harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi ( x )_{\hat{n}}= k \Lambda^{3 / 2}e^{- \frac{i \pi}{2 e}\int d^{3}y \partial_{i}E_{i}}V_{\eta}^{1 / 2}( x , \hat{n}) \Phi ( x ) V_{\eta}^{* 1 / 2}( x , - \hat{n})
E_{\varepsilon}= \big\{(t,x,z) \in E : t - |x-z| < \varepsilon \sqrt{xz}\;\;\textrm{or}\;\; x+z -t < \varepsilon \sqrt{xz}\big\}, F_{\varepsilon}= \big\{(t,x,z) \in F : t - (x+z) < \varepsilon \sqrt{xz}\;\;\textrm{or}\;\; t > \varepsilon^{-1}\sqrt{xz}\big\}.
\sigma^r L'_{j}=\bigcap_{i=0}^{r-1}\sigma^{r-i}(L_{i+j})=\bigcap_{i=0}^{r-1}\sigma^{i+1}(L_{j-(i+1)})=\bigcap_{i=0}^{r-1}\sigma^{i}(L_{j-i})=M'_{j}.
\sum_{\underline{\sigma}\in \{-1,1\}^{V(G)}}\prod_{(u,v)\in E(G)}f_e(s_{u,v}\sigma_u\sigma_v)=2^{|V(G)|}\sum_{F\subseteq E(G) \atop O(G_F)=\emptyset}\prod_{(u,v)\notin F}c_1(e)\prod_{(u,v)\in F}s_{u,v}c_2(e).
D\Bigl(\binom{n+j}{r}g(\lambda)\Bigr) =\sum_{\lambda^{+}}\binom{n+1+j}{r}g(\lambda^{+})-\binom{n+j}{r}g(\lambda) &=\binom{n+1+j}{r}Dg(\lambda)+\binom{n+j}{r-1}g(\lambda).
(E_1'\times_{\mathbb{P}^1}E_2')/\langle s' \rangle \colon\begin{cases}Y_1^2= (X-Z)\left(X-\left(\frac{1-\mu\sqrt{\lambda}}{1+\mu\sqrt{\lambda}}\right)^2Z\right) Y_2^2= (X-Z)\left(X-\left(\frac{1-\sqrt{\lambda}}{1+\sqrt{\lambda}}\right)^2Z\right)\end{cases}.
\mbox{Re}\int_{\tilde \Gamma}((\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})-\widehat{\Psi},\overline{\widehat{\Psi}})+(\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})-\widehat{\tilde\Psi},\overline{\widehat{\tilde\Psi}}))d\sigma=0.
F_{n}( \hat{x}){}_{\stackrel{\mathrm{\Large ~ \sim ~}}{n \gg 1}}\e^{n \beta_{0}({\hat{x}}) / 2}= e^{n \operatorname{l n}\left( \frac{{\hat{x}}+ \sqrt{{\hat{x}}^{2}+ 4}}{2}\right)}
8 a_5^2-81 a_6^2=4 a_4-9 a_6=9 a_1 a_6+a_5=8 a_1 a_5+9 a_6=8 a_1^2-1=0, &3 a_1^2 a_6+ a_1 a_5+2 a_3+ a_4+ a_6=-a_1 a_6+a_2=0.
\psi ( \vec{p}\, ) \, = \,{\frac{1}{\sqrt{2}}}\left( \begin{array}{c c}{\phi_{R}( \vec{p}\, ) \, + \, \phi_{L}( \vec{p}\, )}{\phi_{R}( \vec{p}\, ) \, - \, \phi_{L}( \vec{p}\, )}\end{array}\right) \quad ,
\frac{\partial x^i}{\partial \tilde{x}^j}(p)=\delta^i_j,\frac{\partial^2 x^i}{\partial \tilde{x}^j\partial \tilde{x}^k}(p)=0\frac{\partial^3 x^i}{\partial \tilde{x}^j\partial\tilde{x}^k\partial\tilde{x}^l}(p)=\eta_{ia}c^a_{jkl}.
(\beta+1)\| \zeta - \frac{1}\beta \phi_n\| \leqslant \| \zeta - \frac1\beta \phi_n +\beta (1-H)(\zeta -\frac1\beta\phi_n) \| = \| \zeta -\frac1\beta\phi_n + \beta (1-H) \zeta - (1-H)\phi_n\|.
\ell_{1}^{\mu}( \tau ) = \bar{\ell}_{0}^{\mu}+{\frac{1}{2}}\delta^{\mu}+ V^{\mu}\tau , \ell_{2}^{\mu}( \tau ) = \bar{\ell}_{0}^{\mu}-{\frac{1}{2}}\delta^{\mu}+ V^{\mu}\tau ,
- \frac{i}{2 \mu}\left[ \epsilon^{i a b}k_{a}( - k^{2}\chi_{b}^{j}+ k_{e}k^{j}\chi_{b}^{e}) + \epsilon^{j a b}k_{a}( - k^{2}\chi_{b}^{i}+ k_{e}k^{i}\chi_{b}^{e}) \right] = 0 .
\prod_{n=1}^{\infty}(1-q^{n})(1-zq^{n})(1-z^{-1}q^{n})=\sum_{n=0}^{\infty}(-1)^{n}q^{n(n+1)/2}z^{-n}\left(\dfrac{1-z^{2n+1}}{1-z}\right).
{\cal L}^{(s,t)}:=\left(\begin{array}{cccc}1 E_1^{(s,t)}1 &\ddots \ddots & E_{m-1}^{(s,t)}1\end{array}\right),{\cal R}^{(s,t)}:=\left(\begin{array}{cccc}Q_1^{(s,t)}1 &Q_2^{(s,t)}\ddots & \ddots 1 & Q_m^{(s,t)}\end{array}\right).
a(\,\Theta^{(3)}(S^{(11)}),T)= &[a(\theta^{(3)}(S_1^{(11)};Z),T)/32+a(\theta^{(3)}(S_2^{(11)};Z),T)/72+a(\theta^{(3)}(S_3^{(11)};Z),T)/24] & \cdot [(1/32)+(1/72)+(1/24)]^{-1}.
\mathfrak{m}_{n}= \frac{1}{2}\frac{|1 + \tau_{n}\hat{\alpha}_{n-1}|^2}{1 + \mathcal{R}e(\tau_{n}\hat{\alpha}_{n-1})}= \frac{1}{2}\frac{1 - |\tau_{n+1}\hat{\alpha}_{n-1}|^2}{1 - \mathcal{R}e( \tau_{n+1}\hat{\alpha}_{n-1})}, n \geq 1,
p^{r_0}(t;x,y) = \sum_{n=1}^\infty \left[ \left(\frac{a_n(x)}{\sinh (\mu y)}\sin\left(\frac{n \pi y}{r_0}\right)\right) \exp\left(\frac{-(n^2 \pi^2 / r_0^2 +\mu^2)t}{2}\right) \right].
\frac{1}{2}\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}+\|\Delta u\|_{L^{2}}^{2}=\int_{\mathbb{R}^{3}}(u\cdot\nabla) u\cdot \Delta u dx-\int_{\mathbb{R}^{3}}\Delta\Psi\nabla \Psi\cdot\Delta u dx :=J_{1}+J_{2}.
\biggl( -\frac{1}{2}\frac{d^2}{dr^2}+ \frac{(N^2 -1)(N^2 -3)}{8r^2}+ \frac{r^2}{2}+\sum_{K=1}^{\infty}\frac{c_K g^K}{N^{4K-2}}(r^2 )^{2K}\biggr ) \rho=\epsilon_0\rho .
\sum_{j=1}^n \bigg\| \partial_r L_jf + \mathbf{H}_n\frac{L_jf}{|x|}\bigg\|_{2}^2 = \sum_{j=1}^n \bigg\| L_j \bigg(\partial_r f + (\mathbf{H}_n - 1) \frac{f}{2|x|}\bigg)\bigg\|_{2}^2.
\delta_{\kappa}X^{M}E_{M}{}^{r}= 0 \, \quad \Rightarrow \quad \delta_{\kappa}X^{\mu}= - i \bar{\theta}_{1}\Gamma^{\mu}\delta_{\kappa}\theta_{1}- i \bar{\theta}_{2}\Gamma^{\mu}\delta_{\kappa}\theta_{2}+{\cal O}( \theta^{4}) \, .
\frac{\delta}{\delta \omega^{ab}}~-~ \,{^*{\hskip -1.5pt}}\, (e^c\wedge{\cal D}e^d)\epsilon_{abcd}{}~=~ 0\qquad \hbox{and}\qquad\frac{\delta}{\delta e^a}~+~ \,{^*{\hskip -1.5pt}}\, (e^b\wedge{}F^{cd})\epsilon_{abcd}~=~ 0\ \ ,
V_{e f f}= \int d x \Biggl ( \frac{1}{2}\pi_{c}^{2}+ \frac{m^{2}}{2}\phi_{c}^{2}\Biggr ) + \frac{\hbar}{2}\sum_{n = 1}^{\infty}\Biggl ( m^{2}+ \Bigl ( \frac{n \pi}{L}\Bigr )^{2}\Biggr )^{1 / 2}.
\sigma_{P}^{2}=\mathbb{E}[\sum_{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb{E}[R_{i}])^{2}+\sum_{i=1}^{n}\sum_{j=1,i\ne j}^{n}x_{i}x_{j}(R_{i}-\mathbb{E}[R_{i}])(R_{j}-\mathbb{E}[R_{j}])]
\bar{F}(x) = \rho_{1}f^{*}(\gamma)\bar{F}_{\gamma}(x) + \rho_{2}\bar{V}^{r}(x) + \rho_{1}f^{*}(\gamma) \int_{0^-}^{x}\bar{B}^{r}(x-u)\,dF_{\gamma}(x-u), x \geq 0.
\alpha_{-l}=\begin{pmatrix}f_{-l}(0) f_{-l}(L) f_{-l}'(L) f_{-l}'(0)+f_{-l}'''(0) f_{-l}'''(L) f_{-l}''(L)+f_{-l}^{(4)}(L)\end{pmatrix}, ~l=\overline{1,6},
\begin{array}{lrcl}\pm e_{i}\pm e_{j}(40) \mbox{with}i\neq j \; \mbox{and}\;i,j=1,2,3,4,5 1/2(\pm e_{1}\pm e_{2}\pm e_{3}\pm e_{4}\pm e_{5}\pm \sqrt{3}e_{6}) (32) \mbox{with}\mbox{even number of minus signs}(0,0,0,0)_{i}(6) \mbox{with}i=1,2,3,4,5,6.\end{array}
\phi ( \tau , \sigma ) = q + \frac{p}{2 \pi}\tau + \frac{i}{\sqrt{4 \pi}}\sum_{n \neq 0}\frac{1}{n}\left({a_{n}}e^{- i n ( \tau + \sigma )}+{\bar{a}_{n}}e^{- i n ( \tau - \sigma )}\right) .
b^{\prime}(t) h(t) +b(t) f (t) h(t) =( \ell(t) +1)f(t) , b^{\prime}(t) h(t) =f(t)\left\{\ell(t) +1-\int_{0}^{t}( \ell(s)+1) f(s) \mathrm{d}s-1\right\}\hphantom{b^{\prime}(t) h(t)}{}=f(t) \left\{\ell(t) -\int_{0}^{t}(\ell(s) +1) f(s) \mathrm{d}s\right\}\leq0.
\ln\frac{m_{2l+2}N+n_0-b}{n_0-b}=\ln\frac{m_{2l+2}N+n_0-b}{m_0N+n_0-b}+\ln\frac{m_{0}N+n_0-b}{n_0-b}\leq{\frac{l+2}{98C_k}}.
\Psi_{c}( x , t ) = \left[ \frac{\Delta \hat{q}\Gamma ( 1 / 4 )}{2}\right]^{- \frac{1}{4}}\exp \left\{- \frac{( x - \langle \hat{q}\rangle )^{4}}{8 ( \Delta \hat{q})^{4}}+ \frac{i}{\hbar}\left( \langle \hat{p}\rangle x - \frac{1}{2}\langle \hat{q}\rangle \langle \hat{p}\rangle \right) \right\}\; .
t_{A}({\bf x},{\bf y}; z) =z \cdot\prod_{a \in A-\max(A)}(z +{\bf y}_{\leq a}^{A}+{\bf x}_{> a}^{A}) , s_{A}({\bf x},{\bf y}; z) =\prod_{a \in A}(z +{\bf y}_{\leq a}^{A}+{\bf x}_{> a}^{A}) .
\aligned \frac{4(n-1)}{n-2}\Delta \varphi_i+R\varphi_i= -\frac{n-1}{n}\tau^{2a}\varphi_i^{N-1}+\left(|\sigma+LW_i|^{2}+k_i^{2}\right)\varphi_i^{-N-1}, -\frac{1}{2}L^{*}LW_i= \frac{n-1}{n}\varphi_i^{N}d\tau^a.\endaligned
\phi'^{(r)}_b(x)=\rho^{(r)}_{bA}(\Sigma'(x))\zeta^A =\rho^{(s)}_{bc}(h(x))\rho^r_{cA}(\Sigma(x))\zeta^A =\rho^{(s)}_{bc}(h(x))\phi^{(r)}_c(x).
F_{m,r}(s,\beta) - \frac{1}{m\left(s+2\sin^2\left(\pi\frac{\beta}{m}\right)\right)}=\frac{1}{m}\sum_{j=1}^{m-1}\frac{e^{2\pi i \frac{jr}{m}}}{s+2\sin^2\left(\pi\frac{j+\beta}{m}\right)}.
\widehat{Z}:=\left(s_0- \frac{S_{0}}{9},s_0+ \frac{S_{0}}{9}\right) \times\Omega,\widehat{J}_1:=\left(s_0- \frac{S_{0}}{9},s_0+ \frac{S_{0}}{9}\right) \times \Gamma_1.
\lim_{n\rightarrow\infty}\int_t^T\langle\Phi'(s,(u^n)^+(s)),f^n(s)1_{\{u^n>0\}}\rangle_{1,-1}\,ds = ~ \int_{\mathcal{O}_t}\Phi'(s,x,u^+(s,x))\,\mu(dsdx)~a.s..
\mathbb T_{1,Q}^{\flat}=\{T \in \mathbb T_{j}^{\flat}: T \cap Q \neq \emptyset,~ \omega(T) \in \tau_1 \}, \mathbb T_{2,Q}^{\flat}=\{T \in \mathbb T_{j}^{\flat}: T \cap Q \neq \emptyset,~ \omega(T) \in \tau_2 \}.
b^{2}\left( \frac{1 - A^{2}}{( A^{2}b^{2})}m^{2}- \omega ( \omega \pm 4 ) \pm b^{- 2}m \right) Y_{m n}^{k \pm}+ \partial_{x}( x^{2}\partial_{x}Y_{m n}^{k \pm}) = 0 \ .
L(s,\phi_n\times \psi_m)=\prod_{\substack{1\leq i\leq l 1\leq j \leq s}}L(s, \zeta_i\times \eta_j)\prod_{\substack{1\leq i \leq s 1\leq j\leq k}}L(s+\frac{1}{2},\eta_i\times\xi_j)L(s-\frac{1}{2},\eta_i\times\xi_j).
\mathbf{\Phi}_{\mathrm{r},g}^{H}\mathbf{\Phi}_{\mathrm{r},g}+\mathbf{\Phi}_{\mathrm{t},g}^{H}\mathbf{\Phi}_{\mathrm{t},g}=\mathbf{I}_{\bar{M}},\forall g\in\mathcal{G}.
0 = z^2x^2 w = (1-q^{-2}(q+q^{-1})\nu_y + q^{-4}\nu_y^2) w = (1-q^{-4M-2}(q+q^{-1})\lambda + q^{-8M-4}\lambda^2) w = (1 - q^{-4M-3}\lambda)(1-q^{-4M-1}\lambda) w.
2c_k\varepsilon_k+\|c_kv^k+y^k-x^k-\alpha_k(x^k-x^{k-1})\|^2+ &4\alpha_k\|c_{k-1}v^{k-1}+y^{k-1}-x^{k-1}-\alpha_{k-1}(x^{k-1}-x^{k-2})\|^2 = 2c_k\varepsilon_k\leq 4\gamma \sigma^2\varepsilon_k= & \sigma^2\|x^{k+1}-x^k\|^2+4\alpha_k\sigma^2\|x^k-x^{k-1}\|^2 = \sigma^2\|y^k-x^k\|^2+4\alpha_k\sigma^2\|y^{k-1}-x^{k-1}\|^2.
\Phi^\sharp_{\mathfrak{p}}= \{\Phi_{\mathfrak{p}}\in \mathrm{CM}(E_{\mathfrak{p}}):\; \iota_0\in\Phi_{\mathfrak{p}}\}\;;\; \overline{\Phi}^\sharp_{\mathfrak{p}}= \{\Phi_{\mathfrak{p}}\in \mathrm{CM}(E_{\mathfrak{p}}):\; \overline{\iota}_0\in\Phi_{\mathfrak{p}}\},
\beta_{\alpha,1-\alpha}(t) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{(e^{-i\pi}t)^{\alpha-1}}{(1+e^{-i\pi}t)^{\alpha}}\right\}= \begin{cases}\frac{\sin\pi\alpha}{\pi}\, t^{\alpha-1}(1-t)^{-\alpha}0\le t \le 1 0 t>1 \end{cases}
\langle A + | A - \rangle = \det_{\lambda , \lambda^{\prime}}\langle v_{+}( \lambda ) | v_{-}( \lambda^{\prime}) \rangle , \langle \pm | A \pm \rangle = \det_{\lambda , \lambda^{\prime}}\langle v_{\pm}^{0}( \lambda ) | v_{\pm}( \lambda^{\prime}) \rangle ,
\begin{array}{c}{{\displaystyle ( C \Gamma^{\mu_{1}\dots \mu_{p}}) = \epsilon ( - 1 )^{( p - 1 ) ( p - 2 ) / 2}( C \Gamma^{\mu_{1}\dots \mu_{p}})^{T}\quad , \quad \mu = 0 , 1 , \dots , D - 1}}\\{{\displaystyle \epsilon = - \sqrt{2}\cos{\frac{\pi}{4}}( D + 1 ) \quad .}}\\ \end{array}
\Vert \nabla\varphi \Vert^2_{\mathbb{L}^2(\Omega)}= \int_{\Omega}|\nabla\varphi(\mathrm{x})|^2 d\mathrm{x}= \delta^2 \int_{\mathrm{B}}\delta^{-2}|\nabla_{\xi}\varphi(\delta \xi + z)|^2 d\xi = \Vert \nabla \hat{\varphi}\Vert^2_{\mathbb{L}^2(\mathrm{B})}
\rho_{N}( \nu ) ={\frac{N}{N + 1}}\, \delta^{( 2 )}\bigl ( \nu - 1 / ( N + 1 ) \bigr ) +{\frac{1}{N + 1}}\, \delta^{( 2 )}\bigl ( \nu - 1 / ( N + 1 ) + 1 \bigr )
\frac{\mathcal{E}_{s,\Lambda}^{\rm cp}(N)}{N}=\frac{\mathcal{E}_{s,\Lambda}^{\rm cp}(N)}{N(N-1)}\cdot(N-1)\leq\frac{\mathcal{E}_{s,\Lambda}^{\rm cp}(m^d)}{m^d(m^d-1)}\cdot(N-1)\leq -\frac{2}{d}\log m^d \cdot \frac{N-1}{m^d-1}
a{\mathbin{\rm\vphantom{X}\hskip 1.1pt\hskip -0.5pt\_\hskip -0.3em\_\vrule width 0.5pt\hskip 1.7pt}}(B\wedge C) =(a{\mathbin{\rm\vphantom{X}\hskip 1.1pt\hskip -0.5pt\_\hskip -0.3em\_\vrule width 0.5pt\hskip 1.7pt}}B)\wedge C + \hat{B}\wedge (a{\mathbin{\rm\vphantom{X}\hskip 1.1pt\hskip-0.5pt\_\hskip -0.3em\_\vrule width 0.5pt\hskip 1.7pt}}C) ,
{\bf A_2A_1}f(\bar z_2)=\int d\bar z_0dz_0 exp_q\{-q^k\bar z_0z_0\}\left[ \int d\bar z_1dz_1 exp_q\{-\bar z_1z_1\}A_2(\bar z_2,z_1)A_1(\bar z_1,z_0)\right]f(\bar z_0)\ .
\begin{cases}kx\prod_{i=1}^{\frac{D-1}{2}}(x-\alpha_i)^2-\prod_{i=1}^{\frac{D-1}{2}}(x-\beta_i)^2=k(x-1)\prod_{i=1}^{\frac{D-1}{2}}(x-\gamma_i)^2 kx\prod_{i=1}^{\frac{D-1}{2}}(x-\alpha_i)^2-\lambda\prod_{i=1}^{\frac{D-1}{2}}(x-\beta_i)^2=k(x-\lambda)\prod_{i=1}^{\frac{D-1}{2}}(x-\delta_i)^2\end{cases}
\left\Vert G_\infty \right\Vert =\sup_{t \geq 0}\sup_{s\in\mathbb{N}}\sup_{Z_s \in \mathbb{R}^{2ds}}e^{\mu (t) s}e^{\beta (t) \left( E_s (Z_s) + I_s (Z_s)\right)}\left| g_\infty^{(s)}(t,Z_s)\right|
\pi_{q a}^{r}={\frac{\partial L_{r e l}}{\partial{\dot{\rho}}_{q a}^{r}}}= \sum_{b = 1}^{N - 1}f_{a b}({\dot{\vec{\rho}}}_{q c}\cdot{\dot{\vec{\rho}}}_{q d}){\dot{\rho}}_{q b}^{r},
\psi_{k}^{\prime \prime}+ 2 \frac{z^{\prime}}{z}\psi_{k}^{\prime}+ k^{2}\psi_{k}= 0 \, , ~ ~ \pi_{k}^{\prime \prime}- 2 \frac{z^{\prime}}{z}\pi_{k}^{\prime}+ k^{2}\pi_{k}= 0 .
\left|A_{I_0,V}\vec z\right|\sim\left\|\vec g\right\|_{\dot A^{s-\frac{1}{p},\frac{n}{n-1}\tau}_{p,q}(V,\mathbb{R}^{n-1})}=\left\|\operatorname{Tr}\vec f\right\|_{\dot A^{s-\frac{1}{p},\frac{n}{n-1}\tau}_{p,q}(V,\mathbb{R}^{n-1})}\lesssim\left\|\vec f\right\|_{\dot A^{s,\tau}_{p,q}(W,\mathbb{R}^n)}\lesssim\left|A_{Q(I_0,k_0),W}\vec z\right|.
\nabla_M \nabla^M \phi +2\alpha_1 \Lambda e^{\alpha_1\phi}+\frac{\alpha_2T_3}{2}\frac{1}{\sqrt{|g|}}\int d^4 \xi \sqrt{|\gamma|}\delta^5(x-X) \gamma^{\alpha \beta}\partial_\alpha X^A\partial_\beta X^B g_{AB}e^{\alpha_2 \phi}=0,
\breve\Delta^\mathrm{ehw}_n =\frac{1}{N}\sum_{i=1}^nR_{n,i}X_{n,i}\widehat\varepsilon_{n,i}^2X_{n,i}^{\prime},\widetilde\Delta^\mathrm{ehw}_n=\frac{1}{N}\sum_{i=1}^nR_{n,i}X_{n,i}\varepsilon_{n,i}^2X_{n,i}^{\prime}, \shortintertext{and}\Delta^\mathrm{ehw}_n =\frac{1}{n}\sum_{i=1}^nE[X_{n,i}\varepsilon_{n,i}^2X_{n,i}^{\prime}].
E_{a,b}^{(M)}E_{b,a}^{(N)}=\begin{cases}(-1)^{\bar E_{a,b}\bar E_{b,a}}E_{b,a}E_{a,b}+(q_a-q_a^{-1})^{-1}(K_{a,b}-K_{a,b}^{-1}), M=N=1; \displaystyle \sum_{t=0}^{\min(M,N)}E_{b,a}^{(N-t)}\begin{bmatrix}K_{a,b}; 2t-M-N t \end{bmatrix}E_{a,b}^{(M-t)}, .\end{cases}
d X^{0}= \sum_{i = 1}^{g}\left( \lambda_{i}\omega_{i}+ \bar{\lambda}_{i}\bar{\omega}_{i}\right) + \mathrm{e x a c t}~ ~ , ~ ~ d X^{9}= \sum_{i = 1}^{g}\left( \gamma_{i}\omega_{i}+ \bar{\gamma}_{i}\bar{\omega}_{i}\right) + \mathrm{e x a c t}
W_{a,c}(T,x) = \|x_{0,1}x_{2,3}\ddot c\|^{\dot a}\|x_{0,2}x_{1,3}\|^{\dot b}\|x_{0,3}x_{1,2}/\ddot a\|^{\dot c}\delta_{F}\Big(x_{0,1}x_{2,3}\ddot c +\frac{x_{0,3}x_{1,2}}{\ddot a}-x_{0,2}x_{1,3}\Big)
\omega^{i}= \phi^{j}\, ( \mathrm{1}+ \frac{1}{2 !}A^{2}+ \frac{1}{4 !}A^{4}+ \cdots )_{j}^{\, \, \, i}- d \xi^{\beta}\, ( \frac{1}{2 !}A + \frac{1}{4 !}A^{3}+ \cdots )_{\beta}^{\, \, \, \, i}
\int_\zeta^\pi \frac{Q(\theta)\, \cos^{d-3}(\theta/2)\, \sin\theta\, d\theta}{\sqrt{\cos\zeta-\cos\theta}}= \frac{2^{5/2}\, \sqrt{\pi}\, \Gamma((d+3)/2)}{\Gamma(d/2+2)}\, \cos^{d+2}\bigg(\frac{\zeta}{2}\bigg).
\limsup_{n\to\infty}\frac{1}{n}\log Z_n((G,G_1),\varepsilon) \ge\limsup_{k\to\infty}\frac1{k\, p(\frac\varepsilon{2})}\log \Bigg( \frac{1}{m^{k\, p(\frac\varepsilon{2})}}\sum_{|\underline g| ={k\, p(\frac\varepsilon{2})}}s(\underline g, k\, p(\frac\varepsilon{2}), \varepsilon)\Bigg) & \ge \frac1{p(\frac{\varepsilon}{2})}\log 2.
I((X,Y),\pmb k):=\int_{\Delta(X,Y)}\prod_{\alpha\in X\cup Y}\omega_{\delta(\alpha)}(t_\alpha)\sum_{m_{1j}\in SSYT(h_{r-1})}\frac{1-t_{|X|}^{m_{12}}}{m_{12}^{k_{2}}\cdots m_{1r}^{k_r}},
\phi-\overline{\phi}\in L^\infty([0,T^*]; H^3), \ \phi_t \in L^\infty([0,T^*]; H^2), & u\in L^\infty([0,T^*]; H^3),\phi \nabla^4 u \in L^2([0,T^*] ; L^2), & u_t \in L^\infty([0,T^*]; H^1)\cap L^2([0,T^*] ; D^2),
\sum_{i \neq j}(P_i \overline{Q}_{L,j}g Q_i Q_j + h.c.) = \sum_{i \neq j}P_i \overline{Q}_{L,j}g \left[Q_i Q_j + \omega (P_iP_j +P_iQ_j + Q_i P_j) \right] + h.c. - \sum_{i \neq j}P_i \overline{Q}_{L,j}g\omega (P_iP_j +P_iQ_j + Q_i P_j) + h.c.
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B,B]}\left(t+ \sum_{i=0}^{n-1}g(T^i x) \right)\ge \sum_{n=2^{k-1}}^{2^k-1}1_{[-B+C,B-C]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right) \ge \beta \tau \sum_{n=0}^{2^k-1}1_{[-B-C,B+C]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right) \ge \beta \tau \sum_{n=0}^{2^k - 1}1_{[-B,B]}\left( \sum_{i=0}^{n-1}t + g(T^i x) \right)
G^{(4)}_{\mathcal R}(x,y)= &-\frac{i}{2\pi}\left[\int_{\mathcal I_{\theta_c,\zeta_o}}+ \int_{\mathcal D_{\zeta_o,\frac\pi2+\theta_{\hat{x}}-i\infty}}\right]\frac{2i\sin\zeta \widetilde{\mathcal S}(\cos\zeta,n)}{n^2-1}{e^{ik_{+}\left(-\vert y \vert\cos(\zeta+\theta_{\hat y})+ \vert x \vert\cos(\zeta-\theta_{\hat x})\right)}}d\zeta.
H_{\pm}= \left(\begin{array}{cc}m ie^{\mp i\theta}\left(\partial_{r}\mp\frac{i}{r}\partial_{\theta}\mp\frac{eB}{2}r\right) ie^{\pm i\theta}\left(\partial_{r}\pm\frac{i}{r}\partial_{\theta}\pm\frac{eB}{2}r\right) -m\end{array}\right) .
A_\infty=\frac{k}{8}\begin{pmatrix}1 0 0 -1\end{pmatrix}\left(\frac{dz}{z}-\frac{d\bar z}{\bar z}\right), \Phi_\infty=\begin{pmatrix}0 r^{k/2}z^kr^{-k/2}0\end{pmatrix}\,dz.
\|T_hf\|_p^p =\int_X \Bigl|\int_X f(y)\> K_h(x,dy)\Bigr|^p\> d\omega_X(x) &\le\int_X\Bigl(\int_X |f(y)|^p\> K_h(x,dy)\Bigr) \Bigl(\int_X 1\> K_h(x,dy)\Bigr)^{p/q}d\omega_X(x) =\|f\|_p^p.
\tilde{\varphi}= \tilde{\varphi}(f_0) = p' \circ(K^x+ f_0^x) + (K^y + f_0^y) \, q' \circ(K^x+ f_0^x) + (D_1u +D_1 g) \circ (K + f_0), \tilde{\psi}= \tilde{\psi}(f_0) = q \circ (K^x + f_0^x) +(D_2u + D_2 g) \circ (K + f_0),
H_{2,1}=\int_{-\frac{\pi}{2}}^{\frac{3\pi}{2}}\!\!\int_\gamma^{\frac{3}{2}}\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}=H_{2,1}^1+H_{2,1}^2, H_{2,2}=\int_0^{2\pi}\!\!\int_{\frac{3}{2}}^{R}\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}
\mathfrak{b}_1 =e_1\!+\!4e_2\!-\!3e_3, \mathfrak{b}_2=3e_1\!+\!7e_2\!-\!e_4\!+\!e_5, \mathfrak{b}_3 =4e_2\!-\!e_4\!+\!e_6
H_{\mathrm{BrLie}}^{s}\left( \mathcal{P}_{\mathrm{Br}}^{s}\overline{\chi}_{\mathrm{Br}}^{s}\circ \overline{\eta}_{\mathrm{BrL}}^{s}\Lambda_{\mathrm{Br}}\right) =U_{0,1}\left( \overline{\eta}_{\mathrm{Br}}^{s}\right)_{1}U_{1,2}.
\Upsilon_{mplk}^T\delta_t\Upsilon_{mplk}+\Upsilon_{mplk}^T\sum_{i=1}^3\Lambda_i\delta_{x_i}\Upsilon_{mplk}=\lambda\Upsilon_{mplk}^T{\rm{\bf K}}^{-1}\Upsilon_{mplk}\frac{\Delta W_m^k}{\tau}.
\lim_{t\to T_{-}}\Phi_{2}g(t,x) = \lim_{t\to T_{-}}\int_{y=0}^{L}g(T,y)\frac{e^{-i(x-y)^{2}/(4(T-t))}}{-i\sqrt{T-t}}dy begin{align*}10pt] = i\lim_{t\to T_{-}}\int_{y=0}^{L}g(T,y)\frac{\bar{e^{i(x-y)^{2}/(4(T-t))}}}{\sqrt{T-t}}dy=i\frac{\sqrt{\pi}}{2}e^{-i\pi/4}g(T,x)=\lim_{t\to T_{+}}\Phi_{2}g(t,x),
\frac{\partial x}{\partial z_1}\bigg|_{t , 0, z_2}= \frac{\partial x}{\partial z_1}\bigg|_{t , l, z_2}= \frac{\partial x}{\partial z_1}\bigg|_{t , z_1, 0}= \frac{\partial x}{\partial z_1}\bigg|_{t , z_1, l}= 0 .
\lim_{t \to \infty}\big( \left\| u(\cdot,t;t_0,u_0,v_0)-u^{**}(t) \right\|_{C^0(\bar\Omega)}+\left\| v(\cdot,t;t_0,u_0,v_0)-v^{**}(t) \right\|_{C^0(\bar\Omega)}\big)=0,
\begin{aligned}r_k =\displaystyle\sum^{k}_{j=1}(r_{j}-r_{j-1}) = \displaystyle\sum^{k}_{j=1}\frac 1{\varphi(H(r_{j-1}))}\leq \displaystyle\sum^{k}_{j=1}\frac 1{\varphi(e^{j-1}K)}\leq \displaystyle\int^\infty_{0}\frac{du}{\varphi(e^{u-1}K)}= \displaystyle\int^\infty_{K/e}\frac{dt}{t\varphi(t)}\end{aligned}
\sum_{i= -\infty}^k 2^{-i\frac{q}{p-q}}V_r(x_{i-1}, x_i)^{\frac{pq}{p-q}}\leq \bigg( \sum_{i=-\infty}^{k}2^{-i\frac{r}{p-r}}V_r(x_{i-1},x_i)^{\frac{pr}{p-r}}\bigg)^{\frac{q(p-r)}{r(p-q)}}.
\epsilon_{l}( H_{G}^{2 l}( M ) ) = \; < \left( \begin{array}{c c c}{{1}}\\{{\vdots}}\\{{1}}\\ \end{array}\right) , \left( \begin{array}{c c c}{{\mu_{1}}}\\{{\vdots}}\\{{\mu_{k}}}\\ \end{array}\right) , \ldots , \left( \begin{array}{c c c}{{\mu_{1}^{l}}}\\{{\vdots}}\\{{\mu_{k}^{l}}}\\ \end{array}\right) >
\frac1n \sum_{z \in G}\langle \rho(z) a, b \rangle \langle b', \rho'(z) a' \rangle = \begin{cases}0 \colon (\rho,V) \ncong (\rho',V') \frac1{\dim V}\langle a, a' \rangle \langle b', b \rangle \colon (\rho,V) = (\rho',V'), \end{cases}
{d^{\dagger}}^{( i )}( \eta^{n}\cdot \tau^{n}{\cal{G}}) \; \; = \; \; \eta^{n}\cdot \tau^{n}\,{d^{\dagger}}^{( i )}{\cal{G}}+{\textsf{P}}_{N}( n ) \; \eta^{n - 1}\cdot \tau^{n - 1}\,{d^{\dagger}}^{( i )}{\cal{G}}
\Theta_{n}^{\left( j\right)}\left( x\right) \nabla \mathbb{K}_{n}^{(j)}(x)+\Lambda_{n}^{\left( j\right)}\left( x;2,1\right) \mathbb{K}_{n}^{(j)}(x)=\Lambda_{n}^{\left( j\right)}\left( x;1,1\right) \mathbb{K}_{n-1}^{(j)}(x).
R = \left( \begin{array}{c c c c}{Q^{- 1}}&{0}&{0}&{0}\\{0}&{\lambda^{- 1}}&{0}&{0}\\{0}&{\sigma}&{\lambda}&{0}\\{0}&{0}&{0}&{- Q}\\ \end{array}\right) , \qquad \sigma = Q^{- 1}- Q .
I [\,\phi\,]\,=\,{1\over 2}\int d^{n+1}x \, \partial_\mu \Big( ( x^0 )^{-n+1}\,\phi \partial_\mu \phi \Big) \,-\,{1\over 2}\int d^{n+1}x \, \phi \, \partial_\mu \Big( ( x^0 )^{-n+1}\,\partial_\mu \phi \Big) \,.
(t^{-1}x,t\bar{x},p\rho_{-w_{0}(\lambda)}(t)^{-1},\rho_{-w_{0}(\lambda)}(t)q ) =(\rho_{V}(t)x\rho_{V}(t)^{-1},\rho_{V}(t)\bar{x}\rho_{V}(t)^{-1},\rho_{V}(t)p,q\rho_{V}(t)^{-1}).
\chi(\Gamma_N(QD_{2^\alpha})) =\mbox{max}~\{\chi(K_2+\Gamma_N(D_{2^{\alpha-2}})), \chi(K_2+\Gamma_N(D_{2^{\alpha-3}}))\}&=\mbox{max}~\{2+\chi(\Gamma_N(D_{2^{\alpha-2}})), 2+\chi(\Gamma_N(D_{2^{\alpha-3}}))\}
E \left( \beta \right) \simeq \frac{1}{2 4 a}\left( \frac{2 \pi a}{\beta}\right)^{2}- \frac{1}{2 \beta}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, a / \beta \, \gg \, 1 ,
P(\mathbf{y}_{k})=\sum_{\mathbf{i},\mathbf{s},\mathbf{s'}}P(\mathbf{y}_{k}\mid \mathbf{U}=\mathbf{i},S_{k}=\mathbf{s},S_{k-1}=\mathbf{s'}) \lambda_{k}^{\mathbf{i}}(\mathbf{s},\mathbf{s'}),
-\mathcal Q_{\infty}\varphi(x_{j})\le \frac{F^{2}(\nabla\varphi(x_{j}))\mathcal Q_{2}\varphi(x_{j})}{p_{j}-2}+\left(\frac{\lambda_{2}(p_{j},\Omega)^{\frac{1}{p_{j}-4}}|u_{j}(x_{j})|}{F(\nabla \varphi (x_{j}))}\right)^{p_{j}-4}\frac{u_{j}(x_{j})^{3}}{p_{j}-2}=:\ell_{j}.
V_{0}( \lambda ) ={\frac{( - 1 )^{\eta - 1}}{( 4 \pi )^{\eta}\eta ! z_{-}^{D - 1}}}\left[ \left\{\ln ( \ell \mu z_{-}) +{\frac{1}{2}}\sum_{n = 1}^{\eta}n^{- 1}\right\}\hat{\zeta}( 1 - D ) + \hat{\zeta}^{\prime}( 1 - D ) \right] .
x^{(k,\ell)}_j = \int_\Sigma \left\langle \mathcal{L}_{\Sigma}^{\ell}\partial^{k}_t u,\varphi_{\iota+j}\right\rangle\, d\mu=\int_\Sigma \left\langle \partial^{k}_t u, \mathcal{L}_{\Sigma}^{\ell}\varphi_{\iota+j}\right\rangle\, d\mu=0.
\mathbb{C}_{\omega \times K}=\mathbb{C}_{\omega_1}\underset{\mathbb{C}}{\boxtimes}\cdots \underset{\mathbb{C}}{\boxtimes}\mathbb{C}_{\omega_n}\underset{\mathbb{C}}{\boxtimes}\mathbb{C}_K\simeq\mathcal{L}_{\omega_1}\underset{\mathbb{C}}{\boxtimes}\cdots \underset{\mathbb{C}}{\boxtimes}\mathcal{L}_{\omega_n}\underset{\mathbb{C}}{\boxtimes}\mathbb{C}_K,
\left\|\dfrac{1}{2M}\sum_{n=1}^{2M}(-1)^n \left( P_{\sigma^{-1}(1)}z_{M_{2n}}+ y_{2n}\right)\right\|\geq C^{-1}\left\|\dfrac{1}{2M}\sum_{n=1}^{2M}y_{2n}\right\|> C^{-1}F.
\mathbb{E}\left[\overline{\mathbf{W}}\right] = \mathbb{E}\left[\sum^K_{i=1}\frac{a_i}{p_i}\mathbf{W}_i \right] = \sum^K_{i=1}a_i\frac{p_i}{p_i}\mathbf{\Sigma}_i = \sum^K_{i=1}a_i \mathbf{\Sigma}_i=\sigma^2\mathbf{I}_m\sum^K_{i=1}a_i,
y =\frac{(\sigma s^2t^{\sigma}-2)^2}{(\sigma s^2t^{\sigma}+2)^2}+O(t),~~v=\frac{1}{4s^2t^{\sigma}}-\frac{\sigma^{2}s^2t^{\sigma}}{16}+O(t), u =-r\frac{2+\sigma s^2t^{\sigma}}{2-\sigma s^2t^{\sigma}}(1+o(1))
b_{l}^{n}= ( - \partial_{a_{l}}) ( U_{l}^{n}) S t r \left( \ldots \phi^{j_{l}}\ldots \right) - k ( A_{1}+ A_{2}) - \frac{n}{l}[ k ( B_{1}+ B_{2}) + b_{l}^{n}+ D_{l}^{n}|_{i_{n}\leftrightarrow a_{l + 1}}+ \ldots + D_{l}^{n}|_{i_{n}\leftrightarrow a_{p}}]
X = \frac{1}{2}\left(\begin{array}{c}2\,Y_{N,1}(1-Y_{N,1})+(1-Y_{N,0})\,Y_{N,0}+(1+Y_{N,-2})(-Y_{N,-2}) -Y_{N,0}(1-Y_{N,0})-(1-Y_{N,-2})(Y_{N,-2})-2\,Y_{N,-1}(1-Y_{N,-1})\end{array}\right)=q_N.
L = \exp \left( \begin{array}{c c}{}&{\theta_{N}}\\{\overline{{\theta}}^{N}}&{}\\ \end{array}\right) \exp \left( \begin{array}{c c}{i \frac{1 + i \gamma^{4}}{2}\gamma_{p}x^{p}}&{}\\{}&{0_{4}}\\ \end{array}\right) \exp \left( \begin{array}{c c}{i \frac{\gamma^{4}}{2}\rho}&{}\\{}&{0_{4}}\\ \end{array}\right)
{\cal L}_{\rm eff}= \frac{1}{8\pi}{\rm Im}\!\int\!\!{\rm d}^2\theta \,\tau_D(A_D)\,W^2_D-\frac{1}{4\pi}{\rm Im}\!\int\!\!{\rm d}^2\theta{\rm d}^2\bar\theta\,\bar A_D\,A (A_D)\,.
s''(t+k'\bar{\tau}) = \bar{\nu}^{\kappa+k'-1}\left(\bigwedge_{i = \bar{t}_\textup{p}}^{\bar{t}_{\textup{p}}+\bar{\tau}-1}\bar{s}(i) \right) \wedge \bar{\nu}^{\kappa+k'}\left(\bigwedge_{i = \bar{t}_\textup{p}}^{\bar{t}_\textup{p}+h}s(i)\right) = \bar{\nu}^{k'}s''(t),
(\cdot,\cdot)_h: X_h\times X_h\to \mathbb{R}(u_h,v_h)_h\mapsto \frac{1}{N_{\Omega}^{\gamma_1}}\sum_{i=1}^{N_\Omega}\Big(u_h({\vec{\omega}_i}),v_h({\vec{\omega}_i})\Big)_a+ \frac{1}{N_{\partial\Omega}^{\gamma_2}}\sum_{i=1}^{N_{\partial\Omega}}\Big(u_h({\vec{\beta}_i}),v_h({\vec{\beta}_i})\Big)_b.
\vec{\cal K}U_{E , \hat{q}}= \left( \begin{array}{r}{{\vec{K}^{\prime}\, u_{E , \hat{q}}^{( 1 )}}}\\{{E^{- 1}\sqrt{V}\, \pi \vec{K}^{\prime}\, u_{E , \hat{q}}^{( 1 )}}}\\ \end{array}\right) \, , \quad \vec{K}^{\prime}= \vec{K}+ F \, \frac{\vec{\sigma}}{2}\, ,
\begin{pmatrix}A^T B^T 0 (A^*)^T\end{pmatrix}\begin{pmatrix}S M M^T 0\end{pmatrix}\begin{pmatrix}A 0 B A^*\end{pmatrix}= \begin{pmatrix}A^TSA + A^TMB + (A^TMB)^T A^TMA^* (A^TMA^*)^T 0\end{pmatrix}.
\hat{C}_{0 0 0}^{l_{3}l_{1}l_{2}}= \rho ({l_{3}, l_{1}, l_{2}}) \sum_{k}{\frac{( - 1 )^{k}}{k !}}{\frac{( l_{1}+ k ) ! ( l_{2}+ l_{3}- k ) !}{( l_{1}- k ) ! ( l_{2}- l_{3}+ k ) ! ( l_{3}- k ) !}},