harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\inf_{\psi\in\Psi}\left(f+\psi\circ L\right)^{*}_\Phi\left(\phi\right)+g^{*}_\Psi\left(\psi\right) =\left(f+\bar{\psi}\circ L\right)^{*}_\Phi\left(\phi\right)+g^{*}_\Psi\left(\bar{\psi}\right) & =\left(f+g\circ L\right)^{*}_\Phi\left(\phi\right)\leq r.
\left\vert\int_{\Gamma_t}(\mathcal B_1{\mathbf V}\cdot{\mathbf V})\vert_{x_1=0}\,dx_2ds\right\vert\le C_1\left\{\Vert \dot q^+\vert_{x_1=0}\Vert_{L^2(\Gamma_t)}^2+\Vert\varphi(t)\Vert_{L^2(\mathbb R)}^2\right\}+C_2\Vert\varphi\Vert^2_{L^2(\Gamma_t)}\,.
{\cal J}_{i}^{0}\equiv - \frac \partial{\partial x}\psi^{i}- 2 \varepsilon^{i j k}\psi^{j}\pi_{k}= - \sqrt{1 -{\bf \varphi}^{2}}\; \varpi_{i}- \varepsilon^{i j k}\varphi^{j}\varpi_{k}\equiv J_{i}^{0}.
F\equiv \frac{\xi^2 m^4(\xi m^2 + M^2)}{(\xi m^2 -M^2)^3}\log(\sqrt{\xi}m/T) - \frac{M^2(M^4 + 4 \xi^2 m^4 - 3 \xi m^2M^2)}{(\xi m^2 - M^2)^3}\log(M/T).
D_{0}^{c}= D_{0}^{c}( x_{o u t}, x_{i n}) = \frac{1}{2 ( 2 \pi )^{2}}\int_{0}^{\infty}\frac{d e_{0}}{e_{0}^{2}}\exp \left[ - \frac{i}{2}\left( e_{0}m^{2}+ \frac{\Delta x^{2}}{e_{0}}\right) \right] .
V(x^0+y)=V(x^0)+\sum_{i=1}^n\left.\frac{\partial V(x)}{\partial x_i}y_i\right|_{x=x^0}+\frac{1}{2}\sum_{i,j=1}^n\left.\frac{\partial^2 V(x)}{\partial x_i\partial x_j}y_iy_j\right|_{x=x^0+\theta y},
{f^{[1]}\left(z_0,z_{1}\right)}:=\begin{cases}\frac{f(z_0) - f(z_1)}{z_0 - z_1}, \text{if~$z_0\neq z_1$}\frac{d}{dz}f(z)_{\vert{z=z_0}}\end{cases}, z_0, z_1\in\mathbb{T}.
\sum_{c_1c_2\ldots c_k\in\mathfrak{B}_{s_1s_2\ldots s_p}}\sum_{i_1i_2\ldots i_k\in \mathfrak{A}_{c_1c_2\ldots c_k}}\mathsf{E}(M_{i_1i_2}M_{i_2i_3}\ldots M_{i_{k-1}i_k}M_{i_ki_1})\cr\leq N \frac{(k-1)!}{\prod_{\ell=1}^{p}(s_\ell-2)!!\prod_{\ell=1}^{p-1}(k-s_1-\ldots-s_{\ell})}\prod_{\ell=1}^{p}B(s_\ell)
\frac{\partial{{L}_D}^{(i,j)}}{\partial u^{\alpha (i,j)}}-\Delta_1 (\frac{\partial{{L}_D}^{(i-1,j)}}{\partial (\Delta_1u^{\alpha (i-1,j)})}) -\Delta_2(\frac{\partial{{L}_D}^{(i,j-1)}}{\partial (\Delta_2 u^{\alpha (i,j-1)})})=0,
\psi^{z}_{,TT}-\psi^{z}_{,\sigma\sigma}+\frac{4\alpha}{1+\alpha}(T+\tilde{\Lambda})^{-1}\psi^{z}_{,T}-\frac{2\alpha(1-\alpha}{(1+\alpha)^{2}}(T+\tilde{\Lambda})^{-2}\psi^{z}=0,
\phi_1(x,y) = -\frac{1}{2}\int\limits_a^y \psi(x,t)dt + \psi_1(x)= \widetilde{\phi}_1(x,y) + \psi_1(x)~~ &\phi_2(x,y) = \frac{1}{2}\int\limits_a^x \psi(t,y)dt + \psi_2(y)= \widetilde{\phi}_2(x,y) + \psi_2(y),
\frac{B\zeta_c^{h^T_\tau}n^{-c}}{( \frac{4m}{9n}z_{1 - \alpha})^2 - \frac{8m}{9n}z_{1 - \alpha}\|\Theta_\tau\|_2^2 + \|\Theta_\tau\|_2^4}< \frac{1 - \beta}{4}
\begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{-{\frac{1}{2}}}}&{{{\frac{1}{2}}}}&{{{\frac{1}{2}}}}&{{-{\frac{1}{2}}}}\\{{- 3}}&{{0}}&{{1}}&{{- 2}}\\{{1}}&{{- 1}}&{{0}}&{{0}}\\ \end{matrix}
\max \, \left\{\mathbf{x}^1 \mathbf{R}\colon \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\begin{bmatrix}(1-\beta) \mathbf{I}\mathbf{I}- \beta \mathbf{P}\end{bmatrix}= \mathbf{e}_i, \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\geq \mathbf{0}\right\},
F_{N+1}(\tau) = \, \sum_{j=0}^{N}(-1)^{N+j}{N \choose j}\left[F\left( \tau + \frac{(j+1)\pi}{|t|}\right) - F\left( \tau + \frac{j\pi}{|t|}\right) \right] = \, \sum_{j=0}^{N}(-1)^{N+j}{N \choose j}\int_0^{\frac{\pi}{|t|}}F'\left(\tau + s + \frac{j\pi}{|t|}\right)\,ds.
{\bar{\Delta}^{\rm P}_{\rm Proactive}}= -1 -{\tau_{\rm f}}- \frac{{\left({{\tau_{\rm c}}+{n_m}+ \mathcal{T}}\right){\epsilon_m}}}{{1 -{\epsilon_m}}}+ \frac{2}{{1 -{\epsilon_m}}}\left({{\tau_{\rm c}}+{n_1}+ \mathcal{T}+ \sum\limits_{i = 1}^{m - 1}{\left({{n_{i + 1}}-{n_i}}\right){\epsilon_i}}}\right).
N(1, q) \le \liminf_{\ell \to \infty}N(1, \tilde v_{r_{j_\ell}}) = \liminf_{\ell \to \infty}N(1, v_{r_{j_\ell}}) = \liminf_{\ell \to \infty}N(r_{j_\ell}, v_{*}) = \lambda_*.
\eta^\rightarrow(B_1 \cap B_2) = \eta^\rightarrow((B_1^c \cup B_2^c)^c) & = \{[\eta^\rightarrow(B_1)]^c \cup [\eta^\rightarrow(B_2)]^c\}^c & = \eta^\rightarrow(B_1) \cap \eta^\rightarrow(B_2).
\begin{array}{l}H_{1}=\dfrac{1}{(3h+2\alpha_{1}-2\alpha_{2})}\{2q_{1}q_{2}p_{1}+q_{2}p_{1}^2+2q_{2}^2p_{2}-2q_{1}p_{2}+2t_{2}q_{2}p_{1}+2p_{1}p_{2}begin{align*}10pt]\phantom{H_{1}=}+2\alpha_{1}q_{2}+2t_{1}p_{1}+2t_{2}p_{2}\}.\end{array}
\frac{x'}{x}= \frac{1}{rv}\frac{y\xi(y)}{1+y\xi(y)-y}\Big(1+O\left(\frac{1}{r}\right)\Big) &= \frac{1}{rv}\frac{v\xi}{1+v\xi-v}\Big(1+O\left(\frac{\log v}{r}\right)\Big) &= \frac{\xi'}{r}\Big(1+O\left(\frac{\log v}{r}\right)\Big)
\displaystyle\sum\limits_{e=1}^{l_{r-1}}\sum\limits_{f=1}^{l_n}b^{st}_{ef}p^{r-1}_{\tilde{e}e}p^{n}_{\tilde{f}f}=\sum\limits_{e=1}^{l_{r-1}}\sum\limits_{f=1}^{l_n}b^{ef}_{\tilde{e}\tilde{f}}p^{r-1}_{es}p^{n}_{ft}
U_N(z,k) = \min_{\nu \in \mathcal{A}}\mathbf{E}^\nu\!\left[ \sum_{j=k}^{\tau-1}\Big(r\big(z^{u,N}(t_j), \nu_j(z^{u,N}(t_j))\big)\right. + \left.\int_{t_{j}}^{t_{j+1}}\phi\big(z^{u,N}(s), \nu_j(z^{u,N}(t_j))\big)\, ds\Big) +U_N\big(z^{u,N}(t_\tau), \tau\big) \right],Nz \in \mathcal{S}_k\,.
W_{K_1K_2 K_3 K}(s) =\frac{\pi}{z(s)}\frac{1}{(2\pi)^8}|\beta_h(s)|^2 \beta_h(s) \overline{\beta_{\sigma}(s)}e^{\gamma(s) + \frac{\zeta(s)\cdot \zeta(s)}{4 z(s)}}
{\binom{1}{- 1}}{\binom{1}{1}}{\binom{1}{- 1}}, \, \, \, \, \, \, \, \, \, \, \,{\binom{1}{1}}{\binom{1}{- 1}}{\binom{1}{- 1}}, \, \, \, \, \, \, \, \, \, \, \,{\binom{1}{- 1}}{\binom{1}{- 1}}{\binom{1}{1}},
E\tau=\frac{(R_{2}^{2}-R_{1}^{2})}{D}[log\frac{1}{\epsilon}+log2+2\beta^{2}]+\frac{1}{2}\frac{R_{2}^{2}}{1-\beta^{2}}log\frac{1}{\beta}-\frac{1}{4}R_{2}^{2}+O(\epsilon,\beta^{4})R_{2}^{2}.
\int_{\mathbb R^N}\left(\frac{f(t_{n}v_{n})}{t_{n}}- f(v_{n})\right)v_{n}=\int_{\mathbb R^N}(V_{\infty}- V(\varepsilon x))v_{n}^{2}-\int_{\mathbb R^N}\phi_{\varepsilon, v_n}v_n^2+o_{n}(1),
A\Psi=(\lambda_2-\lambda_1)\Psi, A=\frac{2}{\xi+\eta}\left( \eta B_\xi - \xi B_\eta \right) +\frac{\xi-\eta}{\xi\eta}\tilde{\mathcal{I}}-\frac{\xi-\eta}{\xi+\eta}\gamma-\frac{F}{2}\xi\eta.
\mu(F)= \int_F b_F \,d\mu = \int_{F\setminus \bigcup_i Q^1_i}b_F \,d\mu + \sum_i \int_{Q^1_i}b_F \,d\mu &\leq \mu \Big(F \setminus \bigcup_i Q^1_i \Big)^{1/q'}\Big( \int_F|b_F|^q \,d\mu \Big)^{1/q}+ \frac{1}{2}\sum_i \mu(Q^1_i) & \leq A^{1/q}\mu \Big(F \setminus \bigcup_i Q^1_i \Big)^{1/q'}\mu(F)^{1/q}+\frac{1}{2}\mu(F),
\leq \frac{1}{(\log N)^{2}}\displaystyle\int_{N}^{N^2}\Big\{\frac{1}{(n-1)}+\frac{1}{2(n-1)^2}\Big\}dn & \leq \frac{1}{\log N}+ \frac{N}{2(N^2-1)(\log N)^{2}},
\mathcal{T}^*_{(j,-l)}\mathcal{T}_{(k,-m)}g(\xi)=\int \mathcal{K}_{(j,-l),(k,-m)}(\xi,\eta)g(\eta)d\eta &=\int \mathcal{K}_{(0,j-l),(1,j-m)}(\tilde{\xi},\tilde{\eta})g_{-j}(\tilde{\eta})d\tilde{\eta}=\mathcal{T}^*_{(0,j-l)}\mathcal{T}_{(1,j-m)}g_{-j}(\tilde{\xi}).
E_{l}^{0}\left(\mathbf{k}\right) =\left|\mathbf{k}\right|^{2p},l=1,\dots,n,\quad\mathbf{k}\in\mathrm{clos}\,\widetilde{\Omega}, E_{1}^{0}\left(\mathbf{k}\right) \geqslant r^{2p},\quad\mathbf{k}\in\mathrm{clos}\,\widetilde{\Omega}\setminus\mathcal{B}\left(r\right),\quad0<r\leqslant r_{0}.
Op_h^w(B(x,\xi)(x+p'(\xi))^2)v_{\Lambda}^{\Sigma}= h Op_h^w\Big(B(x,\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})(x+p'(\xi))\Big)\mathcal{L}v^{\Sigma}+ hOp_h^w(r_0-r'_0)v^{\Sigma}\,,
\Sigma_2(u)\le\sum_{j=1}^\infty\frac{S}{S_1}u^{{2}/{\alpha}}\mathcal{H}_{\tilde{Y}_2}^0[S_1,T_1]\exp \bigl(-b(jT_1)^{\beta}\bigr)\Psi(u) \bigl(1+o(1)\bigr).
X_{m n p q r s}= \frac{1}{\prod 2 \omega}\sum_{t}\frac{1}{2 \omega_{t}}( \lambda_{m s r t}\lambda_{t q p n}+ \lambda_{n m s t}\lambda_{t r q p}\vspace{- 6 p t}+ \lambda_{p n m t}\vspace{- 8 p t}\lambda_{t s r q})
a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdot\cdot\cdot+a_{0}a_{1}a_{2}\cdot\cdot\cdot a_{n}=\frac{a_{0}}{1+}\frac{-a_{1}}{1+a_{1}+}\frac{-a_{2}}{1+a_{2}+}\cdot\cdot\cdot\frac{-a_{n}}{1+a_{n}}.
\begin{dcases}\sum_{e\in E(G)}w_i(e)=1, \sum_{e:\,v\in e}B_i(v,e)=1,\ \v\in V(G), w_i(e)^{p_i-r}\prod_{v\in e}B_i(v,e)=\alpha_i,\ \e\in E(G).\end{dcases}
P\Bigl[|\xi|^2\leq \prod_{j=1}^n (1+\epsilon_j)^{-\alpha_j}\Bigr] =1-\exp\bigl(-e^{-\sum\limits_{j=1}^n \alpha_j\log(1+\epsilon_j)}\bigr) \simeq e^{-\sum\limits_{j=1}^n \alpha_j\log(1+\epsilon_j)}\simeq e^{-c\sum\limits_{j=1}^n L_j}\ .
\delta \left( \frac{U( \cdot ) \, \Lambda (t)}{A_1 \big( X(\cdot -r) \big) \, X( \cdot )}\, \mathbb{I}_{[0 \vee (t-r),t]}( \cdot ) \right) = \frac{W(t)}{x \, A_1 ( x )}+ \frac{A_1^\prime (x )}{A_1(x)}\big( W(t)^2 -t \big) - A_1 (x) \, t \, W(t).
\|\phi(a_i)b-b\| =\|\phi(a_i)b-b\|\|k\| &=\bigl\|\bigl(\phi(a_i)b-b\bigr)\otimes k\bigr\| &=\|\phi(a_i)b\otimes k-b\otimes k\| &=\|\psi(a_i\otimes 1)(b\otimes k)-b\otimes k\| &\to 0.
\left( \partial_\mu \phi \right)\left( \partial_\nu \phi \right)= e^{2\phi}\left( \partial_\mu a \right) \left( \partial_\nu a \right)\ ,\qquad\partial_\mu \left( e^{2\phi}\partial_\mu a \right) = 0\ ,\qquad\partial^2 \phi = - e^{2\phi}\left( \partial_\mu a \right) \left( \partial_\mu a \right)\ .
p_{i,j}^{(t+1)}= \begin{dcases}(1-p_0) p_{0,0}^{(t)}\mbox{if}\; i=j=0; (1-p_j^{}) p_{i,j}^{(t)}\,+\, p_{j-1}^{}p_{i,j-1}^{(t)}\mbox{if}0 < i \le j \le n j < t; 0 \mbox{otherwise}; \end{dcases}
\left|D^k\left(\partial_{\xi}^{\beta}f(x,\xi)\right)+D^k\left(\partial_{\xi}^{\beta}f(y,\xi)\right)-2D^k\left(\partial_{\xi}^{\beta}f(\frac{1}{2}(x+y),\xi)\right)\right|\leq M|x-y|, |k|_1=l-1,
\int_{0}^{\infty}d k k^{{\mu}- 1}\mathrm{s i n}( a k + \frac{b}{k}) ={\pi}\left( \frac{b}{a}\right)^{\frac{\mu}{2}}\{\mathrm{c o s}\frac{{\mu}{\pi}}{2}J_{\mu}( 2 \sqrt{a b}) - \mathrm{s i n}\frac{{\mu}{\pi}}{2}N_{\mu}( 2 \sqrt{a b}) \}
\partial_{x_Q}^{e_s}f({\bf x}): = \nabla f({\bf x}) \cdot{\mathbf v}=\frac{1}{\sqrt{|Q|}}\sum_{j\in Q}\frac{\partial f}{\partial x_{j,s}}({\bf x}) =\Big(\frac{1}{\sqrt{|Q|}}\sum_{j\in Q}\partial_{x_{j,s}}f\Big)({\bf x})\,.
[\alpha,\beta]:= \alpha \circ \beta - (-1)^{|\alpha| \cdot |\beta|}\beta \circ \alpha, &(\alpha \circ \beta) (a_0, \ldots, a_{|\alpha| +|\beta|}):= \sum_{r=0}^{|\alpha|}(-1)^{r |\beta|}\alpha (a_0, \ldots, a_{r-1}, \beta(a_{r}, \ldots, a_{r+|\beta|}), a_{r+|\beta|+1}, \ldots, a_{|\alpha| +|\beta|}).
v^{\prime}* v^{\prime \prime}= ( b_{i}^{\prime}+ b_{i}^{\prime \prime}, l_{\pm ( i , i + 1 )}^{\prime}e^{- ( h / 2 ) b_{i + 1 ; n - 1}^{\prime \prime}}+ e^{- ( h / 2 ) b_{i ; n - 1}^{\prime}}l_{\pm ( i , i + 1 )}^{\prime \prime}) .
f_{A_{+}}(x)=\frac{2\sqrt{6}}{x^{2}}\sum_{j=1}^{\infty}v_{j}^{2/3}e^{-v_{j}}U(-\frac{5}{6},\frac{4}{3};v_{j})withv_{j}=\frac{2|a_{j}|^{3}}{27x^{2}}
a(G')\geq a(s^k(G))\geq \frac{\min\left\{\frac{2}{\Delta}a(G),8\right\}}{4^k}\geq \frac{\min\left\{\frac{2}{\Delta}a(G),8\right\}}{4^{\log_2(m+1)+1}}=\frac{\min\left\{\frac{1}{\Delta}a(G),4\right\}}{2 (m+1)^2}.
a^i_p\mapsto \begin{cases}(-1)^{p+1}t^{0,-p-1}, i=0; \eta^{i\beta}f_{\beta,p}, 1\leq i\leq n,\, p\geq 0; (-1)^{p+1}t^{i,-p-1}, 1\leq i\leq n,\, p<0; f_{0,p}, i=n+1. \end{cases}
\| \Phi (t) \| \le C_{8}|t|^3, |t| \le t^0;C_{8}= \beta_{8}\delta^{-1}\bigl( \|X_1\|^4 c_*^{-1/2}+ \|X_1\|^6 c_*^{-3/2}+ \|X_1\|^8 c_*^{-5/2}\bigr).
\frac{4}{n+4}=1-(1+4n^{-1})^{-1}=1-\sum_{i=0}^{\infty}(\begin{matrix}-1\\i\end{matrix})4^{i}n^{-i}=\frac{4}{n}-\frac{16}{n^{2}}+\frac{64}{n^{3}}-\cdot\cdot\cdot
\vec{\tilde{q}}= \left( \begin{array}{c}{\tilde{q}^{( 1 )}}\\{\vdots}\\{\tilde{q}^{( N )}}\\ \end{array}\right) , \vec{\tilde{p}}= \left( \begin{array}{c}{\tilde{p}^{( 1 )}}\\{\vdots}\\{\tilde{p}^{( N )}}\\ \end{array}\right) .
b': Y_\xi\times X_\xi\to k, a': Y_\xi\to k, b'(\lambda,\alpha) =\exp{\Bigg(-\pi i (x_1'(\lambda),\ldots,x_r'(\lambda))S_\xi\mathfrak{d}^{-1}\begin{pmatrix}v_1(\alpha) \vdots v_r(\alpha)\end{pmatrix}\Bigg)}, a'(\lambda) =\exp\Bigg{(-1/2\pi i (x_1'(\lambda),\ldots,x_r'(\lambda))S_\xi'\begin{pmatrix}x_1'(\lambda) \vdots x_r'(\lambda)\end{pmatrix}\Bigg)},
\left( \begin{array}{c c}{{n_{1}^{\prime}}}&{{n_{2}^{\prime}}}\\{{l_{1}^{\prime}}}&{{l_{2}^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a}}&{{c / 2}}\\{{2 b}}&{{d}}\\ \end{array}\right) \left( \begin{array}{c c}{{n_{1}}}&{{n_{2}}}\\{{l_{1}}}&{{l_{2}}}\\ \end{array}\right)
\Omega_{j}( x_{n}) \overline{{\Omega}}_{i}( x_{m}) = \delta_{i , j}\frac{\delta_{m , n}}{\Delta x_{n}}\pm q^{\pm 1}\hat{{\cal R}}_{i k j l}( q ) \overline{{\Omega}}_{l}( x_{m}) \Omega_{k}( x_{n})
\partial_{+}\partial_{-}( e^{- 2 \phi}) = - \lambda^{2}+ \frac{\mu^{4}}{1 6 \beta}{}\mathrm{s e c h}^{4}\left[ \frac{\mu}{2 \sqrt 2}( \alpha ( x^{+}- x_{0}^{+}) - \frac{1}{\alpha}( x^{-}- x_{0}^{-}) ) \right] ,
[L_3 x]_{i,j}= \cos\bigg( \frac{2\pi}{N_1}\bigg)\bigg[(x_{i+1,j}- x_{i,j}) + (x_{i-1,j}- x_{i,j})\bigg] + \cos\bigg(\frac{2\pi}{N_2}\bigg)\bigg[(x_{i,j+1}- x_{i,j}) + (x_{i,j-1}- x_{i,j})\bigg].
S_B(n,k,q) = q^{2k-1}(1+q)\sum\limits_{\pi \in B_{\subseteq}([n-1],k-1)}q^{m(\pi)}+(1+q)\sum_{i=1}^kq^{2i-1}\sum\limits_{\tau' \in B_{\subseteq}([n-1],k)}q^{m(\tau')}+\sum\limits_{\pi \in B_{\subseteq}([n-1],k)}q^{m(\pi)}= [2k+1]_qS_B(n-1,k,q)+q^{2k-1}(1+q)S_B(n-1,k-1,q).
\begin{array}{ll}{\big |}\delta F^{\nu}_{k+1}{\big |}^{t_0,\Delta,s}_{sup,1}={\big |}\delta Q^S(F^{\nu}_{k},F^{\nu}_{k})\ast^g \Gamma^v_{\nu}{\big |}^{t_0,\Delta,s}_{sup,1}\leq\frac{1}{2}{\big |}\delta F^{\nu}_{k}{\big |}^{t_0,\Delta,s}_{sup,1},\end{array}
M_{A_{0}Y_{1}}^{2}= 2 \left( \begin{array}{c c}{- H}&{- 2 i v}\\{2 i v}&{H}\\ \end{array}\right) M_{Y_{8}Y_{9}}^{2}= 2 \left( \begin{array}{c c}{H}&{- 2 i c}\\{2 i c}&{H}\\ \end{array}\right) .
\Gamma_{(S,\xi)}(t) = \phi_{S^{1/2}}(\Gamma_{(I_r,{\rm D}\phi_{S^{-1/2}}(\xi) )}(t)) &= S^{\frac{1}{2}}\exp (t S^{-\frac{1}{2}}\xi S^{-\frac{1}{2}}) S^{\frac{1}{2}}.
R(\kappa) =\sum_{j=-2}^{\beta-6}\kappa^jG_j+\mathcal O(\kappa^{\beta-5}), G_j =G_{0,j}-\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge -2,j_3\ge 0}{j_1+j_2+j_3=j}}G_{0,j_1}vE_{j_2}v^*G_{0,j_3},
\partial_t \partial^{\alpha}\bar{w}- \partial_y^2 \partial^{\alpha}\partial\bar{w}= - \partial^{\alpha}\big((u^s + \tilde{u}_1)\partial_x \bar{w}+ \tilde{v}_2 \partial_y\bar{w}&\qquad\qquad+( u^s_{yy}+ \tilde{w}_{1, y})\bar{v}+(\partial_x\tilde{w}_2) \bar{u}\big).
\eta(z\cdot g) =|\det zg|\cdot h\left(e_r\cdot zg\right)^{-r}= |\det zg|\cdot h(te_r\cdot g)^{-r}&= |\det z|\cdot |\det g|\cdot |t|^{-r}\cdot h(e_r\cdot g)^{-r}= |t|^r\cdot |t|^{-r}\cdot \eta(g) = \eta(g)
\lim_{q \to 1}< ( a^{*})^{k_{1}}( a )^{k_{2}}( c^{*})^{k_{3}}( c )^{k_{4}}>_{q}= \int ( \alpha^{*})^{k_{1}}( \alpha )^{k_{2}}( \gamma^{*})^{k_{3}}( \gamma )^{k_{4}}d u ,
\left( u_{\underline{{{m}}}}^{~ \underline{{{a}}}}\right) = \left( u_{\underline{{{m}}}}^{a}, u_{\underline{{{m}}}}^{~ i}\right) ~ ~ \in ~ ~ S O ( 1 , D - 1 ) ~ \Leftrightarrow ~ u_{\underline{{{m}}}}^{~ \underline{{{a}}}}\eta^{\underline{{{m}}}\underline{{{n}}}}u_{\underline{{{n}}}}^{~ \underline{{{b}}}}= \eta^{\underline{{{a}}}\underline{{{b}}}}
\begin{cases}\mathbf{S}= \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1 \}, \mathbf{n}= (0, 1), \mathbf{P}= \{(x_1, 0) \in \mathbb{R}^2 \mid x_1 \in \mathbb{R}\}.\end{cases}
S_{BH}= 2\pi \left( \sqrt{N_5^{(1)}}+ \sqrt{\bar{N}_5^{(1)}}\right)\left( \sqrt{N_5^{(2)}}+ \sqrt{\bar{N}_5^{(2)}}\right)\left( \sqrt{N_5^{(3)}}+ \sqrt{\bar{N}_5^{(3)}}\right)\left( \sqrt{n}+ \sqrt{\bar{n}}\right)
\Gamma_{\underline{{\alpha}}\underline{{\beta}}}^{a}= \left( \begin{array}{c c}{{\gamma_{\alpha \beta}^{a}\delta_{q}^{r}}}&{{0}}\\{{0}}&{{\gamma^{a \alpha \beta}\delta_{q^{\prime}}^{r^{\prime}}}}\\ \end{array}\right) \, , \quad a = 0 , 1 , . . . , p \, ,
\langle J^{5}( x , y ) \rangle = - \frac{g_{5}^{2}}{3 2 \pi^{2}}F_{\mu \nu}\widetilde{F}^{\mu \nu}\sum_{n}\frac{1}{M_{n}}\phi_{n}^{{}_{L}}( y ) \phi_{n}^{{}_{R}}( y ) \quad .
\left\{R_{\pm 0 A ( u )}, R_{\pm 0 ( v )}^{B}\right\}= 0 \, , \qquad \left\{R_{\pm 0 A ( u )}, R_{\mp 0 ( v )}^{B}\right\}= \delta_{u v}\frac{1}{2}( 1 \mp i \Omega \Pi )_{A}{}^{B}\, ,
1 \otimes r^{\dot{I}}_{\dot{J}}\otimes \Xi^{\lambda_1}_{\lambda_2}\left( \bar{\phi}^{\rho_1}\otimes s^{\dot{K}}\otimes \phi^{\rho_2}\right) = \delta_{\lambda_2}^{\rho_2}\sum_{\dot{K}_1 \dot{K}_2 = \dot{K}}\delta^{\dot{K}_2}_{\dot{J}}\bar{\phi}^{\rho_1}\otimes s^{\dot{K}_1 \dot{I}}\otimes \phi^{\lambda_1}.
\mbox{$\cal L$}_{{\rm eff}}=\frac{|eB |}{2\pi}\sum_{n=0}^{\infty}\int \frac{dk_{0}}{2 \pi i}\left[ \ln (k_{0}-E_{n,+}+\mu +\frac{b}{2})+\ln (k_{0}-E_{n,-}+\mu -\frac{b}{2})\right].
\mathbb{E}_{\eta^{(\beta)}}[\textbf{X}] = \mathbb{E}_{\widetilde{p}_n^{(\beta)}}[{\bf{X}}]~~~ \mathbb{E}_{\eta^{(\beta)}}[(\textbf{X}\textbf{X}^T)]=\mathbb{E}_{\widetilde{p}_n^{(\beta)}}[(\textbf{X}\textbf{X}^T)].
\lambda_{j|v}^{r,s}(z_j,s) \equiv_v -\Lambda_{M_j}^{r,s}(g_j^{v-1}(z_j,s),h^{v-1}(s))+\sum_{\alpha,\beta=1}^n \Lambda_{N_j}^{\alpha,\beta}(z_j,s)\frac{\partial{g_j^r}^{v-1}}{\partial z_j^{\alpha}}\frac{\partial{g_j^s}^{v-1}}{\partial z_j^{\beta}}
\begin{aligned}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}q_{\beta}(0,t) =-\alpha \left(e^{\theta}-1\right)q_{\beta}(0,t), \frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}q_{\beta}(n,t) =-\alpha \left(e^{\theta}-1\right)q_{\beta}(n,t)+\alpha\sum_{j=1}^{n}\frac{\theta^{j}}{j!}q_{\beta}(n-j,t),\n\ge 1,\end{aligned}
{}_3F_2\bigg[\begin{matrix}\alpha \alpha \beta &1 \alpha-\beta+1\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(1+\frac12\alpha)\Gamma(1+\alpha-\beta)\Gamma(1-\frac12\alpha-\beta)}{\Gamma(\alpha+1)\Gamma(1-\frac12\alpha)\Gamma(1-\beta)\Gamma(1+\frac12\alpha-\beta)}.
\big\langle \Phi'(\vec p)^*\vec g,\widehat{\vec p}\big\rangle_{( L^\infty(D)^5)'\times L^\infty(D)^5}= \big\langle F'(V(\vec p))^*\vec g, V'(\vec p)\widehat{\vec p}\big\rangle_{\mathcal{L}(X)'\times \mathcal{L}(X)}begin{align*}1mm] =\int_0^T\! \big\langle V'(\vec p)\widehat{\vec p}\big(u'(t) +Qu(t)\big),w(t)\rangle_X\,\d t
\Psi_{2}(X,Y) =-9 X^3 Y^3-12 X^3 Y^2+X^3 Y+2 X^3-12 X^2 Y^3 +8 X^2 Y^2+10 X^2 Y+X Y^3+10 X Y^2-X Y+2 Y^3 \Psi_{3}(X,Y) =435X^4Y^4+231X^4Y^3+231X^3Y^4+45X^4Y^2-385X^3Y^3+45X^2Y^4 &-39X^4Y-63X^3Y^2-63X^2Y^3-39XY^4+4X^4+9X^3Y+123X^2Y^2+9XY^3 &+4Y^4+15X^2Y+15XY^2-XY.
\frac{\partial}{\partial x_j}\int_D \frac{f(x,\theta)}{r^{m-1}}u(y)dy= \int_D u(y) \frac{\partial}{\partial x_j}\left[\frac{f(x,\theta)}{r^{m-1}}\right]dy-u(x)\int_S f(x,\theta)\cos(r_{x_j})ds.
g_1=(z_1-z_{11}(p))(z_1-z_{12}(p))\dots (z_1-z_{1\,k}(p))\,, & g_2=(z_2-z_{21}(p))(z_2-z_{22}(p))\dots (z_2-z_{2\,k}(p))\,, & \vdots & g_{n}=(z_{n}-z_{n\,1}(p))(z_{n}-z_{n\, 2}(p))\dots (z_{n}-z_{n \,k}(p))\,.
\Pi^*(u) ={1\over 2}u^{2-2ab/(a+b)}\Big ( 1+ u^{2(ab-\min\{a,b\})/(a+b)}\Big ) &={1\over 2}u^{2-2ab/(a+b)}\big ( 1+ o(1) \big ) \textrm{when}u\downarrow 0.
A_{\mu}^{( 3 )}= \mp{\frac{2 i}{g}}( H_{2}{\frac{v_{\mu}^{1}}{\rho_{1}}}+ H_{3}{\frac{v_{\mu}^{2}}{\rho_{2}}}) + \kappa ( E_{2 1}^{\pm}R_{\mu}^{1}+ E_{3 1}^{\pm}R_{\mu}^{2}) \delta ( R^{1}\cdot R^{2}) .
\omega\circ E(\theta(x_\lambda)^*\theta(x_\lambda)) =\omega\circ E(\theta(x_\lambda^*x_\lambda))=\omega\circ \theta\circ\theta^{-1}\circ E \circ \theta(x_\lambda^* x_\lambda)=\omega\circ\theta\circ E(x_\lambda^* x_\lambda)
z^\alpha K_\alpha(z) = 2^{\alpha-1}\Gamma(\alpha) \left[ 1 + \sum_{j=1}^\infty \frac1{j! (1-\alpha)_j}\left(\frac{z}2\right)^{2j}- \frac{\Gamma(1-\alpha)}{\Gamma(1+\alpha)}\left(\frac{z}2\right)^{2\alpha}\sum_{j=0}^\infty \frac1{j! (1+\alpha)_j}\left(\frac{z}2\right)^{2j}\right],
2 d^2 H_{d,k}+ 2d I_{d,k}- d^2 \left(d + \frac{1}{2}(k+1)\right) T_{d,k+1}= 2 d^3 H_{d,k}- d^2 I_{d,k}= -2 I_{d,k}+ \left(d + \frac{1}{2}(k + 1)\right) T_{d,k+1}= ,
\delta_{+,q,q'}:=\max\big(\sup_{(0,+\infty)}|\omega_{+,q,q'}|,\sup_{t\in(0,+\infty)}(1+{|x|\over \sqrt{2}}-|q|+t{|v|\over 2^{3\over 2}})|\dot \omega_{+,q,q'}(t)|\big)\le 2|q-q'|,
m_{\varphi,\overline{\mathcal{O}(1)},X_0}(r) = \frac{1}{2\pi}\int_0^{2\pi}\log \sqrt{1 + \sum_{j=1}^{n}|f_j(re^{i\theta})|^2}d\theta \le \sum_{j=1}^n \frac{1}{2\pi}\int_0^{2\pi}\log \sqrt{1 + |f_j(re^{i\theta})|^2}d\theta = \sum_{j=1}^nm_{f_j}(r)
x_{s,t}\sim\sum_{k=1}^{\infty}\sum_{i_{1},\ldots,i_{k}=1}^{d}V_{(i_{1},\ldots,i_{k})}(x_{s})\int_{s<u_{1}<\cdots<u_{k}<t}dw_{u_{1}}^{i_{1}}\cdots dw_{u_{k}}^{i_{k}},
\sum_{C\in \mathcal P(X)}\int_C K_0 dvol_0 = \sum_{C\in \mathcal P(X)}\int_C -1 \hskip 2pt dvol_0 & = - Vol(X, g_0) & \leq - Vol(X, g_1) & = \sum_{C\in \mathcal P(X)}\int_C -1 \hskip 2pt dvol_1 & < \sum_{C\in \mathcal P(X)}\int_C K_1 dvol_1.
H:=-\mu \sum_{j=1}^{N}c^{\dagger}_jc_j-\sum_{j=1}^{N-1}\,\Big(\tau \, c_j^{\dagger}c_{j+1}+\tau \, c^{\dagger}_{j+1}c_j+\Delta c_jc_{j+1}+\Delta c^{\dagger}_{j+1}c^{\dagger}_j)
\mathcal{A}\!\left[\!\begin{array}{c}\tilde{w}\tilde{p}\end{array}\!\right]\!=\!\left[\begin{array}{c}\frac{1}{\rho}\tilde{p}T\tilde{w}_{zz}-kg(z)\int_{0}^{L}g(z)\frac{1}{\rho}\tilde{p}\mathrm{d}z\end{array}\right]\!=\!\left[\begin{array}{c}f h\end{array}\right],
E[L^2]= \sum_i f(x_i)\delta \big(\log_2{\big(f(x_i)\delta\big)}\big)^2 = \sum_i f(x_i)\delta \big(\log_2{f(x_i)}+\log_2{\delta}\big)^2 = \sum_i f(x_i)\delta\big(\log_2^2{f(x_i)}+2\log_2{f(x_i)}\log_2{\delta}\big)+\log_2^2{\delta}.
K^{(\mathrm{soft})}(\kappa, \pi;u, v) =\begin{bmatrix}DS^{(\mathrm{soft})}(\kappa, \pi;u,v) S^{(\mathrm{soft})}(\kappa, \pi;u,v) - S^{(\mathrm{soft})}(\kappa, \pi;v,u) IS^{(\mathrm{soft})}(\kappa, \pi;u,v) \end{bmatrix}.
\int d^{R}x\int d^{R}y\left[ X^{\alpha},X^{\beta}\right] W[J,K,K^{*}]=i\varepsilon \int d^{R}x\int d^{R}y\left\{X^{\alpha}<<(\Delta O)^{\beta}>>-X^{\beta}<<(\Delta O)^{\alpha}>>\right\}
\int \prod_{a = 1}^{N}\prod_{j = 1}^{n_{a}}d^{D}z_{j}^{( a )}\bigg | \prod_{b < c}^{N}\; f^{( b , c )}\bigg | \; \leq \; \prod_{a < b}^{N}\Bigg [ \int \; \prod_{i = 1}^{n_{a}}d^{D}z_{i}^{( a )}\prod_{j = 1}^{n_{b}}d^{D}z_{j}^{( b )}\; \bigg | f^{( a , b )}\bigg |^{N - 1}\Bigg ]^{\frac{1}{N - 1}}\; .
v_0 = w_1 = la_1 b_1 m_1 c_1 m_2d_1e_1f_1r, &v_1 = la_2s_1^{-1}b_1m_1c_1m_2d_1e_1f_1r, &v_2 = la_2b_2m_1c_1m_2d_1e_1f_1r,
\beta = - \frac{Q}{2}+ \sqrt{\frac{Q^{2}}{4}- 2 + 2 h}, d = 1 - \frac{\frac{Q}{2}- \sqrt{\frac{Q^{2}}{4}- 2 + 2 h}}{\frac{Q}{2}- \sqrt{\frac{Q^{2}}{4}- 2}},
\left. \begin{array}{l l l l}{{T r \, [ \, Q_{0}^{3}\, ] \, = 4 \,}}&{{\, T r \, [ \, Q_{1}^{3}\, ] \, = 4 \,}}&{{\, T r \, [ \, Q_{2}^{3}\, ] \, = - 8 \,}}&{{\, T r \, [ \, Q_{3}^{3}\, ] \, = 0}}\\ \end{array}\right.
L = \lim_{j\rightarrow \infty}\frac{C_{dN_{i_{j}}}(H_{dN_{i_j}})}{dN_{i_{j}}}&= \lim_{j\rightarrow \infty}\int_{\mathbb{R}}\log \left( 1+ Px\right) F^{H_{dN_{i_j}}(\Phi_{N_{i_{j}}}\otimes{\rm I}_{N_{i_j}})H_{dN_{i_j}}^{\ast}/N_{i_{j}}}(x) &= \int_{\mathbb{R}}\log \left( 1+Px\right) F^{\mathbf{H}\Phi_{\infty}\mathbf{H}^{\ast}}(x).
\zeta_{A}^{\prime}( 0 ) = \sum_{n}\left\{- \beta ( E_{n}+ \mu ) - \beta ( E_{n}- \mu ) - 2 \ln \left( 1 - e^{- \beta ( E_{n}+ \mu )}\right) - 2 \ln \left( 1 - e^{- \beta ( E_{n}- \mu )}\right) \right\}
\nabla \Phi_{\gamma_1 \ldots \gamma_k \dot\gamma_1 \ldots\dot\gamma_k}= ie^{\alpha\dot\alpha}\Phi_{\gamma_1 \ldots \gamma_k \alpha\dot\gamma_1 \ldots \dot\gamma_k \dot\alpha}- i k^2 e_{(\gamma_k|(\dot\gamma_k|}\Phi_{|\gamma_1 \ldots \gamma_{k-1}) |\dot\gamma_1 \ldots\dot\gamma_{k-1})}.
W(\Delta,z)p_{\Gamma^{\prime}}=\sum_{\Gamma\vdash d}\frac{d!}{|C_{\Gamma^{\prime}}|}z^{d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)}\mu_{0}^{h_+,d}(\Gamma^{\prime},\Delta,\Gamma)p_{\Gamma}.
U_k'\cdot \varphi^{r_1'\cdots, r_s'}(X_1,\cdots, X_n, Y_1,\cdots, Y_n) &= \varphi^{r_1'\cdots r_s'}(X_1,\cdots, X_n, Y_1,\cdots, Y_k+2\pi ,\cdots, Y_n) \cdot U_k'.
( - 1 )^{\frac 1 2 ( n + 1 )}{\frac{1}{n !}}\epsilon^{\mu_{1}\cdots \mu_{2 n + 1}}\partial_{\mu_{n + 1}}B_{\mu_{n + 2}\cdots \mu_{2 n + 1}}- m \sqrt{- g}B^{\mu_{1}\cdots \mu_{n}}= 0 .
B=\left[\begin{array}{c}B_0 B_{-1}B_1 \overline B_1 \vdots B_n \overline B_n\end{array}\right],U=\left[\begin{array}{ccccccc}U_0 U_{-1}U_1 \overline U_1 \cdots U_n \overline U_n\end{array}\right].
N\rho ([\mathfrak{A}_c i_1 (F), \mathfrak{A}_c i_2 (G)]_{\mathfrak{A}_c(\mathcal{I})}) = [\mathfrak{A}\chi_1 \circ N \rho_1 (F), \mathfrak{A}\chi_2 \circ N \rho_2 (G)]_{\mathfrak{A}(\mathcal{M})}= 0.
\left(\frac{i}{2}\right) dz^{(it_1)}\wedge \left(\left(\frac{1}{2h}+t_2\right)dh-id\theta\right) = \left(\frac{i}{2}\right)\left(\frac{(p-iq)(1+t_1)}{p^2+q^2}+t_2(p-it_1q)\right) \omega.
\Gamma^1_{11}= \frac{1}{2\Delta}(GE_1-2FF_1+FE_2), \Gamma^2_{11}= -\frac{1}{2\Delta}(FE_1-2EF_1+EE_2), & \Gamma^1_{12}= -\frac{1}{2\Delta}(FG_1-GE_2), \Gamma^2_{12}= \frac{1}{2\Delta}(EG_1-FE_2), & \Gamma^1_{22}= -\frac{1}{2\Delta}(GG_1-2GF_2+FG_2), \Gamma^2_{22}= \frac{1}{2\Delta}(FG_1-2FF_2+EG_2),
\frac{d}{dt}\Gamma+\Big(| u|^2_{D^2}+\frac{1}{\sigma}| H|^2_{D^2}\Big) \leq C\frac{d}{dt}\int_{\Omega}\Big(\Big(P+\frac{1}{2}|H|^2\Big) u- H\cdot \nabla u\cdot H\Big)x &+C\Big(\Gamma+\frac{1}{\sigma}|\nabla H|^2_2\Big)(|\nabla u|^2_2+|\nabla \theta|^2_2+|D(u)|_\infty+1).