mathfrak{p}^{\, 12d+3-5s})^2 \, (1 - q_\mathfrak{p}^{\, 18d+11-9s}) \, (1-q_\mathfrak{p}^{\, 30d-11s}) \, (1-q_\mathfrak{p}^{54d+7-21s})},
\omega=a(r(t))\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+\left(b(r(t))+1\right)\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right).
c_1\big(\Omega_X^1(\log (D_1+D_2))\big) =D_1+D_2, c_2\big(\Omega_X^1(\log (D_1+D_2))\big) =c_2(X) + D_1^2 + D_2^2 + D_1\cdot D_2, c_3\big(\Omega_X^1(\log (D_1+D_2))\big) =c_2(X)\cdot (D_1+D_2) -c_3(X) &+ D_1^3+D_2^3+D_1^2\cdot D_2+D_1\cdot D_2^2,
\begin{array}{rcl}T_+ = -{\sqrt 3\over2}(X_{(11)}+X_{(-1-1)})+{1\over 2}(X_{(31)}+X_{(-3-1)})\,, T_- ={\sqrt3\over 2}(X_{(21)}+X_{(-2-1)})-{1\over2}(X_{(01)}+X_{(0-1)})\,, T_3 = -{\sqrt{3}\over 2}(X_{(10)}-X_{(-10)})+{1\over2}(X_{(32)}-X_{(-3-2)})\,, \end{array}
v(x, y) = \left(\sup_{(x', y') \in B_{r \sqrt{s}}^n(0, 0)}\{u(x',y')\}\right) \max\left\{1 - \frac{|(x, y)|^{2-n}- r^{2-n}}{(sr)^{2-n}- r^{2-n}}, 0\right\}.
J^{L,\lambda_L}(\xi) =\widehat R_{\bigl(\pi_G(\xi)\bigr)^{-1}}(\xi) +\lambda_L\theta\bigl(\pi_G(\xi))\,, J^{R,\lambda_R}(\xi) =\widehat L_{\bigl(\pi_G(\xi)\bigr)^{-1}}(\xi) +\lambda_R\theta\Bigl(\bigl(\pi_G(\xi)\bigr)^{-1}\Bigr)
\bar{A}=\left(\begin{array}{cccccc}a_{11}a_{12}\cdots a_{1M}\tilde{a}_{21}\tilde{a}_{22}\cdots \tilde{a}_{2M}\cdots \cdots \cdots \cdots \tilde{a}_{N1}\tilde{a}_{N2}\cdots \tilde{a}_{NM}\end{array}\right),
I_{m,p}^{(1)}(s,x)=\frac{(-1)^{m}}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\Gamma\left(z\right)\,\Gamma\left(s+z-\frac{1}{2}\right)\,(\pi\,x\,m)^{-2z}\,dz=2\,(-1)^{m}\,\left(\pi x\,m\right)^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}\left(2\pi x\,m\right)
S_j^\ell u[x] :=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(\varphi_a(x,\xi)-y\cdot\xi)}s_j^\ell(x,\xi) u[y] d\xi , P_j^\ell(t) u[x] :=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(x\cdot\xi-\varphi_a(y,\xi))}p_j^\ell(y,\xi;t) u[y] d\xi ,
f_k(r) = c_k e^{\frac{r^2}{4}}r^{2l_k}F[l_k + \frac{m}{2}, 2l_k + \frac{m}{2}; -\frac{r^2}{4}] &=c_k\frac{\Gamma(2l_k + \frac{m}{2})}{\Gamma(l_k)}e^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big) &= C_ke^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big),
( \bar{\varepsilon}_{+}^{a})^{\dagger}( \bar{\varepsilon}_{-}^{a}) = 0 ; \hspace{1 . 0 c m}( \bar{\varepsilon}_{+}^{a})^{\dagger}( \bar{\varepsilon}_{+}^{a}) = ( \bar{\varepsilon}_{-}^{a})^{\dagger}( \bar{\varepsilon}_{-}^{a}) = 1 .
\mathbb{E}\left( \left. \sum_{i\in \Omega}X_{i}\right\vert S_{n}>\mathrm{VaR}_{q}(S_{n})\right) =\sum_{i\in \Omega}\mathbb{E}\left( \left.X_{i}\right\vert S_{n}>\mathrm{VaR}_{q}(S_{n})\right) ,\varnothing\neq \Omega \subset \{1,\ldots ,n\}.
\tilde{A}_{i}({\bf k}) = \Gamma ( \nu ) \left({\frac{|{\bf k}|}{2}}\right)^{- \nu}\left({\frac{1}{2}}a_{i}({\bf k}) - i{\frac{k_{i}}{|{\bf k}|^{2}}}c a_{0}({\bf k}) \right) .
A_i(x)A_i(y) = A_i(y)A_i(x), t A_j(x)A_i(y)-\frac{tx-y}{x-y}\Big( A_j(x)A_i(y) - A_j(y)A_i(x) \Big) = A_i(x) A_j(y), S A_i(q x) = q^{i}A_i(x)S,
0 = \langle A_2(u_2)+\lambda_2 B_2(u_2),(u_2-\overline{u}_2)_+\rangle_{\mathcal{H}_2}&- \langle F_2(T_1u_1,\overline{u}_2,\nabla (T_1u_1),\nabla \overline{u}_2)+G_2(T_1u_1,\overline{u}_2),(u_2-\overline{u}_2)_+\rangle_{\mathcal{H}_2}.
\begin{aligned}p^{+}(x_1) = \left(1-u^3\right)x_1^6 -12\left(1-u^3\right)x_1^5 +16\left(1-3u^3\right)x_1^4-64\left(3-u^3\right)x_1^3 +1536x_1^2- 6144x_1+8192, p^{-}(x_1) = \left(1+u^3\right)x_1^6 -12\left(1+u^3\right)x_1^5 +16\left(1+3u^3\right)x_1^4-64\left(3+u^3\right)x_1^3 +1536x_1^2- 6144x_1+8192. \end{aligned}
\gamma=\frac{4}{3\pi^{2}}K^{2}(m)(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})=-\frac{1}{6}+\frac{1}{64}m^{2}+\frac{1}{64}m^{3}+\cdot\cdot\cdot,
\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot z)]+\varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^2(y)] &+\varepsilon(x,y+z)\big(2[\alpha(h)\cdot[y,z],\alpha^2(x)]-\varepsilon(x,y+z)[[h\cdot y,\alpha(z)],\alpha^2(x)]\big)=0.
0=\varepsilon(\{a, b\})1_A=S(\{a, b\}_{(1)})\{a,b\}_{(2)}=(-1)^{|b_{(1)}||a_{(2)}|}(S(\{a_{(1)},b_{(1)}\})a_{(2)}b_{(2)}+S(a_{(1)}b_{(1)})\{a_{(2)}, b_{(2)}\}).
1 = \int{\cal D}g ( \sigma , \tau ) \delta ( \gamma_{+ +}^{g}) \delta ( \gamma_{--}^{g}) \mathrm{d e t}\left( \frac{\delta \gamma_{+ +}^{g}}{\delta g}\right) \mathrm{d e t}\left( \frac{\delta \gamma_{--}^{g}}{\delta g}\right)
\begin{array}{c}{{M_{A B C}= \frac 1{1 2}( M^{a b}\left( \sigma_{A B C}\right)_{b a}+ M_{a b}( \widetilde{\sigma}_{A B C})^{b a})}}\\{{M^{a b}= M^{A B C}( \widetilde{\sigma}_{A B C})^{a b}\ , \quad M_{a b}= M^{A B C}\left( \sigma_{A B C}\right)_{a b}}}\\ \end{array}
\begin{matrix}F_{D}&=\frac{1}{2}C_{D}\rho_{w}u^{2}A_{D}\\F_{S}&=\frac{1}{2}C_{F}\rho_{w}u^{2}A_{S}\\F_{L}&=\frac{1}{2}C_{L}\rho_{w}u^{2}A_{L}\end{matrix}]F\sim\rho_{w}u^{2}d^{2}
\mathcal{I}_{\mathrm{D}}(\delta_1, 0) = \widehat{\mathcal{I}}_{\mathrm{D}}(\delta_1, \delta_2) = \inf_{\lambda \in \mathbb{R}_{+}^2}\left[ \langle\lambda, \delta \rangle+\sup_{\varpi \in \Pi\left(\mu_1, \mu_2\right)}\int_{\mathcal{V}}g_\lambda \left(s_1, s_2\right) d \varpi(s_1,s_2)\right],
\sum_{u_1, \ldots, u_k}\Big(\prod_{i = 1}^k g(u_i)\Big) \sideset{}{''}\sum_{s_{1, 2}, \ldots, s_{k-1, k}}\Big(\prod_{\substack{1 \leq i, j \leq k i \neq j}}\mu(s_{i, j})\Big) \times \Big(\prod_{i=1}^k \frac{\mu(a_i)}{g(a_i)}\Big)\Big(\prod_{j=1}^k \frac{\mu(b_j)}{g(b_j)}\Big) y^{(m)}_{a_1, \ldots, a_k}y^{(m)}_{b_1, \ldots, b_k},
\ \sum_{n_{r}= 1}^{\infty}\sum_{k_{r}= 1}^{n_{r}}{\binom{n_{r}}{k_{r}}}\frac{( - )^{k_{r}}}{n_{r}}k_{r}E_{r}^{k}= E_{r}\cdot \sum_{n^{\prime}= 0}^{\infty}( E - E_{r})^{n^{\prime}}= E .
\frac{1}{2}x^{2}+C_{2}= -\int \frac{a\sec^{2}(\theta)}{a^{3}(1+\tan^{2}(\theta))^{3/2}}d\theta &= -\int \frac{1}{a^{2}}\cos(\theta)d\theta &= -\frac{1}{a^{2}}\sin(\theta) \frac{a^{2}}{2}(x^{2}+2C_{2}) = -\frac{z'(x)}{\sqrt{a^{2}+z'(x)^{2}}}\frac{1}{2}(1+C_{1}^{2})(x^{2}+2C_{2}) = -\frac{z'(x)}{\sqrt{1+C_{1}^{2}+z'(x)^{2}}}
F(W)c_{(\vartheta,\varphi)}=\int_{\gamma}F (Z )E_{(\vartheta, \varphi)}(Z , W )d\sigma_{(\vartheta, \varphi)}(Z ) - \int_{\Gamma}E_{(\vartheta , \varphi)}(Z ,W ) \frac{\partial F}{\partial Z_{(\vartheta,\varphi)}}dZ\wedge dZ^{*},
\frac{d^2\psi}{dz^2}+\bigg[\frac{\frac{1}{\gamma}}{z}+\frac{\frac{1}{2}\big(\delta-\frac{1}{\gamma}\big)}{z+1}+\frac{\frac{1}{2}\big(\delta-\frac{1}{\gamma}\big)}{z-1}\bigg]\frac{d\psi}{dz}+\frac{-4R^{2}E z+2R}{z(z+1)(z-1)}\psi=0.
\Gamma_{t}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) - g_{t}\sim g_{t}^{2}\gamma^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) + O ( g_{t}^{3}) \; .
E(|\|X_{1}-X_{2}\|_{2}^{2}-p\tau_{1}|^{3}) \le\surd{E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau_{1})^{2}\}}\times \surd{E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau_{1})^{4}\}}=\surd{O(p)}\times \surd{O(p^{2})}=O(p^{\frac{3}{2}}).
J_{2k-1}\left(\frac{4\pi \sqrt{ILJM}}{c}\right)=\sum_{\ell=0}^{\infty}\frac{(-1)^\ell}{\ell! \,\ \Gamma(\ell+2k)}\left( \frac{2\pi \sqrt{IJLM}}{c}\right)^{2k+2\ell-1}.
\|[S(t+h)-S(t)]\xi\|_{\mathbb H^\mu}\le h\int_0^1\|S'(t+\zeta h)\xi\|_{\mathbb H^\mu}d\zeta & \le h\|\xi\|_{\mathbb H^\mu}\int_0^1\frac{d\zeta}{t+\zeta h}= \|\xi\|_{\mathbb H^\mu}\ln \left(1+\frac ht\right) & \le \|\xi\|_{\mathbb H^\mu}\gamma^{-1}t^{-\gamma}h^\gamma.
\Big\langle : e^{i\varphi(f)}:_M e^{i\varphi(g)}:_M \Big\rangle_{C_{m=0}}\; := \;\lim_{m \rightarrow 0}\Big\langle : e^{i\varphi(f)}:_M e^{i\varphi(g)}:_M \Big\rangle_{C_{m}}\; .
\left. \begin{array}{c c c}{S_{\sigma \sigma^{\prime}}\nonumber}\\{S_{\Delta}\nonumber}\\{\tilde{S}_{\Delta}}\\ \end{array}\right\}\varpi = ( \varpi + 2 ) \left\{\begin{array}{c c c}{S_{\sigma \sigma^{\prime}}\nonumber}\\{S_{\Delta}\nonumber}\\{\tilde{S}_{\Delta}}\\ \end{array}\right. .
Y:= Y_0 + \nabla Y_1 := \{(R_0(t) + \nabla^{{g}_0(t)}_i R^i_1(t))_{t\geq 0}| (R_0(t))_{t\geq 0}\subset S^2T^*(N\times\mathbb{R}) ; (R_1(t))_{t\geq 0}\subset S^2T^*(N\times\mathbb{R}) \otimes \Gamma(T(N\times\mathbb{R}))\},
\mathcal{E}_2= \bigcup_{\substack{(m_0,\ldots,m_N) \in\, \mathbb{Z}^{N+1}\forall n,~ |m_n| \leq \Delta}}\bigcup_{\substack{N<n_1,n_2<L n_1 \neq n_2}}\left\{\alpha \in \mathcal{A}~\middle|~ \prod_{n=0}^{N}(n+\alpha)^{m_n}=\frac{n_2+\alpha}{n_1+\alpha}\right\}.
\mathcal{N}(t,0,\xi,v,\overline v) = e^{t \mathcal{L}}f(v,\overline v) \quad \mathcal{N}(t,t,\xi,v,\overline v) = f \left(e^{t\mathcal{L}}v, e^{t \mathcal{L}}\overline v\right) \xi.
\hat{r}( p )^{a n}= - 2 - \sqrt{1 - \frac{4 m^{2}}{p^{2}}}\, \log \frac{\sqrt{1 - \frac{4 m^{2}}{p^{2}}}- 1}{\sqrt{1 - \frac{4 m^{2}}{p^{2}}}+ 1}\, .