harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{p}_j(x)=\theta(p_j(x)-\bar{u}_j)+\bar{u}_j, \theta=\min\Big\{\Big|\frac{M-\bar{u}_j^n}{M_j-\bar{u}_j^n}\Big|,\Big|\frac{m-\bar{u}_j^n}{m_j-\bar{u}_j^n}\Big|,1\Big\},
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{M}J_1^{(k)}|\leq C\big(h^{2r+2}+(\Delta t)^{4}\big)+ C\|\theta_{\Psi}^{M}\|_{\mathcal{L}^2}^{2}+C\Delta t \sum_{k=1}^{M-1}{\|\theta_{\Psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
s_\gamma(x)s_\delta(x)-1 =\frac{\cosh(x_\gamma)\cosh(x_\delta)-\sinh(x_\gamma)\sinh(x_\delta)}{\sinh(x_\gamma)\sinh(x_\delta)}=\frac{\cosh(x_\gamma-x_\delta)}{\sinh(x_\gamma)\sinh(x_\delta)}&=\cosh[i\pi(\gamma-\delta)]r_\gamma(x)r_\delta(x)=\cos[\pi(\gamma-\delta)]r_\gamma(x)r_\delta(x)
\bar{\beta}^2 = q_1^2+q_2^2\alpha^2+q_3^2\bar{\alpha}^2+q_4|\alpha|^4+q_5^2 +2(q_1q_2+q_3q_4\bar{\alpha}^2)\alpha+2(q_1q_3+q_2q_4\alpha^2)\bar{\alpha}+2(q_1q_4+q_2q_3)|\alpha|^2+ &\, +2q_1q_5\beta+2(q_2q_5\alpha+q_3q_5\bar{\alpha})\beta+2q_4q_5|\alpha|^2\beta,
P ( i \rightarrow f ) = \sum_{n_{1}, \cdots , n_{\ell}= 0}^{\infty}\, e^{- i m ( n_{1}r_{1}+ \cdots + n_{\ell}r_{\ell}+ D_{G}/ 2 ) ( t_{f}- t_{i})}\, < \psi_{f}| n_{1}, \cdots , n_{\ell}> < n_{1}, \cdots , n_{\ell}| \psi_{i}> \ \ \ .
y\in \widetilde{D}_{s,j}:= \left\{z:\; c\left( \frac{1}{2}+j\right) (t-s)^{1/2}<\left\vert_{\!_{\!_{\,}}}z-c\,(t-s)^{1/2}\right\vert\right. &\left.<c\left( 2+\frac{1}{2}+j\right)(t-s)^{1/2}\right\},
\Psi^R_{\delta_i}\equiv \Psi^R_{l_i},\Psi^R_{q_i}\qquad\Psi^R_{l_i}= \left( \begin{array}{c}\psi^R_{\nu_i}\psi^R_{e_i}\end{array}\right)\qquad\Psi^R_{q_i}= \left( \begin{array}{c}\psi^R_{u_i}\psi^R_{d_i}\end{array}\right)
\int_{\mathbb{R}^2}\partial_1^j\partial_2^l\psi_T=0 \quad\mbox{provided}\;(j,l)\not=(0,0), \int_{\mathbb{R}^2}x_1\partial_1^j\partial_2^l\psi_T=0 \quad\mbox{provided}\;(j,l)\not=(0,0),(1,0).
({\rm A}) \mathfrak{G}_{\sigma_{\lambda}}^{(\beta)}(X_n)=\operatorname{DET}\big|h_{\lambda_j +i,j}^{(\beta)}(X_n)\big|_{1 \le i,j \le n}={\operatorname{DET}\big| x_i^{\lambda_j +n-j}(1+\beta x_i)^{j-1}\big|_{1 \le i,j \le n}\over \prod\limits_{1 \le i < j \le n}(x_i-x_j)},
\gamma\mathcal{M}_{am-a}\mathcal{M}^{am-a}+(\gamma-\gamma^{-1})\sum_{e<m-a,e\neq a}\mathcal{M}_{m-ee}\mathcal{M}^{m-ee}+\beta_{am-a}^{am-a}\mathcal{M}_{m-aa}\mathcal{M}^{m-aa}=C.
\Delta P_{( A )}^{\mu}= - \frac{2 e^{4}}{3 m^{2}c^{5}}\frac{e^{2}}{A^{2}m^{2}}\int ( F_{\sigma \gamma}\dot{x}^{\gamma}) F_{\lambda}^{\sigma}F^{\lambda \beta}( F_{\beta \alpha}\dot{x}^{\alpha}) d x^{\mu}\,
E_{n}(f;[0,1]) \le M_u\omega_\varphi^u(f,1/n)_\infty \frac{M_u}{n^u}\sum_{l=0}^n (l+1)^{u-1}E_{l}(f;[0,1]) \ge \omega_\varphi^u(f,1/n)_\infty.
\begin{array}{c}\mathbf t^1=(t_1^1, \cdots, t^1_{3d+1})\in sym^{3d+1}(\mathbf P^1) \mathbf t^2=(t_1^2, \cdots, t^2_{2d+1})\in sym^{2d+1}(\mathbf P^1) \mathbf t^3=(t_1^3, \cdots, t^3_{2d+1})\in sym^{2d+1}(\mathbf P^1).\end{array}
\sum_{j = 1}^{k}{\|e_{{j}}\xi \|_p^p \leq \| \xi\|_p^p = 1}\,\,\,\, \,\,\,\, \sum_{j = 1}^{k}{\|e_{{j}}^* \eta \|_q^q \leq \| \eta\|_q^q =1}.
\frac{1}{2 \pi}\int_{0}^{\infty}d \omega^{\prime \prime}\tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}\tilde{\beta}_{\omega^{\prime \prime}\omega}^{R}= \langle a_{\omega}^{R \dag}a_{\omega}^{R}\rangle_{\mathrm{{i n}}}.
\mathrm{t r}\left[ P \, \Gamma^{, \rho \sigma}\right] = 2 i \delta^{\rho \sigma}t \, ( \mathrm{t r}P ) - i \frac{1}{k^{2}}P^{\rho \sigma}\left[ e^{k^{2}( 1 - \xi^{- 1}) t}- 1 \right] + i \frac{1}{k^{2}}P^{\rho \sigma}\left[ e^{k^{2}( - 1 + \xi^{- 1}) t}- 1 \right]
\hat{\omega}_{1 1}= \ln | k_{2}| , \quad \hat{\omega}_{2 2}= \ln | k_{1}| , \quad \hat{\omega}_{1 2}= \hat{\omega}_{2 1}= - \ln \left| \frac{( u_{1}- u_{2}) ( v_{1}- v_{2})}{( u_{1}- v_{2}) ( v_{1}- u_{2})}\right| .
\frac{(-q;q)_n^4 (-q;q)_{2n}^2 (q^6;q^6)_n^2}{(q^2;q^2)_n^2}\sum_{k=0}^{n}\frac{(q;q^2)_k^2(q;q^2)_{2k}}{(q^2;q^2)_{2k}(q^6;q^6)_k^2}[8k+1]q^{2k^2}.
\begin{array}{c}{h^{m ( s )}= \varphi^{m ( s )}+ \eta^{m ( 2 )}\chi^{m ( s - 2 )};{\varphi_{l}}^{l m ( s - 2 )}= 0 ,}\\{\epsilon^{m ( s - 1 )}= \varepsilon^{m ( s )}+ \eta^{m ( 2 )}\zeta^{m ( s - 3 )};{\varepsilon_{l}}^{l m ( s - 3 )}= 0 ,}\\ \end{array}
\mathbf{H}^{'}=\begin{bmatrix}0.40e^{i1.3972}1.12e^{i0.7737}0.43e^{i1.2874}0.84e^{i0.3067}1.24e^{-i0.9872}1.70e^{i0.9784}0.83e^{-i0.2156}0.67e^{-i1.6414}\end{bmatrix}.
\begin{aligned}G^{\alpha{\beta}}_0 U_{\alpha{\beta}}- \psi [u] - \varphi_t\, - G^{\alpha{\beta}}_0 U_{\alpha{\beta}}(x_0, t_0) + \psi [u] (x_0, t_0) + u_t (x_0, t_0) \geq \, G [U_{\alpha{\beta}}] - \psi [u] - u_t - m_R \geq 0\end{aligned}
A_{1}=\frac{1}{r_{\epsilon}}\left[\int_{\mathbb{T}}\frac{r(x+y)}{\sqrt{\lambda+\rho(x+y)}}\nabla\phi_{\epsilon}(y)dy\right]^{2}A_{2}=\frac{1}{r_{\epsilon}}\left[\int_{\mathbb{T}}\left(\frac{1}{\sqrt{\lambda+\rho_{\epsilon}(x)}}-\frac{1}{\sqrt{\lambda+\rho(x+y)}}\right)r(x+y)\nabla\phi_{\epsilon}(y)dy\right]^{2}.
F_{\beta_i(t)}\mid_{\alpha_{i,1}(t) \leftarrow \frac{\alpha_{i-1,2}(t) \beta_{i+1}(t) \beta_{i+1}(t+1)}{\alpha_{i,1}(t+1)}}=\frac{\left(\begin{aligned}\beta_{i-1}(t) \alpha_{i-1,2}(t) \beta_{i+1}(t) \beta_{i+1}(t+1) + +\alpha_{i,1}(t+1) \alpha_{i-1,1}(t) \beta_{i+1}(t) \end{aligned}\right)}{\alpha_{i,1}(t+1)}.
f( x, y ) = 4x^2 + 4y^2 + 2( \sqrt{5}- 1 ) x y - \tfrac{( 5 + \sqrt{5})}{2}g( x, y ) = 4x^2 + 4y^2 - 2( \sqrt{5}- 1 ) x y - \tfrac{( 5 + \sqrt{5})}{2}.
\begin{aligned}|\{u_{k}\geq 1 + \lambda^2 F \}\cap (B_3 \times (-2a,0))| \leq |\{u_{k}\geq 1 + F\}\cap (B_3 \times (-2a,0))|-\gamma \leq|\{u_{k-1}\geq 1+\lambda^2F\}\cap (B_3 \times (-2a,0))|-\gamma \leq |B_3\times[-2,0]| - k \cdot \gamma. \end{aligned}
\sum_{i=r+1}^{t}\prod_{j=r+1}^{i-1}(q-1-d_j-2j+2)\prod_{j=i+1}^{t}(q-1-d_j-2j+4) =\, \sum_{i=r+1}^{t}\prod_{j=r+1}^{i-1}(q-d_j-2j+1)\prod_{j=i+1}^{t}(q-d_j-2j+3).
|\Gamma_{0,0,0}\cap\Gamma_{1,0,0}|=(q-1)^3(q^2+q+1)(q+1)q^6 &|\Gamma_{0,0,0}\cap\Gamma_{1,1,0}|=(q-1)^3(q+1)^2q^6 &|\Gamma_{0,0,0}\cap\Gamma_{1,1,1}|=(q-1)^3(q^2+q+1)(q+1)q^6.
\frac{B_{n}}{n!}=\sum_{\pi\in\mathcal{C}\left(n\right)}\frac{\left(-1\right)^{\vert \pi \vert}}{\left(\pi+1\right)!}= \sum_{p=1}^{n}\binom{n+1}{p+1}\left(-1\right)^{p}\sum_{k_{1}+\dots+k_{p}=n}\frac{1}{\left(k_{1}+1\right)!\dots\left(k_{p}+1\right)!}
\mathbb E_x\big[(\mathbf1_{D^c}g)(X_{\tau_{D}})\mathbf1_{\{X_{\tau_D-}\in D\}}\mathbf1_{\{\tau_{V_n}=\tau_D\}}\big]\to \mathbb E_x\big[(\mathbf1_{D^c}g)(X_{\tau_{D}})\mathbf1_{\{X_{\tau_D-}\in D\}}\big]
S^\sqsubset=\mathop{\sqsupset}[S]=\{b\in B:\exists s\in S\ (s\sqsubset b)\}. \intertext{Likewise, the \emph{image}of any $T\subseteq B$ is the preimage of the opposite relation $\sqsupset$, i.e.}T^\sqsupset=\mathop{\sqsubset}[T]=\{a\in A:\exists t\in T\ (a\sqsubset t)\}.
\gamma B_{a}^{0 i}= \partial_{j}\eta_{a}^{( 2 ) i j}, \; \gamma B_{a}^{i j}= \eta_{a}^{( 1 ) i j}, \; \gamma H_{i}^{a}= C_{i}^{( 1 ) a}, \; \gamma H_{0}^{a}= - \partial^{i}C_{i}^{( 2 ) a},
v^- = f(v^+)= \begin{pmatrix}0\ \ -I_{l-h}\ \ 0 I_{h-1}\ \ 0\ \ 0 0\ \ 0 \ \ -I_{n-l+2}\end{pmatrix}v^+ +\begin{pmatrix}\beta_{l-h}0_{h-1}2\cdot\beta_{n-l+2}\end{pmatrix}
t_{\alpha}{}^{i}: = \; \stackrel{\mathrm{B}}{\Sigma}_{\alpha}{}^{\! i}( \stackrel{\mathrm{B}}{X}) \qquad \mathrm{a n d}\qquad \stackrel{\mathrm{B}}{\tau}_{\alpha \beta}{}^{i}= 0 ={\tau}_{\alpha \beta}{}^{\! i}- 2 \stackrel{\mathrm{B}}{X}_{[ \alpha \beta ]}{}^{\! i}
\mathcal{E}_{n}(s) =\sum_{i=1}^d \left| \frac{n_i \pi}{a_i}-\frac{(1-s)\pi}{2a_i}\right|^{2s}\!\!\!+\sum_{i=1}^{d}O(\frac{1}{n_i}), \, \, n\in{\mathbb Z_+^d}
\sup_{{\bar{\tau}}< t \le T}|{\bar{B}}^n_k(t) -{\bar{B}}^n_k({\bar{\tau}})| ={\bar{B}}^n_k(T) -{\bar{B}}^n_k({\bar{\tau}}) ={\bar{X}}^n_k({\bar{\tau}}) -{\bar{X}}^n_k(T) \le{\bar{X}}^n_k({\bar{\tau}}),
\boxtimes_{j \in J}K_j = \boxtimes_{i \in I}\boxtimes_{j \in J_i}K_j \xrightarrow{\ \boxtimes_{i \in I}\psi_i \}\boxtimes_{i \in I}\Delta^{(J_i)}_* L_i = \Delta^{(\pi)}_*\left(\boxtimes_{i \in I}L_i\right) \xrightarrow{\ \Delta^{(\pi)}(\varphi) \}\Delta^{(\pi)}_*\Delta^{(I)}_* M = \Delta^{(J)}_* M,
\| \varphi (f) - (\varphi^{\prime}(f) + \sigma (f)) \| \leq \| p\varphi (f)- p\varphi (f)p \| + \| (1-p) \varphi (f) - \sum_{i=1}^l f(z_i)p_i \| & < \frac{2\varepsilon}{3}+ \frac{\varepsilon}{3}= \varepsilon
g_{-}^{-1}(u) E^{(n)}_{jj}g_{-}(u) = g_{-}^{-1}(u) \left( \frac{\partial}{\partial u^{(n)}_j}g_{-}(u) \right)+\left( \frac{\partial}{\partial u^{(n)}_j}g_{+}(u) \right) g_{+}^{-1}(u) \, .
\Phi_2(t) \leq \frac{1}{2}|\sqrt{\rho}u_{tt}(t)|^2_{2}+C(1+|\nabla u_t|^2_{2})|\nabla u_t|^2_{2}C|\nabla u|^2_{\infty}+C(|P_{tt}|^2_2+|\rho_{tt}|^2_2+|H_{tt}|^2_2).
\Sigma_{\bar{Y}^j}= \mbox{cov}(Y_0^j + P^{'}(P P^{'})^{-1}\bar{V}^j , Y_0^k + P^{'}(P P^{'})^{-1}\bar{V}^k ) & = \Sigma_{\bar{V}^j}P^{'}(P P^{'})^{-1}(P P^{'})^{-1}P^{'},
\begin{aligned}\ d_0(\Psi)(X_0\xrightarrow{f}X_1\xrightarrow{g}X_2)=G(X_1\xrightarrow{g}X_2)\circ \Psi(X_0\xrightarrow{f}X_1) &d_1(\Psi)(X_0\xrightarrow{f}X_1\xrightarrow{g}X_2)=\Psi(X_0\xrightarrow{gf}X_2) &d_2(\Psi)(X_0\xrightarrow{f}X_1\xrightarrow{g}X_2)=\Psi(X_1\xrightarrow{g}X_2)\circ F(X_0\xrightarrow{f}X_1)\end{aligned}
\textbf{r}_k=A_k\{\textit{A}^2\textbf{r}_{k-2}+B_k \textit{A}\textbf{r}_{k-2}+C_k \textbf{r}_{k-2}+F_k \textit{A}\textbf{r}_{k-3}+G_k \textbf{r}_{k-3}\},
\frac{S O ( 4 , n_{A}n_{B})}{S O ( 4 ) \times S O ( n_{A}n_{B})}\to \frac{S O ( 2 , n_{1}n_{3}+ n_{2}n_{4})}{S O ( 2 ) \times S O ( n_{1}n_{3}+ n_{2}n_{4})}\times \frac{S O ( 2 , n_{1}n_{4}+ n_{2}n_{3})}{S O ( 2 ) \times S O ( n_{1}n_{4}+ n_{2}n_{3})}.
\begin{cases}u_t+uu_x+vu_y+wu_z+\hat{f}w-fv=-\frac{1}{\rho}P_x, v_t+uv_x+vv_y+wv_z+fu=-\frac{1}{\rho}P_y, w_t+uw_x+vw_y+ww_z-\hat{f}u=-\frac{1}{\rho}P_z-g. \end{cases}
\bar D_{\dot\gamma}\bar D_{\dot\alpha}\Phi^{i*}=J^{\bar\imath}\!_{j\dot\alpha}\!^\beta{}_{,\bar k}\bar D_{\dot\gamma}\bar\Phi^k D_{\beta}\Phi^j +2i J^{\bar\imath}\!_{j\dot\alpha}\!^\beta\delta_{\beta\dot\gamma}\partial_t\Phi^j\ .
\min_{x\in [-1/\delta,-\zeta]}a(x) = a\left(x_0\right)= \mu(c_A - \mu) - \mu^2 \frac{\delta^2(1-\mu)^2}{4\mu^2}&> \mu(c_A - \mu) - \frac{1}{4}(1 - \mu)^2\frac{2\mu}{1-\mu}&= \mu(c_A - 1/2 - \mu/2) &>(c_A - 1)\mu ,
PP^{(\ell,0)}(q) = \frac{1}{M_{\ell}(q)^2}\sum_{\mu, \atop \ell(\mu) \le \ell}(- q)^{|\mu|}q^{n(\mu)+n(\mu')}\big( \dim_{q}V_{\mu}^{{\mathfrak{gl}}(\ell)}\big)^2,
f_{F}^{( 1 )}( x ; \omega ) = \frac{1 + i \gamma^{1}}{2}f_{F}( x ; \omega ) = \frac{1}{2}\left( \begin{array}{c}{f_{F}^{+}( x ; \omega ) - f_{F}^{-}( x ; \omega )}\\{- f_{F}^{+}( x ; \omega ) + f_{F}^{-}( x ; \omega )}\\ \end{array}\right)
{\rm Tr}\ln \hat{\cal F}= \lim_{L \rightarrow \infty}\frac{i}{(4 \pi)^{\frac{d}{2}}}\int_{1/L^2}^{\infty}\frac{ds}{s^{\frac{d}{2}+1}}\int d^dx{\rm Tr}\sum_{n=0}^{\infty}(i s)^n \hat{a}_n(x) ,
\begin{array}{r c l}{{x^{i}x^{j}}}&{{=}}&{{q x^{j}x^{i}~ , ~ ~ \mathrm{f o r}~ i < j ~ ~ \mathrm{a n d}~ ~ i \not = - j ~ ,}}\\{{x^{i}x^{- i}}}&{{=}}&{{x^{- i}x^{i}+ \lambda q^{i - 3 / 2}L_{i - 1}}}\\{{}}&{{=}}&{{q^{- 2}x^{- i}x^{i}+ \lambda q^{i - 3 / 2}L_{i}~ , ~ ~ \mathrm{f o r}~ i > 0 ~ ,}}\\ \end{array}
a = a_{M}\left( \left[ A_{i}^{a}\right] , \left[ \partial_{i}B_{a}^{0 i}\right] , \left[ \pi \right] , \left[ \mathcal{P}_{\Gamma}\right] \right) e^{M}\left( \eta_{2 a}^{i}, \partial^{\left( j \right.}\eta_{2 a}^{\left. i \right)}\right) + \gamma_{I}b ,
\bar{\nabla}_{M}{\cal H}^{M P Q}- \bar{\nabla}_{M}h_{N}^{~ P}\bar{H}^{M N Q}+ \bar{\nabla}_{M}h_{N}^{~ Q}\bar{H}^{M N P}- ( \bar{\nabla}_{M}\hat{h}_{~ N}^{M}) \bar{H}^{N P Q}- h_{~ N}^{M}( \bar{\nabla}_{M}\bar{H}^{N P Q}) = 0 .
u^*_\varepsilon(t)-u_0^*(t)= G_\varepsilon(t)\left(\| B^*\widehat\varphi_0(t)\|_U\, B^* \widehat \varphi_\varepsilon(t) - \| B^*\widehat\varphi_\varepsilon(t)\|_U\, B^* \widehat\varphi_0(t)\right).
( \ln a_{1}^{2}e^{\phi})^{\prime}( \ln a_{2}^{2}e^{\phi})^{\prime}+ ( \ln a_{2}^{2}e^{\phi})^{\prime}( \ln a_{3}^{2}e^{\phi})^{\prime}+ ( \ln a_{3}^{2}e^{\phi})^{\prime}( \ln a_{1}^{2}e^{\phi})^{\prime}+ ( 2 \sum V_{i}- ( H^{\ast})^{2}) a^{6}e^{2 \phi}= \phi^{\prime 2},
{\hat M}=-\frac{1}{2\pi gP}\frac{1}{R^{\prime}}\frac{d}{dz}\frac{R{R^{\prime}}^2(1+D^2{R^{\prime}}^2)}{(1+{R^{\prime}}^2)^{3/2}}\frac{d}{dz}\frac{1}{R^{\prime}}
\lambda_{qrr}\lambda_{iij}\Big((1-\xi^2)\chi_{(k\,p)}(g_{ij}) +\xi\chi_{r}(g_{(i\,p)})\chi_{i}(g_{(k\,p)})\chi_{(i\,q)}(g_{r}) +\xi\chi_{(k\,p)}(g_{(j\,r)})\chi_{(j\,q)}(g_{(i\,r)})\Big)=\lambda_{qrr}\lambda_{iij}\chi_{(k\,p)}(g_{ij})(1-2\xi^2)
p(\boldsymbol{s},\boldsymbol{s}^{r},\boldsymbol{s}^{c},\boldsymbol{x}^{r},\boldsymbol{x}^{c})=p(\boldsymbol{s})\prod_{m}\mathop{p(s_{m}^{r}|s_{m})}\prod_{m}\mathop{p(s_{m}^{c}|s_{m})}\prod_{m}p(x_{m}^{r}|s_{m}^{r})p(x_{m}^{c}|s_{m}^{c}).
\begin{array}{rcl}{\sf \Omega}^1_D(U) := \left\{\beta\,:\, \beta\in{\sf \Omega}^{1}(U)\,:\,\beta(\xi)=0,\,\xi\in{\frak X}(\partial U)\,\right\}, \mathfrak{H}_D^1(U) := \mathfrak{H}^1(U)\cap{\sf \Omega}^k_D(U). \end{array}
J = [ \operatorname*{d e t}({\frac{\partial \bar{g}_{\mu \nu}}{\partial l_{i}}},{\frac{\partial \bar{g}_{\mu \nu}}{\partial l_{j}}}){\cal D}\mathrm{e t}^{\prime}( \bar{F}^{\dagger}\bar{F}) ]^{\frac{1}{2}}/ V \; ,
\Delta K_s ={}s(s-1)(16|X|^6-8|X|^2\langle J_ZX,X\rangle+|J_ZX|^2)K_{s-2}+s\left(4p|X|^2+8|X|^2+\frac{1}{2}\sum_{j=1}^p\langle J_{[X,S_j]}X,S_j\rangle\right)K_{s-1}
\big \{f , g \big \}= \Big ( \frac{\partial f}{\partial x_{i}}\frac{\partial g}{\partial p_{i}}- \frac{\partial g}{\partial x_{i}}\frac{\partial f}{\partial p_{i}}\Big ) + \theta \Big ( \frac{\partial f}{\partial x_{1}}\frac{\partial g}{\partial x_{2}}- \frac{\partial g}{\partial x_{1}}\frac{\partial f}{\partial x_{2}}\Big ) .
S_N = - \sum_{\xi=1}^{N-1}\mu_\xi [ (1+\gamma_\xi) \sigma_\xi^1 \sigma_{\xi+1}^1 + (1-\gamma_\xi) \sigma_\xi^2 \sigma_{\xi+1}^2] - \sum_{\xi=1}^N \nu_\xi \sigma_\xi^3
\varepsilon_{ab}=\left( \begin{array}{cc}0 1 -1 0 \end{array}\right) = \varepsilon_{\dot{a}\dot{b}},\quad \varepsilon^{a b}= - \varepsilon_{a b}, \quad \varepsilon^{\dot{a}\dot{b}}= - \varepsilon_{\dot{a}\dot{b}}.
+ \int d^{2}z \left( i \bar{c}^{a}D_{\bar{z}}^{a b}c^{b}+ A_{\bar{z}}^{a}\pi^{a}- A_{0}^{a}\Phi^{a}- i \bar{b}^{a}b^{a}- \epsilon^{a b c}A_{0}^{a}c^{b}\bar{b}^{c}\right) \Biggr ]
\hat{s}^{(K,J,\mathbf{R})}= \underset{s_\xi^{(K,J,\mathbf{R})}\in{S}^\mathcal{B}_{(K,J,\mathbf{R})}}{\operatorname{argmin}}\left[ -\frac{1}{n}\sum_{i=1}^n \ln\left\{s_\xi^{(K,J,\mathbf{R})}(\mathbf{y}_i|\mathbf{x}_i) \right\}\right].
\rho^{2}= 2 R \, \left( | \vec{x}_{1}- \vec{x}_{2}| + | \vec{x}_{2}- \vec{x}_{3}| + \cdots + | \vec{x}_{N - 1}- \vec{x}_{N}| + | \vec{x}_{N}- \vec{x}_{1}| \right) .
f_{(3,1,0,2)}=(1-q t) (1-q^{2}t^{2}) x_1 x_2 x_4\left({\rm Tr}[k^{2+2u}]{\rm Tr}[k^{1+u}]f_{(2,0,0,1)}\right. &+q^2 t\{\rm Tr}[a a^\dag k^{2+2u}]{\rm Tr}[k^{1+u}]f_{(0,0,2,1)}&+\left.{\rm Tr}[k^{2+2u}]{\rm Tr}[a a^\dag k^{1+u}]f_{(2,0,1,0)}+q^2 t^2\{\rm Tr}[a a^\dag k^{2+2u}]{\rm Tr}[a a^\dag k^{1+u}]f_{(1,0,2,0)}\right).
b\; =\cdots\big(\big((x_{i+1}\cdots x_{j})(x_{j+1}\cdots x_k)\big)(x_{k+1}\cdots x_l)\big)\cdots b' =\cdots\big((x_{i+1}\cdots x_{j})\big((x_{j+1}\cdots x_k)(x_{k+1}\cdots x_l)\big)\big)\cdots
{\cal L}= -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}+ ( D_{\mu}\phi_{a})^{*}D^{\mu}\phi_{a}- m^{2}\phi_{a}^{*}\phi_{a}-{\frac{i e}{2}}\epsilon^{\mu \nu}F_{\mu \nu}\phi_{a}^{*}( \sigma_{3})_{a b}\phi_{b},
A^n\sum_{r = 1}^n\frac{p_{nr}(K)}{(1-K)^{\rho(n,r)}}\theta_K^r + \sum_{k = 0}^{n-2}A^{n-k}\sum_{2a+3b = n-k}\frac{c_{ab}}{(1-K)^{a+b}}\left(A^k\sum_{r = 1}^k\frac{p_{kr}(K)}{(1-K)^{\rho(k,r)}}\theta_K^r\right)
|V_{Z_1}(t, x; \xi)|\leq \begin{cases}Ct^{(\gamma-1)\frac\alpha2}p(t, x-\xi)\,, \d=1\,; Ct^{\nu_0 \alpha-1}|x-\xi|^{-d+\gamma-\nu_1+2-\nu_0}p(t, x-\xi)\,, \d\geq 2 \end{cases}
{\mathbb{E}}(R_{2}^{\ast})=\frac{1}{2}\sum_{j,\ell=1}^{p}{\mathbb{E}}(g_{iu}^{(j)}g_{iu}^{(\ell)}){\mathbb{E}}\big (\partial_{j}\partial_{\ell}s_{iu}({\mathbf{0}})\big )\,.
1-\mu(r,0,\gamma A+(1-\gamma)B) =1-(p-q)(1-2\pi_0) &\sim1-(1-2/\sqrt{r})\bigg(1-\frac{2(1-1/\sqrt{r})}{2(1-1/\sqrt{r})+\sqrt{r}}\bigg) &\sim\frac{4}{\sqrt{r}},
x^1+ix^2=\frac{2(F-\bar{F}\xi^2)+2\xi(1+\xi\bar{\xi})r}{(1+\xi\bar{\xi})^2}x^3=\frac{- 2(F\bar{\xi}+\bar{F}\xi)+(1-\xi^2\bar{\xi}^2)r}{(1+\xi\bar{\xi})^2}.
|a_{\alpha}|\leq \left(\frac{\alpha!\Gamma(L)}{\Gamma(L+|\alpha|)}\right)^{1/2}\frac{\Gamma(n+|\alpha|)}{\Gamma(n)\alpha!r^{|\alpha|}}\left(\max_{\xi\in S}|\xi^{\alpha}|\right) \left(\max_{|z|=r}|f_L|\right)
q_{1t}= -\frac{2}{3}\frac{u_t}{u}q_1q_2 +\frac{1}{3}q^2_1+\frac{2}{3}\left(t - \frac{u^2_t}{u^2}\right) + \frac{2}{3}\left(\frac{u_t}{u}\right)_tq_2.
\begin{array}{ll}\tau\left(v^{1},\delta_{t}v^1\right)+ \tau\sum\limits^{\mathcal{N}}_{n=2}\left(v^{n-\frac{1}{2}},\delta_{t}v^{n}\right)\geq \frac{1}{2}\left(\left\|v^{\mathcal{N}}\right\|^{2}- \left\|v^{0}\right\|^{2}\right). \end{array}
p_{}\left( x_{1},\ldots,x_{l}\right) ={\displaystyle\prod\limits_{j=1}^{l}}p_{j}\left( x_{j}\right) \overset{}{\longrightarrow}p_{}^{\prime}\left( x_{1},\ldots,x_{l}\right) \neq{\displaystyle\prod\limits_{j=1}^{l}}p_{j}\left( x_{j}\right)
R_{\ \nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R + 2 l^{2}\left[ \tau_{\ \nu}^{\mu}+ \left( \alpha -{\frac{1}{4}}\right){\cal S}_{\ \nu}^{\mu}+{\frac{\beta}{3}}{\cal K}_{\ \nu}^{\mu}\right] ={\frac{\kappa^{2}}{l}}T_{\ \nu}^{\mu}\ .
\dot \mu_{k}(0)=-\frac{1}{2}\frac{\left< D_{\phi \phi}^2 F(0,\mu^*_{k})[\phi^*_{k}, \phi^*_{k}], \phi^*_{k}\right>_{L^2}}{\left< D_{\phi \mu}^2 F(0,\mu^*_{k})\phi^*_{k},\phi^*_{k}\right>_{L^2}},
\mathbf{u}^{s'}(A_i)= \mathbf{u}^{s}(X)+((A_i-X)^T \otimes I_2) \emph{Vec}\left( \nabla\mathbf{u}^{s}(X) \right) & + ((A_i-\widetilde{Y}_i)^T \otimes I_2) (M^s - I_4) \emph{Vec}\left( \nabla\mathbf{u}^{s}(X) \right)+ \mathbf{R}^s_{i}(X), ~ i\in \mathcal{I}^{s'}, ~ \forall X\in T_{\ast}^s,~ s=\pm,
E_n^{-1}(z) \frac{A_n^{(1)}(s) - A_n^{(1)}(0)}{s}E_n(z)= \frac{1}{2\pi i}\oint_{|t|= 2Cn^{-\frac{5}{2}}}\frac{E_n^{-1}(z) A_n^{(1)}(t) E_n(z)}{t(t-s)}dt
Z_{1 2}= \left( \begin{array}{c c c c}{{\alpha}}&{{\cdot}}&{{\cdot}}&{{\cdot}}\\{{\cdot}}&{{\gamma}}&{{\delta}}&{{\cdot}}\\{{\cdot}}&{{\mu}}&{{\beta}}&{{\cdot}}\\{{\cdot}}&{{\cdot}}&{{\cdot}}&{{\nu}}\\ \end{array}\right) \, .
z e_k = \sum_{j=0}^{k-2}g_{k,j}. y_k^j e_k &= \sum_{j=0}^{k-2}g_{k+1,j}. y_k^j e_k + \sum_{j=0}^{k-2}(-1)^{k-j}g_{k+1,k-1}E_{k-1-j}(x_1, \ldots, x_{k-1}). y_k^j e_k &= \sum_{j=0}^{k-1}g_{k+1,j}. y_k^j e_k.
\frac{I}{I_{0}}=\frac{(\alpha_{n}-\alpha_{p})e^{(\alpha_{n}-\alpha_{p})d}}{\alpha_{n}-\alpha_{p}e^{(\alpha_{n}-\alpha_{p})d}}\Rightarrow\frac{I}{I_{0}}\cong\frac{e^{\alpha_{n}d}}{1-(\alpha_{p}/\alpha_{n})e^{\alpha_{n}d}}
\mathcal{I}_{\mathrm{D}}(\delta) = \sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\inf_{\lambda \in \mathbb{R}_{+}^2}\left[\langle\lambda, \delta\rangle+\int_{\mathcal{V}}g_\lambda \, d \pi\right] = \inf_{\lambda \in \mathbb{R}_{+}^2}\sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\left[\langle\lambda, \delta\rangle+\int_{\mathcal{V}}g_\lambda \, d \pi\right].
\sum_{I\subset[N]}(-1)^{|I|}\tilde{\phi}(m+n+2\chi^I) =\sum_{I\subset[N]}(-1)^{|I|}\dot{\phi}\left(|m|+|n|+2|I|\right) &=\sum_{k=0}^N\tbinom{N}{k}(-1)^{k}\dot{\phi}\left(|m|+|n|+2k\right) &= \mathfrak{d}_2^N \dot{\phi}\left(|m|+|n|\right).
\frac{M_n(i,j)}{n^{\delta_{ij}+1/2}}= \sum_{l=0}^j\sum_{k=0}^i \frac{b_{k,l,n}^{(i,j)}}{n^{\delta_{ij}}}\left(\frac{N_n(k,l)-\nu_n(k,l)}{\sqrt{n}}\right) \sim \Gamma(1-\delta_{ij})\sum_{l=0}^j\sum_{k=0}^i\xi_{k,l}^{(i,j)}\left(\frac{N_n(k,l)-\nu_n(k,l)}{\sqrt{n}}\right)
\left( \partial_{\mu}1 \otimes 1 + \frac{1}{4}\epsilon_{\alpha \beta \delta}\omega_{\mu}^{\alpha \beta}1 \otimes \gamma^{\delta}+ \frac{i}{2}A_{\mu}^{c}\sigma_{c}\otimes 1 - \frac{1}{2 \ell}e_{\mu \alpha}1 \otimes \gamma^{\alpha}\right) \varepsilon_{r}= 0 ,
a_{n,(n-1)/2}=\frac{q^{\frac{n^2-4n+3}{4}}(q;q^2)_n}{(1-q^{n})(q^2;q^2)_{n-1}}{n-1\brack \frac{n-1}{2}}_{q^2}begin{align*}7pt] \equiv\frac{(-1)^{\frac{n-1}{2}}q^{-n+1}(q;q^2)_n (-q;q)_{n-1}^2}{(1-q^{n})(q^2;q^2)_{n-1}}\pmod{\Phi_n(q)^2}.
\psi_{\tau}= ( \psi_{\mu}^{a}\, , \, \psi , \, \psi^{a}) \quad , \quad \gamma_{\tau}= ( \gamma_{\mu}^{a}\, , \, \gamma , \, \gamma^{a}) \, , \, \, \xi = ( \gamma_{\tau}\, , \, \omega^{a}) .
\forall i\in\{1,...,n\}:u_{x}(i)^{-1}=(g_{\pi_{x}(i)}^{-1}xg_{i})^{-1}=g_{i}^{-1}x^{-1}g_{\pi_{x}(i)}=g_{i}^{-1}x^{-1}g_{\rho_{x^{-1}}(i)}=w_{x^{-1}}(i).
|E_\alpha(t)|\leq C\eta \sum_{|\alpha|\leq N-1}\|\partial^{\alpha}(\widetilde{\rho},\widetilde{u},\widetilde{\theta})\|^2+C_{\eta}\varepsilon^2\sum_{|\alpha|=N}(\|\partial^{\alpha}(\widetilde{u},\widetilde{\theta})\|^2+\|\partial^{\alpha}f\|^2)+C_{\eta}\varepsilon\varepsilon^{2-2a}.
T_{w(1,m)}(E_{k,1}(s+1)E_{N,k}(-s)) \qquad{}= (-1)^{m-1}\times\begin{cases}E_{1,1}(s+1)E_{2,1}(-s+m-1) , E_{3,1}(s-m+2)E_{2,3}(-s+2m-2) , E_{k+1,1}(s+1)E_{2,k+1}(-s+m-1) , E_{2,1}(s+m)\Big( E_{2,2}(-s) + \delta_{s,0}(m-1)c \Big) .\end{cases}
c_{2}( V^{1}) - \frac{1}{2}c_{2}( T X ) = ( \eta^{( 1 )}- 6 c_{1}( B ) ) \sigma - \frac{1}{2}\left( 1 7 2 - \frac{1 5}{\lambda^{( 1 )}}\left( \lambda^{( 1 ) 2}- \frac{1}{4}\right) \right) F ,
\begin{aligned}\left| H[b_0]-H[b_1]-H[b_2]\right| \lesssim \left(\max_{0\leq i\leq 2}||b_i||_{L^{2}(\mathbb{R}^{3})}^{5/3}\right) ||b_0-b_1-b_2||_{L^{2}(\mathbb{R}^{3})}, \end{aligned}
A = \frac{1}{N^2}\sum_{k =1}^{\infty}\frac{2}{\pi^2 k^2}\left| \sum_{n=1}^{N}{e^{2 \pi i k x_n}}\right|^4 - \frac{1}{N}\sum_{k \in \mathbb{N}\setminus (2\mathbb{N})}\frac{8}{k^2 \pi^2}\left| \sum_{n=1}^{N}{e^{2 \pi i k x_n}}\right|^2 + 1.
\rho_E^{\mu\oplus\nu+t\tilde\mu\oplus\tilde\nu}=\Phi_2^{\mu\oplus\nu+t\tilde\mu\oplus\tilde\nu}( \rho_{0}(\gamma) z),e)\rho_E^{0\oplus0}(\gamma) \Phi_2^{\mu\oplus\nu}(z,e)^{-1}=\Phi_2^{\mu\oplus\nu+t\tilde\mu\oplus\tilde\nu}((\Phi_1^{\mu})^{-1}( \rho_{\mu}(\gamma) z),e)\rho_E^{0\oplus0}(\gamma) \cdot \Phi_2^{\mu\oplus\nu}((\Phi_1^{\mu})^{-1}(z),e)^{-1}.
M'_{k, \ell}= \lambda_{q+1}^{k - (m+1)q - r_\ell + f}\sum_{0 \leq s_1, \dots, s_q \leq m+1}\left(\prod_{i=1}^q\binom{m+1}{s_i}\right)\left(\prod_{i=1}^q(-\lambda_i)^{s_i}\right) & \times \lambda_{q+1}^{(m+1)q-\sum_{i=1}^q s_i}\sum_{s_{q+1}= 0}^r(-1)^{s_{q+1}}\binom{r}{s_{q+1}}\binom{(k - \sum_{i=1}^q s_i + f) - s_{q+1}}{r_\ell}.
\Upsilon^{(\alpha)}(z) = \begin{pmatrix}1 0 \displaystyle \frac{i}{8}(4\alpha^2+3) 1\end{pmatrix}\pi^{\sigma_{3}/2}\begin{pmatrix}\displaystyle I_{\alpha}(z^{1/2}) \displaystyle \frac{i}{\pi}K_{\alpha}(z^{1/2}) begin{align*}0.2cm]\displaystyle \pi i z^{1/2}I_{\alpha}^{\prime}(z^{1/2}) \displaystyle -z^{1/2}K_{\alpha}^{\prime}(z^{1/2})\end{pmatrix}H_{0}(z),
\left(L_0^kf\right)(y) =\int_0^\pi \sum_{n=0}^{2B(\alpha)-1}K_{E,\alpha}^{k}(y,x+n\pi) f(x)dx &=\sum_{n=0}^{2B(\alpha)-1}\int_0^\pi K_{E,\alpha}^{k}(y,x+n\pi)\tilde f(x+n\pi)dx &=\int_{\mathbb{T}_\alpha}K_{E,\alpha}^{k}(y,x)\tilde f(x)dx.
H_{\mu_\lambda}\big(\xi| \pi_{\lambda}^{-1}(\mathcal R_n(\lambda))\big) = H_{\mu_\lambda}(\xi \vee \pi_{\lambda}^{-1}\mathcal R_n(\lambda)) - H_{\mu_\lambda}(\pi_{\lambda}^{-1}(\mathcal R_n(\lambda)) \le \log k
\frac{d}{dt}\left( e^{-(\gamma_* +\mathbf{i}\beta_i)t}\left(\xi_{i,1}(t)+\mathbf{i}\xi_{i,2}(t) \right)\right)= e^{-(\gamma_* +\mathbf{i}\beta_i)t}\left(\mathcal{E}_{i,1}(t)+\mathbf{i}\mathcal{E}_{i,2}(t) \right) .
\langle Z_n,\sum_{i=1}^m \lambda_i u_i \rangle = \sum_{i=1}^m \lambda_i \langle Z_n,u_i \rangle\ sum_{i=1}^m \lambda_i \langle Z ,u_i \rangle = \langle Z, \sum_{i=1}^m \lambda_i u_i \rangle,
f_{Y}(y)=\frac{\delta^{\sigma^{-1}}}{a\chi(\sigma^{-1})}I^{m,n}_{p,q}\Bigg[\frac{\delta^{\sigma^{-1}}y}{a}\left|\begin{matrix}(a_i,\sigma^{-1}A_i,\alpha_i)_{1,p}(b_j,\sigma^{-1}B_j,\beta_j)_{1,q}\end{matrix}\right.\Bigg],\ \y>0.
S ={\frac{T r_{0}^{D}}{2^{D}}}\int \sqrt{g}\; \left( ( 1 - q ) \phi^{{\frac{2 D}{D - 2}}}+{\frac{8}{( D - 2 )^{2}}}[ ( \partial \phi )^{2}+{\frac{D - 2}{4 ( D - 1 )}}\phi^{2}R ] +{\cal O}( \phi^{{\frac{2 ( D - 4 )}{D - 2}}}) \right) .
R = - \frac{1}{2 Q^{2}}\left[ \frac{3 5}{1 6}+ Q^{2}\frac{\Lambda_{7}}{M_{7}^{5}}+ \frac{7}{8}\sqrt{( \frac{5}{2})^{2}- 8 Q^{2}\frac{\Lambda_{7}}{M_{7}^{5}}}\right] ~ .
\beta_{00}(t)=\frac{\pi g{\rm e}^{-\pi g t/4}}{16\tilde{\omega}\tilde{\tilde{\omega}}}\left[\left(2\pi g-i\tilde{\tilde{\omega}}\right){\rm e}^{i\tilde{\tilde{\omega}}t}-\left(2\pi g-i\tilde{\tilde{\omega}}\right){\rm e}^{-i\tilde{\tilde{\omega}}t}\right]\;.
g\left(\begin{array}{cc}n_1 n_2 m_1 m_2 \end{array}\right)=\frac{G}{2^{(n_1+m_1+n_2+m_2)/2}}\sqrt{\frac{\Gamma(n_1+m_1+1)\Gamma(n_2+m_2+1)}{\Gamma(n_1+1)\Gamma(m_1+1)\Gamma(n_2+1)\Gamma(m_2+1)}}.
\left\vert \frac{1}{N}\sum_{j=1}^{N}\sum_{m\in\mathbb{Z}^{2}}h\left( t\left(j\right) +m\right) -\int_{\mathbb{R}^{2}}h\left( t\right) ~dt\right\vert\leq V(f)\D\left( \left\{t(j)\right\}_{j=1}^{N}\right) \;.