harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
v_t+(-1)^m\sum_{|\alpha|=|\beta|=m}\mu^{\alpha_d+\beta_d-2m}D^\alpha (\tilde{A}^{{\alpha}{\beta}}D^\beta v){+\mu^{-2m}\lambda v}&=\sum_{|\alpha|\le m}\mu^{\alpha_d-2m}D^\alpha \tilde{f}_\alpha
\begin{aligned}A Q_{1,N}+ W Q_{2,N}= Q_{1,N}F_N + f_{N+1,N}q_{1,N+1}e_N^{\mathrm T}H_N Q_{2,N}= Q_{2,N}F_N + f_{N+1,N}q_{2,N+1}e_N^{\mathrm T}. \end{aligned}
\zeta_\delta(x)=PW_{\delta}(x)-W_{\delta}(x)+\delta^{-\frac{n-4}{2}}\Big(\varphi_0\Big(\frac{e_n-x}{\delta}\Big)-\varphi_0\Big(\frac{e_n+x}{\delta}\Big)\Big) x\in{B}=B_1(0),
=\rho_{t}+\rho_{r}\frac{x}{r}\frac{Fx}{r}+\rho\left( F_{r}\frac{x}{r}\right) \frac{x}{r}+\rho\frac{F}{r}-\rho Fx\frac{x}{r^{3}}& +\rho_{r}\frac{y}{r}\frac{Fy}{r}+\rho\left( F_{r}\frac{y}{r}\right)\frac{y}{r}+\rho\frac{F}{r}-\rho Fy\frac{y}{r^{3}}
\int_{1}^{\infty}\frac{1}{w_1^{1+sp}}\mathrm{d}w_1\int_{0}^{\infty}\frac{y^{n-2}}{\left(1+y^2\right)^{\frac{n+sp}{2}}}\mathrm{d}y=\frac{1}{sp}\int_{0}^{\infty}\frac{y^{n-2}}{\left(1+y^2\right)^{\frac{n+sp}{2}}}\mathrm{d}y.
(s_{j}\hdots s^{2}_{i}\hdots s_{j})(s_{j-1}\hdots s^{2}_{i}\hdots s_{j-1}) =(s_{j}\hdots s_{i+1})s^{2}_{i}\hdots s_{j-1}s_{j}s_{j-1}\hdots s^{2}_{i}\hdots s_{j-1}&=(s_{j}\hdots s_{i+1})s^{2}_{i}\hdots s_{j}s_{j-1}s_{j}\hdots s^{2}_{i}\hdots s_{j-1}&=(s_{j}\hdots s_{i+1})s_{j}s^{2}_{i}\hdots s_{j-2}s_{j-1}s_{j-2}\hdots s^{2}_{i}\hdots (s_{j}s_{j-1}).
\mathcal{M}_{\omega,c}^*(\eta)=\begin{cases}\{\Psi \in \mathcal{M}_{\omega,c}\ |\ \omega Q(\Psi)+cP(\Psi)\ge \eta\}, \d=1, \{\Psi \in \mathcal{M}_{\omega,\mathbf{c}}\ |\ \mathbf{c}\cdot \mathbf{P}(\Psi)\ge \eta\}, \d=2, \emptyset, \d=3.\end{cases}
\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 \sum_{n=0}^{2^k-1}1_{[-B-C,B+C]}\left(t+ \sum_{i=0}^{n-1}f(T^i x) \right)
(J_m^k f) (x, r \xi) = \frac{r^{m-k}}{|r|}(J_m^k f)(x, \xi) \mbox{for}r\neq 0 (J_m^k f)(x+ s \xi, \xi) = \sum\limits_{l=0}^k \binom{k}{l}(-s)^{k-l}(J_m^l f) (x, \xi) \mbox{for}s\in \mathbb{R}.
q^{k_{m}^{(s)}}>m^{2}(T_{m}+1)^{m-1}A_{m}^{(s)}B_{m}^{(s)}q^{m-1}\cdot 2(B_{m}^{(s)}q^{m})^{m}=:q^{m^{2}+m-1}D_{m}^{(s)}
\Phi ( z , \vec{x}, t ) \, = \, \sum_{p = 1}^{\infty}\, \int{\frac{d \vec{k}}{( 2 \pi )^{n - 1}}}\,{\frac{z^{n / 2}\, J_{\nu}( u_{p}z )}{R w_{p}( \vec{k}) \, J_{\nu \, + \, 1}( u_{p}R )}}\lbrace{{\bf a}_{p}( \vec{k}) \}e^{- i w_{p}( \vec{k}) t + i \vec{k}\cdot \vec{x}}\, \, + \, \, c . c . \rbrace \, ,
\hat \varrho_{ij}=\frac 1{a_i}\tilde \varrho_{ij}-\frac{b_i+1}{a_i}\hat \varrho_{i+1,j}-\frac{c_i}{a_i}\hat \varrho_{i+2,j}, i=N-3,N-2,\cdots, 1, &\hat \varrho_{N-1,j}= \frac 1{a_{N-1}}\tilde \varrho_{N-1,j},\hat \varrho_{N-2,j}=\frac 1{a_{N-2}}\tilde \varrho_{N-2,j}-\frac{b_{N-2}+1}{a_{N-2}}\hat \varrho_{N-1,j},
(1.g) = -\ \frac{i}{6(2\pi)^4}\mbox{tr}\int d^dk\ \frac{\stackrel{\displaystyle (Sk)}{\scriptscriptstyle (-p)\phantom{+}}\stackrel{\displaystyle (S(k+q))}{\scriptscriptstyle (-q)\phantom{+qqqqqqq}}\stackrel{\displaystyle (S(k-p))}{\scriptscriptstyle (p+q)\phantom{+qqqqqqq}}}{k^{2l}(k-p)^{2l}(k+q)^{2l}}
(\star) \leq \left( 1-c'\gamma_{n}\right)\sum_{k=2}^{p-1}\binom{p-1}{k}2^{p-2+k}\gamma_{n}^{k}\left( \frac{\sqrt{K_{p-1}K_{p+1-k}}}{n^{(p-k/2)\alpha}}+ \gamma_{n}^{2(p-1-k)}\frac{\sqrt{K_{k}K_{2}}}{n^{\frac{(k +2)\alpha}{2}}}\right) & = O \left( \frac{1}{n^{(p+1)\alpha}}\right),
\mathcal P^{\mathcal S}_\rho[u][\varphi]:=\int_\Omega D^2u:D^2\varphi+\tau \nabla u\cdot\nabla\varphi dx,\ \forall u\in{{H}}^{2,\mathcal S}_\rho(\Omega),\varphi\in{H}^2(\Omega).
\sum_{a=d_1+1}^{\infty}q^{-a}Z_{\tilde{g}_{1, a}}(s, \chi, (\mathcal{O}_K^{\times})^2)= q^{-e_1s}\sum_{a=d_1+1}^{\infty}q^{-a}c_1(\chi, a) := c_1(\chi)q^{-e_1s},
c^{\left(0,J\right)}_{ij}=K\left(d^{\left(I',J'\right)}_{i,j},\dots\right),\quad\mbox{where $K\left(d^{\left(I',J'\right)}_{i,j},\dots\right)$ is a constant defined by $d^{\left(I',J'\right)}_{i,j},\dots$.}
X(t) = \begin{cases}\eta(t) (-r \le t \le 0), \displaystyle \eta(0) + \int_0^t \hat{A}_0 \big( s, \{X(u) \, ; \, -r \le u \le s \}\big) \, \mbox{d}s + \int_0^t \hat{A}\big( s, X(s-r) \big) \, \mbox{d}W(s) (0 \le t \le T), \end{cases}
\div\langle Q(\lambda, \mu), \nu_{(t,x)}^\varepsilon\otimes\sigma_{(t,x)}^\varepsilon\rangle=\langle Q(\lambda, \mu), \nabla \nu_{(t,x)}^\varepsilon\otimes\sigma_{(t,x)}^\varepsilon\rangle+\langle Q(\lambda, \mu), \nu_{(t,x)}^\varepsilon\otimes\nabla \sigma_{(t,x)}^\varepsilon\rangle.
\left| e^{f(z_i)}- e^{f(z_{i-1})}\right|= e^{f(z_i)}\left| 1 - e^{f(z_{i-1}) - f(z_i)}\right|\leq \bigl\| e^f \bigr\|_{\infty}|f(z_i) - f(z_{i-1})| \, .
Z(s,f,\chi,\Delta_3)= \sum_{n=1}^{\infty}\int_{\pi^{n}O_v^\times \times \pi^{n}O_v^\times}\chi(ac \f(x,y))|f(x,y)|^s |dxdy|= \sum_{n=1}^{\infty}q^{-2n-4ns}\int_{O_v^{\times 2}}\chi(ac \ (\pi^{n}y^3-x^2)^2+\pi^{4n}x^4y^4) |(\pi^{n}y^3-x^2)^2+\pi^{4n}x^4y^4|^s|dxdy|.
\omega=-\frac 12+u-v,\quad\lambda_n=\frac{2n^2 + 2n + 1}{4}-\frac{(u-v)^2}{8}, &\kappa_n=-\frac{n^2 \left(- n^2 + \left(\frac{u-v}{2}\right)^2\right)^2}{4 (2 n - 1) (2 n + 1)}.
\chi( \widetilde{\mathfrak{m}}_{1}) \chi( \widetilde{\mathfrak{m}}_{2})=\chi( \widetilde{\mathfrak{m}}_{3})\chi( \widetilde{\mathfrak{m}}_{4})+\chi( \widetilde{\mathfrak{m}}_{5}) \chi( \widetilde{\mathfrak{m}}_{6}).
\sum_k \| \Gamma^{A,B}(a_k)(X)\|_2^2 = \sum_k \left\langle \Gamma^{A,B}(a_k)(X), \Gamma^{A,B}(a_k)(X) \right\rangle & = \sum_n \left\langle \Gamma^{A,B}(\overline{a_k})\Gamma^{A,B}(a_k)(X), X \right\rangle & = \left\langle \Gamma^{A,B}(\vert a \vert^2)(X), X \right\rangle & \leq \|\vert a \vert^2\|_\infty \|X\|_2^2 = \|a\|_\infty^2 \|X\|_2^2.
q^\mu \left\{\tilde{\Gamma}_\mu (q;p,p^\prime) -ie \mu^{\frac{\epsilon}{2}}(p^\prime +p)_\mu \tilde{D}\right\}= -e \mu^{\frac{\epsilon}{2}}\left\{( \tilde{\Gamma_2}(p^\prime)+i p^{\prime 2}\tilde{D})- (\tilde{\Gamma_2}(p)+i p^{2}\tilde{D})\right\}\; .
\lambda(p(\zeta),t) =\begin{cases}\dfrac{1}{4\pi t}\left(e^{-4\pi t\sin^{-1}p(\zeta)}-e^{-4\pi t (\pi-\sin^{-1}p(\zeta))}\right) \; t\neq 0 \pi - 2\sin^{-1}(p(\zeta)) t=0.\end{cases}
\frac{\omega^{\rho+1}-\omega^{\rho}}{\tau}+\frac{\kappa_{i_1+1}^{(1)}-\kappa_{i_1}^{(1)}}{\Delta x}+\frac{\kappa_{i_2+1}^{(2)}-\kappa_{i_2}^{(2)}}{\Delta y}+\frac{\kappa_{i_3+1}^{(3)}-\kappa_{i_3}^{(3)}}{\Delta z}=0,~a.s.,
\frac{\sigma_{\hat{g}}^{2}}{\hat{g}^{2}}\approx\frac{1}{\hat{g}^{2}}(\frac{\partial\hat{g}}{\partial L})^{2}\sigma_{L}^{2}+\frac{1}{\hat{g}^{2}}(\frac{\partial\hat{g}}{\partial T})^{2}\sigma_{T}^{2}+\frac{1}{\hat{g}^{2}}(\frac{\partial\hat{g}}{\partial\theta})^{2}\sigma_{\theta}^{2}
\rho^{\omega}_{K,\mathbf{a},f}(Y_{\mathbf{u}})=\rho^{\omega}_{K,\mathbf{a},f}(Y_0+\sum_{i=1}^{m}u_iY_i)\le (m+1)\max\{\rho^{\omega}_{K,\mathbf{a},f}(Y_i)\mid i\in \{0,1,\ldots,m\}\}.
d z = \frac{1}{2 \pi}\sum_{i}N_{i}\frac{d}{d S_{i}}G d x = - \frac{1}{2 \pi i}\sum_{i}N_{i}\frac{d}{d S_{i}}\Big [ R \Big ( x + i \frac{m}{2}\Big ) - R \Big ( x - i \frac{m}{2}\Big ) \Big ] \ ,
M(z) := \frac{1}{2}\begin{pmatrix}1 1 i -i\end{pmatrix}\begin{pmatrix}a(z) 0 0 a^{-1}(z)\end{pmatrix}\begin{pmatrix}1 -i 1 i\end{pmatrix}, z \in \mathbb C\setminus [-1,1] ,
h_{L}(x):=\begin{cases}g_{l,u}x \mbox{if}x\in[-\delta,0] g_{r,l}x \mbox{if}x\in[0,\delta]\end{cases}\mbox{and}h_{U}(x):=\begin{cases}g_{l,l}x \mbox{if}x\in[-\delta,0] g_{r,u}x \mbox{if}x\in[0,\delta].\end{cases}
\int^t_0 \Lambda\big(\mathbf{D},\mathrm{d}z\big)=\nabla^\ast_t\big( \mathrm{D}^\bullet_\bullet\nabla_t+\mathrm{D}^\bullet_+\big)+ \int_0^t\big[\mathrm{D}^-_\bullet\nabla+\mathrm{D}^-_+\big](z)\mathrm{d}z
\|\Psi_{p,i}(g^1,\cdots,g^p)(t)\|_{\mathcal{G}_\lambda^*}+\|\Phi_{p,i}(g^1,\cdots,g^p)(t)\|_{\mathcal{G}_\lambda^*}\leq C^pt^{4p}\prod_{j=1}^p\|g^j(t)\|_{\mathcal{G}_{\lambda}^*}
|K_p(r)|=\frac{|u_{p,+}(r)^p-\tilde{u}_{p,+}(r)^p|}{p|u_{p,+}(r)-\tilde{u}_{p,+}(r)|}\leq \left(\max\{u_{p,+}(r),\tilde{u}_{p,+}(r)\}\right)^{p-1},
(xyz)^8|_{H_1c_{m-3}}=\left(\prod_{u\in U\cap H_1}(a^2uc_{m-4}c_{m-3},a^{-1}uc_{m-4}c_{m-3},a^2buc_{m-4}c_{m-3})\right) &\times\left(\prod_{u\in U\cap H_1}(buc_{m-4}c_{m-3},abuc_{m-4}c_{m-3},a^{-1}buc_{m-4}c_{m-3})\right).
r=\frac{n(\sum_{i=1}^{n}X_{i}Y_{i})-(\sum_{i=1}^{n}X_{i})(\sum_{i=1}^{n}Y_{i})}{n[\sqrt{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}]\times\sqrt{\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}}]}
A(t)^{1/2}F(t) = |t| S^{1/2}P + t |t| G + \Phi (t),|t| \le t^0, G := Z S^{1/2}P + S^{1/2}P Z^* + \frac{1}{2}N_0 S^{-1/2}P + S^{-1/2}N_* P + N_* S^{-1/2}P + \mathcal{I}_* (1).
\begin{cases}||.||_{k,\alpha}\leq ||.||_{k',\alpha}||.||_{-k',\alpha}\leq ||.||_{-k,\alpha}, \mathcal{W}^{k,\alpha}_{0}\hookrightarrow \mathcal{W}^{k,\alpha'}_{0}\mathcal{W}^{-k,\alpha'}_{0}\hookrightarrow \mathcal{W}^{-k,\alpha}_{0}, \end{cases}
\delta_{\kappa}Z^{\underline{{M}}}E_{\underline{{M}}}^{~ \underline{{a}}}= 0 , \quad \delta_{\kappa}Z^{\underline{{M}}}E_{\underline{{M}}}^{~ \underline{{\alpha}}}= ( 1 + \bar{\Gamma})_{~ \underline{{\beta}}}^{\underline{{\alpha}}}\kappa^{\underline{{\beta}}}( \xi ) ,
\langle \xi(t),z \rangle =\langle \xi_0,z\rangle + \int_0^t \langle \xi(s),\Delta z \rangle\,{\rm d}s+\int_0^t \langle v(s)\xi(s) , \nabla z\rangle \,{\rm d}s+\langle\int_0^t \tilde G(v(s))\,{\rm d}W(s),z\rangle
\begin{array}{l l l}{\Pi_{t a d}^{\mu \nu , \alpha \beta}\left( k \right) \equiv \displaystyle{\frac{1}{4}}}&{\left( \eta^{\mu \alpha}\tilde{\Gamma}^{\nu \beta}+ \eta^{\mu \beta}\tilde{\Gamma}^{\nu \alpha}+ \eta^{\nu \alpha}\tilde{\Gamma}^{\mu \beta}+ \eta^{\nu \beta}\tilde{\Gamma}^{\mu \alpha}\right)}\\{}&{- \eta^{\mu \nu}\Delta^{\alpha \beta}- \eta^{\alpha \beta}\Delta^{\mu \nu}}\\ \end{array}.
D =\partial_{\mu}\otimes\gamma^{\mu}+A_{\mu}\otimes\gamma^{\mu}D^2 =\partial_{\mu}\partial_{\lambda}\otimes\gamma^{\mu}\gamma^{\lambda}+\partial_{\mu}(A_{\lambda})\otimes\gamma^{\mu}\gamma^{\lambda}+ A_{\mu}\partial_{\lambda}\otimes\gamma^{\mu}\gamma^{\lambda}+A_{\mu}A_{\lambda}\otimes\gamma^{\mu}\gamma^{\lambda}.
\mathrm{t r}^{\prime}e^{- ( \Delta + m^{2}) t}= \sum_{\delta \neq{\bf 1}}\sum_{p = 1}^{\infty}\int_{- \infty}^{\infty}d u \left[{\frac{{\sqrt{2}}l ( \delta )}{( 4 \pi t )^{3 / 2}}}\int_{\rho_{H}}^{\infty}d b{\frac{b e^{- b^{2}/ 4 t - M^{2}t}}{{\sqrt{\mathrm{C o s h}b - \mathrm{C o s h}\rho_{H}}}}}\right] ,
(\xi_i, (x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n)) \to (\xi_i^{q_0}x_0, \ldots , \xi_i^{q_{i-1}}x_{i-1}, \xi^{q_{i+1}}x_{i+1}, \ldots , \xi_i^{q_{n}}x_n),
\liminf_{s \searrow 0}\int_0^1 |\phi_s'|^2 s^{-1}\dot P_s - \lambda_{1,2}(\sigma_s) |\phi_s|^2 s^{-1}\dot Q_s \,dt &\ge \int_0^1 | \phi_\sigma'|^2 \ddot P_0(t) - \lambda_{1,2}(\sigma) | \phi_\sigma |^2 \ddot Q_0(t) \,dt
T^{{\underline{{a}}}}={\cal D}E^{{\underline{{a}}}}= -{\frac{i}{2}}\hat{E}^{\hat{\underline{{\alpha}}}}\wedge \hat{E}^{\hat{\underline{{\beta}}}}\Gamma_{{\underline{{{\hat{\alpha}\hat{\beta}}}}}}^{{\underline{{a}}}}= -{\frac{i}{2}}( \hat{E}\Gamma^{{\underline{{a}}}}\hat{E})
\langle \nabla_i\nabla_i \Psi_t-\nabla_l\Psi_t\Gamma_{ii}^l-\frac{\sum_k(\nabla_k\nabla_k\Psi_t-\nabla_l\Psi_t\Gamma_{kk}^l)}{n}, \nabla_m \Psi_t\rangle= -\Gamma^m_{ii}+\frac{\sum_k \Gamma^k_{ii}}{n}+O(t).
\big(x_{2,1}^{-}(x_{2,0}^{-})^2 +x_{2,0}^{-}x_{2,1}^{-}x_{2,0}^{-}+(x_{2,0}^{-})^2 x_{2,1}^{-}\big)x_{1,0}^{-}v^{+}&=\frac{1}{4}\big((a+\frac{5}{2})^2-(a-\frac{1}{2})^2+(a+\frac{3}{2})^2-(a-\frac{3}{2})^2\big)(x_{2,0}^{-})^3 x_{1,0}^{-}v^{+}&=(3a+\frac{3}{2})(x_{2,0}^{-})^3x_{1,0}^{-}v^{+}.
\langle\mu,\phi\rangle:=\int\phi(\boldsymbol{\xi})\mu(d\boldsymbol{\xi}),\quad\mathcal{R}_{\boldsymbol{\theta}}[\phi]:=\mathbb{E}_{\boldsymbol{\theta}}^\emptyset\left[\int_0^{T_\emptyset}\phi(X_t)\:dt\right].
\int_{2}^{x}\frac{\pi(t;m,1)}{t^2}dt =\int_{m}^{2m}\frac{\pi(t;m,1)}{t^2}dt+\int_{2m}^{x/2}\frac{\pi(t;m,1)}{t^2}dt+\int_{x/2}^{x}\frac{\pi(t;m,1)}{t^2}dt &\ll \int_{m}^{2m}\frac{dt}{mt}+\int_{2m}^{x/2}\frac{dt}{\varphi(m)t\log (t/m)}+\int_{x/2}^{x}\frac{dt}{mt}&\ll \frac{1}{m}+\frac{\log\log x-\log\log m}{\varphi(m)}
A^{0}( x ) = \int_{- \infty}^{\infty}d \tau \a^{0}( x , \tau ) = \ - \frac{e}{4 \pi R}\ \frac 1 2 \int_{- \infty}^{\infty}d \tau \ \Bigl ( \varphi ( t - \tau - R / c ) + \varphi ( t - \tau + R / c ) \Bigr ) = - \frac{e}{4 \pi R}
|\eta_t^N(\vec{f})|\leq \varepsilon_5^N(\vec{f})+\sum_{l=1}^{+\infty}\int_0^t\ldots\int_0^{t_{l-1}}\sum_{\mathcal{C}_{t_j}^N\in \{A_{t_j}^N, \mathcal{B}_{11}\}\atop 1\leq j\leq l}\varepsilon_5^N\left(\mathcal{C}_{t_{l}}^N\ldots\mathcal{C}_{t_1}^N\vec{f}\right)dt_1dt_2\ldots dt_{l}
\left( \Psi + n^{\mu}\, \nabla_{\mu}\right) \varphi\longrightarrow \Omega^{-{\frac{{\cal D}-1}{2}}}\left( \Omega\overline{\Psi}\varphi +n^{\mu}\nabla_{\mu}\varphi-{\frac{{\cal D}-2}{2}}{\frac{n^{\mu}\partial_{\mu}\Omega}{\Omega}}\varphi \right) ,
\lambda (r_-,p_+)= - \begin{pmatrix}\ell_1 \ell_2 \end{pmatrix}\begin{pmatrix}-K 1 1 -M\end{pmatrix}^{-1}\begin{pmatrix}\ell_1^t \ell_2^t \end{pmatrix}; \mu (r_-,p_+)= \frac 1{24}\begin{pmatrix}m_1 m_2 \end{pmatrix}\begin{pmatrix}-K 1 1 -M\end{pmatrix}^{-1}\begin{pmatrix}m_1^t m_2^t \end{pmatrix}.
E[V_{[0,\gamma_t]}] = \sum_{\ell=0}^{\gamma_t}E[G(0,S_\ell) 1_{A_t}] \le C_3 P(A_t) \sum_{\ell=0}^{\gamma_t}(1+\sqrt{\ell})^{-1}\le C_4 P(A_t) \sqrt{\gamma_t}\,.
\varphi(z,t):=\sum_{\lambda=1}^n \varphi_j^{\lambda}(z,t) \frac{\partial}{\partial z_{\lambda}}=\sum_{\lambda=1}^n\varphi^\lambda(z,t)\frac{\partial}{\partial z_\lambda}=\sum_{v,\lambda=1}^n \varphi_v^{\lambda}(z,t) d\bar{z}_v \frac{\partial}{\partial z_{\lambda}}
\lambda\nabla^{LC,\lambda}_X(Y)= \lambda\nabla^{LC}_X(Y)+(\partial_Y\lambda) X + (\partial_X\lambda)Y - g(X,Y)Z = \nabla^{LC}_X(\lambda Y)+g(Y,Z)X - g(X,Y)Z = \nabla^{LC}_X(\lambda Y)+\frac14\big(-(YZ+ZY)X-X(YZ+ZY) &\phantom{\nabla^{LC}_X(\lambda Y)+\frac14\big(}+(XY+YX)Z+Z(XY+YX)\big) = \nabla^{LC}_X(\lambda Y)+\frac14(YXZ+ZXY-YZX-XZY)
-\displaystyle\sum_{j=1}^{d}\mbox{ess}\,\displaystyle\liminf_{s\to 0-}\int_{\Gamma}\,q_{j}(u(r +s\nu(r)))\,\nu_{j}(r)\,\beta(r)\,dr \leq M\,d\,\int_{\Gamma}\,\eta\left(u(r)\right)\,\beta(r)\,dr.
W(B,\varepsilon,\delta)= \{f\in L^0(\Omega,\Sigma,\mu): \mathrm{there~exists~a~set}~E\in\Sigma~\mathrm{such~that}&E\subset B, \mu(B\backslash E)\leq\delta,f_{\chi_{E}}\in L^\infty(\Omega,\Sigma,\mu),\|f_{\chi_{E}}\|_{L^\infty(\Omega,\Sigma,\mu)}\leq\varepsilon\},
q_{2}(x)\ast\left(h_{g}(x)l(x),2h_{g}(x)a(x)\right) =q_{2}(x)\ast\left[k(x)\ast\left(l_1(x),q(x)\right)+\mu(x)\ast\left(f(x),0\right)\right] &=q_2(x)k(x)\left(l_1(x),q(x)\right)+q_2(x)\mu(x)\left(f(x),0\right).
{d(\mu_k+i\nu_k) \over dt}= 4\nu_0 \left[\langle\vec{n}_{k}, \vec{n}_{k-1}\rangle e^{q_{k}-q_{k-1}}-\langle\vec{n}_{k+1}, \vec{n}_k\rangle e^{q_{k+1}-q_{k}}\right],
[t^n] (f_1\circ y) = \sum_{k=1}^n \lambda^{k}B_{n,k}(1!y_1,2!y_2,\dots), [t^n] (f_2\circ z) = \sum_{k=1}^n (\lambda)_k\, B_{n,k}(1!z_1,2!z_2,\dots) = \sum_{k=1}^n \lambda\tbinom{\lambda-1}{k-1}(k-1)!B_{n,k}(1!z_1,2!z_2,\dots).
e^{\sqrt{2}t_1 a^\dagger}\vert 0 \rangle *_{\rm point}e^{\sqrt{2}t_2 a^\dagger}\vert 0 \rangle = \pi^{-1/4}\sqrt{2\over 3}e^{{1\over 3}(t_1^2+t_2^2)-{4\over 3}t_1 t_2}e^{-{2\sqrt{2}\over 3}(t_1+t_2) a^\dagger +{1\over 6}(a^\dagger)^2}\vert 0 \rangle.
_{2}\phi_{1}\left[\begin{array}{ccc}\alpha, \beta\; ; z \gamma \end{array}\right]=\frac{(\beta)_{\infty}(\alpha z)_{\infty}}{(\gamma)_{\infty}(z)_{\infty}}\;_{2}\phi_{1}\left[\begin{array}{ccc}\frac{\gamma}{\beta}, z\; ; \beta \alpha z \end{array}\right].
\frac{d}{dx}\bigg(\frac{(1-a-ax+b+bx)_p}{(1-a-ax)_p}\bigg)\bigg|_{x=0}= \frac{a-b}{a}\cdot \prod_{\substack{1\leq j\leq p j\neq a-b}}(j-a+b)\cdot\prod_{\substack{1\leq j\leq p j\neq a}}\frac1{j-a}\cdot \bigg(\sum_{\substack{1\leq j\leq p j\neq a}}\frac{a}{j-a}-\sum_{\substack{1\leq j\leq p j\neq a-b}}\frac{a-b}{j-a+b}\bigg)\equiv 0\pmod{p}.
\mathbb E\left[ \left\|\sum_{i=1}^{n}\int_{0}^{(i-1)\Delta_n}(\mathcal S(\Delta_n)-I)\Sigma_s^{\mathcal S_n}(\mathcal S(\Delta_n)-I)^* ds\right\|_{\mathcal H}^2\right]\geq C n^{1-4H}.
(\frac{d}{dt}e^{X(t)})e^{-X(t)}=\frac{d}{dt}X(t)+\frac{1}{2!}[X(t),\frac{d}{dt}X(t)]+\frac{1}{3!}[X(t),[X(t),\frac{d}{dt}X(t)]]+\cdot\cdot\cdot
\check R_i( u )=\sum_{p=0}^k (-q)^{k-p}\left[\begin{array}{c}k p\end{array}\right]^2_q\ \frac{(q^{-2}\, ;\, q^{-2})_{k-p}}{(uq^{-2p}\,;\, q^{-2})_{k-p}}\ \Sigma_i^{(p)}\ ,
Z_{1}X_{2}Y_{3}=[\begin{matrix}c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}\\c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}\\ -c_{2}s_{3}&s_{2}&c_{2}c_{3}\end{matrix}]
\frac{\partial \Gamma_{\Lambda}[ \phi ]}{\partial \Lambda}= - \frac{\Omega}{2}\mathrm{t r}\int_{0}^{\infty}\! \! \! \! \! d q \frac{q^{D - 1}}{C ( q , \Lambda )}\frac{\partial C ( q , \Lambda )}{\partial \Lambda}\left\langle \left[ 1 + C . \frac{\delta^{2}\Gamma_{\Lambda}}{\delta \phi \delta \phi}\right]^{- 1}\! \! \! ( \mathbf{q}, \mathbf{- q}) \right\rangle
{\bf h_{\varphi}}= \left( \begin{array}{c c}{e^{\varphi / 4}}&{{\frac{1}{2}}\varphi_{w}e^{- \varphi / 4}}\\{0}&{e^{- \varphi / 4}}\\ \end{array}\right) , \qquad{\bf \bar{h}_{\varphi}}= \left( \begin{array}{c c}{e^{\varphi / 4}}&{0}\\{-{\frac{1}{2}}\varphi_{\bar{w}}e^{- \varphi / 4}}&{e^{- \varphi / 4}}\\ \end{array}\right) .
\mathbf{e}_i \,(\mathbf{I}-\beta \,\mathbf{P}^{1})^{-1}\, \mathbf{h}^{1}+ \mathbf{x}^0 \, \widehat{\mathbf{h}}^0 + \nu \, \mathbf{x}^1 \,\boldsymbol{\theta}^1 = v_i^{N^{\{0, 1\}}}+ \mathbf{x}^0 \, \widehat{\mathbf{h}}^0 + \nu \, \mathbf{x}^1 \,\boldsymbol{\theta}^1,
|j+k|^\zeta \le (|j|+|k|)^\zeta = \frac{|j|+|k|}{(|j|+|k|)^{1-\zeta}}= \frac{|j|}{(|j|+|k|)^{1-\zeta}}+ \frac{|k|}{(|j|+|k|)^{1-\zeta}}\le \frac{|j|}{|j|^{1-\zeta}}+ \frac{|k|}{|k|^{1-\zeta}}= |j|^\zeta +|k|^\zeta\, .
c+o(1) =I[u_n] &= \frac{1}{p}([u_n]_{s,p}^p+a\|u_n\|_p^p) &\phantom{=}-\frac{b}{2}\int_{\mathbb{R}^N}(K\ast F(u_n))F(u_n)dx-\frac{\varepsilon_g}{p_g}\|u_n\|_{p_g}^{p_g}.
\begin{aligned}\begin{rcases}u_t+Au\geq 0, \u \geq \psi\ \ \ & (u_t+Au)(u-\psi) =0 \ \ \ \end{rcases}\ \{\rm{in}}\ \ \Omega \times (0,T] u=\phi\ \ \ \ \ \ \ \{\rm{on}}\ \ \partial_p(\Omega \times (0,T]) \end{aligned}
\mu_t\;= -\frac{1}{27\mu^2}\mu_{\sigma^4}+\frac{8}{27\mu^3}\mu_{\sigma}\mu_{\sigma^3}+\frac{2}{9\mu^3}\mu_{\sigma^2}^2-\frac{4}{27\mu^4}\left(9\mu_{\sigma}^2+\mu^3\right)\mu_{\sigma^2}\qquad\qquad\qquad\qquad\qquad+\frac{8\mu_{\sigma}^4}{9\mu^5}+\frac{2\mu_{\sigma}^2}{9\mu^2}+\frac{2}{27}(\sigma\mu_{\sigma}+2\mu).
h_R(x_i^{2k+1}) =h_R(x_i^{2k})x_i+(-1)^{|x_i^{2k}||x_i|}h_R(x_i)x_i^{2k}&=\bigtriangleup (2k)y_ix_i^{2k-1}x_i+(-1)^{|x_i^{2k}||x_i|}y_ix_i^{2k}&=(k+1+k(-1)^{(2k-1)|x_i|^2})y_ix_i^{2k}&=\bigtriangleup (n+1)y_ix_i^{n}.
X^-(x_{\alpha,k}\xi_{k+1}^{i}\otimes_{k+1}\xi_{k+2}^{j}) = x_{\alpha,k}X^-(\xi_{k+1}^{i}\otimes_{k+1}\xi_{k+2}^{j}), X^-((s_{\alpha,k+1}+ \xi_{k+1}s_{\alpha+1,k+1}) \xi_{k+1}^{i}\otimes_{k+1}\xi_{k+2}^{j}) = (s_{\alpha,k+1}+ \xi_{k+1}s_{\alpha+1,k+1}) X^-(\xi_{k+1}^{i}\otimes_{k+1}\xi_{k+2}^{j}) ,
\|v_{m}\|_{H}^2 + h^2\sum_{n=0}^{m-1}\|z_{n+1}\|_{H}^2 + h\sum_{n=0}^{m-1}\|B_{1}^{1/2}v_{n+1}\|_{H}^2 + \|\varphi_{m}\|_{V}^2 + h^2\sum_{n=0}^{m-1}\|v_{n+1}\|_{V}^2 + \|B_{2}^{1/2}\theta_{m}\|_{H}^2 + h^2\sum_{n=0}^{m-1}\|B_{2}^{1/2}\delta_{h}\theta_{n}\|_{H}^2 \leq C_{4}
\begin{cases}1\leq x_{r-k}\leq x_{r-k-1}\quad\mbox{for}0\leq k\leq k^*-1, 1\leq x_{r-k^*}\leq \left(\frac{\Xi}{x_1\cdots x_{r-k^*-1}}\right)^{1/(k^*+1)}\mbox{for}k=k^*, x_{r-k}\geq \left(\frac{\Xi}{x_1\cdots x_{r-k-1}}\right)^{1/(k+1)}\quad\mbox{for}k^*+1\leq k\leq r-2, x_{1}\geq \Xi^{1/r}\quad\mbox{for}k= r-1.\end{cases}
L(t,x,y)dt \! =\! \frac{1}{2\sigma_B^2}\|h(t,x,y)\|^2 dt\!+\!\frac{1}{\sigma_B^2}Y_t^{\prime}dh(t,x,y) \Phi(x,y) = -\frac{1}{\sigma_B^2}Y_T^{\prime}h(T,x,y).
\| |\nabla_x|^{\frac{1}{2p}}|\nabla_y|^{\frac{1}{2p}}Q_L u \|_{L_t^p L_{x,y}^q}\lesssim L^{\frac{1}{2}}\| Q_L u\|_{L^2_{x,y,t}}, \textnormal{if}\ \ \frac{2}{p}+ \frac{2}{q}= 1, \p >2, \| Q_L u \|_{L_t^p L_{x,y}^q}\lesssim L^{\frac{1}{2}}\| Q_L u\|_{L^2_{x,y,t}}, \textnormal{if}\ \ \frac{3}{p}+ \frac{2}{q}= 1, \p >3.
A=\sum_{i=1}^n\left[\alpha^i\frac{\partial}{\partial t^j_i}+\overline{\alpha}^i\frac{\partial}{\partial \overline{t}^j_i}\right]B=\sum_{i=1}^n\left[\beta^i\frac{\partial}{\partial t^j_i}+\overline{\beta}^i\frac{\partial}{\partial \overline{t}^j_i}\right].
w = \begin{pmatrix}1 0 \end{pmatrix}P Q(t) P^{-1}\begin{pmatrix}W_2 \overline{W_2}\end{pmatrix}+ O(t^{-1/4 + C_1 \varepsilon^2_1 + 2\mu C_2 \varepsilon^2_1 + C_4 \varepsilon_1^2})
\rho^{\tilde{A}}_{2t}(S_{(A,\gamma_1 \gamma_2)})= e^{2\pi\sqrt{-1}2t}S_{(A,\gamma_1 \gamma_2)}= P_A e^{2\pi\sqrt{-1}t}S_{\gamma_1}e^{2\pi\sqrt{-1}t}S_{\gamma_2}= P_A \rho^Z_t(S_{\gamma_1}) \rho^Z_t(S_{\gamma_2}) = \rho^Z_t(P_A S_{\gamma_1}S_{\gamma_2})
Q(x_1\odot \ldots \odot x_n)=\sum_{i=1}^n \, \sum_{\sigma \in Sh(i,n-i)}\epsilon(\sigma)q_i\left(x_{\sigma(1)}, \ldots, x_{\sigma(i)}\right)\odot x_{\sigma(i+1)}\odot \ldots \odot x_{\sigma(n)},
\int_{0}^{1}f(x)dx=lim_{N\rightarrow\infty}\sum_{m=1}^{M}\sum_{n=0}^{N}\frac{(-1)^{n}+1}{{(2M)}^{n+1}(n+1)!}f^{(n)}(x)|_{x=\frac{m-1/2}{M}},
{\hat{\Psi}}\ = \ \sum_{j = | k | - 1 / 2}^{\infty}\sum_{| m | \le j}[ a_{k m}^{j +}{\hat{D}}_{k - 1 / 2 , m}^{j}a \ + \a_{k m}^{j -}{\hat{D}}_{k + 1 / 2 , m}^{j}a^{*}] \ \in \{\hat{\cal S}}_{k}\ ,
|E (t,x) |^2 = \frac{\cos^2 t(x- x_2)}{c_2^2 w(s)^2}+ \frac{\sin^2 t(x- x_0)}{c_1^2 w(s)^2}+ \epsilon \frac{\psi_2(x)^2}{c_2^2}+ 2\sqrt{\epsilon}\frac{\psi_2(x) \cos t(x-x_2)}{c_2^2 w(s)}+ \epsilon \frac{\psi_1(x)^2}{c_1^2}+ 2\sqrt{\epsilon}\frac{\psi_1 (x) \sin t(x-x_0)}{c_1^2 w(s)}.
q<1-e^{2s}\;\;\;\Rightarrow\;\;\;\exists \beta_2<1:\; \begin{cases}\phi_2(\beta)<0,\;\;\;{\rm if}\;\;\; 0<\beta<\beta_2, \phi_2(\beta)=0,\;\;\;{\rm if}\;\;\; \beta=\beta_2, \phi_2(\beta)>0,\;\;\;{\rm if}\;\;\;\beta>\beta_2,\end{cases}
\begin{aligned}g(p(x),|\nabla u(x)-R|) \le g\big(p(x),|\nabla u(x)-\nabla v(x)|+|\nabla v(x)-R|\big) &\le 2\left(|\nabla u(x)-\nabla v(x)|^{p(x)}+|\nabla v(x)-R|^2\right).\end{aligned}
\delta=2arcsin(\frac{\sqrt{1-\frac{p_{e}}{R_{SOI}}}\sqrt{1+\frac{p_{e}}{R_{SOI}}-\frac{2\mu p_{e}}{v_{\infty}^{2}R_{SOI}^{2}}}}{1+\frac{v_{\infty}^{2}p_{e}}{\mu}-\frac{2p_{e}}{R_{SOI}}})
h^{-1}=\left(\begin{matrix}h^1 h^2 h^3 h^4\end{matrix}\right) =\left(\begin{matrix}(h^1)_{i\bar{j}}(h^2)_{i\bar{p}\bar{q}}(h^3)_{pq\bar{i}}(h^4)_{pq\bar{m}\bar{n}}\end{matrix}\right), p\leq q, m\leq n,
\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (s_k)_{k=0}^{m}\right) \stackrel{a\to 0+}{\longrightarrow}\mathcal{D}\left((s_{k+1})_{k=0}^{2m-1}, (s_k)_{k=0}^{m}\right) = (-1)^{m}\mathcal{D}\left((s_{k})_{k=0}^{2m-1}, (s_k)_{k=m}^{2m}\right),
\mathcal{C}_{s}^{m}=\sum_{k=1}^{m-s}(-1)^{k}\sum_{a_{1},\cdots,a_{k}=1}^{\infty}\delta_{a_{1}+\cdots+a_{k},m-s}\frac{m!}{(m-a_{1}-\cdots-a_{k})!}\prod_{j=1}^{k}\frac{[2(a_{j})]!}{a_{j}!},
\begin{array}{cccc}R_{ijkh}=\frac{1}{n-2}(g_{ih}\varrho_{jk}+g_{jk}\varrho_{ih}-g_{ik}\varrho_{jh}-g_{jh}\varrho_{ik}) -\frac{\tau}{(n-1)(n-2)}(g_{ih}\varrho_{jk}-g_{ik}\varrho_{jh}).\end{array}
\rho_{\infty}(\lambda) = \frac{p_0'(\lambda)}{2\pi}+ \sum_{i}\int_{\lambda_{iL}}^{\lambda_{iR}}\frac{d\nu}{2\pi}K(\lambda-\nu)\rho_{\infty}(\nu), \rho_{ia}(\lambda) = \frac{K'(\lambda-\lambda_{ia})}{2\pi}+ \sum_{i}\int_{\lambda_{iL}}^{\lambda_{iR}}\frac{d\nu}{2\pi}K(\lambda-\nu)\rho_{ia}(\nu).
\sum_{k = 0}^{\infty}\left( \frac{1}{32}\right)^{k}\binom{2k}{k}^2 \frac{H_{2k}}{2k-1}= \frac{\sqrt{\pi}\left( \pi + 3 \ln(2) - 4\right)}{2 \Gamma^{2}\left( \frac{1}{4}\right)}- \frac{\Gamma^{2}\left( \frac{1}{4}\right) \left( \pi - 3 \ln(2) - 2\right)}{16 \pi^{3/2}}
B_{2nk}-B_{2n}=A_{nk}-A_{n}-\left(\sum_{p-1 \mid 2nk}\frac{1}{p}-\sum_{p-1 \mid 2n}\frac{1}{p}\right)=A_{nk}-A_{n}-\sum_{\substack{p-1 \mid 2nk, p-1 \nmid 2n}}\frac{1}{p}.
Z[\zeta^{\mu}, \xi^{\mu}, \eta^{\mu}] ={\cal N}\, \int \prod_x \{{\rm d}v_{\mu}\,{\rm d}j^{\mu}\,{\rm d}|h|^2 \,{\rm d}\alpha \}\, e^{{\rm i}\int{\rm d}^4 x \,{\cal L}[\zeta^{\mu}, \xi^{\mu}, \eta^{\mu}; v_{\mu},j^{\mu},|h|,\alpha]}\ ,
T_{B}^{( i )}( z ) = \displaystyle \sum_{\mu = 0}^{n - 1}\phi_{( a , a + \bar{\varepsilon}_{\mu})}( z ) K^{( i )}\left( \left. a + \bar{\varepsilon}_{\mu}\begin{array}{c}{{a}}\\{{a}}\\ \end{array}\right| z \right) \phi^{( a + \bar{\varepsilon}_{\mu}, a )}( z )
\nu_j(t) = t^{-\frac{1+\frac{n-2s}{2s}\bar{\sigma}_jb_j^{2-2s}}{n-4s}}\left[d_j + \int_{t_0}^t\tau^{\frac{1+\frac{n-2s}{2s}\bar{\sigma}_jb_j^{2-2s}}{n-4s}}(Ph)_j(\tau)d\tau\right],
\alpha_i={\cal{G}}(\alpha'_1, \ldots , \alpha'_N) \alpha'_i,\hspace{10mm}\mbox{with}\hspace{5mm}{\cal{G}}(\alpha'_1, \ldots , \alpha'_N)=\frac{\sum(\alpha'_i)^2+2\!\begin{array}{c}{}begin{align*}-3mm]{\sum\sum}begin{align*}-3mm]{}_{j<l}\end{array}\! \alpha'_j \alpha'_l c_{jl}}{\sum\alpha'_i}.