harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
K_{B}= \frac{\lambda_{2}}{\lambda_{1}}= \left( \frac{i k - 2 \cos \rho}{i k + 2 \cos \rho}\right)^{2}\ \frac{i k - 2}{i k + 2}\ \frac{i k - 2 c_{+}}{i k + 2 c_{+}}\ \frac{i k + 2 c_{-}}{i k - 2 c_{-}},
\left\{{\rm e}^{-{A\over 2}}\not\hspace{-0.1cm}{\partial}_x+\Gamma^r\left(\partial_r+{D_1\over4}A'+{D_2\over 4}B'\right)+\lambda\chi+{\rm e}^{-{B\over2}}\not\hspace{-0.1cm}{\Delta}_y\right\}\Psi=0.
\Omega_{i}^{( 2 ; 3 )}= \int_{0}^{\tau_{0 i}}\frac{\mathrm{d}\theta}{\operatorname{c o s}\theta}= \operatorname{l n}\left( \frac{1 + \operatorname{s i n}\tau_{0 i}}{1 - \operatorname{s i n}\tau_{0 i}}\right) .
\epsilon_{D}(\lambda,a)=F(D)\sum_{n=1}^{\infty}\int^{\infty}_{0}r^{D-3}\left(r^{2}+\left(\frac{n\pi}{a}\right)^{2}\right)^{\frac{1}{2}}exp\left[-\lambda\left(r^{2}+ \left(\frac{n\pi}{a}\right)^{2}\right)^{\frac{1}{2}}\right].
f = (xy + b^{1/2}x)^2 + (x^2 + a^{1/2}x)^2 x &= (x^2 + a^{1/2}x)^4 + (x)^4 x + (y + b^{1/2}+ b^{1/4})^4 x^2 + (a^{1/4})^4 x^3
\begin{aligned}F_1(t)=i\int_0^M e^{i(t-s)\Delta^2}(|u|^2u)(s) \, ds, F_2(t)=i\int_M^{t-M}e^{i(t-s)\Delta^2}(|u|^2u)(s) \, ds, F_3(t)=i\int_{t-M}^t e^{i(t-s)\Delta^2}(|u|^2u)(s) \, ds. \end{aligned}
\int_\Omega \Bigl\{a u_{tt}w + c^2 (\nabla u +\varepsilon |\nabla u|^{q-1}\nabla u)\cdot \nabla w +b \nabla u_t \cdot \nabla w \Bigr\}\, dx +\alpha \int_{\hat{\Gamma}}u_t w \, dx = \ -\int_0^t \int_\Omega fu_t w\, dx+\int_{\Gamma}g w \, dx,\forall w\in W^{1,q+1}(\Omega).
\rho(z_1, z_2) = \rho(-z_1, -z_2), z_2 > 0 &= \frac{e^{-2AB}}{\sqrt{\pi}}\int_{-\infty}^{A-B}e^{-t^2}dt +\frac{e^{2AB}}{\sqrt{\pi}}\int^{\infty}_{A+B}e^{-t^2}dt,
{J^{z}}_{[ \kappa , \lambda ]}( z , \bar{z}) = 2 \left( 2 \omega_{\bar{z}z}- \frac{1}{\mu}p \right) \lambda_{\bar{z}}- 2 \left( 2 \lambda_{\bar{z}z}- \frac{1}{\mu}\kappa \right) v_{\bar{z}},
{\cal L}= -\frac{1}{4}(\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c})^{2}-\frac{m^{2}}{2}A_{\mu}^{a}A^{a\mu}
\begin{cases}u_t=\Delta (\gamma (v)u) x\in\Omega,\;t>0 v_t-\Delta v+v=u x\in\Omega,\;t>0 \partial_\nu u=\partial_\nu v=0, x\in\partial\Omega,\;t>0 u(x,0)=u_0(x),\;\;v(x,0)=v_0(x), x\in\Omega,\end{cases}
\tilde{S}_{2}(f)(x, t) = \sum_{n = 1}^{\infty}\left(\int_{0}^{t}e^{L_{n}(t - s)}\exp_{\tau}(D_{n}, t - \tau - s) \left(\tfrac{2}{l}\int_{0}^{l}f(\xi, s) \sin(\tfrac{\pi n}{l}\xi) \mathrm{d}\xi\right) \mathrm{d}s\right) \sin(\tfrac{\pi n}{l}x).
M(x,y,z) =\left( \begin{array}{cccc}0 z+\l_0 \l_1 \l_2 -z-\l_0 0 \l_3 \l_4 -\l_1 -\l_3 0 z- \l_0 -\l_2 -\l_4 -z+\l_0 0 \end{array}\right), \l_0, \dots, \l_4 \in \frak m_{x,y}.
\begin{aligned}AG\geq \left(\frac{2}{n}-C_{3}-\frac{4(p-1)}{np}\right)G^{2}\phi-\frac{C_{1}^{2}}{R^{2}}\frac{G}{C_{3}}-2K\phi G-2\lambda_{1}G\phi &+\lambda_{2}e^{b\omega}\phi G\left(C_{3}-b+\frac{b(p-1)}{p}\right) \geq \left(\frac{2(2-p)}{np}-C_{3}\right)G^{2}\phi-\frac{C_{1}^{2}}{R^{2}}\frac{G}{C_{3}}-(2K+2\lambda_{1})\phi G. \end{aligned}
(q^{2}+y^{2}M_{\ast}^{2})^{k}\,C_{2k}^{\frac{1}{2}d-1}\left( \frac{y\left(q^{2}+M_{\ast}^{2}\right)}{\sqrt{q^{2}+y^{2}M_{\ast}^{2}}}\right)=\sum_{l=0}^{2k}B_{l}\,(q^{2},M_{\ast}^{2})\,C_{l}^{\frac{1}{2}d-1}(y)
\sum_{j=0}^{m}\sum_{k=0}^{m}\tilde{\alpha}_{j+k+1}\xi_{j}\xi_{k}&\ \ \ \ = \int_0^\infty \cdots \int_0^\infty \left( \sum_{j=0}^{m}(x_1\cdots x_d)^{j}\xi_{j}\right)^2 (x_1 \cdots x_d) p({\pmb x}) d\mu \cdots d\mu
\phi (T) =\phi_T(T) = \sum_{i=1}^{n_T}\phi \big(T_{e^T_i,T(e^T_i)}\big)=\sum_{i=1}^{n_T}f_{T,i}\big(T(e^T_i)\big) =\Psi\Big(\big(\sum_{i=1}^{n_T}e^T_i\otimes f_{T,i}\big)_{\mathcal U}\Big)(T),
\mathcal{S}= (x_1,x_2, \dots,x_m,\, y_1, y_2, \dots, y_m,\, z_1,z_2) \mathcal{S}^* = (x_1, y_m,\, x_2, y_{m-1},\, \dots,\, x_m, y_1, \, z_2,z_1).
\langle{\cal O}\rangle_{R}= \sum_{n = 0}^{\infty}\frac{1}{n !}\frac{1}{( 2 \pi )^{n}}\int \left[ \prod_{i = 1}^{n}d \theta_{i}g ( \theta_{i}, R ) e^{- m R \cosh \theta_{i}}\right]{\cal F}_{2 n}^{\cal O}( \theta_{1}, \ldots , \theta_{n}) ,
\beta_{\ell}(t, u^{1}, \dotsc, u^{m-d-1}) ={}\gamma(t) + u^{1}X_{1}(t) + \dotsb + u^{m-d-1}X_{m-d-1}(t) &+ u_{\ell}^{m-d}(t, u^{1}, \dotsc, u_{\ell}^{m-d-1}) X_{m-d}(t) + \dotsb &+ u_{\ell}^{m-1}(t, u^{1}, \dotsc, u_{\ell}^{m-d-1}) X_{m-1}(t),
\chi_{l-1}(g_{(1\,l)}g_{(1\,l-2)}) a_{l-1}[a_{(1\,l-2)},a_{(1\,l)}]_c &=-3\xi^2\lambda_{112}\chi_{(1\,l)}(g_{(2\,l-2)})\chi_{l-1}(g_{(1\,l)}g_{(1\,l-2)})a_{l-1}a_{(3\,l)}a_{(2\,l-2)}&+3\lambda_{112}\chi_{1}(g_{(3\,l-2)})\chi_{l-1}(g_{(1\,l)}g_{(1\,l-2)})a_{l-1}a_{(3\,l-2)}a_{(2\,l)}.
{\cal G}^{( 2 )}= 2 \partial d^{( 1 )}+ \frac{1}{2 \pi \alpha^{\prime}}( i_{h}i_{k}C^{( 4 )}) + 2 ( 2 \pi \alpha^{\prime}) \frac{( k . h )}{h^{2}}\partial \omega^{( 0 )}\partial{\tilde{\omega}}^{( 0 )},
{\frac{[ n^{\prime}]_{0}}{n_{0}b_{0}}}= -{\frac{8 \pi G_{5}}{3 c^{4}}}\left[ \sigma - 2 \varrho_{m}c^{2}( 1 + \zeta e^{- 2 \psi}) - 3 \varrho_{r}c^{2}e^{- 2 \psi}- 5 \varrho_{\psi}c^{2}\right] ,
\left(u-{vt}\right)\Gamma^\pm(u)\Gamma^\pm(v)+ \left(v-ut \right)\Gamma^\pm(v)\Gamma^\pm(u) =0, \left(v-ut\right)\Gamma^+(u)\Gamma^-(v)+\left(u-vt\right)\Gamma^-(v)\Gamma^+(u) =\delta(u,v)(1-t)^2.
\Theta_p^u= T_1(\Phi_\lambda^{-1}(u\Theta u^*))=T_1(\Phi_\lambda^{-1}(uv^*v\Theta v^*vu^*))=T_1[\Phi_\lambda^{-1}(uv^*)\Phi_\lambda^{-1}(v\Theta v^*)\Phi_\lambda^{-1}(vu^*)] = T_1(\Phi_\lambda^{-1}(uv^*))\cdot\Theta_p^v\cdot T_1(\Phi_\lambda^{-1}(uv^*))^*.
\left.{\frac{1 6 \pi^{2}}{g_{i , \mathrm{e f f}}^{2}}}\right\vert_{\mathrm{\f i e l d \t h e o r y}}^{\ \overline{{D R}}}= \frac{1 6 \pi^{2}}{g_{i}^{2}( \mu )}+ b_{i}( 2 \gamma + 2 ) ;
|\Gamma_{m+1,v,0}| =\frac{s_1-1}{s_1}\frac{s_2-1}{s_2}\cdots\frac{s_m-1}{s_m}\frac{l}{s_{m+1}}&=\frac{d-k}{d+1-k}\frac{d+1-k}{d+2-k}\cdots\frac{d+m-1-k}{d+m-k}\frac{l}{d+m+1-k}&=\frac{d-k}{d+m-k}\frac{l}{d+m+1-k}.
H^{1}({\cal S},{\cal O}_{{\cal S}}( - 1 ) \oplus \bigoplus_{i = 3}^{7}{\cal O}_{{\cal S}}( - i ) ) = H^{0}({\cal S},{\cal O}_{{\cal S}}( - 1 ) \oplus \bigoplus_{i = 3}^{7}{\cal O}_{{\cal S}}( - 2 + i ) )^{*}.
F_1 = \alpha_1 y x^8 + \beta_1 z x^6 + \gamma_1 s x^5 + \delta_1 t x^5 + \varepsilon_1 u x^4 + \cdots, F_1 = \alpha_2 y x^{10}+ \beta_2 z x^8 + \gamma_2 s x^7 + \delta_2 t x^7 + \varepsilon_2 u x^6 + \cdots,
\mathcal{I}_n=\frac{1}{V_n(\hat{\theta})}\left(\binom{n}{2}^{-1}\sum_{t=1}^{\binom{n}{2}}\left(U_t^2-2U_t\frac{e^{\hat{\theta}}}{1+e^{\hat{\theta}}}+e^{\hat{\theta}}\left(\frac{e^{\hat{\theta}}-1}{(1+e^{\hat{\theta}})^2}\right)\right)\right)^2
m^{2}\geq 4 \pi v^{2}( n_{e}, n_{m}) \frac{1}{\mathrm{I m}\tau}\left( \begin{array}{c c}{1}&{- \mathrm{R e}\tau}\\{- \mathrm{R e}\tau}&{| \tau |^{2}}\\ \end{array}\right) \left( \begin{array}{c}{n_{e}}\\{n_{m}}\\ \end{array}\right)
K_{A,\theta}(x,y) := (-1)^{\theta_1 \mathbf{1}\{x_1 = m_1-1, y_1 = 0\}+ \theta_2 \mathbf{1}\{x_2 = m_2 - 1, y_2 = 0\}}w_A(x,y), x,y \in \mathbb{T}_{m_1,m_2}.
f(x_1,\dots,x_d) =(g\circ h)(x_1,\dots,x_d)=g\left(h((x_1,\dots,x_d)\right) =g\left(h^2(x_i,h^{d-1}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_d))\right) =(g\circ h^2)(x_i,h^{d-1}((x_1,\dots,x_{i-1},x_{i+1},\dots,x_d)))
\hspace{5 . 0 c m}\exp \left\{- i \frac{2 \pi}{e L}\vec{\cal D}\vec{n}\right\}\exp \left\{i \frac{2 \pi}{L}\int d^{3}x \tilde{\rho}_{\mathrm{m a t t}}^{3}\vec{n}\vec{x}\right\}\ ,
\frac{-1}{4}\left(\overrightarrow{\partial}_{x}^{2}-\overrightarrow{\partial}_{X}^{2}\right) K_{i\left( P-p\right)}(e^{x+X})\,K_{i\left(P+p\right)}(e^{x-X})=-e^{2x}\,K_{i\left(P-p\right)}^{\prime}(e^{x+X})\,K_{i\left(P+p\right)}^{\prime}(e^{x-X})\;,
\Lambda_{\pm}( q_{0}) = \Lambda ( q_{0}) \pm \frac{2 \kappa \bigg ( \frac{2}{\pi}\bigg )^{\frac{1}{2}}}{[ \frac{1}{2}( q_{0}- 1 ) ] !}\bigg ( \frac{1 + k}{1 - k}\bigg )^{- \frac{\kappa}{k}}\bigg ( \frac{8 \kappa}{1 - k^{2}}\bigg )^{\frac{1}{2}q_{0}}\bigg [ 1 + O \bigg ( \frac{1}{\kappa}\bigg ) \bigg ]
F \left( \beta \right) = E_{0}+ \widetilde{F}\left( \beta \right) = - \frac{\pi^{2}a}{6 \beta^{2}}- \frac{1}{2 \beta}\log \left( \frac{\beta}{2 \pi a}\right) - \frac{1}{\beta}\sum_{p , \, n = 1}^{\infty}\frac{1}{n}\, e^{- 4 \pi^{2}\, n \, p \, a / \beta}.
\sup_{x_1^2+\hdots+x_n^2 \leq r_0^2, t=t_0}H \, \langle \omega,\nu \rangle^{-1}\leq \max \Big \{\sup_{x_1^2+\hdots+x_n^2 = r_0^2, t_j \leq t \leq t_0}H \, \langle \omega,\nu \rangle^{-1}, \, \sup_{x_1^2+\hdots+x_n^2 \leq r_0^2, t=t_j}H \, \langle \omega,\nu \rangle^{-1}\Big \}.
\bigoplus_{i=0}^{x-1}H_x(i,\phi(i)) =\bigoplus_{i=0}^{x-1}\left( T_x(i)\oplus F_n(T_x(i))\oplus F_n (T_x(\phi(i)))\right) &=\bigoplus_{i=0}^{x-1}T_x(i) \bigoplus_{i=0}^{x-1}F_n(T_x(i)) \bigoplus_{i=0}^{x-1}F_n(T_x(\phi(i)))
\langle{\cal E}\rangle ={\frac{1}{\Omega}}\int_{\Omega}d t \, d^{3}\! k \,{\cal E}_{\bf k}( t ) ={\frac{1}{\Omega}}\sum_{M \in \Omega}\delta t \int_{\Omega}d^{3}\! k \,{\cal E}_{\bf k}^{M},
| \mathcal{B}_{p}\rangle = \exp \left( \sum_{n = 1}^{\infty}\, \mp \frac{1}{n}\alpha_{- n}\cdot \tilde{\alpha}_{- n}- b_{- m}\tilde{c}_{- m}- \tilde{b}_{- m}c_{- m}\right) \int d^{2 5 - p}q_{\bot}e^{- i q_{\bot}\cdot y_{\bot}}| q_{\bot}, q_{\parallel}= 0 \rangle ,
a^{+}(\hat{n})_{k,\lambda}a = a^{+}\bigg(\sum_{p=0}^{k}S_{2,\lambda}(k,p)(a^{+})^{p}a^{p}\bigg)a &=\sum_{p=0}^{k}S_{2,\lambda}(k,p)(a^{+})^{p+1}a^{p+1}.
|D_{\tau_n^\mathfrak{B}}-D_{2n}| \le |\tau_n^{\mathfrak{B}}-2n|=|D_{\tau_n^{\mathfrak{B}}}|, \shortintertext{and, thus,}|D_{2n}|\le |D_{2n}-D_{\tau_n^\mathfrak{B}}| +|D_{\tau_n^\mathfrak{B}}|\le |\tau_n^{\mathfrak{B}}-2n|+|D_{\tau_n^\mathfrak{B}}|=2|D_{\tau_n^\mathfrak{B}}|.
\frac{1}{\binom{n}{k}}\sum_{S \subset [n], \, |S|=k}\mathcal{W}_2^2(P_{S},Q_{S}) = \frac{1}{|\Pi|}\sum_{(S_1,\ldots,S_m) \in \Pi}\frac{1}{m}\sum_{i=1}^m \mathcal{W}_2^2(P_{S_i},Q_{S_i}).
\check R_i(u)=\Sigma_i^{(2)}- (q+q^{-1})\frac{(q^2-q^{-2})}{1-uq^{-2}}\Sigma_i^{(1)}+q^2 \frac{(1-q^{-2})(1-q^{-4})}{(1-u)(1-uq^{-2})}\ .
\prod_{-1\leq i<j}\left(q^{i-j}\right)^{\frac{1}{2}n(i)n(j)}=\left[\frac{q^k}{q^0}\cdot\frac{q^l}{q^0}\cdot\frac{q^l}{q^k}\cdot\frac{q^m}{q^0}\cdot\frac{q^m}{q^k}\cdot\frac{q^m}{q^l}\right]^{\frac{1}{2}}=q^\frac{3m+l-k}{2}.
\lim_{\epsilon\rightarrow 0}\int_{\mathbb{T}^d}m_\epsilon(x ) \phi_\epsilon (x )dx=\lim_{\epsilon\rightarrow 0}\int_{\mathbb{T}^d}F^{-1}_\epsilon(w_\epsilon (x ) ) \phi_\epsilon (x )dx=\int_{\mathbb{T}^d}\int_{\mathcal{Y}^d}F^{-1}(w (x,y ))\phi (x,y )dydx
T_{+ +}^{\prime}= T_{+ +}- \partial_{+}X ( \delta_{N}( \sigma , \pi ) - \delta_{N}( \sigma , 0 ) ) \ , T_{--}^{\prime}= T_{--}+ \partial_{-}X ( \delta_{N}( \sigma , \pi ) - \delta_{N}( \sigma , 0 ) ) \ ,
\Phi^{\alpha}(x^{i},y^{i},f_{j_{1}}^{i},f_{j_{2}j_{1}}^{i},...,f_{j_{k+1}...j_{1}}^{i})=01\leq \alpha \leq \dim \mathcal{U}_{k+1}-\dim \varepsilon \mathcal{G}_{k}
\begin{aligned}\mathcal{J}_{3}= \sum_{j\in I,(\alpha_i,\Lambda)>0}\langle e(i,i+1,j) \rangle_{\mathcal{R}_{3}^{\Lambda}}&+\sum_{(\alpha_{i_j},\Lambda)>0j=1,2,3}\langle e(i_1,i_2,i_3) \rangle_{\mathcal{R}_{3}^{\Lambda}}.\end{aligned}
F^{0}( N , g_{j}) = \sum_{n = 0}^{\infty}a_{j_{1}\cdots j_{n}}\delta g_{j_{1}}\cdots \delta g_{j_{n}}+ \sum_{i}\left( V_{i j}\delta g_{j}\right)^{2 - \gamma_{0}^{( i )}}\sum_{n = 0}^{\infty}b_{( i ) j_{1}\cdots j_{n}}\delta g_{j_{1}}\cdots \delta g_{j_{n}},
\left\|\delta\tilde{g}\right\|^2_{\tilde{g}}=2(1+2u)\int{d^2}\xi\sqrt{\tilde{g}}{{(\delta\varphi)}^2}+\int{d^2}\xi\sqrt{\tilde{g}}{\tilde{g}^{ac}}{\tilde{g}^{bd}}{\tilde{P}_{ab}}(\delta\theta){\tilde{P}_{cd}}(\delta\theta).
D_{\psi}^{j}(z)(f) = \int_{\psi^{-1}(z)}f(x,y)~dD_{\psi}^{j}(z)(x,y)= \int_{\varphi^{-1}(z)}\int_{p_{Y}^{-1}(y)}f(x,y)~dD_{p_{Y}}^{j}(y)(x)~dD_{\varphi}(z)(y).
\frac{d}{dx}e^{x(e_{\lambda}(t)-1)}=(e_{\lambda}(t)-1)e^{x(e_{\lambda}(t)-1)}&=\sum_{n=0}^{\infty}\bigg(\sum_{k=0}^{n}\binom{n}{k}(1)_{n-k,\lambda}\phi_{k,\lambda}(x)-\phi_{n,\lambda}(x)\bigg)\frac{t^{n}}{n!}.
d s^{2}={\frac{2^{5}}{Q \log{u / \epsilon}}}\left[{\frac{u^{2}}{2 R_{0}^{2}}}d x_{| |}^{2}+ R_{0}^{2}u^{2}d u^{2}+ R_{0}^{2}d \Omega_{5}^{2}\right] ,
\eta_k^2 (\xi_1-\eta_1)^2 - \eta_1^2 (\xi_k -\eta_k)^2 = \; \eta_k^2 \xi_1^2 -2 \eta_k^2 \eta_1 \xi_1 - \eta_1^2 \xi_k^2 + 2 \eta_1^2 \eta_k \xi_k.
\begin{aligned}t^0 = \frac{r_0}{2}\alpha_0^{1/2}\alpha_1^{-1/2}\left( \| h \|_{L_{\infty}}\| h^{-1}\|_{L_{\infty}}\|f \|_{L_\infty}\|f^{-1}\|_{L_\infty}\right)^{-1}.\end{aligned}
\sigma^*\begin{cases}\le \d 1-\frac{5}{4(a_1+a_2+a_3)}\mbox{if $3(a_2+a_3)\le 7a_1$,}begin{align*}1em]\le \d 1-\frac{3}{a_1+3a_2+3a_3}\mbox{if $3(a_2+a_3)> 7a_1$, $3 a_3+a_1\le 5 a_2$ and $3 a_3< a_1+3 a_2$,}begin{align*}1em]= \d 1-\frac{1}{2a_3}\mbox{otherwise.}\end{cases}
\theta(g)=\psi_{\beta,\rho}(g) =\tau\left(\varpi^{-l}B(X+2^{-1}\varpi^{l-1}X^2,\beta) -2^{-1}\varpi^{-1}B(X,\beta)\right)\cdot \rho(\overline X) =\tau\left(\varpi^{-l}B(X,\beta)\right)\cdot\rho(\overline X).
R^e_n(s_0) =(-e+1,1)(-e+2,2)\cdots (0,e) R^e_n(s_1) =(1,e+1)(2,e+2)\cdots (e,2e) R^e_n(s_m) =((m-1)e+1,me+1)((m-1)e+2,me+2)\cdots (me,(m+1)e),
\lim_{\begin{subarray}{c}M \to \infty M' \to -\infty \end{subarray}}\prod_{i=1}^{k}\frac{z_{i}^{M'-1}}{(1+z_{i})^{M}}\langle \prod_{1\le i \le k}^{\curvearrowright}C_{\mu_{i}}^{[M', M]}(z_{i})\prod_{1\le i \le k}\beta_{\nu_{i}, x_{i}}^{*}\rangle_{[M', M]}.
\left<W(C)\right>_{\rm low-energy}=\int Dh_{\mu\nu}\exp\left\{-\int d^3x\left[\frac{1}{6\zeta}H_{\mu\nu\lambda}^2+g^2h_{\mu\nu}^2-2\pi ih_{\mu\nu}\Sigma_{\mu\nu}\right]\right\}.
\frac{B}{A}=\frac{\Gamma(1-|\alpha|)\left(|\alpha|-\frac{m\bar{\lambda}_{\theta}}{2\pi}\right)}{\Gamma(1+|\alpha|)\left(|\alpha|+\frac{m\bar{\lambda}_{\theta}}{2\pi}\right)}\left(\frac{kr_0}{2}\right)^{2|\alpha|}.
f(n,1,2) = \begin{cases}\displaystyle\sum_{i=n/2}^n{n \choose i}\mbox{if}n\mbox{is even,}{n-1 \choose \frac{n-3}{2}}+\displaystyle\sum_{i=(n+1)/2}^n{n \choose i}\mbox{if}n\mbox{is odd.}\end{cases}
F^{\rm od}_{\rm ed}(q) =\left(-q;q^2\right)_\infty\sum_{n=1}^\infty \frac{\left(-q^2;q^2\right)_{n-1}q^{2n}}{\left(-q;q^2\right)_n}&=\frac{\left(-q;q^2\right)_\infty}{2}\left(-1+\sum_{n=0}^\infty \frac{\left(-1;q^2\right)_{n}q^{2n}}{\left(-q;q^2\right)_n}\right).
\phi_{3}( \vec{q}_{1}, \vec{q}_{2}, \vec{q}_{3}, \vec{n}) = \frac{g \phi_{2}( q^{\prime}, z^{\prime})}{( s_{1 2 3}- m^{2}) ( 1 - x_{2})}+ \frac{g \phi_{2}( q^{\prime \prime}, z^{\prime \prime})}{( s_{1 2 3}- m^{2}) ( 1 - x_{3})}\ .
\beta_{s,a^1}^1=\frac{r^1(s,a^1,a_s^2)-r^1(s,a_s^1,a_s^2)}{r^1(s,a^1,a_s^2)-\sum_{s'\in S}\left(\frac{\mu(s',s,a^1,a_s^2)}{||\mu||}+\delta(s,s')\right) r^1(s',a_{s'}^1,a_{s'}^2)},
b_1 =-1-z_1-z_2-z_3, b_2 =\frac{3}{5}+(z_1+z_2+z_3)+(z_1z_2+z_2z_3+z_3z_1), \left(z_1+z_2+\frac{2}{3}+z_1z_2 \right)z_3 =-\frac{1}{3}-\frac{2}{3}(z_1+z_2)-z_1z_2.
q_1+q_2 = 2(\hat W_{r_0}(q-1)-\hat W_{r_0}(q)-\hat W_{r_0}(q+1) + \hat W_{r_0}(q+2)) &=2(\hat W_{r_0}(q) - 2\hat W_{r_0}(q+1) + \hat W_{r_0}(q+2)) &>0.
\Psi(u)=\,\sum^N_{k=1}\,e^{\imath\,\vartheta_{k\Psi}(\varphi^{-1}_{k}(x),z_k)}\frac{1}{\sqrt{N}}\,n_{k\Psi}(\varphi^{-1}_{k}(x),z_k)\,|\varphi^{-1}_{k}(x),z_k\rangle.
( - \Delta_\Omega)^{\alpha / 2}\varphi ( x) = c_{d, \alpha}\mbox{P.V.}\int_\Omega [ \varphi( x) - \varphi ( y)] J ( x, y) \d y + c_{d, \alpha}\, \kappa ( x) \varphi ( x), \mbox{for}x \in \Omega
\sum_{j=0}^{n}(-1)^{j}\mathrm{dim\,}H_{b,m}^{j}(X)=\sum_{j=0}^{n}(-1)^{j}\mathrm{dim\,}H^{j}(M,L^{m})=\int_{M}\mathrm{Td\,}(T^{1,0}M)\mathrm{ch\,}(L^{m}),
\frac{{{\rm{D}}^{2}}}{\partial{{z}^{i}}\partial\overline{{{z}^{j}}}}=\frac{{{\partial}^{2}}}{\partial{{z}^{i}}\partial \overline{{{z}^{j}}}}+\frac{\partial K}{\partial \overline{{{z}^{j}}}}\frac{\partial}{\partial{{z}^{i}}}+\frac{\partial K}{\partial{{z}^{i}}}\frac{\partial}{\partial \overline{{{z}^{j}}}}+{{\psi}_{i\overline{j}}}
\times \prod_{a = 1}^{N}\Big [ c^{( a )}\Big ]^{n_{a}+ m_{a}}\; \; \prod_{b = 1}^{N}\left[ \left( \frac{2 e^{- \gamma}}{M^{( 1 )}}\right)^{\frac{\pi}{\pi + g N}\frac{1}{N}}\prod_{I = 2}^{N}\Big ( \frac{2 e^{- \gamma}}{M^{( I )}}\Big )^{( U_{I b})^{2}}\right]^{n_{b}+ m_{b}}\; .
w_{1}[p,q](i\mu+1) = w_{1}[p,q+p+\frac{1}{2}](i\mu), w_{1}^{(n)}[p,q](i\mu+1) = w_{1}^{(n)}[p,q+p+\frac{1}{2}](i\mu), w_{2}[p,q](i\mu+1) = w_{3}[p,q+p+\frac{1}{2}](i\mu), w_{2}^{(n)}[p,q](i\mu+1) =w_{3}^{(n)}[p,q+p+\frac{1}{2}](i\mu), w_{3}[p,q](i\mu+1) = w_{2}[p,q+p+\frac{1}{2}](i\mu), w_{3}^{(n)}[p,q](i\mu+1) =w_{2}^{(n)}[p,q+p+\frac{1}{2}](i\mu).
\begin{array}{c}\frac{3}{4n}C_{n}\int_{\Sigma}\varphi_{e_{2n}}\Delta_{b}\varphi d\Sigma_{p}=\frac{3}{4n}\int_{D}(\Delta_{b}\varphi )^{2}d\mu +\frac{3}{4n}\int_{D}\left \langle \nabla_{b}\Delta_{b}\varphi ,\nabla_{b}\varphi\right \rangle d\mu =0.\end{array}
\textnormal{Area}(\textnormal{Sect}_{\alpha, \epsilon}(R)) = \frac{R^2}{2}\left(\tan^{-1}(\alpha + \epsilon)-\tan^{-1}(\alpha - \epsilon)\right)= \frac{\epsilon R^{2}}{1+\alpha^{2}}+ O\left(\epsilon^{3}R^{2}\right).
|p_{(L_1,\cdots,L_k)}|^2-|p_{(L_1,\cdots,L_k,L_{k+1})}|^2-|p_{(L_1,\cdots,L_k,L_{k+2})}|^2+|p_{(L_1,\cdots,L_k,L_{k+1},L_{k+2})}|^2=0
A_{t}( \mid \vec{p}\mid , \lambda , m , \theta ) = - \frac{\lambda^{2}}{3 2 \pi}\int_{0}^{\Lambda_{0}^{2}}d ( \vec{k}^{2}) \frac 1{w_{k}( \vec{k}^{2}+ \vec{q}^{2}/ 4 + m^{2})},
e''_k:=\mathcal{L}'({\mathbf V}_k,\Psi_k)(\delta{\mathbf V}_k,\delta\Psi_k)-\mathcal{L}'(S_{\theta_k}{\mathbf V}_k,S_{\theta_k}\Psi_k)(\delta{\mathbf V}_k,\delta\Psi_k),
v^2 =\mathfrak{Pf}(A,\sigma)(v,v)=\mathfrak{b}_K(h(v),h(v))=\mathfrak{b}_K(u\otimes1+w\otimes\sqrt\alpha,u\otimes1+w\otimes\sqrt\alpha) &=\mathfrak{b}(u,u)+\sqrt{\alpha}\mathfrak{b}(u,w)+\sqrt{\alpha}\mathfrak{b}(w,u)+\alpha\mathfrak{b}(w,w)=\mathfrak{b}(u,u)+\alpha\mathfrak{b}(w,w)\in F,
E(A, B\,C) = \det T(A)^{-1}T(B_{-})\,T(A) \, T(B_{-})^{-1}\det T(B_{-}) T(A)^{-1}T(C_-)\,T(A)\, T(C_{-})^{-1}\,T(B_{-})^{-1}& = \det T(A)^{-1}T(B_{-})\,T(A) \, T(B_{-})^{-1}\det T(A)^{-1}T(C_-)\,T(A)\, T(C_{-})^{-1}& = E(A, B) \, E(A,C) .
f(u):= \min_{i \in [n]}\left\{i: B_{u}\leq B_{w(i)}\leq E_{w(i)}\leq E_{u}\right\}, l(u):= \max_{i \in [n]}\left\{i: B_{u}\leq B_{w(i)}\leq E_{w(i)}\leq E_{u}\right\}.
e=\frac{p}{2(phc-s)},a=\frac{phc}{phc-s},b=\frac{p}{2}\bigg[\frac{c(c-1)+s^2}{(phc-s)(1-c)}\bigg], c=\frac{1}{4}\bigg[\frac{\exp(-ph)(1-c)+s(\exp(-ph)-1)}{(phc-s)(1-c)}\bigg], d=\frac{1}{4}\bigg[\frac{\exp(ph)(c-1)+s(\exp(ph)-1)}{(phc-s)(1-c)}\bigg].
\lim_{\lambda\rightarrow0}\frac{\epsilon_{quad}}{(\frac{\lambda}{T})^2}=\sum_{i\neq 1}[\frac{16\sin^2(\frac{T\lambda_i}{2})}{T^2\lambda_i^5}+\frac{4}{\lambda_i^3}-\frac{8\sin(T\lambda_i)}{T\lambda_i^4}]|\langle1|\Delta H|i\rangle|^2.
\sum_{q=r+1-\sum_{d=j}^{i+1}\ell_{d}}^{r-\sum_{d=j+1}^{i+1}\ell_{d}}a_q^2\leq\frac{\beta^2}{g\big(\lfloor(1-(1+\tilde\varepsilon)^{-j})r\rfloor\big)^2}\leq \frac{h^2}{16^2}\cdot 16^{j-i}.
u=\sum_{s,t=1}^{d_1}\alpha_{\mathrm{st}}X_s\otimes X_t=\sum_{s,t=1}^{d_1}\big(\alpha_{\mathrm{st}}^1X_s\big)\otimes\big(\alpha_{\mathrm{st}}^2 X_t\big)=:\sum_{n=1}^{d_1^2}X_{n1}\otimes X_{n2},
S_b^{(2)}={1\over 4\pi}\int d\tau \int_0^{2\pi \alpha}d\sigma \Big[{1\over z_0^2}(\partial z^{(1)})^2 +{1\over z_0^2}(\partial x_4^{(1)})^2+{3(z^{(1)})^2\over z_0^4}(\dot{z_0}^2 + \dot{r}_0^2 )
S\,\simeq\,\frac{M_0^2}{2}\,\int\limits_{t_1}^{t_2}\!dt\int\limits_{0}^{R}\!dr\,\left[\dot{\bf{u}}^2(t,r)\,-\,{\bf{u}}^{'2}(t,r)\right]\,+\,\sum_{a=1}^{2}\,\frac{m_a}{2}\,\int\limits_{t_1}^{t_2}\!dt\,\dot{\bf{u}}^2(t,r_a)\,{,}
A_k =[(\partial_{x_k}\alpha , \partial_{x_k}\beta , \partial_{x_k}\rho )\wedge (\alpha , \beta , \rho ) + (\delta \partial_{x_k}\alpha - \alpha\partial_{x_k}\delta ,\delta \partial_{x_k}\beta - \beta \partial_{x_k}\delta , \delta\partial_{x_k}\rho-\rho \partial_{x_k}\delta )].\sigma \, ,
d \, R_{g}( A |_{h}) = a^{M}( h ) \frac{\partial x^{N}( h g )}{\partial x^{M}( h )}\frac{\partial}{\partial x^{N}( h g )}= a^{M}( h ) R_{M}^{N}( h g , h ) \frac{\partial}{\partial x^{N}( h g )}\, ,
\varepsilon^{abc}~\phi_a~\frac{\partial\phi_b}{\partial x}~\frac{\partial\phi_c}{\partial y}=\varepsilon^{abc}~\phi_a~W_b~V_c~n~\frac{\partial f}{\partial x}~\frac{\partial\theta}{\partial y}+\varepsilon^{abc}~\phi_a~V_b~W_c~n~\frac{\partial\theta}{\partial x}~\frac{\partial f}{\partial y}
\frac{\partial^2 f_l}{\partial z_i \partial z_j}(t\alpha) =\delta_{ij}h(t\alpha)+t\alpha_i\frac{\partial h_l}{\partial z_j}+t\alpha_j\frac{\partial h_l}{\partial z_i}&=\sqrt{2}a_l\delta_{ij}+2\sqrt{2}\alpha_i\alpha_ja_l(a^2_{n+1}+\cdots+a^2_m) &=\frac{\partial^2 g_l}{\partial z_i \partial z_j}(t\alpha)
D\left(\beta_{x,1}\parallel\beta_{x,2}\right)=\sup_{n}H_{\beta_{x,1}\parallel\beta_{x,2}}(\mathcal{P}_{x,n}),\quad\textrm{where}H_{\beta_{x,1}\parallel\beta_{x,2}}(\mathcal{P}_{x,n})=\sum_{A\in\mathcal{P}_{x,n}}\beta_{x,1}(A)\log\frac{\beta_{x,1}(A)}{\beta_{x,2}(A)}.
Z=\int d\mu\,{\cal D}_{e^{\phi}\hat{g}}X{\cal D}_{e^{\phi}\hat{g}}\phi{\cal D}_{e^{\phi}\hat{g}}b{\cal D}_{e^{\phi}\hat{g}}c\,\exp{\left\{-S_{P}[X,\hat{g}]-S_{gh}[b,c,\hat{g}]\right\}}
|N_{2}(v_{5})|(t,r) \leq \begin{cases}\frac{C r}{(r^{2}+\lambda(t)^{2}) t^{7}\log^{7b-6+5N}(t)}, r \leq \frac{t}{2}\frac{C \log^{8}(t)}{r^{7/2}\log^{b}(t) t^{7/2}}, r \geq \frac{t}{2}\end{cases}
\delta^2\|Z\|_F^2 \ge \sum_{i=1}^m\|A_iZ-Z^TA_i\|_F^2\ge \sum_{i=1}^m |x^*(A_iZ-Z^TA_i)x|^2 &=\sum_{i=1}^m |\lambda-\bar{\lambda}| \ |x^*A_i x|^2=|\lambda-\bar{\lambda}| \sum_{i=1}^m |x^*A_i x|^2.
|x(t,\tau,\xi')-x(t,\tau,\xi)|\leq |\xi-\xi'|e^{\tilde{p}_{1}|t-\tau|}\textnormal{with}\tilde{p}_{1}=\ell_{1}+\frac{(M_{0}+\ell_{2})e^{\ell_{1}\theta}}{1-\tilde{v}_{0}},
D\mathbf{B}=\left({\begin{array}{cc}D_{x_1}\mathbf{B}_1 D_{x_1}\mathbf{B}_2 D_{x_2}\mathbf{B}_1 D_{x_2}\mathbf{B}_2 \end{array}}\right)=\left({\begin{array}{cc}\mathbf{K}_1\star f_1 \mathbf{K}_2 \star f_2 \mathbf{K}_0 \star\mu \mathbf{K}_3\star f_3 \end{array}}\right)~~~x=(x_1,x_2),~~\mathbf{B}=(\mathbf{B}_1,\mathbf{B}_2),
\Pi_{\pm q}^{m}={\cal D}_{\pm q}X^{m}- i \left({\cal D}_{\pm q}\Theta^{\alpha 1}\sigma_{\alpha \beta}^{m}\Theta^{\beta 1}+{\cal D}_{\pm q}\Theta_{\alpha}^{2}\tilde{\sigma}^{m \alpha \beta}\Theta_{\beta}^{2}\right) = 0 ,
\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\alpha})^{-1}\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\beta})\,=\,\widetilde{\Lambda}^{\epsilon,\kappa}(\tilde{\alpha},\tilde{\beta})\,\widetilde{\Omega}^{\epsilon,\kappa}(\tilde{\alpha},x,\tilde{\beta})\,,
\begin{aligned}H( P(x), 0) = (Q, 0)\left(H(x, 0)\right) &\Psi\left(t, H(x,y)\right) = H\left(\Phi(t,x,y)\right) &\Psi_s\left(t, H(x,0)\right) = H\left(\Phi_s(t,x,0)\right)\end{aligned}
\mathcal{G}_\mathfrak{g}^{(0)}= X_\mathfrak{g}\mathcal{G}_\mathfrak{g}^{(1)}= \big\{(x,n-m,y) \in X_\mathfrak{g}\times \mathbb Z \times X_\mathfrak{g}\bigm| \sigma^n(x)=\sigma^m(y)\big\}\subset X_\mathfrak{g}\times \mathbb Z \times X_\mathfrak{g}.
\frac{1}{p_{s}^{2}}~{\cal I}_{\mu_{1}\cdots \mu_{s},\alpha_{1}\cdots\alpha_{s}}^{(s)}(x)~{\cal I}_{\alpha_{1}\cdots \alpha_{s},\nu_{1}\cdots\nu_{s}}^{(s)}(x)=\Im_{\mu_{1}\cdots \mu_{s},\nu_{1}\cdots \nu_{s}}^{(s)},
\int_{B_{\rho}(x_0)}\frac 1 n |\nabla u|^n \leq \liminf_{i\to\infty}\int_{B_{\rho}(x_0)}\frac 1 n |\nabla u_i|^n \,dv\leq\limsup_{i\to\infty}\int_{B_{\rho}(x_0)}e_{\varepsilon_i}(u_i)\,dv &\leq \limsup_{i\to\infty}\int_{B_{R}(x_0)}e_{n,\varepsilon_i}(u_{\varepsilon_i}) (\cdot , \tau )\,dv= \int_{B_{R}(x_0)}\frac 1 n |\nabla u|^n \,dv
\rho^{2}\frac{\partial^{2}V}{\partial r^{2}}=4\rho^{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1,\textrm{}j\neq i}^{n}m_{i}m_{j}\sin^{2}{\left(\frac{1}{2}|\alpha_{i}-\alpha_{j}|\right)}g'\left(2r\sin{\left(\frac{1}{2}|\alpha_{i}-\alpha_{j}|\right)}\right)-2\rho^{2}\sum\limits_{i=1}^{n}m_{i}A^{2}.
\frac{1}{2}\frac{d}{dt}\left({\rho}_4^2(t) \|y_t(t)\|^2 dx\right) + \int_I a(0,t,x){\rho}_4^2 |y_{xt}|^2 dx \leq - \int_I D_2 a(0,t,x) \rho_4^2 y_x y_{xt}dx + \int_I \rho_4 \rho_{4,t}|y_t|^2 dx &+ C \left(\int_I{\rho}_4^2 |G_{t}|^2 dx + \int_\mathcal{O}{\rho}_4^2 |f_t|^2 dx + \int_I{\rho}_4^2 |y_t|^2 dx + \sum_{i=1}^2 \int_I{\rho}_4^2 |p^i_t|^2 dx \right).
h_{\beta,\gamma}= \frac{zb_jf_{j+1}+ \overline\beta(\beta\alpha_jb_j+\gamma\rho_j) + \overline\gamma(\beta\rho_jb_j-\gamma\overline\alpha_j)f_{j+1}}{1 + \gamma(\overline\beta\rho_j-\overline\gamma\alpha_jb_j)z + \beta(\overline\beta\overline\alpha_j+\overline\gamma\rho_jb_j)zf_{j+1}},
\hat{v}=\begin{pmatrix}0 -(id_{E_0(P_0)\oplus E_0(P)}\oplus (\otimes v)^\ast) id_{E_0(P_0)\oplus E_0(P)}\oplus (\otimes v) 0 \end{pmatrix}\colon \Gamma(V(P_0, P, \mathfrak{s})\oplus V(P_0, P, \mathfrak{s}\otimes L)) \to \Gamma(V(P_0, P, \mathfrak{s})\oplus V(P_0, P, \mathfrak{s}\otimes L)).
R^{(1)}(d) =\mathsf{D}_{d}\log(\Gamma(d)^{-1}(r-s)^{d-1})=-\psi(d)+\log(r-s), R^{(m+1)}(d) =\mathsf{D}_{d}R^{(m)}(d)+R^{(1)}(d)R^{(m)}(d),m=1,2,\dots.
TBV_{i}^{T}=|q^{T}_{i-1}(x_{i-\frac{1}{2}})-q^{T}_{i}(x_{i-\frac{1}{2}})|+|q^{T}_{i}(x_{i+\frac{1}{2}})-q^{T}_{i+1}(x_{i+\frac{1}{2}})|.