harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta_{I}\mathcal{P}_{2 i}^{a}= - \frac{1}{2}\epsilon_{0 i j k}F^{j k a}+ \partial_{i}\pi^{a}, \delta_{I}\mathcal{P}_{1}^{a}= - \pi^{a}+ \pi^{( 1 ) a}, \delta_{I}\mathcal{P}_{2}^{a}= \pi^{( 2 ) a},
\pi_{i}= \frac{\partial L}{\partial v^{i}}= \frac{{m}\left[ v^{i}- u^{i}u^{0}( 1 + u^{0}\vec{u}\vec{v}) \right]}{\sqrt{( 1 + u^{0}\vec{u}\vec{v})^{2}- ( \vec{v})^{2}}}= - m \omega_{i}
A = \begin{bmatrix}p^3 + 1 p 2 p^4 p^2 \end{bmatrix}= \begin{bmatrix}1 0 -p^4 1 \end{bmatrix}\begin{bmatrix}1 - p^2 + p^5 \end{bmatrix}\begin{bmatrix}1 p -p^2 -1 - p^3 \end{bmatrix}
c_{X,n}^{(i,j)}(t) = \sum_{\pi \in T_n^{(i,j)}}\mathrm{sgn}(\pi) ~ t^{\delta(\pi)/2}\prod_{\mathclap{\substack{k\in [n] k+\pi(k) \;}}}o_{1,X,k+\pi(k)}(t) \prod_{\mathclap{\substack{r \in [n], ~ r\neq i r+\pi(r) \; \;}}}e_{1,X,r+\pi(r)}(t),
J_n \left( \frac{r_1 r_2}{t}\right)\sim\sqrt{\frac{2 t}{\pi r_1 r_2}}\exp\left[- i \left(\frac{r_1 r_2}{t}+ \frac{n^2 t}{2 r_1 r_2}- \frac{n \pi}{2}\right)\right].
\begin{array}{rcl}\|e_{i+1}\| = \|\tilde{x}_{i}-\tilde{x}_{i+1}+e_{i+1-(m+1)}\|\leq\|\tilde{x}_{i}-\tilde{x}_{i+1}\|+\|e_{i+1-(m+1)}\| \leq \|\tilde{x}_{i}-\tilde{x}_{i+1}\|+\underset{k=1}{\overset{i+1-(m+1)}{\sum}}\|\tilde{x}_{k-1}-\tilde{x}_{k}\|+M_{1}\leq\underset{k=1}{\overset{i+1}{\sum}}\|\tilde{x}_{k-1}-\tilde{x}_{k}\|+M_{1},\end{array}
\begin{gathered}\int_0^\infty C_1(u) x^{-2\alpha}\int_0^x \rho^{\nu+1}(\frac{q^2\rho^2}{x^2}; q^2)_{-\alpha}J_\nu (u\rho; q^2) \,d_q \rho\, d_q u = \int_0^x g_1 (\rho) x^{-2\alpha}\rho^{\nu+1}(\frac{q^2\rho^2}{x^2}; q^2)_{-\alpha}\, d_q\rho.\end{gathered}
\rho(l_0,l,d) = \lim_{\tau\to 0}{\displaystyle{\left({\partial \over \partial \tau}\right)^n \left[ N_{l_0,l}(d) G_l/l \right]}\over \displaystyle{\left({\partial \over \partial \tau}\right)^n \left[ G_{l_0}/l_0 \right]}}\ .
\sum_{t=1}^{\infty}\frac{g(t)\Big(\sum_{s=1}^{t}g(s)f(s)\Big)^p}{\Big(\sum_{s=1}^{t}g(s)\Big)^\gamma}\leq\bigg(\frac{p}{\alpha-1}\bigg)^p\sum_{t=1}^{\infty}\frac{g(t)f^p(t)\Big(\sum_{s=1}^{t}g(s)\Big)^{\gamma(p-1)}}{\Big(\sum_{s=1}^{t-1}g(s)\Big)^{p(\gamma-1)}},
C=x_{0}^{2}+y_{0}^{2}+\frac{a^{2}}{b^{2}}z_{0}^{2}-a^{2}=p^{2}C_{1}^{2}+\frac{a^{2}}{b^{2}}z_{0}^{2}-a^{2}=\frac{1}{p^{2}}(d-nz_{0})^{2}+\frac{a^{2}}{b^{2}}z_{0}^{2}-a^{2}
+ \frac{1}{2}F_{ab}\xi^a \partial_{\tau}\xi^b-\frac{1}{4}D_a D_b F_{cd}\xi^a\xi^b \psi^c \psi^d + \sum_{n=3}^{\infty}\left( \frac{1}{n!}D_{a_1}\ldots D_{a_{n-1}}F_{{a_n}b}\xi^{a_1}\ldots \xi^{a_n}\partial_{\tau}\overline{X}^b\right.
S = -{\frac{1}{2 g_{Y M}^{2}}}\int d^{n}x \left( ( \partial \phi_{1})^{2}+ ( \partial \phi_{2})^{2}+{\frac{( n - 2 )}{4 ( n - 1 )}}\tilde{R}( \phi_{1}^{2}+ \phi_{2}^{2}) \right) \ ,
{{\mathbf{\ddot F}}_k}=\left[{{\mathbf{U}}_k^{[ 1 ]}{\mathbf{P}}_k^{[ 1]}, \ldots ,{\mathbf{U}}_k^{[ J]}{\mathbf{P}}_k^{[ J]}}\right]\in\mathbb{C}^{M\times c_\Sigma}
C_{p,0}^{(\ell)}=\sum_{j=1}^{p}(-1)^{j-1}\sum_{\beta=0}^{p-j}2^{p-j-\beta}\binom{n-\ell}{\beta}\binom{\ell}{p-j-\beta}\binom{\beta}{p-j}&=\sum_{j=1}^{p}(-1)^{j-1}\binom{n-\ell}{p-j}=(-1)^{p-1}\sum_{j=0}^{p-1}(-1)^{j}\binom{n-\ell}{j}=\binom{n-\ell-1}{p-1}=0
\boldsymbol{\varepsilon}^{\perp}(\boldsymbol{W}_{\mathcal{T}})\subset\boldsymbol{\Sigma}_{\mathcal{T}}^{\textsf{HHJ}}\;\;\textrm{and}\;\; b_{\mathcal{T}}(\boldsymbol{\varepsilon}^{\perp}(\boldsymbol{\varphi}), v)=0 \forall~\boldsymbol{\varphi}\in\boldsymbol{W}_{\mathcal{T}}, v\in V_{\mathcal{T}}.
\mathbb{H}^{r}= \begin{cases}H_{0}^{r}(\Omega), 0 \leq r<1 / 2, H_{00}^{1 / 2}(\Omega) \varsubsetneqq H_{0}^{1 / 2}(\Omega), r=1 / 2, H_{0}^{r}(\Omega), 1 / 2<r \leq 1, H_{0}^{1}(\Omega) \cap H^{r}(\Omega), 1<r \leq 2,\end{cases}
\eqref{eq-to}\times ( v/mj ) < (mju^4v - m^2ju^mv^2 + mju^2v^m - mu^2v + 2uv^2)a^2 & + (mju^2v - m^2juv^2 + mjv^m - 2u + mv)u^2a^2 &- 2(mju^mv - m^2ju^2v^2 + mjuv^m - u^2 + v^2)ua^2 =0.
\frac{\epsilon_2(s)}{\zeta(s)}=\frac{\left(1-p_2^{-s}\right)^{-1}\left(1-p_3^{-s}\right)^{-1}\left(1-p_4^{-s}\right)^{-1}\textnormal{...}}{\left(1-p_1^{-s}\right)^{-1}\left(1-p_2^{-s}\right)^{-1}\left(1-p_3^{-s}\right)^{-1}\textnormal{...}}
\dot{r}^{2}+ \tilde{V}( r ) = 0 , \; \dot{t}= \frac{- E}{M - \frac{J^{2}}{4 r^{2}}- \frac{r^{2}}{l^{2}}}, \; \dot{f}= \frac{E}{\frac{l}{r^{2}}( M - \frac{J^{2}}{4 r^{2}}) - \frac{1}{l}}.
\frac{\partial^2 p}{\partial t^2}- c_1c_2\frac{\partial^2 p}{\partial x^2}+(c_1-c_2) \frac{\partial^2 p}{\partial x\partial t}= -\frac{2\alpha}{t}\Bigl[ \frac{\partial p}{\partial t}+\frac{(c_1-c_2)}{2}\frac{\partial p}{\partial x}\Bigr]
\begin{aligned}F_{p,s}(a+b+c) \ge t(F_{p,r}(a+b),F_{r,s}(c))\ge \ge t(t(F_{p,q}(a),F_{q,r}(b)),F_{r,s}(c))= =t_*(F_{p,q}(a),F_{q,r}(b),F_{r,s}(c)).\end{aligned}
\rho^2 \asymp \begin{cases}n^{-\frac{2s}{2s+1/2}}, b \geq n^{\frac{1}{2s+1/2}}, \left({\sqrt{b}n}\right)^{-\frac{2s}{2s+1}}, n^{\frac{1}{2s+1/2}}/m^{\frac{2s+1}{2s+1/2}}\leq b < n^{\frac{1}{2s+1/2}}, (n/\sqrt{m})^{-\frac{2s}{2s+1/2}}, b < n^{\frac{1}{2s+1/2}}/m^{\frac{2s+1}{2s+1/2}}. \end{cases}
\kappa^{n-2}_{01}=-(n-2)!\,\check{\chi}_{H_{n}\cup H_-}\,\displaystyle\sum_{j=1}^{n-1}(-1)^{j}\varphi_{j}\,\bar{\partial}\varphi_1 \wedge \dots \wedge\widehat{\bar{\partial}\varphi_j}\wedge\cdots\wedge\bar{\partial}\varphi_{n-1}.
\|\partial_ru\|^2_{L^2(S^1)}\le\|f\|^2_{L^2(S^1)}+\|\partial_r(dist_N(u))\|^2_{L^2(S^1)}\le\|f\|^2_{L^2(S^1)}+C\delta\|\nabla u\|^2_{H^{1/2}(B)}.
\, =\, \Big \lvert \mathbb E^{1,2}\big [ \widetilde{b}( v, x, \omega_{1}) - \widetilde{b}( v, x, \omega_{2}) \big ] \vert_{\{x = X_{v}^{\mathrm m_{2}}\}}\Big \rvert \, \le \, C_{T}\mathbb E^{1,2}\big [ \lvert \omega_{1,v}- \omega_{2,v}\rvert \wedge 1 \big] \, ,
u_j\left(A_j\left(p_j,\frac{\epsilon}{2}\right)\right) = \dfrac{r_j^{n-2}G\left( X_0,A\left(\widetilde p_j,\frac{\epsilon}{2}r_j \right)\right)}{\omega(B(q_j,r_j))}& \geq C \dfrac{r_j^{n-2}G\left( X_0,A(q_j,r_j) \right)}{\omega(B(q_j,r_j))}= Cu_j(A_j(0,1)) \geq C' >0,
( D_y(\varphi(\cdot)D\psi(\cdot))(x),\,z)=\, D_y\varphi(x)\, D_z\psi(x)+\varphi(x)\,D^2\psi(x)(y,z) =\, \partial_y\varphi(x)\, \partial_z\psi(x)+\varphi(x)\,D^2\psi(x)(y,z).
\beta_{1}= \frac{1}{2}\left\langle \sum_{X_{i}\in ({\it G}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}< j_{\alpha_{1}}< i_{\alpha_{2}}< j_{\alpha_{2}}}{\left\{i_{\alpha_{1}}, j_{\alpha_{1}}, i_{\alpha_{2}}, j_{\alpha_{2}}\right\}\in J_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf 1}_{( w ( t_{i_{\alpha_{1}}}) = w ( t_{j_{\alpha_{1}}}) )}\right. \times
\mathcal{H}^r_{\alpha}\left(I\right)=\{u\in L^2(I);~~~ x^{(\alpha+k+1/2)}\left(\frac{1}{x}\frac{d}{dx}\right)^{(k)}\left(x^{-(\alpha+1/2)}u\right)\in L^2(I)~\,,\forall k\leq r\}.
\Psi_k^\prime := \bigl \{(\bar{A}_{t+\tau}, Z_{t}, W_{t}) \in \Psi_{t,t+\tau}^{A_t;Z_t,W_t}~:~ \hat{d}((\bar{A}_{t+\tau}, Z_{t}, W_{t}), (\bar{A}_{t+\tau}^k, \bar{Z}_{t}^{k}, \bar{W}_{t}^{k})) < \epsilon \bigr \}.
{\cal S}^{U ( q )}= \frac{1}{g_{S Y M}^{2}}\int d t \int d^{3}\sigma^{\prime}~ \sqrt{\det ( G^{i j})}~ \mathrm{t r}_{q}\left( \frac{1}{2}G_{i j}^{\prime}{\cal F^{\prime}}^{0 i}{\cal F^{\prime}}^{0 j}- \right.
\hat{d}_{z,y,x}= \mathbf{W}_{z,y,x}^T \mathbf{U}_{z,y,x}= \mathbf{W}_{z,y,x}^T \left[\begin{array}{c}4s_{z,y-1,x}-\sigma_{z,y,x}4s_{z,y,x-1}-\sigma_{z,y,x}4s_{z,y-1,x-1}-\sigma_{z,y,x}4s_{z-1,y,x}-\sigma_{z-1,y,x}\vdots 4s_{z-P,y,x}-\sigma_{z-P,y,x}\end{array}\right]
(1-|y^{-1}x\cdot o|^2)^{(s-\epsilon)/2}= |d_{y^{-1}x}|^{-(s-\epsilon)}= (1-|x\cdot o|^2)^{(s-\epsilon)/2}\, (1-|y\cdot o|^2)^{(s-\epsilon)/2}(1-(x\cdot o,y\cdot o))^{-(s-\epsilon)}.
L'(t) =\frac{cG_{3}}{p}G_{1}f(x,y,v)v-\frac{acG_{3}}{p}\varphi_{1}(y)-cG_{3}\varphi_{1}(y)\varphi_{2}(z)+cG_{3}\varphi_{1}(y)\varphi_{2}(z)-b\varphi_{2}(z) &\leq \frac{cG_{3}G_{1}}{p}M_{3}-\frac{acG_{3}}{p}k_{1}y-bk_{2}z &\leq \frac{cG_{3}G_{1}}{p}M_{3}-\tilde{\mu}\left(\frac{cG_{3}}{p}y+z \right) &=\frac{cG_{3}G_{1}}{p}M_{3}-\tilde{\mu}L(t),
\int_0^t t^{\frac{\beta}{\alpha}}(t-\tau)^{-1+\frac{\beta}{\alpha}}\tau^{-\frac{2\beta}{\alpha}}\, d\tau = \int_0^1 (1-\tau')^{-1+\frac{\beta}{\alpha}}(\tau')^{-1+(1-\frac{2\beta}{\alpha})}\, d\tau' \lesssim \ 1
\mathcal{S (}\Sigma \mathcal{) =}\int d^{4}x \frac{\delta \Sigma}{\delta \varphi^{i}}\frac{\delta \Sigma}{\delta \varphi^{* i}}\; + \; \frac 1 2 C_{\; A B}^{C}\varepsilon^{A}\varepsilon^{B}\frac{\partial \Sigma}{\partial \varepsilon^{C}}\; ,
N=\prod_{\substack{p|W p\nmid D\Delta_L}}(p-\omega(p)-1)\prod_{\substack{p|W p|D\Delta_L}}(p-\omega(p))\prod_{\substack{p|V/W p\nmid Da_0}}(p-1)\prod_{\substack{p|V/W p|Da_0}}p.
\hat{z}_{k}=\frac{l_{k}^{z}}{l_{1}^{z}}\frac{\sum\limits_{i=1}^m \sum\limits_{j=1}^n{e_{ij1}x_{i}y_{j}}}{\sum\limits_{i=1}^m \sum\limits_{j=1}^n{e_{ijk}x_{i}y_{j}}}.
T_n^*(\mathcal F_2^c)=\min\left\{\left[\frac{1}{\theta_{1,K_2^c}}\sqrt{\frac{a_n\theta_{2,K_2^c}}{\nu^*}}-\frac{\theta_{2,K_2^c}}{\theta_{1,K_2^c}}\right]^+,1\right\}, \n\in \mathcal F_2^c,
\langle\psi,D^2\psi\rangle= \langle\psi,D(\mu|\psi|^2\psi)\rangle =\mu\underbrace{\langle\psi,(\nabla|\psi|^2)\cdot\psi\rangle}_{=0}+\mu\langle\psi,|\psi|^2D\psi\rangle =\mu^2|\psi|^6,
\Bigg\|\sum_{i=1}^n a^j_i e_i + x' + \sum_{i=1}^{m}y_i\Bigg\| \geq \theta \Bigg(\|x'\| + \sum_{i=1}^{m}\|y_i\|\Bigg) > \theta \big(\|x'\| + m - m\varepsilon\big);
v_i(\tau, \xi) = \int_0^\tau \sum_{j=1}^n \Big( C_{ij}(x_i(s, \tau, \xi)) v_j(s, x_i(s, \tau, \xi)) + D_{ij}(x_i(s, \tau, \xi)) v_j(s, 0) begin{align*}6pt] + f_i\big(s, x_i(s, \tau, \xi) \big) \Big) \, ds + v_{0, i}\big(x_i(0, \tau, \xi) \big),
\Bigl(\mathbb E \Bigl\| \sum^n_{j=1}\varepsilon_j d_j\Bigr\|^p\Bigr )^{\frac 1p}\leq C \Bigl(\mathbb E \Bigl \| \sum^n_{j=1}d_j\Bigr\|^p\Bigr )^{\frac 1p}.
\aligned \sum_{i=1}^{n-1}\langle i_{e_i}R^{\nabla},i_{e_i}R^{\nabla}\rangle+\langle i_{\frac \partial{\partial r}}R^{\nabla},i_{\frac \partial{\partial r}}R^{\nabla}\rangle = \sum_{1\le j_1\le n}\sum_{i=1}^{n}\langle R^\nabla (e_i,e_{j_1}), R^\nabla (e_i,e_{j_1})\rangle &= 2 ||R^\nabla||^2\, ,\endaligned
-\frac{2}{(1-f)^{2}}\left\{f^{2}_{ij}+\frac{2f_{i}f_{ij}f_{j}}{(1-f)}+\frac{f^{2}_{i}f^{2}_{j}}{(1-f)^{2}}\right\}=-\frac{2}{(1-f)^{2}}\Big(f_{ij}+\frac{f_{i}f_{j}}{1-f}\Big)^{2}.
W ( \lambda_{i}) = \frac{\eta}{\lambda_{i}}+ \frac{\sqrt{a \lambda_{i}+ \eta^{2}}}{\lambda_{i}}+ \frac{a}{2 N}\sum_{j}\frac{1}{\sqrt{a \lambda_{j}+ \eta^{2}}}\, \frac{1}{\sqrt{a \lambda_{j}+ \eta^{2}}+ \sqrt{\vphantom{\lambda_{j}}a \lambda_{i}+ \eta^{2}}}.
\begin{aligned}\bigg\|\sum_{j=1}^n L_j^2 f \bigg\|_{2}^2=\sum_{k = 1}^\infty\mu_k^2\bigg\| \frac{P_k f}{|x|^2}\bigg\|_{2}^2, \sum_{j=1}^n \bigg\|\frac{L_j f}{|x|}\bigg\|_{2}^2 =\sum_{k = 1}^\infty\mu_k\bigg\| \frac{P_k f}{|x|^2}\bigg\|_{2}^2.\end{aligned}
\zeta_{{\mathrm{R o b i n}}, u = 1}^{\prime}( 0 ) = -{\frac{4 1}{8 6 4}}-{\frac{7}{4 5}}\ln ( 2 ) -{\frac{1}{2}}\ln ( \pi ) -{\frac{3 1}{9 0}}\ln ( a ) +{\frac{1}{3}}\zeta_{R}^{\prime}( - 3 ) +{\frac{1}{2}}\zeta_{R}^{\prime}( - 2 ) -{\frac{1 1}{6}}\zeta_{R}^{\prime}( - 1 ) .
\varphi(x;\tau) =72g_{2}x^{2}(6x^{2}-g_{2})+216g_{3}x(4x^{2}-g_{2})-g_{2}^{3}-108g_{3}^{2}& \geq72g_{2}\times \frac{g_{2}}{4}\times \frac{g_{2}}{2}-g_{2}^{3}-108g_{3}^{2}& =8g_{2}^{3}-108g_{3}^{2}>0,
\frac{c_6}{q_{\sigma,t}}\left((c_2 \ln_2 t - c_1 \ln_3 t)^{-2\rho_\sigma-1}\right)<\frac{c_7}{q_{\sigma,t}}(\ln_2 t)^{-2\rho_\sigma-1}= c_8\frac{(\ln_2 t)^{\mu-1-2\rho_\sigma-1}}{\ln t}
\left[\operatorname{Ext}\theta^{(\lambda')}_I\right](x):= \,\frac{[\ell(I)]^{\frac12}}{\varphi(-k_0)}\left[\theta^{(\lambda')}\otimes\varphi\right]_{Q(I,k_0)}(x) = \,\frac{[\ell(I)]^{\frac12}}{\varphi(-k_0)}\theta^{((\lambda',0))}_{Q(I,k_0)}(x) = \,\frac{1}{\varphi(-k_0)}\theta^{(\lambda')}_I(x')\varphi\left(\frac{x_n}{\ell(I)}-k_0 \right),
H_{1 / 2}^{R p e r t}\equiv H_{\beta}- H_{1 / 2}+ ( \beta \! - \! 1 / 2 ){\frac{\delta ( x )}{x}}= ( \beta \! - \! 1 / 2 ){\frac{\delta ( x )}{x}}+{\frac{( \beta \! - \! 1 / 2 )^{2}}{x^{2}}}~ .
\bigl[\widehat{T}(w_1,w_2)\bigr](\theta x) = \bigl\langle \bigl[T(\theta(x\otimes w_1))\bigr](\cdotp), w_2\bigr\rangle & = \bigl\langle \theta(\cdotp)\,\bigl[T(x\otimes w_1)\bigr](\cdotp), w_2\bigr\rangle & = \theta\bigl[\widehat{T}(w_1,w_2)\bigr](x),
{D}_{-1}= \{\left( p-1, (p-1)^2, \cdots , (p-1)^l, [\xi_1(p)], \cdots , [\xi_k (p)] \right) | \, p \in\mathcal{P}\},{D}_{1}= \{\left( p+1, (p+1)^2, \cdots , (p+1)^l, [\xi_1(p)], \cdots , [\xi_k (p)] \right) | \, p \in\mathcal{P}\}.
\int_0^L \int_0^L |r(t_1,t_2)|^2dt_1dt_2= \int_0^L \int_0^L\frac{1}{N^2}\sum_{(\mu,\mu')\in\mathcal{E}^2}e^{2\pi i(t_1-t_2)\langle\mu-\mu',\alpha\rangle}dt_1dt_2= \frac{1}{N^2}\sum_{(\mu,\mu')\in\mathcal{E}^2}\left|\int_0^L e^{2\pi it\langle\mu-\mu',\alpha\rangle}dt\right|^2,
\partial_j\Big[\big(Q^{(r)}_{\alpha\gamma}\big)_t(x')\Big]=\big(\partial_j\partial_r E_{\gamma\beta}\big)(x',t)=\big(\partial_r\partial_j E_{\gamma\beta}\big)(x',t)=\partial_r\Big[\big(Q^{(j)}_{\alpha\gamma}\big)_t(x')\Big],
{\cal H}_G[\gamma, -i\hbar(\delta/\delta \gamma)]\Psi = (-\hbar^2/2)\frac{\delta}{\delta \gamma_{ij}}G_{ijkl}[\gamma]\frac{\delta}{\delta \gamma_{kl}}\Psi - 2\,{}^{(3)}\!R[\gamma](\det\gamma)^{1/2}\Psi = 0 ,
<\bar{0}|\eta_{\hat{\alpha}\hat{\beta}}:\bar{T}^{\hat{\alpha}\hat{\beta}}:|_{P_0}|\bar{0}> = -<\bar{0}|:\bar{T}^{\hat{1}\hat{1}}:|_{P_0}|\bar{0}> = -\frac{1}{\sinh^2(\chi)}<\bar{0}|:T^{\hat{0}\hat{0}}:|_{P_0}|\bar{0}>
d s^{2}= - d t^{2}+ k [ d x^{2}+ \frac{s i n^{2}x}{c o s^{2}x + R^{2}s i n^{2}x}d \theta_{1}^{2}+ \frac{R^{2}c o s^{2}x}{c o s^{2}x + R^{2}s i n^{2}x}d \theta_{2}^{2}]
\chi_\gamma(a,\alpha,z) = \sum_{b\in A}\langle \pi_\gamma(a,\alpha,z) \delta_b, \delta_b \rangle begin{align*}5 pt] = \sum_{b\in A}[ \pi_\gamma(a,\alpha,z) \delta_b ](b) begin{align*}5 pt] = \sum_{b\in A}\gamma(z \langle b-\tfrac{1}{2}a, \alpha \rangle ) \delta_b(b-a).
L_{2k}=-\frac{2y_{1}y_{2}}{\zeta}\cdot[R_{k}+\frac{y_{1}^{2}+y_{2}^{2}}{\zeta y_{2}}P_{k}-\frac{(y_{1}^{2}-y_{2}^{2})}{2y_{1}y_{2}}\cdot \frac{\partial_{2}P_{k}}{2k}-\frac{\partial_{1}P_{k}}{2k}+\frac{y_{1}^{2}+y_{2}^{2}}{y_{1}y_{2}^{2}}\cdot\frac{P_{k}}{4k}] ,
\widehat{{\cal M}}^{( B = 0 )}= \hat{V}^{( B = 0 )}( x , t ) ( \hat{V}^{B = 0 )})^{T}( x ,{\frac{1}{t}}) = \left( \begin{array}{c c}{{\cal M}( \omega )}&{0}\\{0}&{{\cal M}^{- 1}( \omega )}\\ \end{array}\right)
\bar{\phi}^{\bar{i}}{''}+ F(r) \bar{\phi}^{\bar{i}}{'}= - \bar{\Gamma}^{\bar{i}}_{\bar{j}\bar{k}}\bar{\phi}^{\bar{j}}{'}\bar{\phi}^{\bar{k}}{'}+ g^{k\bar{i}}\left(G(r)\partial_{k}V_{BH}+ H(r)\partial_{k}V\right) \:,
a''(n;i) =\sum_{j=i+1}^na(n;i,j)=b(n;i)+b'(n;i)+\sum_{j=i+1}^{n-2}a(n-1;i,j), a''(n-1;i) =\sum_{j=i+1}^{n-1}a(n-1;i,j)=b'(n-1;i)+\sum_{j=i+1}^{n-2}a(n-1;i,j),
\sum_{i=1}^N |r_i| =\sum_{i=1}^N \sqrt{\omega_i}\left( |r_i|/\sqrt{\omega_i}\right) \le\left( \sum_{i=1}^N \omega_i \right)^{1/2}\left( \sum_{i=1}^N r_i^2/\omega_i \right)^{1/2}\le 2
g(A_{ntX}X,Z) =-g(\widetilde\nabla_{Z}nX, tX)-g(\widetilde\nabla_{X}\varphi Z-\varphi\widetilde\nabla_{X}Z, tX) = g(\widetilde\nabla_{Z}tX, nX)-g(\widetilde\nabla_{X}\varphi Z, tX)-g(\widetilde\nabla_{X}Z, \varphi tX).
- \partial_{\beta}\partial^{\beta}F + F^{\prime \prime}+ \frac{1}{r}F^{\prime}+ \left[ \big ( A_{\beta}+ \partial_{\beta}\vartheta \big )^{2}+ \frac 1 2 \big ( 1 - F^{2}\big ) - \left( \frac{\chi^{2}}{r^{2}}+ \vartheta^{\prime}{}^{2}\right) \right] F = 0 ,
{\bf E}\bigl[\beta^q_{2, 2}(\delta)\bigr] = \exp\Bigl(\frac{1}{4}\int\limits_0^\infty (e^{-tq}-1) e^{-(\delta-1/2)t}\,{\bf E}\bigl[e^{-t^2\,C_2/32}\bigr] \frac{dt}{t}\Bigr).
e^{- 2 \phi}= \left| \frac{Q_{1}\sqrt{c_{2}}}{Q_{2}\sqrt{c_{1}}}\frac{\sinh \left[ \sqrt{c_{1}}( I ( A ) + \tilde{c}_{1}) \right]}{\sinh \left[ \sqrt{c_{2}}( I ( A ) + \tilde{c}_{2}) \right]}\right| e^{\sqrt{2}( f_{0}I ( A ) + f_{1})}.
\int_{\mathbb{R}^m}u^2dV_g=\int_{\mathbb{R}^m}u^2 e^{-\frac{|x|^2}{4(m-2)}m}dV_{g_0}; &\int_{\mathbb{R}^m}|\nabla_g u|_g dV_g = \int_{\mathbb{R}^m}|\nabla_{g_0}u|_{g_0}^2 e^{-\frac{|x|^2}{4}}dV_{g_0}.
\varphi^{\prime}( x ) = \left( \begin{array}{c}{\chi^{\prime}( x )}\\{\xi^{\prime}( x )}\\ \end{array}\right) , \beta_{\mu}^{\prime}= - \left( \begin{array}{c c}{0}&{\alpha_{\mu}^{\prime}}\\{\overline{{\alpha}}_{\mu}^{\prime}}&{0}\\ \end{array}\right) .
\quad\{c_{2,\tau}(\widetilde{X}_{1,2})(\|X_{1}-X_{2}\|_{2}^{2}-p\tau)^{2}\}=\{c_{2,\tau}(\widetilde{Y}_{1,2})(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau)^{2}\}=\{c_{2,\tau}(\widetilde{Z}_{1,1})(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau)^{2}\}=Q_{2}.
g_{n,k}(r)=A\rho^{\gamma}e^{-\rho/2}(Z\alpha\rho L_{n-|k|-1}^{(2\gamma+1)}(\rho)+(\gamma-k)\frac{\gamma\mu c^{2}-kE}{\hbar cC}L_{n-|k|}^{(2\gamma-1)}(\rho))
- \frac{1}{2}\bar{\Omega}_{\bar{i}\bar{j}\bar{k}\bar{l}}F^{( 2 , 0 ) \bar{k}\bar{l}}= \lambda F_{\bar{i}\bar{j}}^{( 0 , 2 )}, \quad - \frac{1}{2}\Omega_{i j k l}F^{( 0 , 2 ) k l}= \lambda F_{i j}^{( 2 , 0 )}.
I_\epsilon=\int_{\rho>\epsilon}\frac{1}{n+1}f_1\bigl(\Delta^\prime u_2-(n+1)f_2\bigr)\omega^{n+1}\hphantom{I_\epsilon=}{}+\operatorname{Re}\int_{\rho=\epsilon}i(f_1\overline{\partial}u_2-u_1\partial f_2)\wedge\omega^n+\int_{\rho>\epsilon}({\rm cpt\supp}).
\int_M\frac{|\nabla R+R\nabla f+\frac{\alpha'}{2}R\nabla R|^2}{R}d\mu=\int_M\big( |\nabla R|^2\big(\frac{\alpha'}{2}\sqrt{R}+\frac{1}{\sqrt{R}}\big)^2+R|\nabla f|^2 &-2\big(R+\frac{\alpha'}{4}R^2\big)\big(R+\frac{\alpha'}{4}R^2-r\big)\big)d\mu.
0 = \rho_j^{n}-\rho_j^{n+1}&+ \frac{\sigma_2 \Delta t}{\sigma_0 \Delta x}\left(\frac{E_{j-\frac 12}}{\kappa}\frac{\rho_{j-1}^n-e^{-E_{j-\frac 12}\Delta x/\kappa}\rho_j^n}{1-e^{-E_{j-\frac 12}\Delta x/\kappa}}- \frac{E_{j+\frac 12}}{\kappa}\frac{\rho_j^n-e^{-E_{j+\frac 12}\Delta x/\kappa}\rho_{j+1}^n}{1-e^{-E_{j+\frac 12}\Delta x/\kappa}}\right).
I_{2p}~=~\left( \begin{array}{ccc}1_p 0 0 0 0 0 0 0 1_p \end{array}\right)\!~~,~~ I_q~=~\left( \begin{array}{ccc}0 0 0 0 1_q 0 0 0 0 \end{array}\right)~.\vspace{2mm}
\tilde{\tau}= \tau - \eta ( \tau ) \ \ , \ \ \delta_{\eta}x^{\mu}= \tilde{x}^{\mu}( \tau ) - x^{\mu}( \tau ) = \eta \dot{x}^{\mu}( \tau ) \ \ , \ \ \delta_{\eta}e ( \tau ) = \frac{d}{d \tau}\left( \eta ( \tau ) e ( \tau ) \right) \ ,
\left(J_a^\alpha f_n\right)(x)= & \sum_{k=0}^n (-\alpha)^kC_n^k x^{n-k}\left[\frac{1}{k+1}\left(\frac{x-a}{\alpha}\right)^{k+1}E_1\left(\frac{x-a}{\alpha}\right)-\frac{k!}{k+1}e_k\left(\frac{x-a}{\alpha}\right)e^{\frac{-(x-a)}{\alpha}}+1\right]
\sum_{\chi \in X_p}\chi(q_1)\overline{\chi}(q_2)\sum_{k=1}^{p-1}\vert S(k,\chi)\vert^2= \sum_{\chi \in X_p}\chi(q_1)\overline{\chi}(q_2)\sum_{k=1}^{p-1}\left\vert\sum_{l=1}^k\chi (l)\right\vert^2 = \sum_{\chi \in X_p}\sum_{k=1}^{p-1}\sum_{1\leq l_1,l_2\leq k}\chi(q_1 l_1)\overline{\chi(q_2 l_2)}=(p-1)^2{\mathcal{A}}(q_1,q_2,p),
\pi_{n}(x_n|y^{n-1}_{n-J}) &=\exp{\Big\{\sum_{y_{n}}\log\big(r_{n}(x_n|y_n,y^{n-1}_{n-M}\big)q_n(y_n|y^{n-1}_{n-M},x_n)-1-s\gamma_n(x_n,y^{n-1}_{n-N})+\lambda_n(y^{n-1}_{n-J})\Big\}},~\forall{x_n}\in{\cal X}_n.
\lim_{\Lambda}\omega_{\beta,\mu,\Lambda}^{0}(\frac{1}{V}\int_{\Lambda}dx b^*(x)\ \frac{1}{V}\int_{\Lambda}dx b(x))=\lim_{\Lambda}\omega_{\beta,\mu,\Lambda}^{0}(\frac{b^{*}_{0}b_{0}}{V}) = \rho_{0}(\beta) \ ,
A_{\tilde{\varphi}}\rightarrow L^{- 1}e^{\phi_{0}}\frac{\hat{B}\rho^{2}}{2 \tilde{\Lambda}}, B_{\tilde{\varphi}}\rightarrow L e^{- \phi_{0}}\frac{\hat{E}\rho^{2}}{2 \tilde{\Psi}},
\frac{\partial\phi}{\partial W}(W^*_{k}(n),\mu_k) =1(W^*_{k}(n+1)>0), \frac{\partial \phi}{\partial \mu}(W^*_{k}(n),\mu_k) =-1(W^*_{k}(n+1)>0)\frac{J_{k}(-n)}{\mu_k^2}.
g^{\ast}x =\frac{x}{1+bx}=x-bx^{2}+b^{2}x^{3}-\cdots+h_{1}(x), g^{\ast}x^{2}=\left( \frac{x}{1+bx}\right)^{2}=x^{2}-2bx^{3}+3b^{2}x^{4}-\cdots+h_{2}(x), g^{\ast}x^{3}=\left( \frac{x}{1+bx}\right)^{3}=x^{3}-3bx^{4}+6b^{2}x^{5}-\cdots+h_{3}(x),\ & \cdots,
c_{n,k,m}\left(Y,\mathcal{Z}\right)=\begin{cases}\frac{2k+m-1}{m+1}{\frac{k+m-2}{2}\choose \frac{m-1}{2}}{\rm for\;}k\;{\rm odd}, 2{\frac{k+m-1}{2}\choose \frac{m+1}{2}}{\rm for\;}k\;{\rm even.}\end{cases}
\widetilde{F}_w:=U \circ F_w \circ U^{-1}\begin{pmatrix}z x \end{pmatrix}= \widetilde{F}\begin{pmatrix}z x \end{pmatrix}+ \begin{pmatrix}\frac{\pi^2}{4}w+ \mathcal{O}(wx, wz^2) 0 \end{pmatrix}.
\omega_{\mu}^{ab}=e_{\nu}^{a}\Gamma_{\sigma\mu}^{\nu}e^{\sigma b}+e_{\nu}^{a}\partial_{\mu}e^{\nu b}=e_{\nu}^{a}\Gamma_{\sigma\mu}^{\nu}e^{\sigma b}-e^{\nu b}\partial_{\mu}e_{\nu}^{a},
\int_{\mathbb R}[\zeta_k \nu_{n_k}+\varphi_k \nu_{n_k}]D(b_k) dx\leqq 2\|D(b_k)\|\|\nu_{n_k}\|\leqq C \|{\bf{w}}_{k}\|_{1\times 1}\|{\bf{h}}_{n_k}\|_{1\times 1}=O(\epsilon).
d \langle E_{A B}, w_{{\cal M}_{0}}^{( a )}\rangle = \frac{g^{2}\lambda}{8 \pi}d \langle E_{A B}, w_{\mathrm{f l a t}}^{( a )}\rangle - \frac{\bar{\mu}}{2}d \left\langle E_{A B}, \epsilon^{a b c}\left( \sum_{A}d r_{A}^{b}\right) \wedge \left( \sum_{B}d r_{B}^{c}\right) \right\rangle .
\begin{aligned}\int_M(\tilde{u}_n^{(1)}-\tilde{u}_n^{(2)})\mathrm{d}\mu= -(\lambda_{n,j}^{(1)}-\lambda_{n,j}^{(2)}) +O\left(\sum_{i=1}^2\lambda_{n,j}^{(i)}e^{-\lambda_{n,j}^{(i)}}\right),\end{aligned}
\sum_{k=0}^\infty[8k+1]\frac{(aq;q^2)_k(q/a;q^2)_k (q;q^2)_{2k}}{(q^2;q^2)_{2k}(aq^6;q^6)_k(q^6/a;q^6)_k}\,q^{2k^2}=\frac{(q^5;q^6)_\infty(q^7;q^6)_\infty(aq^3;q^6)_\infty(q^3/a;q^6)_\infty}{(q^2;q^6)_\infty(q^4;q^6)_\infty(aq^6;q^6)_\infty(q^6/a;q^6)_\infty}.
\rho^5_{m_0\rightarrow 0+0+0}\!=\frac{1}{64\pi^4}\int_0\!\!\int_0\!\!\int_0 \!\!d\mu_1 d\mu_2 d\mu_3 \frac{\mu_1\mu_2\mu_3\theta(\lambda_E)}{\lambda_E(\mu_1^2,\mu_2^2,\mu_3^2)}\left( 1\!-\frac{\mu_1+\mu_2+\mu_3}{m_0}\right).
S^{\prime}\left[ \xi^{\prime}\right] \left( z_{0}\right) =S\xi\left(z_{0}\right) +\left( \xi_{0}-S\xi\left( z_{0}\right) \right) \left\{1-S\left[ \varphi_{z_{0}}\mid_{Z}\right] \left( z_{0}\right) \right\}& =\xi_{0}-S\left[ \varphi_{z_{0}}\mid_{Z}\right] \left( z_{0}\right)\left( \xi_{0}-S\xi\left( z_{0}\right) \right) .
\textit{H}( e , m , | \widetilde{q}|^{2}, \widetilde{q \hspace{- 0 . 2 c m}/}) = - \frac{4 e^{2}\widetilde{q \hspace{- 0 . 2 c m}/}}{( 4 \pi )^{2}}\int_{0}^{1}d x \int_{0}^{1 - x}d y \biggl \{\frac{( M_{a}^{2})}{Z_{a}}K_{- 1}( Z_{a}) + \frac{( M_{b}^{2})}{Z_{b}}K_{- 1}( Z_{b}) \biggr \}
\chi_{top}\; := \left( \frac{g^2}{16 \pi^2}\right)^2 \; \;\int d^4x \;\langle T \; q(x) \; q(0) \rangle \; \; , \; \;q(x) \; := \;\mbox{Tr}\Big( F(x) \tilde{F}(x)\Big) \; ,
\int \limits_{\mathbb{S}(\mathbb{H}_{n}^{1})}1_{B}\left( \xi \right) \pi_{n}\left( \mathrm{d}\xi \right) =\int \limits_{\mathbb{S}(\mathbb{C}_{n})}1_{\phi_{n}^{-1}\left( B\right)}\left( u\right) \pi \left( \mathrm{d}u\right) B\in \mathcal{B}\left( \mathbb{S}(\mathbb{H}_{n}^{1})\right),
\mbox{{\em H\"{o}l}}(T(h_{1},h_{2},p,\cdot ),z_{0})\leq u(h_{1},h_{2},p,\tilde{\lambda}_{h_{1},h_{2},p})=-\frac{p\log p+(1-p)\log (1-p)}{p\log (\deg (h_{1}))+(1-p)\log (\deg (h_{2}))}<1
\bar{Q}_i=\int d^3 x\: \biggl\{\varepsilon_{ijk}\left( F_{0j}-\tilde{F}_{0j}\right) \chi_k+\psi_0 D_i\lambda+\psi_i D_0\lambda-\left( F_{0i}-\tilde{F}_{0i}\right) \eta+\varepsilon_{ijk}\psi_j D_k\lambda+{1\over 2}\left[\phi,\lambda\right]\chi_i\biggl\}.
\sum_{j}e^{(\phi+t\psi)(x_j)}=\sum_{n\ge1}\sum_{n-1< \psi_j\le n}e^{(\phi+t\psi)(x_j)}\le C^2\sum_ne^{n|t|}\mu(n-1<|\psi|\le n) &\le C^2\sum_ne^{n(|t|-\beta)}<\infty
t_R^{(n)}=s_R^{(n)}\left\{\theta(s_1-t){t\over s_1}+[\theta(s_2-t)-\theta(s_1-t)]{s_2-t\over s_2-s_1}+\cdot\cdot\cdot+\theta(t_n-s){t-s_n\over s-s_n}\right\}.
E_s^{i*}=\alpha_i^*\left[\frac{1}{\ln2(\sum_{j=i}^M\lambda_j^*+\gamma_i^*h_{sp}^i)}-\frac{1}{\theta_i}\right]^+\alpha_i^*=\theta_iE_s^{i*}\left[\frac{1}{\ln2\left(\log_2(1+\theta_i\beta_i^*)-\sum_{j=i}^M\eta P_p\lambda_j^*+\gamma_i^*P_{int}-\mu_i^*\right)}-1\right]^+
f_{k}^{( D )}( \Omega^{({D})}) = F \mathrm{\boldmath \large ~ \left( ~ \right. ~}\! \! \! k , \operatorname{l n}\left( k^{2}/ \mu^{2}\right) , \Omega^{({D})}\mathrm{\boldmath \large ~ \left. ~ \right) ~}\! \! \; ,
\mathrm{End}(W) = \bigoplus_{k=0}^d \mathrm{End}_k(W) \textrm{if}\; d\; \textrm{is even}, \mathrm{End}(W) = \bigoplus_{k=0}^{\frac{d-1}{2}}\mathrm{End}_k(W) \textrm{if}\; d\; \textrm{is odd}.
\int_{0}^{\infty}d r J_{0}( q r ) J_{0}( q^{\prime}r ) = \frac 1{q + q^{\prime}}F \left[ \frac 1{2}; \frac 1{2}; 1 ; \frac{4 q q^{\prime}}{( q + q^{\prime})^{2}}\right] ,
\leq \omega^{\sum_{a = 1}^{N}2 n_{a}}\left\langle \prod_{b = 1}^{N}\left[ \chi_{+}^{( b )}[ Q^{M , S}] \right]^{q_{b}}\left[ \chi_{-}^{( b )}[ Q^{M , S}] \right]^{2 n_{b}- q_{b}}\right\rangle_{Q^{M , S}}.
\int_{S^2}(\tilde{x}^i \frac{\partial \tilde{x}^j}{\partial u^a}-\tilde{x}^j\frac{\partial \tilde{x}^i}{\partial u^a})\tilde{\sigma}^{ac}\pi_{rc}d\mu_{S^2}=-\int_{S^2}\epsilon^{ij}_{\,\,\,l}\tilde{x}^l\tilde{\epsilon}^{bc}\partial_b\pi_{rc}d\mu_{S^2}.
\sum_{|j| \leq m_0}q_{\delta,j,m}= \sum_{|j| \leq m_0}\frac{1}{2 \pi i (j + \theta_1/2 ) + z}- \int_{-1/2}^{1/2}\sum_{|j| \leq m_0}\frac{\mathrm{du}}{2 \pi i (j + \theta_1/2+ u ) + z}+ O(m_0^2/m).