harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\operatorname{Cov}(f(X_{\alpha,\beta}),g(X_{\alpha,\beta})) = \int_{[-1,1]}(1-x^2) g'(x) \tilde{f}'(x) \mu_{\alpha,\beta}(dx), \tilde{f}= \int_{0}^{+\infty}Q_t^{\alpha,\beta}(f)dt,
\sum_{q\geq 1}\frac{1}{2q+|x|}\frac{1}{2^{2q+|x|}}\frac{(2q+|x|)!}{q!(q+|x|)!}\leq C_{1}\sum_{q\geq 1}\frac{1}{2q+|x|}\frac{1}{2^{2q+|x|}}\frac{2^{2q+|x|}}{\sqrt{q}}\leq \frac{C_{1}}{2}\sum_{q\geq 1}\frac{1}{q^{\frac{3}{2}}}=: C_{1}'<\infty,
\mathcal{M}^\infty_n=\left(\begin{array}{ccc}{-\mathcal{A}^\infty_{1,n}}{\mathcal{B}^\infty_n}{0}{\left(\mathcal{B}^\infty_n\right)^*}{\mathcal{A}^\infty_{2,n}}{0}{0}{0}{Pl^\infty}\end{array}\right)
dX_t =b\left(t,X_t,\mathcal{L}(X_t),\hat{\alpha}\left(t,X_t,\mathcal{L}(X_t),\sigma^{-1}Z_t\right)\right)dt+\sigma dW_t X_0 =\xi , dY_t =-f\left(t,X_t,\mathcal{L}(X_t),\hat{\alpha}\left(t,X_t,\mathcal{L}(X_t),\sigma^{-1}Z_t\right)\right)dt+Z_t dW_t Y_T =g(X_T,\mathcal{L}(X_T)),
S_{e f f}={\frac{1}{4 \pi}}\mathrm{I m}\, \left( \int d^{2}\theta d^{2}\bar{\theta}\Phi_{D}^{i}\overline{{\Phi}}_{i}+{\frac{1}{2}}\int d^{2}\theta \tau^{i j}W_{i}W_{j}\right) ,
X(a,\zeta)X(b,\delta) = X(ab,\zeta\delta) X(a,\zeta)Y(b,\delta) = Y(ab,\zeta\delta) Y(b,\delta)X(a,\zeta) = Y(ab,\zeta\delta) Y(a,\zeta)Y(b,\delta) = X(ab,\zeta\delta).
\begin{array}{l l}{{T_{N}^{\mu \nu}}}&{{= \displaystyle{\frac{\partial{\cal L}}{\partial A_{\lambda , \mu}^{a}}A_{\lambda}^{a , \nu}- \eta^{\mu \nu}{\cal L}}}}\\{{{}}}&{{{}}}\\{{{}}}&{{= - \Pi^{a \mu \lambda}A_{\lambda}^{a , \nu}- \eta^{\mu \nu}{\cal L}}}\\ \end{array}
\nabla_{P(p,1,1)}(z) = 1+z + \frac{p-1}{2}z = \frac{(p+1)}{2}z, \nabla_{P(p,1,r)}(z) = \nabla_{P(1,1,r)}(z) + \frac{(p-1)(r+1)}{4}z = 1+ \frac{(p+1)(r+1)}{4}z,
\mathrm{T r}e^{i F_{5 i}}- 2 \mathrm{T r}_{A}e^{i F_{5 i}}= m_{i}- 2 \mathrm{t r}F_{5 i}^{2}+ 2 \Big ( \mathrm{t r}F_{5 i}\Big )^{2}+{\frac{2}{3}}\mathrm{t r}F_{5 i}^{4}-{\frac{2}{3}}\mathrm{t r}F_{5 i}\mathrm{t r}F_{5 i}^{3}+ \dots
0\leq e^{-\psi(r)}u(r) \leq\frac{\max_{\lbrack0,r_{0}]}e^{-\psi}}{\min_{[0,r_{0}]}e^{-\psi}}\cdot\frac{1}{|H_{0}|}=e^{\sup_\Omega \psi-\inf_\Omega\psi}\frac{1}{|H_0|}.
\theta^{\alpha 1}= v_{\dot{A}}^{\alpha +}\theta_{\dot{A}}^{-}, \ \theta^{\alpha 2}= v_{A}^{\alpha -}\theta_{A}^{+}, \ \lambda_{+}^{\alpha}= v_{A}^{\alpha -}\lambda_{A}, \ \lambda_{-}^{\alpha}= v_{\dot{A}}^{\alpha +}\lambda_{\dot{A}}
\left(\sum_{k=1}^{n}\sum_{j=1}^{d_{k}}\lambda_{j}^{k}g_{\ell}^{i_{j}^{k}}\right)+g_{\ell}^{j_{0}}=\left(\sum_{j=1}^{d_{\ell}}\lambda_{j}^{\ell}g_{\ell}^{i_{j}^{\ell}}\right)+g_{\ell}^{j_{0}}=\left(\lambda^{\ell}-g_{\ell}^{j_{0}}\right)+g_{\ell}^{j_{0}}=\lambda^{\ell}\in\mathbb{Q}.
\frac{d \psi}{d I}=\frac{d \phi}{d \gamma}\,\frac{1}{2\,\sqrt{I-2}}= \frac{1}{\,\gamma}\,\frac{d \phi}{d \gamma}\qquad\frac{d^2 \psi}{d I^2}= \frac{1}{4\,\gamma^2}\,\frac{d^2 \phi}{d \gamma^2}- \frac{1}{4\,\gamma^3}\,\frac{d \phi}{d \gamma}\,,
(B_+-A_+)\alpha_{+j}= \frac{\nu_+ i\xi_j}{\mu_{+}(A_++B_+)}(i\xi'\cdot \alpha'_+ - A_+\alpha_{+N}), (B_+-A_+)\alpha_{+N}= -\frac{\nu_+A_+}{\mu_{+}(A_++B_+)}(i\xi'\cdot \alpha'_+ - A_+\alpha_{+N}), (B_--A)\alpha_{-j}= -\frac{i\xi_j}{A}(i\xi'\cdot\beta'_- + B_-\beta_{-N}),(B_- - A)\alpha_{-N}= -(i\xi'\cdot\beta'_- + B_-\beta_{-N}).
x_1^2 + x_2^2 + \cdots + x_i^2 = (\varpi_1 x_{2}- \varpi_{0}x_1) + (\varpi_2 x_3 - \varpi_1x_2) + \cdots + (\varpi_i x_{i+1}- \varpi_{i-1}x_i) = - \varpi_{0}x_1 + \varpi_i x_{i+1}= (x_1 + x_2 + \cdots + x_i) x_{i+1}
\le C_{p,\nu,\epsilon}N^{\frac{\kappa_p}{2}(\frac12-\frac1p)+\epsilon}\|(\prod_{i=1}^{10}\sum_{\atop{l(\Delta)=N^{-1/2}}}|E_{\Delta}g_i|^2)^{\frac1{20}}\|_{L^{p}(w_{B_R})}^{1-\kappa_p}(\prod_{i=1}^{10}\sum_{\atop{l(\tau)=N^{-1/4}}}\|E_{\tau}g_i\|_{L^{p}(w_{B_R})}^p)^{\frac{\kappa_p}{10p}}.
\exp\left(\frac{-1}{2}(y+r_N - Ny_{N,\vec{h}}(\sigma))^2\right)=(1+o(1)) \e^{- \frac{N}{2}\big(c_1-y_{N}(\sigma)\big)^2}e^{- (c_1-y_{N}(\sigma))z}\N^{c_1c_2 - c_2 y_N(\sigma)}\ .
\left(\begin{array}{c}\bar{x}_{3}\bar{x}_{4}\vdots \bar{x}_{n}\end{array}\right)= O(u_1^5)B^{-1}\left(\begin{array}{c}1 1 \vdots 1\end{array}\right)=O(u_1)\left(\begin{array}{c}1 1 \vdots 1\end{array}\right).
S_n^\diamond= \sum_{k=K_0}^{h_n-1}\sum_{j=1}^{q_k}B_{k,j}+ \sum_{j=1}^{\tau_n}B_{h_n, j},\mbox{where}\tau_n = \biggl\lfloor{{n-3^{h_n-1}}\over{3 m_{h_n}}}\biggr\rfloor- 2.\vadjust{\goodbreak}
\lim_{p\to+\infty}p \left(x^{(k, i)}_{p}- 1\right) < \lim_{p\to+\infty}p \left(x^{(k, i)}_{p -2}x^{(k, i)}_{p-1}x^{(k, i)}_{p}- 1\right) = \lim_{p\to+\infty}p \left(\frac{\sqrt{(i+pk-k)! (i+pk+k)!}}{(i+pk)!}\, x^{(k, i)}_{p-2}\right. &\left.- \sqrt{\frac{(i+pk-k)!}{(i+pk)!}}\, x^{(k, i)}_{p-2}x^{(k, i)}_{p}- 1\right) = \frac{k}{2}
f_{\mathbf{Z}|H_1}(\mathbf{z}|H_1) = \frac{e^{\frac{M (\sigma_{\rm w}^2+\sigma_{\rm a}^2)}{\zeta}}}{\pi^n}\prod_{m=1}^M \int_{\sigma_{\rm w}^2+\sigma_{\rm a}^2}^{\infty}\left(\frac{1}{v}\right)^{\frac{n}{M}}\qquad\qquad\qquad\qquad\cdot e^{-\frac{z_m}{v}}e^{-\frac{v}{\zeta}}dv.
P ( q , q^{\prime}) = \sum_{k = 0 , k^{\prime}= 0}^{\infty}\frac{1}{k ! k^{\prime}!}q^{n_{1}}. . . q^{n_{k^{\prime}}}q^{\prime}{}^{m_{1}}. . . q^{\prime}{}^{m_{k}}P_{\{m_{1}. . . m_{k}| n_{1}. . . n_{k^{\prime}}\}}( \partial_{l})
d\sqrt{n}2^{\sum_{l=1}^dJ_{n,l}\left(\frac{1}{2}-\frac{\alpha^{*}}{2d}\right)}\epsilon_n^{1/2}\lesssim\underbrace{n^{-\frac{2m(\alpha^{*})^2-d(m-1)\alpha^{*}-md^2}{2d(2\alpha^{*}+d)}}(\log{n})^{\kappa}}_{\tau_n(m)}\sqrt{n\log{n}},
A^{0}(U^{k})V^{k}_{t}+A^{i}(U^{k})\partial_{i}V^{k}- B^{ij}(U^{k})\partial_{i}\partial_{j}V^{k}+D(U^{k})V^{k}=F(U^{k};D_{x}U^{k}), \left.U^{k}\right\rvert_{t=0}= U_{0}
\dfrac{1}{E_{1}}=\dfrac{E_{25}^{5}}{E_{5}^{6}}\Bigg(\dfrac{1}{R(q^{5})^{4}}+\dfrac{q}{R(q^{5})^{3}}+\dfrac{2q^{2}}{R(q^{5})^{2}}+\dfrac{3q^{3}}{R(q^{5})}+5q^{4}\quad\quad\quad\quad-3q^{5}R(q^{5})+2q^{6}R(q^{5})^{2}-q^{7}R(q^{5})^{3}+q^{8}R(q^{5})^{4}\Bigg).
M_{T_{1}}\, N_{T_{2}}( t ) = \sum_{{v_{0}\in \Delta^{0}( t ) , \varepsilon \in \Delta^{1}( t ) \atop v_{0}\in R_{\varepsilon}( t ) , P_{\varepsilon}( t ) = T_{1}}}\ \delta_{R_{\varepsilon}( t ) \, \cup_{v_{0}}\, T_{2}}\, .
q^{-3}\nu_y^2\nu_x-(q+q^{-1})\nu_y\nu_x\nu_y + q^{3}\nu_x\nu_y^2 = (q^2-q^{-2})(q-q^{-1})\nu_y, q^{-3}\nu_z^2\nu_y-(q+q^{-1})\nu_z\nu_y\nu_z + q^{3}\nu_y\nu_z^2 = (q^2-q^{-2})(q-q^{-1})\nu_z, q^{-3}\nu_x^2\nu_z-(q+q^{-1})\nu_x\nu_z\nu_x + q^{3}\nu_z\nu_x^2 = (q^2-q^{-2})(q-q^{-1})\nu_x.
\alpha\beta+\beta\alpha=\begin{cases}-b/4, a,c\equiv0\mod 4, \beta-b/4, a\equiv3\mod 4c\equiv0\mod 4, \alpha-b/4, a\equiv0\mod 4c\equiv 3\mod 4, \alpha+\beta-(2+b)/4, a,c\equiv3\mod 4. \end{cases}
{\mathcal D}^{- 1}={\mathcal D}_{0}^{- 1}-{\mathcal D}_{0}^{- 1}{\mathcal V}{\mathcal D}_{0}^{- 1}+{\mathcal D}_{0}^{- 1}{\mathcal V}{\mathcal D}_{0}^{- 1}{\mathcal V}{\mathcal D}_{0}^{- 1}- . . .
\lim_{\delta\rightarrow 0}\frac{1}{\delta^{\frac{2}{n+1}}}\frac{\|y^{\delta}\|-\|y\|}{\|y\|}=n^{\frac{2}{n+1}}c(S^{\circ},n)\frac{\kappa(y)^{\frac{1}{n+1}}}{\langle y, u(y) \rangle}=n^{\frac{2}{n+1}}c(S^{\circ},n)\frac{\langle x, u(x)\rangle}{\kappa(x)^{\frac{1}{n+1}}}=\frac{\tilde{c}(S,n)}{G(x)}\quad.
\mathbf{E}w_1^3 w_2^3 = 9 \left( \mathbf{E}w_1^2 \right)\left( \mathbf{E}w_2^2 \right) \left( \mathbf{E}w_1 w_2 \right) + 6\left( \mathbf{E}w_1 w_2 \right)^3.
Z_{cat}(\mu(M),t) = \prod_{n=1}^{\infty}\left( \frac{1}{1-t^n}\right)^{\mu(M)}= \prod_{n=1}^{\infty}\mu \left( \frac{1}{1-t^n}\right)^{\mu(M)}= \prod_{n=1}^{\infty}\mu \left( \left(\frac{1}{1-t^n}\right)^{[M]}\right) = \prod_{n = 1}^{\infty}\mu(Z_{mot}(M,t^n)).
\max_{y \in [-H, H)}\ \Bigl\vert\sum_{n=N-H}^{N+y}e^{-n/N}\Bigl(R(n) - (2\psi(n)-n)\Bigr)\Bigl(1- \frac{\vert n- N \vert}{H}\Bigr)\Bigr\vert\ll N (\log N )^2 \log (2H)
Z = \int \exp \{- \frac{1}{2 \pi}\sum_{\alpha}\int_{- \frac{\pi}{b}}^{\frac{\pi}{b}}d k | \tilde{\phi}_{\alpha}( k ) |^{2}\sin ( k b ) b^{- 1}\}d \tilde{\phi}_{\alpha}^{*}( k ) d \tilde{\phi}_{\alpha}( k )
(\varphi_j|\varphi_k)_{L^2(\mathrm{M}_0)}=\mu_j^{-1/2}\mu_k^{-1/2}(T\psi_j|T\psi_k)_{L^2(\mathrm{M})}&=\mu_j^{-1/2}\mu_k^{-1/2}(T^\ast T\psi_j|\psi_k)_{H_\Gamma^{3/2}(\partial \mathrm{M})}=\delta_{jk}.
\phi(t,x) = \e^{- i \psi_{\pm}}_{<0}(t, x, D) |D|^{-1}e^{\pm it|D|}e^{i \psi_{\pm}}_{<0}(D, y, 0) ( |D| \phi_0 \pm \phi_1) \ + \frac12 \int_{0}^t e^{- i \psi_{\pm}}_{<0}(t, x, D) |D|^{-1}e^{\pm i(t-s)|D|}e^{i \psi_{\pm}}_{<0}(D, y, s ) Q_{\pm}f(s) ds
W_{\alpha}= - i \lambda_{\alpha}+ \theta \cdot \sigma^{\mu}\cdot \bar{\theta}\, \partial_{\mu}\lambda_{\alpha}- \frac{i}{4}\theta^{2}\bar{\theta}^{2}\partial^{2}\lambda_{\alpha}+ \theta_{\alpha}D - \frac{i}{2}\theta^{2}\sigma^{\mu}{}_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\partial_{\mu}D
\triangle(\langle H, \xi\rangle)=\sum_{i,j,k}h_{ij}^{k^{*}}\langle x, e_{i}\rangle\langle x, e_{j}\rangle\xi^{k^{*}}+\langle H, \xi\rangle-\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}\xi^{k^{*}}(H-\xi)^{l^{*}}, &\mathcal{L}(\langle H, \xi\rangle)=\langle H, \xi\rangle-\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}\xi^{k^{*}}(H-\xi)^{l^{*}}.
g = \frac{\Lambda^{2}}{( 1 + r A x )^{2}}\biggl \{- F ( r ) \, d s^{2}+ F ( r )^{- 1}d r^{2}+ r^{2}\, G ( x )^{- 1}d x^{2}+ r^{2}\, G ( x ) \, \Lambda^{- 4}d \phi^{2}\biggr \}.
\big ( A , ~ B , ~ \psi , ~ F , ~ G \big ) = \big ( ~ Y^{2}, ~ Y^{1}, ~ \rho , ~ - 2 N + \widehat{\cal D}_{\dot{5}}Y^{3}, ~ + 2 W^{\dot{5}}+ 1 2 ( t^{1}Y^{2}- Y^{1}t^{2}) \big ) ,
R_{x}^{\alpha}f(x)\ =-\frac{1}{2\Gamma(2-\alpha)\cos\alpha\pi/2}\times\left[ \frac{d^{2}}{dx^{2}}\int_{-\infty}^{x}(x-x^{\prime})^{-\alpha+1}f(x^{\prime})dx^{\prime}+\frac{d^{2}}{dx^{2}}\int_{x}^{\infty}(x^{\prime}-x)^{-\alpha+1}f(x^{\prime})dx^{\prime}\right] ,1<\alpha<2.
\begin{aligned}I_2= -[(h+h_e\phi(y))\partial_x+g\partial_y]D^\alpha u &+[D^\alpha,[(u+u_e\phi(y)+u_b(1-\phi(y)))\partial_x +v\partial_y ]h-[D^\alpha,(h+h_e\phi)\partial_x+g\partial_y ]u &+D^\alpha (-g(u_e-u_b)\phi'(y)+vh_e\phi'(y)) =: I^1_2+I^2_2+I^3_2.\end{aligned}
\begin{array}{r c l}{\overline{{D_{m}( k_{1}+ p )}}}&{=}&{\displaystyle D_{m}( k_{1}) \left[ - 2 k_{1}\cdot p D_{m}( k_{1}) \right.}\\{}&{}&{\displaystyle \left. - p^{2}D_{m}( k_{1}) + 4 ( k_{1}\cdot p )^{2}D_{m}^{2}( k_{1}) + O ( p^{3}) \right] .}\\ \end{array}
\begin{array}{c}x^\mu (\sigma ,\tau )=x_R^\mu (\sigma ,\tau )+x_L^\mu (\sigma ,\tau ) x_{R,L}^\mu (\sigma ,\tau )=\frac 12x_0^\mu +\frac 12l^2p^\mu (\tau \mp\sigma )+\frac{il}2{\sum}\frac 1n\alpha_{n,\pm}^\mu e^{-2in(\tau \mp\sigma )}\end{array}
\frac{1}{2m}(\frac{dS_{r}}{dr})^{2}+U_{r}(r)+\frac{1}{2mr^{2}}[(\frac{dS_{\theta}}{d\theta})^{2}+2mU_{\theta}(\theta)]+\frac{1}{2mr^{2}sin^{2}\theta}[(\frac{dS_{\phi}}{d\phi})^{2}+2mU_{\phi}(\phi)]=E.
[\Delta^{2},i\Gamma_{\varphi_{R}}]=\Delta[\Delta,i\Gamma_{\varphi_{R}}]+[\Delta,i\Gamma_{\varphi_{R}}]\Delta=2\partial_{k}[\Delta,i\Gamma_{\varphi_{R}}]\partial_{k}+[\partial_{k},[\partial_{k},[\Delta,i\Gamma_{\varphi_{R}}]]].
\int_{\mathbb{R}^{d}}\|\mathbf{z}\|^{-d+\beta}\overline{\psi(\mathbf{z})}d\mathbf{z}= \int_{\mathbb{R}^{d}}\nu(\beta )\|\mathbf{z}\|^{-\beta}\overline{\mathcal{F}(\psi )(\mathbf{z})}d\mathbf{z},\forall \psi \in\mathcal{S}(\mathbb{R}^{d}),
\gcd(F_n,F_m)=F_{\gcd(n,m)}\mbox{for all $m,n\ge 0$, $(m,n)\ne(0,0)$}; & F_{n+m}=F_{n-1}F_m+F_{n}F_{m+1}\mbox{for all $m\ge 0$ and $n\ge 1$}; & F_n^2-F_{n-r}F_{n+r}=(-1)^{n-r}F_r^2\mbox{for all $n\ge r\ge 0$}\mbox{(Catalan identity)}; &F_n^2-F_{n-1}F_{n+1}=(-1)^{n-1}\mbox{for all $n\ge 1$}\mbox{(Cassini identity)}.
\int_{\mathbb R}\sum_j 2^{2j}\big\vert f(\cdot,y_2,y_3)\ast_1\psi^{(1)}_{j}(y_1)\big\vert^2 w_{y_2,y_3}(I_{y_1,j}) dy_1 &=\int_{\mathbb R}S^2_{\psi^{(1)}}\big(f(\cdot,y_2,y_3)\big)(y_1) w_{y_2,y_3}(y_1)dy_1,
\begin{aligned}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}\hat{q}_{\beta}(0,t) =-\lambda \hat{q}_{\beta}(0,t), \frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}\hat{q}_{\beta}(n,t) =-\lambda \hat{q}_{\beta}(n,t)-\frac{\lambda}{\ln p}\sum_{j=1}^{n}\frac{(1-p)^{j}}{j}\hat{q}_{\beta}(n-j,t),\n\ge 1,\end{aligned}
\sim \int \frac{d^{3}p}{( 2 \pi )^{3}}D_{0 0}( p^{0}= 0 , \vec{p}) e^{i \vec{p}\cdot \vec{R}}= \int \frac{d^{3}p}{( 2 \pi )^{3}}\frac{e^{i \vec{p}\cdot \vec{R}}}{\vec{p}^{2}+ m_{e l}^{2}}= \frac{e^{- m_{e l}R}}{4 \pi R}\; .
M_{ij}(h_i-h_j)^2+M_{ik}(h_i-h_k)^2+M_{il}(h_i-h_l)^2 +M_{jk}(h_j-h_l)^2+M_{jl}(h_j-h_l)^2+M_{kl}(h_k-h_l)^2 \geq M((h_i-h_k)^2+(h_i-h_l)^2+(h_j-h_k)^2+(h_j-h_l)^2 -(h_i-h_j)^2-(h_k-h_l)^2) =M(h_i^2+h_j^2+h_k^2+h_l^2+2h_ih_j+2h_kh_l-2h_ih_k-2h_ih_l-2h_jh_k-2h_jh_l) =M(h_i+h_j-h_k-h_l)^2\geq 0.
CC_{n}(\vec{r},\tau_{1...n-1}=0)=\prod_{j<l}^{n}U(\frac{\vec{r}_{j}-\vec{r}_{l}}{\sqrt{n}})\cdot\sum_{i=1}^{N}U^{n}(\vec{r}_{i}-\frac{\sum_{k}^{n}\vec{r}_{k}}{n})\epsilon_{i}^{n}w_{i}(0)
- 2 \bar{R}_{\mu \nu}F^{\rho \mu}F_{\rho}~^{\nu}- \frac 1 2 ( F_{\mu \nu}F^{\mu \nu})^{2}+ \frac 1 2 F^{\lambda \nu}F^{\mu \rho}F_{\lambda \mu}F_{\nu \rho}- 2 ( F_{\rho}~^{\nu}F^{\rho \mu})_{; \nu \mu}+ ( F_{\nu \rho}F^{\nu \rho})^{; \mu}~_{\mu}.
\sum_{j=0}^{10^J-1}(M_{t}^k)_{1,j}=\sum_{i_1,\dots,i_{k-1},j\in\{0,\dots,10^J-1\}}m_{1,i_1}m_{i_1,i_2}\cdots m_{i_{k-1},j}&=\sum_{\substack{a_1,\dots,a_k\in\{0,\dots,9\}a_{k+1}=\dots =a_{k+J}=0}}G(a_1,\dots,a_{J+1})^t\cdots G(a_k,\dots,a_{k+J})^t &\ge \sum_{a=0}^{10^k-1}F_Y\Bigl(\frac{a}{10^k}\Bigr)^t.
\Theta(h,k) =\max \left\{\begin{array}{cc}d(T_2h,T_2k),d(T_2h,T_1h),d(T_2k,T_1k), \frac{d(T_1h,T_2k)+d(T_1k,T_2h)}{2}, \frac{d(T_1h,T_2 h)d(T_1k,T_2k)}{1 + d(T_2 k, T_2 h)}, \frac{d(T_1 h, T_2 k)d(T_1 k, T_2h)}{1 + d(T_2 k, T_2 h)},\frac{d(T_1 h, T_2 k)d(T_1 k,T_2 h)}{1 + d(T_1h, T_1k)}\end{array}\right\},
\begin{aligned}\Vert \partial_x^{\tilde{\beta}+1}\partial_y^ju\cdot \partial_x^{\beta-\tilde{\beta}}\partial_y^{k-j-1}u\Vert_{L^2_y}\leq C\Vert (u+u_e\phi(y)+u_b(1-\phi(y)))^{\frac{1}{2}}\langle y\rangle^{l+k}D^\alpha u \Vert_{L^2_y}^2,\end{aligned}
\begin{pmatrix}\lambda_1 m_1 (\lambda_2-\lambda_1)X_{1 2}\cdots (\lambda_k-\lambda_1)X_{1 k}(\lambda_2-\lambda_1)X^*_{1 2}\lambda_2 m_2 \cdots (\lambda_k-\lambda_2) X_{2 k}(\lambda_k-\lambda_1)X^*_{1 k}(\lambda_k-\lambda_2)X^*_{2 k}\cdots \lambda_k m_k\end{pmatrix}.
[\psi_i^\epsilon(z),\psi_j^{\epsilon'}(w)]=0,\ \psi^\pm_{i,\mp b^\pm_i}\cdot (\psi^\pm_{i,\mp b^\pm_i})^{-1}= (\psi^\pm_{i,\mp b^\pm_i})^{-1}\cdot \psi^\pm_{i,\mp b^\pm_i}=1,
\frac{\partial \psi}{\partial t}= \left( \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) \, \frac{\partial}{\partial x}+ \left( \begin{matrix}{{0}}&{{- i}}\\{{- i}}&{{0}}\\ \end{matrix}\right) \right) \psi \ .
\int_{r_1}^{r_2}b_2(\xi)\,d\xi=\int_{r_1}^{r_2}(r_2-\xi)^2(\xi-r_1)^2\,d\xi=\int_0^{r}(r-\zeta)^2\zeta^2\,d\zeta=\frac{r^5}{30}
\sum_{a=1}^{m}\sum_{\sigma\in \mathrm{Im}(\phi_k)}\mathrm{Pf}_{\sigma}\big(G(\gamma_a,\gamma^a,\gamma_1,\dots,\gamma_{2l})\big)= \mathrm{Pf}\big(G(\gamma_1,\dots,\gamma_{2l})\big).
f(H_k)a_k^\dagger = a_k^\dagger f\left( H_k+\sum_{m=0}^{\lambda_k-1}\beta_{k,m}T_k^m \right), a_kf(H_k) = f\left( H_k + \sum_{m=0}^{\lambda_k-1}\beta_{k,m}T_k^m \right) a_k.
F_j(x_1,\ldots,x_n) = g_j(x_n) \prod_{k=\max\{j,1\}}^{n-1}\exp\bigg(i(-1)^{j}\frac{1}{k!}\binom{k}{j}N_k(x_n)\Big(\sum_{l=1}^{k}l x_l\Big)^k\bigg)
\begin{array}{ll}\theta\left(\alpha(x),[y,z] \right)+\theta\left(\alpha(y),[x,z] \right)+\theta\left(\alpha(z),[x,y] \right) +\rho(\alpha(x))\theta(y,z)+\rho(\alpha(y))\theta(x,z) +\rho(\alpha(z))\theta(x,y)=0\end{array}
\begin{array}{lll}A^{+3}_{0}|h,\ell^{+}_{3},\ell^{-}_{3},u \rangle = \ell^{+}_{3}|h,\ell^{+}_{3},\ell^{-}_{3},u \rangle \ \ A^{-3}_{0}|h,\ell^{+}_{3},\ell^{-}_{3},u \rangle = \ell^{-}_{3}|h,\ell^{+}_{3},\ell^{-}_{3},u \rangle\end{array}
X_1:=\bigcap\limits_{j=1}^r[e_j^*]^\bot~~\operatorname{and}~~X_2:=\operatorname{span}\{h_1,\ldots,h_r\}:=\Big\{\sum\limits_{j=1}^r\lambda_jh_j~|~\lambda_j\in\mathbb R\Big\},
\alpha_{2}= \frac{1}{2}\alpha_{a b}\eta^{a}\eta^{b}+ \alpha_{\; \; a b}^{\mu \nu}\eta^{a}\eta_{\mu \nu}^{b}+ \frac{1}{2}\alpha_{\; \; a b}^{\mu \nu \; \lambda \rho}\eta_{\mu \nu}^{a}\eta_{\lambda \rho}^{b},
\langle \Delta_r u, \varphi_k \rangle = \Delta_r \langle u, \varphi_k \rangle, \langle \Delta_\theta u, \varphi_k \rangle = \langle u, \Delta_\theta \varphi_k \rangle = -\lambda_k \langle u, \varphi_k \rangle, \langle \frac{\partial u}{\partial r}, \varphi_k \rangle = \frac{\partial \langle u, \varphi_k \rangle}{\partial r}.
y_i = G_i x_i + \frac{\beta_i}{M_i}+ \frac{(1-G_i^2)^{1/2}}{\tilde{\lambda}_i}\nu_i &= G_i \left( m_i + \frac{1}{\lambda_i}\xi_i \right) + (1-G_i) \tilde{m}_i + \frac{g_i^{1/2}}{\tilde{\lambda}_i}\nu_i &= \tilde{m}_i + G_i \hat{r}+ \frac{G_i}{\lambda_i}\xi_i + \frac{g_i^{1/2}}{\tilde{\lambda}_i}\nu_i
A_k P_k=( (P_k)_{1}\lambda_{1}\,\ldots \, (P_k)_{\frac{n}{k}}\lambda_{1}\, \ldots \, (P_k)_{\frac{(n-1)n}{k}}\lambda_{\frac{n}{2}}\, \ldots \,(P_k)_{n}\lambda_{\frac{n}{2}}).
\|\pi_{n-}^{\mu}-\pi_{n-}^{\gamma}\|_{TV}= &\frac{E^{\gamma}\left[\left.\left|E^{\gamma}[\frac{d\mu}{d\gamma}(X_{0})|Y_{[0,\infty)},X_{[n,\infty)}]-E^{\gamma}[\frac{d\mu}{d\gamma}(X_{0})|Y_{[0,n-1]}]\right|\right|Y_{[0,n-1]}\right]}{E^{\gamma}\left[\left.\frac{d\mu}{d\gamma}(X_{0})\right|Y_{[0,n-1]}\right]}
\|z(0)\|_X =\left(\sum_{k\in\mathbb{Z}}\left|a_k(0)-\tilde{a}_k(0)\right|^2\right)^{1/2}&\le\left(\sum_{|k|\le N}\left|a_k(0)-\tilde{a}_k(0)\right|^2\right)^{1/2}+\left(\sum_{|k|> N}|a_k(0)|^2\right)^{1/2}.
c_q \le q(c_{q-1}(q+1)+q+1)=q(q+1)(c_{q-1}+1) & \le 2q(q+1)c_{q-1}\le \ldots \le 2^{q-3}\cdot\frac{q!}{6}\cdot\frac{(q+1)!}{24}\cdot c_3.
\begin{aligned}-A_1^T\lambda^k+\beta^kA_1^T(A_1{x}_1^{k+1}+A_2\hat{x}_2^k-b) = -A_1^T\lambda^k+\gamma (1-\tau^k)\beta^kA_1^T(Ax^k-b) +\tau^k\beta^kA_1^T(A_1\bar{x}_1^{k+1}&+A_2\bar{x}_2^k-b)+(1-\gamma)(1-\tau^k)\beta^kA_1^T(Ax^k-b) = -A_1^T\widetilde{\lambda}^k+(1-\gamma)(1-\tau^k)\beta^kA_1^T(A\breve{x}^{k-1}-b),\end{aligned}
{}\sum_{i=1}^{p}\sum_{j=1}^{q}B_{ij}^{(u)}\lambda_{i}^{(u)}z_{j}^{(u)}\leq\sum_{i=1}^{p}\lambda_{i}^{(u)}c_{i}^{(u)}+\sum_{v=1}^{u-1}\sum_{i=1}^{p}\sum_{j=1}^{q}\left (t_{v}-t_{v-1}\right )K_{ij}^{(u,v)}\lambda_{i}^{(u)}z_{j}^{(v)}.
\int_{( \infty )}\frac{d^{D}p^{\mu}}{( 2 \pi )^{D}}\, e^{i \Delta x \cdot p}\int_{0}^{\infty}\frac{d \gamma}{2 \pi}\, e^{- \frac{1}{2}i \gamma ( p^{2}+ m^{2})}= \frac{1}{\pi}\int_{( \infty )}\frac{d^{D}p^{\mu}}{( 2 \pi )^{D}}\, e^{i \Delta x \cdot p}\, \frac{i}{p^{2}+ m^{2}- i \epsilon}\ ,
\begin{aligned}c_j u_{j-1}(\theta_i)+ a_j u_j(\theta_i)+ b_j u_{j+1}(\theta_i)=\theta_i u_j(\theta_i)~(1\leq j\leq D-1) &c_Du_{D-1}(\theta_i)+a_D u_{D}(\theta_i)=\theta_i u_D(\theta_i)\end{aligned}
gxg=y=gh x gh\implies \ (xg)= h(xg)h \implies \ (xg)^{-1}h^{-1}(xg) = h \implies \h = (xg)^{-1}h^{-1}(xg) = (xg)^{-1}[(xg)^{-1}h^{-1}(xg)]^{-1}(xg) = (xg)^{-2}h (xg)^2 \implies
\begin{array}{l c l}{\partial{\bar{\partial}}\phi_{1}}&{=}&{A e^{- \frac{1}{2 \theta}( \phi_{1}+ \frac{1}{2}\phi_{2})}}\\{\partial{\bar{\partial}}\phi_{2}}&{=}&{B e^{- \frac{1}{2 \theta}( \phi_{1}+ 2 \phi_{2})}}\\ \end{array}
\sum\limits_{n = 2}^K{\sum\limits_{k = 2,k \ne n}^K{{{\left[{\bf{G}}\right]}_{n1}}{{\left[{\bf{G}}\right]}_{1k}}{{\left[{\bf{G}}\right]}_{kn}}}}\approx \frac{{K - 2}}{{cM}}\sum\limits_{k = 2}^K{{{\left|{{{\left[{\bf{G}}\right]}_{1k}}}\right|}^2}},
\sum_{i=1}^Ni^{1+2\alpha+2p}\xi_i^2 = \sum_{i=1}^Ni^{1+2\alpha+2p}\cdot i^{-2p}f_{0,i}^2 = \sum_{i=1}^Ni^{1+2\alpha-2\beta}f_{0,i}^2i^{2\beta}\leq N^{(1+2\alpha-2\beta)\wedge 0}\|f_0\|^2_\beta.
F_k^{\mu, \sigma}(\mathbf{x},\mathbf{y}, \mathbf{s}):=\begin{bmatrix}(H+ \rho I )\mathbf{x}-A^T \mathbf{y}+\mathbf{g}-\rho{\mathbf{x}_k}- \begin{bmatrix}0 \mathbf{s}\end{bmatrix}A\mathbf{x}+\delta (\mathbf{y}-\mathbf{y}_k) -\mathbf{b}SX_{\mathcal{C}}\mathbf{e}-\sigma \mu \mathbf{e}\end{bmatrix}.
\begin{gathered}|f(x) - f(y)|^{r_1}\le \frac 2 N \|f\|_{V^{r_1}([s,t])}^{r_1}, |(f-g)(x) - (f-g)(y)|^{r_1}\le \frac 2 N \|f-g\|_{V^{r_1}([s,t])}^{r_1}.\end{gathered}
\begin{array}{ll}\alpha(t,y) := \frac{u^s_{yy}(t,y)}{u^s_y(t,y)}, \alpha_1(t,y) := \frac{u^s_{yt}(t,y)}{u^s_y(t,y)}, \alpha_2(t,y) := \frac{u^s_{yyt}(t,y)}{u^s_y(t,y)}.\end{array}
T_{n,i}^{\left(h,r\right)}\left(w|q\right)=\sum_{j_{1},\cdots,j_{r}=0}^{w-1}\left[j_{1}+\cdots+j_{r}\right]_{q}^{n-i}q^{\sum_{l=1}^{r}\left(i+h-l+1\right)j_{l}}.
\tilde{H}_{n,k,m}=\begin{pmatrix}\tilde{P}_{n,k}(0)a_0 \tilde{P}_{n,k}(1)a_{1}\ldots \tilde{P}_{n,k}(m)a_{m}\tilde{P}_{n,k}(1)a_{1}\tilde{P}_{n,k}(2)a_{2}\ldots \tilde{P}_{n,k}(m+1)a_{m+1}\ldots \tilde{P}_{n,k}(m)a_{m}\tilde{P}_{n,k}(m+1)a_{m+1}\ldots \tilde{P}_{n,k}(2m)a_{2m}\end{pmatrix}
{\cal A}= \frac 1{\sinh |v_1 - v_2|}\int_0^\infty d\tau \int_0^\infty dl^\prime \int \frac{d^2 \vec k_{T}}{(2 \pi)^2}e^{i \vec k\cdot \vec b}e^{- \frac{q^2}2 \tau}e^{- \frac{k^2}2 l^\prime}<e^{i p \cdot X}>_{osc}{\cal N}
\Delta u\wedge\Delta v_1\wedge\dots\wedge \Delta v_{m-1}\wedge\beta_n^{n-m}(\omega) =\int u\Delta v_1\wedge\dots\wedge \Delta v_{m-1}\wedge\beta_n^{n-m}\wedge\Delta \omega, \omega \in C_0^{\infty}(\Omega),
v^{(n)}_{m_{1}m_{2}\ldots m_{n}}= v^{(n)}_{m_{2}\ldots m_{n}m_{1}},\;\;\;\;\;\; (Tr \; V)^{\dagger}= Tr \; V, \;\;\;\;\;\;v^{(n) \; *}_{m_{1}\ldots m_{n}}= v^{(n)}_{m_{n}\ldots m_{1}}
\hat{\lambda}\equiv \hat{a}+ i e^{- \hat{\phi}}, \tilde{\hat{F}^{I}}\equiv e^{- \hat{\phi}}\{}^{\star}\hat{F}^{I}+ \hat{a}\hat{F}^{I}= \hat{\lambda}\hat{F}^{I \ +}+ \mathrm{c . c .},
a_j (\ell, n, \lambda) = \binom{j+n-1}{n-1}- \ell \left [ \binom{j + k - d+ n-1}{n-1}+ \binom{j - k + n-1}{n-1}\right ] & + \sum_{i=2}^\ell (-1)^i \binom{\ell}{i}\binom{j - i k + n-1}{n-1}
z_{1}^n q_1\cdots q_{n}=1, q_1 q_2 z_{n-1}^{2}-q_2 z_{n-2}=1, q_1 q_2 q_3 z_{n-2}^{3}-2 q_1 q_2 q_3 z_{n-1}z_{n-2}z_{n-3}+q_2 q_3 z_{n-3}^2+q_3 z_{n-4}=1.
\|u\|^2_E = \sum_{i=1}^d\||D_{q_i}|u\|_{L^2(\mathbb{R}^{2d})}^2+\||\partial_{q_{i}}V(q_{i})|u\|_{L^2(\mathbb{R}^{2d})}^{2}+\|u\|^2_{L^2(\mathbb{R}^{2d})}~.
S_{{\rm boson}}=\int d^{4}x \sqrt{-g}\left \{-\frac{1}{4}(F_{\mu \nu}^{a})^{2}+\frac{1}{2}\vert D_{\mu}\Phi^{a}\vert^{2}-\frac{\lambda}{2}\left (\vert \Phi^{a}\vert^{2}-v^{2}\right )^{2}\right \},
b_A(\emptyset) = b_2^{(0,01,01,0)}= a_2^0 + 2a_2^1 + a_2^{2''}= 6+2(9)+12=36, b_B(\emptyset) = b_1^{(0,0,0,0)}= a_1^0 = 2, b_C(\emptyset) = b_2^{(0,01,0,0)}= a_2^0 + a_2^1 = 6+9=15, b_D(\emptyset) = b_4^{(0,0,0,0)}= a_4^0 = 66,
S = - 2 \pi R T \int d X^{0}\left[ 1 -{\frac{1}{2}}\sum_{k = - \infty}^{\infty}\left({\frac{d X_{- k}^{I}}{d X^{0}}}{\frac{d X_{k}^{I}}{d X^{0}}}-{\frac{k^{2}}{R^{2}}}X_{- k}^{I}X_{k}^{I}\right) \right]
Z=\sum_{N=1}^\infty e^{-\beta \frac{N\hbar}{2R}}\int\left(\prod_{k,\lambda}d \Pi_\lambda^k\right)\left(\prod_{l,\mu}d \theta_\mu^l\right)<\Pi^k_\lambda \theta_\mu^a|e^{-\beta H\left(\Pi_\lambda^k,Y_\lambda^k,\theta_\mu^a\right)}|\Pi^k_\lambda \theta_\mu^a>
\mathbf{H}^{r}=\mathbf{A}(\boldsymbol{x}^{r})\mathbf{A}^{H}=\sum_{m=1}^{\widetilde{M}}x_{m}^{r}\boldsymbol{a}\left(\overline{\theta}_{m}\right)\boldsymbol{a}^{H}\left(\overline{\theta}_{m}\right), \mathbf{h}^{c}=\mathbf{A}\boldsymbol{x}^{c}=\sum_{m=1}^{\widetilde{M}}x_{m}^{c}\boldsymbol{a}\left(\overline{\theta}_{m}\right),
\varphi_j^{(n)}={e}_j^{(n)}-\frac{1}{n}\sum_{i=1}^{n}{e}_i^{(n)}, 1\leq{j}\leq{n}, \varphi_{n+1}^{(n)}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}{e}_i^{(n)},
\Phi_{-\frac 12+i\varepsilon ,m,\bar m}\approx \left( \frac 1{2i\varepsilon}+\frac \phi{\alpha_{+}}\right) \gamma^{-\frac 12+i\varepsilon -m}\bar\gamma^{-\frac 12+i\varepsilon -\bar m}e^{-\frac \phi{\alpha_{+}}}+
\begin{aligned}\bigg|\int \left(\bar{f}(x, y)-\tilde{f}(x,y)\right)g(x, y)dxdy\bigg| &\leq \left|\int\left(\bar{f}(x, y)-\bar{f}_n(x,y)\right)g(x, y)dxdy\right|+\left|\int\left(\bar{f}_n(x, y)-\tilde{f}(x,y)\right)g(x, y)dxdy\right|.\end{aligned}
\int_0^\infty\left\langle L_a\big(\widetilde v(t)-v(t)\big), \varphi\right\rangle\,dt =-a(u-\widetilde u)^\perp\cdot\int_{\mathbb R^2}\varphi(x)\,dx &=a(u-\widetilde u)\cdot\int_{\mathbb R^2}\varphi^\perp(x)\,dx &=-\langle u-\widetilde u, L_{-a}\varphi\rangle,
2\beta \langle -\Delta_2 u'(t),u(t)\rangle_2 \leq 4\beta ||u'(t)||_{1,2}||u(t)||_{1,2}&\leq \frac{4\beta}{2\epsilon G(t)^\alpha}||u'(t)||_{1,2}^2 + \frac{4\beta\epsilon G(t)^\alpha}{2}||u(t)||_{1,2}^2 &=2\beta\epsilon G(t)^{-\alpha}G'(t) + 2\beta\epsilon G(t)^\alpha||u(t)||_{1,2}^2.
- 2 \partial_{y}^{j}\left[ F \left( x \right) , \pi_{j i}\left( y \right) \right] + M \left[ F \left( x \right) , \Pi_{i}\left( y \right) \right] - \partial_{i}^{y}\left[ F \left( x \right) , \pi \left( y \right) \right] \approx 0 ,
\Leftrightarrow \bar{\xi}={\frac{1}{2^{\frac{1}{4}}}}\left( \begin{array}{l l}{0}{- \Psi^{T}}\end{array}\right) , \, \hspace{2 m m}\xi \equiv{\frac{1}{2^{\frac{1}{4}}}}\left( \begin{array}{l}{\Psi}{0}\end{array}\right)