harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar\delta_+ := \bar\partial_+ + \bar\partial^+ : \Omega^{p,q}(\bar L_-) \to \Omega^{p+1,q}(\bar L_-) \delta_- := \partial_- + \partial^- : \Omega^{p,q}(\bar L_-) \to \Omega^{p,q+1}(\bar L_-)
(q^{r-k+1}c_{q,1,k-1}+q^{r-k}c_r+c_{q,k,r-1})-\big((q^{r-k+1}-q^{r-k})c_{q,1,k-1}-(q^{r-1}-q^{r-k})c_r\big) &=q^{r-k}c_{q,1,k-1}+c_{q,k,r-1}+q^{r-1}c^r=c_{q,1,r-1}+q^{r-1}c_r=c_{q,r-1}+q^{r-1}c_r,
\left\| \sum_{n=0}^{N}\frac{\gamma_n}{(n+\alpha)^s}-\sum_{n=0}^{N}\frac{\gamma_n}{(n+c)^s}\right\| \leq \sum_{n=0}^{N}\left\|(n+\alpha)^{-s}-(n+c)^{-s}\right\| &= \sum_{n=0}^{N}\left\|\int_{c}^{\alpha}(-s) (n+u)^{-s-1}\,du \right\|
(\mathcal{S},\Lambda)=(\mathcal{V}\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}^2, (A+t_1)\xi\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u})\,\,\,\bigcup \,\,\, (\mathcal{V}_1\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}^2, (A+t_1)\xi_1\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1})
T_{2}^{1}= 6 ( - 3 a_{1}^{0}a_{2}^{2}c_{2}^{3}+ 3 a_{2}^{1}a_{1}^{2}c_{2}^{2}+ a_{1}^{0}a_{1}^{2}c_{1}^{1}+ a_{1}^{0}a_{2}^{1}c_{2}^{4}- a_{2}^{0}a_{1}^{2}c_{2}^{3}- 3 c_{1}^{0}a_{2}^{2}a_{2}^{3}
I ( \tau ) \; = \; \exp \left( \sum_{i , j = 1}^{M}V_{n}( w_{i}- z_{j}) - \frac{1}{2}\sum_{i \neq j}^{M}V_{n}( w_{i}- w_{j}) - \frac{1}{2}\sum_{i \neq j}^{M}V_{n}( z_{i}- z_{j}) \right) \; .
\zeta_k(\lambda \cdot a_1) = (-1)^{n-1}\zeta_k(\lambda\cdot \zeta_1(a_1)) & = \frac{(-1)^{n-1}(-1)^{n-1}}{n-1}\zeta_k \zeta_1 x_\lambda \zeta_2(a_1) & = \frac{(-1)^{n-1}}{n-1}\zeta_k x_\lambda \zeta_2(a_1) & = \lambda \cdot \zeta_k (a_1)
D_{i \alpha}\lambda^{j \beta}( z ) = \delta_{i}^{~ j}( \textstyle{\frac{1}{2}}\hat{\lambda}( z ) \delta_{\alpha}^{~ \beta}- \tilde{\hat{\omega}}_{\alpha}{}^{\beta}( z ) ) - \hat{T}_{i}^{~ j}( z ) \delta_{\alpha}^{~ \beta}
s_i s_{i+1}s_{i+2}(2 \rho) = 2\rho - 6 \alpha_{i}-4 \alpha_{i+1}- 2\alpha_{i+2}, s_{i+2}s_{i+1}s_{i}(2 \rho) = 2 \rho - \alpha_{i}- 4 \alpha_{i+1}- 6 \alpha_{i+2}, s_is_{i+2}s_{i+1}( 2 \rho) = 2 \rho - 4 \alpha_i -2 \alpha_{i+1}- 4 \alpha_{i+2}, s_{i+1}s_{i}s_{i+2}(2 \rho) = 2 \rho - 2 \alpha_i -6 \alpha_{i+1}-2 \alpha_{i+2}.
F_{i}^{- n}F_{j}= q^{- n ( \alpha_{i}, \alpha_{j})}F_{j}F_{i}^{- n}+ \sum_{j = 1}^{\infty}q^{( - n - j ) ( \alpha_{i}, \alpha_{j})}\left[ \begin{array}{c}{{- n}}\\{{j}}\\ \end{array}\right]_{( \alpha_{i}, \alpha_{i}) / 2}( a d_{q}F_{i})^{j}( F_{j}) F_{i}^{- n - j},
{\widehat P}_{ab}(p)={\widehat P}_{ba}(p)= (n+1-a)\;p_{a-1}\;p_{b -1}- \sum_{i=a+b-n-1}^{a}\,(a+ b - 2i)\;p_{i-1}\;p_{a+b - 1-i}\,,\forall\,a\leq b=1,\ldots,n\,,
\begin{aligned}\boldsymbol{n}(\delta\beta)=(-1)^{n-1}\boldsymbol{n}(\ast d\ast\beta)=(-1)^{n-1}\ast\mathrm{tr}(d\ast\beta)=(-1)^{n-1}\ast d\mathrm{tr}(\ast\beta)=0, \boldsymbol{n}(\alpha)=(-1)^{n-1}\ast\mathrm{tr}(\ast\alpha)=0. \end{aligned}
\int_{\partial B_1}\frac{(\rho^2-\rho_0^2)^s}{|\rho e-\rho_0\omega|^{n}}\,dH^{n-1}_\omega\,G(\rho_0 e,0)=\rho^{2s-n}\int_{\partial B_1}G(\rho_0 e, \rho_0\omega)\,dH^{n-1}_\omega.
\begin{aligned}|\underline{\vec z_{k+d}}| = 3\sqrt2|\underline{\vec z_{k+d-1}}|^{1+\beta}+O\big(|\underline{\vec z_{k+d-1}}|\big), |\underline{\vec z_{k+d-1}}\wedge\underline{\vec z_{k+d}}| = 3|\underline{\vec z_{k+d-1}}|^{2+\beta}+O\big(|\underline{\vec z_{k+d-1}}|^2\big). \end{aligned}
CS(A)-CS(A_{0}) =-\int_{M}Tr\big{(}d_{A_{0}}a\wedge{a}+\frac{2}{3}a\wedge a\wedge a\big{)}\wedge\ast_{M}P &=-\frac{1}{n-3}\int_{M}Tr(F^{2}_{A}-F^{2}_{A_{0}})\wedge\ast_{M}Q,
e^{2i(m+1)\xi_j}= \left( \frac{1-q_0 e^{i\xi_j}}{e^{i\xi_j}-q_0}\right) \left( \frac{1-q_1 e^{i\xi_j}}{e^{i\xi_j}-q_1}\right) & \times\prod_{\substack{1\leq k\leq n k\neq j}}\left( \frac{1-q e^{i(\xi_j+\xi_k)}}{e^{i(\xi_j+\xi_k)}-q}\right) \left( \frac{1-q e^{i(\xi_j-\xi_k)}}{e^{i(\xi_j-\xi_k)}-q}\right) (j=1,\ldots ,n).
F_{m}(x, \frac{y}{2}) = 2 H_{j_{m}}( 4^{m-1}x, \dfrac{y}{2}) \left( \prod_{k=m+1}^{K}2 H_{j_{k}}(4^{k-1}x, 2^{k-(m+1)}y) \right)= 2 H_{j_{m}}( 4^{m-1}x, \dfrac{y}{2}) F_{m+1}(x, y).
\frac{1}{\kappa}\Omega^\rho_{WZ}(g)=- \frac{1}{2}\int_0^{2\pi}dx\, \langle (g^{-1}dg) \stackrel{\wedge}{,}(g^{-1}dg )'\rangle-\frac{1}{2}\langle (g^{-1}dg)(0)\stackrel{\wedge}{,}dM{\scriptstyle\,}M^{-1}\rangle + \rho(M)
\bigl( \epsilon^{\rm hom}_{\rm stiff}\bigr)^{-1}\xi\cdot\xi = \inf_{U\in[H^1_{\#}(Q)]^3}\int_{Q_1}\epsilon_1^{-1}\bigl({\rm curl}(P_{V^\perp}U)+\xi\bigr)\cdot \bigl({\rm curl(}P_{V^\perp}U)+\xi\bigr)=A^{\rm hom}\xi\cdot\xi.
g = T^{*}T g + (I - T^{*}T) g & = T^{*}(I - T T^{*})^{\frac{1}{2}}\widetilde g + (I - T^{*}T) g & = (I - T^{*}T)^{\frac{1}{2}}T^{*}\widetilde g + (I - T^{*}T) g \in \operatorname{Ran}(I - T^{*}T)^{\frac{1}{2}}
\pi_*\mathrm{PD}(\omega) = \bigg( \int_{\{k^0_x \}\times S^1 \times S^1}\omega \bigg) [\overline{\alpha}_x]- \bigg( \int_{\S^1 \times \{k^0_y \}\times S^1}\omega \bigg) [\overline{\alpha}_y] &- \sum_{i = 1}^{n-1}\bigg( \sum_{j = 1}^{m_i}\int_{\partial \overline{D(w_i^j)}}\omega \bigg) [\overline{\alpha}_i].
\varphi(\theta_{L_{0}}^{Q_{0}}(x))=\theta_{L_{0}}^{Q_{0}}(\varphi(x))=\theta_{L_{0}}^{Q_{0}}(\theta_{M}^{Q}(x)) = \theta_{M}^{Q}(\theta_{L_{0}}^{Q_{0}}(x)) = \theta_{L_{0}}^{Q_{0}}(x)
D_1(s) = \sum_{n \geq 1}\frac{A(1,n)\overline{A(1,n)}}{n^s}= L(s,f \times f) T_1(s) D_2(s) = \sum_{n \geq 1}\frac{A(1,n)^2}{n^s}= L(s, f \times \overline{f}) T_2(s),
\left( -{\frac{3}{4}}\omega -{\frac{9}{3 2}}\right)^{2}>{\frac{9}{1 0 2 4}}\left( - 6 4 \omega{\frac{z_{0}^{2}}{l^{2}}}+ 9 - 6 4{\frac{z_{0}^{2}}{l^{2}}}\omega^{2}\right) ,
\sum_{s \in S}{f_{s \cdot \widetilde{\tau}}\cdot s \cdot \gamma^{-1}\otimes \gamma \cdot \widetilde{\tau}}=\biggl(\sum_{s \in S}f_{s \cdot \widetilde{\tau}}\cdot s\biggr) \cdot \gamma^{-1}\otimes \gamma \cdot \widetilde{\tau}\stackrel{(\ref{eq:liftofzero})}{=}0.
F_{g}={\frac{1}{2}}\epsilon_{i j}^{1}\epsilon_{i j}^{2}\int_{0}^{1}d x \left[{\frac{( 1{-}k_{0}^{2}{-}\vec{k}\cdot \vec{p})^{2}}{x^{2}}}+ 2{\frac{k_{0}^{4}{-}( \vec{k}\cdot \vec{p})^{2}}{x}}+ ( 1{-}k_{0}^{2}{+}\vec{k}\cdot \vec{p})^{2}\right] x^{\vec{k}\cdot \vec{p}+ k_{0}^{2}}( 1 - x )^{- 2 k_{0}^{2}}\, .
\begin{array}{rl}\displaystyle R(3,\mathbb{Z}_n,2) \displaystyle \leq 9({(n+9)/12 \choose 2}+3{(n-3)/12 \choose 2})+\frac{n}{3}& \displaystyle = 9(3{(n+3)/12 \choose 2}+{(n-9)/12 \choose 2})+ \frac{n}{3}& \displaystyle = \frac{n^2}{8}-\frac{7n}{6}+\frac{27}{8}.\end{array}
\begin{array}{lll}\int_{\mathbb R}\nu(x)\Big(a- \frac{1}{\gamma^2}\coth^2(\sqrt{\mu_2}|D|)\Big ) \partial_x^2 \nu (x)dx &=-\Big(a- \frac{1}{\gamma^2}\Big )\|\nu'\|^2 +\frac{1}{\gamma^2}\int_{\mathbb R}\nu(x)[1- \coth^2(\sqrt{\mu_2}|D|)]\partial_x^2\nu (x)dx &\geqq -\Big(a- \frac{1}{\gamma^2}\Big )\|\nu'\|^2, \end{array}
\widehat b_{T,h}(x) :=\frac{\displaystyle{\int_{0}^{T}K\left(\frac{X(s) - x}{h}\right)\delta X(s)}}{\displaystyle{\int_{0}^{T}K\left(\frac{X(s) - x}{h}\right)ds}}\textrm{$;$}x\in\mathbb R
\psi_{1}( x ) = \lim_{\Lambda \rightarrow \infty}\frac{k \Lambda}{2 \pi}\int d \hat{\eta}e^{- i \frac{\theta ( \eta )}{2}}\psi_{1}^{\eta}( x ) ; \p s i_{2}( x ) = \lim_{\Lambda \rightarrow \infty}\frac{k \Lambda}{2 \pi}\int d \hat{\eta}e^{i \frac{\theta ( \eta )}{2}}\psi_{2}^{\eta}( x )
\sum_{k = 1}^{\infty}T_{2 n , 2 k - 1}\frac{v_{2 k - 1}^{( \kappa )}}{\sqrt{2 k - 1}}= - \frac{v_{2 n}^{( \kappa )}}{\sqrt{2 n}}\tanh \left( \frac{\kappa \pi}{4}\right) ,
\Pr\{\mathcal{E}_{k,1}\cap \mathcal{E}_{k,2}\}= \Pr\{\mathcal{E}_{k,1}\}\Pr\{\mathcal{E}_{k,2}| \mathcal{E}_{k,1}\}&= \frac{{{|S_k|}\choose r}}{{n \choose r}}\Pr\{\mathcal{E}_{k,2}| \mathcal{E}_{k,1}\},
\left[ q ( t , \vec{x}) , p ( t , \vec{x}^{\prime}) \right] = \delta ( \vec{x}- \vec{x}^{\prime}) , \quad \left[ \Psi ( t , \vec{x}) , \bar{\Psi}( t , \vec{x}^{\prime}) \right] = - i \delta ( \vec{x}- \vec{x}^{\prime})
F_{ij}=\frac{Gm_{i}m_{j}}{\|q_{j}-q_{i}\|^{2}}\cdot\frac{(q_{j}-q_{i})}{\|q_{j}-q_{i}\|}=\frac{Gm_{i}m_{j}(q_{j}-q_{i})}{\|q_{j}-q_{i}\|^{3}},
\int_{Q_t}\Delta \kappa^* \partial_t\kappa_{\varepsilon}\leq \frac{1}{4}\int_{Q_t}| \partial_t\kappa_{\varepsilon}|^2 + \int_{Q_t}|\Delta \kappa^*|^2 = \frac{1}{4}\int_{Q_t}| \partial_t\kappa_{\varepsilon}|^2 + c,
\varLambda_j=\left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{S_j}\right\rceil,\varLambda'_j=\left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{S'_j}\right\rceil, &S_j=\sum_{i=1}^j\mathcal E_i,S'_j=\sum_{i=1}^j\mathcal E_i'.
T ={\frac{1}{2 \pi M}}\quad{\frac{\sqrt{( M^{2}- | z_{1}|^{2}) ( M^{2}- | z_{2}|^{2})}}{\left[ \sqrt{M^{2}- | z_{1}|^{2}}+ \sqrt{M^{2}- | z_{2}|^{2}}\right]^{2}}}\ .
\frac{d^{2}\tilde{F}_{0}}{d \tilde{z}^{2}}+ \tilde{z}\frac{d \tilde{F}_{0}}{d \tilde{z}}+ ( 1 - \frac{1}{4 \tilde{z}^{2}}+ \frac{\mu^{2 / ( 7 - p )}}{( 7 - p )^{2}\tilde{z}^{2 + 2 / ( 7 - p )}}) \tilde{F}_{0}= 0
R_{11}y^{\prime}_{g_1,1}(\lambda,l_{g_1,1})+R_{21}u_{g_1,1}^{\prime}(\lambda, l_{g_1,1})+R_{12}y^{\prime}_{g_2,2}(\lambda,l_{g_2,2})+R_{22}u^{\prime}_{g_2,2}(\lambda, l_{g_2,2})=0.
\{\mathbf{S}_Q, \mathbf{S}_Q \}= - 2 \mathbf{Z},\qquad\{\mathbf{S}_Q, \mathbf{\bar{Q}}_\mu \}= - i \mathbf{P}_\mu,\qquad\{\mathbf{\bar{Q}}_\mu, \mathbf{\bar{Q}}_\nu \}= - 2 \delta_{\mu\nu}\mathbf{\bar{Z}}.
\sum_{k=1}^n (D_k +D_{k+1})c_k (H_{k+1}-H_k) &= 2D_{n+1}c_{n+1}H_{n+1}-2D_{1}c_1 H_1 -\sum_{k=1}^n (D_{k+1}-D_k)c_k (H_{k+1}+H_k)+2D_{k+1}(c_{k+1}-c_k)H_{k+1}= \bigg(-\sum_{k=1}^n (D_{k+1}-D_k)c_k (H_{k+1}+H_k) \bigg) +O(D_\ast \log(n)^{7/2}n^{-3/2}).
0=\frac{\partial^2 (\phi_r \phi_{r+1})}{\partial x_{a_r}\partial x_{a_{r+1}}}=\frac{\partial \phi_r}{\partial x_{a_r}}\frac{\partial \phi_{r+1}}{\partial x_{a_{r+1}}}+\frac{\partial \phi_r}{\partial x_{a_{r+1}}}\frac{\partial \phi_{r+1}}{\partial x_{a_r}},
P(\cdot, t)= \sup_h\frac{{\rm Rm}_{g(t)}(\cdot)(h,h)}{|h|^2}=\sup_h\frac{{\rm R}_{t, ijkl}(\cdot)h^{ik}h^{jl}}{h_{ik}h^{ik}}, ~\forall ~t\in ~(0, T].
w_1(x) =\sum_{y\neq x}\lambda_0(x,y)F_1(x,y) &=\lambda_0(x,x^{1-})F_1(x,x^{1-})+\lambda_0(x,x^{1+})F_1(x,x^{1+})+\lambda_0(x,x^{N+})F_1(x,x^{N+})+\lambda_0(x,x^{N-})F_1(x,x^{N-}).
\mathcal{F}_{\Omega}(u) = P(\chi^2_1 > u) + \frac{1}{2\pi}e^{-\frac{u}{2}}\ln \left. \left[\frac{1+\omega}{1-\omega}\right] \right\vert_{\omega_{\mathcal{L}}}^{\omega_{\mathcal{U}}}
\rho_{\binom{2}{-}}= R_{\binom{2}{-}}, \rho_{\binom{2,0}{1}}=\frac{1}{2}\left[R_{\binom{2,1}{0}}+R_{\binom{2,0}{1}}+R_{\binom{1,0}{2}}+R_{\binom{-}{2,1,0}}\right], \rho_{\binom{2,1}{0}}=\frac{1}{2}\left[R_{\binom{2,1}{0}}+R_{\binom{2,0}{1}}-R_{\binom{1,0}{2}}-R_{\binom{-}{2,1,0}}\right].
\lambda(\hat{\sigma}, \hat{\tau}/\hat{\sigma}) \, P \left( - \rho \frac{\hat{\tau}}{\hat{\sigma}}+ \sqrt{\frac{(m + (\hat{\tau}/\hat{\sigma})^2)(1 - \rho^2)}{m+1}}T_{m+1}\le w\right) &+ \left(1-\lambda(\hat{\sigma}, \hat{\tau}/\hat{\sigma}) \right) \,P \left( T_m \le w \right) = \eta
- \frac{1}{4}e^{c \phi - \psi}\delta^{(d_t) ij}F^{(d_l)}_i \cdot F^{(d_l)}_j - \frac{1}{4}e^{c \phi -2 \psi}\delta^{(d_t) ij}\delta^{(d_t) kl}D_{\alpha}B_{ik}D^{\alpha}B_{jl}) .
\sum_{l=1}^{h(m,n)}C_{m}^{l}(G^{0})(\lambda)(C_{m}^{l}(G^{0})(\lambda))^{*}=d_{m}^{0}(\lambda )^{\top}\left\{\alpha_{mk}^{2}\delta_{kl}\right\}_{k,l=1}^{\infty}\overline{d_{m}^{0}(\lambda )},
{\rm D}\vartheta_c (Y,\mu, \Gamma)^*=\left (\begin{array}{c}-c^{-1}Y+c^{-1}{\rm D}\theta_c(F(x_c(Y,\mu, \Gamma))+Y/c))^* begin{align*}2mm]h(x_c(Y,\mu, \Gamma)) begin{align*}2mm]-c^{-1}\Gamma+c^{-1}\Pi_{{\cal S}^p_+}(\Gamma-c g(x_c(Y,\mu, \Gamma)))\end{array}\right ), \,y =(Y,\mu,\Gamma) \in \mathbb{B}_{\delta_0}(\overline Y,\overline{\mu},\overline{\Gamma})\, .
\left|\left\{\sum_{k=1}^n\phi_k>\lambda\right\}\right| \le e^{-\gamma \lambda}{\textbf E}\left[\exp\left(\gamma\sum_{k=1}^n\phi_k\right)\right] &\le (1+\mu)e^{-\gamma \lambda}{\textbf E}\left[\exp\left(\gamma\sum_{k=1}^n\xi_k\right)\right] &=(1+\mu)e^{-\gamma \lambda}\prod_{k=1}^n{\textbf E}\left[\exp\left(\gamma \xi_k\right)\right].
\langle \mathbf{D}, \hat{\mathbf{D}}\rangle_{\mathbb{R}^{k \times d}}\equiv \mathbf{D}:\hat{\mathbf{D}}:= \sum_{i = 1}^{k}\sum_{j = 1}^{d}D_{ij}\hat{D}_{ij}\mathbf{D}, \hat{\mathbf{D}}\in \mathbb{R}^{k \times d}.
M_{odd}\begin{pmatrix}n_1 n_2 m_1 m_2 \vdots m_h \vdots m_{(n-1)/2}\end{pmatrix}= \begin{pmatrix}2n -1 - \displaystyle{\sum_{i=n+3}^{2n}}c^{n+3-2i}P_{i-n-3,-i}0 0 \vdots 0 0 \end{pmatrix}.
{}^{i}\!s^{j,h'}_{j,h}=\begin{cases}\left(\begin{matrix}h-1 h'-1\end{matrix}\right)c_{i+1}^{h-h'}, h\neq j-i P(i,h',j)+\left(\begin{matrix}h-1 h'-1\end{matrix}\right)c_{i+1}^{h-h'}, 1\leq h'\leq h, h=j-i\end{cases}
n_{k+1}:= n_{k}+ 2 \lfloor n_k^{1/2}\rfloor, d_k:= \frac{n_{k+1}- n_{k}}{2}, \rho_{k}:= n_k^{\frac{1}{1+\delta}}, \rho_{kj}:=\rho_{k}+ j \, \frac{\rho_{k+1}-\rho_{k}}{4}, j=1,\dots,4, a_0:= 1, a_{k+1}:= \rho_k^{2d_k}a_k.
\mathsf{H}[F] := \frac{\|F\|^2}{2}+ \delta \langle \mathsf{A}F,F \rangle \,,\quad\mbox{with}\mathsf{A}:= (1+(\mathsf{T}\Pi)^*\mathsf{T}\Pi)^{-1}(\mathsf{T}\Pi)^* \,,
(E_{r-1}A^* E_r)_{0,0}=\frac{s q^{d-2r+1}(1-s^{-2}q^{2r-2}) (q:q)_d}{2 (1+q^{d-2r+1}) (q^2;q^2)_{r-1}(q^2;q^2)_{d-r}}, (E_{r}A^* E_{r-1})_{0,0}= - \frac{s (1-s^{-2}q^{2d-2r}) (q:q)_d}{2 (1+q^{d-2r+1}) (q^2;q^2)_{r-1}(q^2;q^2)_{d-r}}.
= - \left( 1 - \frac{2 M G}{r}\right) \frac{d^{2}H \left( r \right)}{d r^{2}}+ \left( \frac{2 r - 3 M G}{r^{2}}\right) \frac{d H \left( r \right)}{d r}- \frac{4 M G}{r^{3}}
g^{-1}={4\over \xi^n_i \xi^m_j \lambda^i_{,\bar z}\lambda^j_{,z}\{\sigma_m,\sigma_n\}}[d\bar z \otimes d\bar z + dz \otimes dz +\xi^s_k({\partial \sigma_s \over \partial p}dp -{\partial \sigma_s \over \partial q}dq) \otimes d\lambda^k].
\| \rho(t)\|_{\mathcal{C}_h}= \left \|\mathcal{P}_h \, u(t) - u(t)^I \right \|_{\mathcal{C}_h}\leq C h^2 |u(t)|_{H^2(\Omega)}= C h^2 \left( |u(0)|_{H^2(\Omega)}+ \int_0^t |u_t(s)|_{H^2(\Omega)}{\rm d}\, s \right) & \leq C h^2 \left( |u_0|_{H^2(\Omega)}+ |u_t|_{L^1(0,t, H^2(\Omega))}\right).
\begin{aligned}4d^2 =(27f^4-14f^2g^2+3g^4)u^4-32fg(3f^2-g^2)u^3v & \;\;-(126f^4-108f^2g^2+14g^4)u^2v^2+96fg(3f^2-g^2)uv^3 & \;\; +(243f^4-126f^2g^2+27g^4)v^4. \end{aligned}
1 + \sum_{a = 1}^{N}\frac{q_{a}^{2}}{\lambda - r_{a}}= \frac{\displaystyle \prod_{a = 1}^{N}( \lambda - \lambda_{a})}{\displaystyle \prod_{a = 1}^{N}( \lambda - r_{a})}\equiv \frac{\Lambda ( \lambda )}{A ( \lambda )}
\| S_{j}f \|^2 = \int |\sqrt{4}m_{j}(x,y) f(R(x,y))|^2 \d(\mu_{4}\times \lambda) &= \dfrac{1}{4}\sum_{k=0}^{3}\int 4 | m_{j}(\Upsilon_{k}(x,y)) f(R(\Upsilon_{k}(x,y))) |^2 \d(\mu_{4}\times \lambda) &= \int \left( \sum_{k=0}^{3}| m_{j}(\Upsilon_{k}(x,y))|^2 \right) |f(x,y)|^2 \d(\mu_{4}\times \lambda).
[Q_{V}^{i},Q_{V}^{j}]=i\epsilon^{ijk}Q_{V}^{k},[Q_{A}^{i},Q_{A}^{j}]=i\epsilon^{ijk}Q_{V}^{k},[Q_{V}^{i},Q_{A}^{j}]=i\epsilon^{ijk}Q_{A}^{k}.
{\frac{d}{d r}}( u - w ) = 0 , \qquad{\frac{d}{d r}}u +{\frac{1}{( 1 + r )^{2}}}{\frac{d}{d r}}( 1 + r )^{2}v = 0 , \qquad{\frac{d}{d r}}( w / A ) = u - v \, .
H =\nabla_{E_1}E_1+\nabla_{E_2}E_2=\frac{1}{2}\nabla_{X_1}X_1+\frac{1}{2}\nabla_{X_2}X_2 &=-\frac{\sqrt{2}}{2}(e^{i\theta_1},0)-\frac{\sqrt{2}}{2}(0,e^{i\theta_2})=\frac{1}{2}(JX_1+JX_2)=-\frac{X^{\perp}}{2},
X = \; \{1, 2, 3, 4, 5, 6, 7, 8\}, \mathcal{A}= \; \{\{1, 2, 3, 4\},\{5, 6, 7, 8\},\{1, 2, 5, 6\}, &\{3, 4, 7, 8\},\{1, 3, 5, 7\},\{2, 4, 6, 8\}\},
\partial \left(\sum_{i=1}^n cp_i\max\{-x_i-y_i,-d_i\}\right) = \sum_{i=1}^n cp_i \partial \left(\max\{0,-x_i-y_i+d_i\}\right) &= \sum_{i=1}^n cp_i \Gamma(-x_i-y_i+d_i)\nabla_v(-x_i) &= \sum_{i=1}^n cp_i \Gamma(-x_i-y_i+d_i)(e_{i\bullet}- e_{\bullet i})
C =\begin{pmatrix}{\frac{(m+1)(-m-n-2+i)}{-n-m-1+i}}{\frac{-n+i}{-n-m-1+i}}-{\frac{m+1}{-n-m-1+i}}{\frac{-{m}^{2}+2i-2n+mi-2m-mn-1}{-n-m-1+i}}\end{pmatrix},U=\begin{pmatrix}n+m+2 0 -1 n+m+3 \end{pmatrix}, V =\begin{pmatrix}0 0 0 n+m+1-i \end{pmatrix}.
\frac{\pi_{c,\kappa}(\mu,\sigma)}{\pi_\mathrm{P}(\mu,\sigma)}=\frac{1}{\displaystyle \frac{\frac{a^2\tilde{b}}{b^2\tilde{a}}\mu^2 + \sigma^2 + \kappa^2}{2\sigma \kappa}+c}\propto\frac{\kappa/2}{\displaystyle \frac{\frac{a^2\tilde{b}}{b^2\tilde{a}}\mu^2 + \sigma^2 + \kappa^2}{2\sigma \kappa}+c}\rightarrow\sigma
\mu ={\frac{2 \pi \epsilon ( 4 - 5 \epsilon + 2 \epsilon^{2})}{( 1 - \epsilon ) ( 2 + \epsilon - 2 \epsilon^{2})}}, \alpha ={\frac{\epsilon ( 1 - \epsilon )}{( 1 - \epsilon + \epsilon^{2})^{3 / 2}}}.
\begin{minipage}[c]{35pc}If $(S_k)_{k\in K}$ is a family of pairwise disjoint subsets of $I,$and we let $S=\bigcup_{k\in K}S_k,$ then themap $\pi(B_S)\to\prod_{k\in K}\pi(B_{S_k})$ determined by theprojections $\pi(B_S)\to\pi(B_{S_k})$ \textup{(}whichby~\eqref{d.prod}is an isomorphism if $K$ is finite\textup{)}is always surjective.\end{minipage}
\varphi_{i}(\mathbf{f})=\frac{l_{i}^{x}}{l_{1}^{x}}\frac{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{1jk}y_{j}z_{k}}}{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{ijk}y_{j}z_{k}}},\thinspace i=1,\thinspace \mathellipsis ,\thinspace m,
\bar y := x_{j_1}^2x_{j_2}\ldots x_{j_{k-3}}x_j = \sum_{s=2}^{k-3}x_{j_s}^2x_{j_1}x_{j_2}\ldots \hat x_{j_s}\ldots x_{j_{k-3}}x_j +\sum_{t=1}^m y_t + Sq^1(g_0+\tilde y) + \sum_{u\geqslant 1}Sq^{2^u}(g_u).
\mathbb{E}(Y)= \sum_{R\in\mathcal{R}}\mathbb{P}(R\mbox{is bad})\leq \sum_{r=1}^{t}\binom{n}{r}\binom{\binom{r}{2}}{\lceil rs/2\rceil}p^{\lceil \frac{rs}{2}\rceil}< \sum_{r=1}^{t}n^{r}(erp)^{\frac{rs}{2}}<t\big(e^{\frac{s}{2}}t^{\frac{s}{2}}n p^{\frac{s}{2}}\big)^{t}=t^{\frac{ts}{2}+1}e^{\frac{ts}{2}}n.
v_{-,a}(x)=2\phi_1(x)\frac{\int_x^\infty dy\thinspace\phi^{-2}_1(y)\int_a^xdt\phi^{-2}_1(t)\int_t^y\phi_1(z)dz}{\int_a^\infty\phi^{-2}_1(y)dy},\x\ge a.
S(\{\xi\})=\det(\Phi_{ij})=\prod_{i=1}^D{h_i(\{\xi\})\over f_i(\xi_i)}\ ,\M_i(\xi_1,\dots,\xi_{i-1},\xi_{i+1},\dots,\xi_D) ={\partial S\over\partial\Phi_{i1}}={S(\{\xi\})\over h_i^2(\{\xi\})}\enspace,
K \left( x', x'', N,(\tau''- \tau') \right) = i N \int_{\tau'}^{\tau''}d t\int_{\Sigma(\tilde{x}^0)}d^3{\bf \tilde{x}}\hspace{0.25em}\left\{K \left( x',x, N, t \right)\stackrel{\leftrightarrow}{\partial_0}K \left( x, x'', N,(\tau''-\tau'-t ) \right) \right\}\vline_{x=\tilde{x}}.
\int_0^{r_\varepsilon}|(\Delta\bar{w}_\varepsilon)| rdr= O\left(\|\bar{w}'_\varepsilon\|_{L^\infty([0,r_\varepsilon])}\int_0^{r_\varepsilon/\mu_\varepsilon}\frac{\mu_\varepsilon r^2 dr}{(1+r^2)^{1+\varepsilon_0+o(1)}}\right) &~~~~~+o\left(\int_0^{r_\varepsilon/\mu_\varepsilon}\frac{rdr}{(1+r^2)^{1+\varepsilon_0+o(1)}}\right)\,.
A_\alpha^*=-\sqrt{2}\begin{pmatrix}S^a_{\alpha p}\end{pmatrix},B_\alpha^*=-1/\sqrt{2}\begin{pmatrix}S^a_{\alpha\mu}\end{pmatrix},C_\alpha^*=-1/\sqrt{2}\begin{pmatrix}S^p_{\alpha\mu}\end{pmatrix}.
< \mu | \tilde{\mu}_{I}> = \frac{( 1 - \tilde{\mu}^{*}\tilde{\mu})^{j + k / 2}| \mu |^{( 2 j + k - 1 ) / 2}e^{\mu^{*}\tilde{\mu}}}{\sqrt{\Gamma ( 2 j + k ) I_{2 j + k - 1}( 2 | \mu | )}}
N(x_\mu;q)=\frac{1}{2}\left[\{(L^+)^{-\frac{1}{2}}+(L^-)^{-\frac{1}{2}}\}\cdot 1+\{(L^+)^{-\frac{1}{2}}-(L^-)^{-\frac{1}{2}}\}\cdot\hat x\right],
\theta_k=(I-\mathcal{H})\partial_{\alpha}^k\tilde{\theta}, \sigma_k=(I-\mathcal{H})\partial_{\alpha}^k\tilde{\sigma},\tilde{\theta}:=(I-\mathcal{H})(\zeta-\bar{\zeta}), \tilde{\sigma}:=(I-\mathcal{H})D_t\tilde{\theta}.
\boxed{\begin{array}{l}\nabla f (z^) = \sum_{i=1}^m \sigma_i^(z^) \nabla g_i(z^), \sigma_i^(z^), g_i(z^) \geqslant 0, ~~~ i = 1,\hdots,m, \sigma_i^(z^) g_i(z^) = 0, ~~~ i = 1,\hdots,m.\end{array}}
\int \frac{d k^{\prime \prime}}{2 \pi}\, \, \frac{1}{\sqrt{k^{2}+ 4}}\, \, \frac{2 i k^{\prime \prime}\epsilon}{( i k^{\prime \prime}- 2 \sigma_{1})^{2}}\, \, \frac{1}{( k + k^{\prime}- 2 k^{\prime \prime}- i \tau )}.
\sum_{t=1}^{T}t^{-\frac{1}{\beta}-1}\leq 1+\int_{1}^{T}u^{-\frac{1}{\beta}-1}\d u=1-\beta u^{-\frac{1}{\beta}}\Big|_{1}^{T}= 1+\beta - \beta T^{-\frac{1}{\beta}}\leq 1+\beta\enspace.
128 +12(k -2)+24(k -1) +4(k^2-k -3)+4(\frac{3k (k +1)}{2}-9) &+\frac{k (k -1)(5k +14)}{6}+\frac{k (k -1)}{2}+\frac{(k -1)k (k +1)}{2}= \frac{4(k +2)(k +3)(k +4)}{3}.
\mathbb{P}_{G \sim G_{n,p}}\left[ \mathcal{H}^c_{\rho,C}\right] \leq \mathbb{P}_{G \sim G_{n,p}}\left(|e(G)-p| > \frac{C \sqrt{\rho \log n}}{2\sqrt{n}}\right)+\mathbb{P}_{G \sim G_{n,p}}\left(\bigcup_{\emptyset \not = S \subseteq [n]}A^c_{\rho,S,\frac{C}{2}}(p)\right).
\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=-\tau_{ij}\overline{S}_{ij}-\frac{C_{*}}{\Delta}k^{3/2}+\frac{\partial}{\partial x_{j}}(D\Delta\sqrt{k}\frac{\partial k}{\partial x_{j}})+\nu\frac{\partial^{2}k}{\partial x_{j}\partial x_{j}}
(\gamma_{\mu \mu_{1}\mu_{2}\ldots \mu_{p}})^{(c)}= \delta_{C}^{p+1}\gamma_{\mu_{p}\ldots \mu_{2}\mu_{1}\mu}= \delta_{C}^{p+1}(-1)^{\frac 12(p+1)p}\gamma_{\mu \mu_{1}\mu_{2}\ldots \mu_{p}},
\begin{aligned}\|v_{2}(t)-v_{2}(\tau_{k})\|_{\dot{H}^{\nu}}&\leq C\int_{0}^{T}\|u(t)\|_{H^{1+\nu}}&+C(\|v_{1}\|_{L^{2q_{0}'}([\tau_{k},\tau_{k+1}];L^{2p_{0}})}^{2}+\|v_{1}\|_{L^{2q_{0}'}([\tau_{k},\tau_{k+1}];L^{2p_{0}})}\|v_{1}\|_{L^{2}([0,T];H^{1})})\sup_{[\kappa\in [\tau_{k},t] ]}\|v_{2}\|_{H^{\kappa}},\end{aligned}
\sigma^{-1}_{\tilde{p}_{i_0}}\circ \gamma_1\circ \sigma_{\tilde{p}_i}\circ B(0;\frac{\epsilon_0}{2}) \cap \sigma^{-1}_{\tilde{p}_{i_0}}\circ \gamma_2\circ \sigma_{\tilde{p}_j}\circ B(0;\frac{\epsilon_0}{2})=\emptyset.
\Pi(x;y)=\prod_{k=1}^m y_k^{\beta n}\prod_{i=1}^n \prod_{j=1}^m{(q^{1/2}t^{1/2}y_jx_i;q)_\infty \over (q^{1/2}t^{-1/2}y_jx_i;q)_\infty}{(q^{1/2}t^{1/2}y_j/x_i;q)_\infty \over (q^{1/2}t^{-1/2}y_j/x_i;q)_\infty}.
\begin{aligned}C^x_i= \xi_i L+\sum_{j=1}^{s}\left(W^j_i\frac{\delta L}{\delta u_{j,x}}+D_x(W^j_i)\frac{\delta L}{\delta u_{j,xx}}+D^2_x(W^j_i)\frac{\delta L}{\delta u_{j,xxx}}+\cdots\right), C^t_i= \tau_i L+\sum_{j=1}^{s}\left(W^j_i\frac{\delta L}{\delta u_{j,t}}\right),\end{aligned}
\tilde{\mathcal{R}}_*:=\mathcal{T}^{-1}\mathcal{R}_* \mathcal{T}+\varepsilon^2 \Pi_S^{\perp}\partial_x (\mathcal{R}_2-\mathcal{T}^{-1}\mathcal{R}_2 \mathcal{T})+\varepsilon^2\Pi_S^{\perp}\partial_x (\overline{\mathcal{R}}_2-\mathcal{R}_2),
\Phi_{p}( y ) = \frac{( 2 \pi )^{2}}{l^{2}\tau_{2}}\sum_{j_{1}j_{2}}< \psi | e^{\frac{2 i \pi}{l}( j_{1}( \hat{y_{1}}- y_{1}) + \frac{j_{2}- \tau_{1}j_{1}}{\tau_{2}}( \hat{y_{2}}- y_{2})}| \psi >
|1 \rangle, \,\, J_{+}(\vec{k}) |1 \rangle, \,\,J_{+}(\vec{k}) J_{+}(\vec{k}') |1 \rangle \,\,\mbox{for}\,\, \vec{k}\not= \vec{k}', \,\,(J_{+}(\vec{k}))^2 |1 \rangle, \,\, \cdots
\sigma_{AB}(x,\xi) =u_{\xi}^{-1}(x)(A(Bu_{\xi}))(x) &=u_{\xi}^{-1}(x)\int_{\Omega}\Big(\Big[\int_{\Omega}F(x,y,z)k_{A}(x,z)dz\Big] &\times\int_{\Omega}\Big[\int_{\Omega}F(y,s,t)k_{B}(y,t)dt\Big]u_{\xi}(s)ds\Big)dy.
\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Tr}\;SD_a(P_\pm-D_aP_\pm D_a)^{2n+2}&=\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Tr}\;\lim_{\epsilon\downarrow 0}\lim_{R\nearrow \infty}I_\pm(\epsilon,R) &=\lim_{\epsilon\downarrow 0}\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Ind}^{(2n+1,\epsilon)}(D_a,S,U)={\rm Ind}^{(2n+1)}(D_a,S,U),
\frac{N-2\alpha}2\int_{\partial B_n}y^{1-2\alpha}v_n\frac{\partial v_n}{\partial\nu}\,dS &=\frac 12 \int_{\partial B_n}y^{1-2\alpha}|\nabla v_n|^2(X-z_0,\nu)\,dS - \int_{\partial B_n}y^{1-2\alpha}(\nabla v_n, X-z_0)\frac{\partial v_n}{\partial\nu}\,dS.
\partial_t \rho + \partial_x (u \rho ) = 0, &\partial_t (\rho u) + \partial_x (\rho u^2) =- \partial_x p(\rho) + \partial_x(\mu(\rho) \partial_xu)+\rho f, &(\rho,u)|_{t = 0}= (\rho_0, u_0)
\langle \nu_{23}(\tilde{x}_2), \nu_{234}(\tilde{x}_1), \nu |\nu_{12}(\tilde{n}_1+\tilde{n}_2), \nu_{123}(\tilde{n}_1),\nu \rangle=\sqrt{\omega_{{\bf \tilde x}}\sigma_{\bf \tilde{n}}}{R}_2 ({\bf \tilde{n}; \tilde{x}; \tilde{\beta};}N ),
\Psi ( x ) = 3 D \left( \begin{array}{c}{{\Psi_{L}( x )}}\\{{\Psi_{R}( x )}}\\ \end{array}\right) \hspace{. 5 c m}\Psi_{\pm}( x ) = 3 D \frac{1 \pm \gamma_{5}}{2}\Psi ( x )
\Delta(\underline{k},0,z) = \det\left(\begin{array}{ccc}A_{k_1}(z) B_{k_1}(z) R_{k_1}(z) A_{k_2}(z) B_{k_2}(z) R_{k_2}(z) A_{k_3}(z) B_{k_3}(z) R_{k_3}(z) \end{array}\right),
\begin{aligned}|u^o_{n,k+1}- \pi_x\circ \widetilde{F}(z^o_{n,k}, u^o_{n,k})| < |z^o_{n,k+1}- \pi_z\circ \widetilde{F}(z^o_{n,k}, u^o_{n,k})| < \frac{|u^o_{n,k}-x^o_{n,k}|}{4}+ \frac{D}{n^2},\end{aligned}
(x+y)^{m+n-1}= x^n(x+y)^{m-1}+ y^m(x+y)^{n-1}\leq x^ns^{m-1}+ y^ms^{n-1}&\leq z(x+z)^{n-1}s^{m-1}+ w (y+w)^{m-1}s^{n-1}&\leq (z+w)s^{m+n-2}= (z+w)(x+y+z+w)^{m+n-2}
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{\ell_1}0 0 0 1 0 0 0 d^{- \ell_3}\end{array}\right)(\, \ell_1 > 0 > \ell_3 \, ) .
\begin{aligned}x=\frac{x_0}{\sin r_0}\sin (r+r_0),\ \tau=\cos (r+r_0),\ |\mu|=\sin (r+r_0), (y,\hat\mu)=\exp(rH_{\frac{1}{2}h})(y_0,\hat\mu_0),\r\in(-r_0,-r_0+\pi), \end{aligned}
|\varpi|^{2m+k}N_3(\varpi^k) \ll |\varpi|^{m+nk}+|\varpi|^{m+nk}\sum_{l=0}^{k-1}|\varpi|^{(n/6-2)(l-k)}\ll |\varpi|^{m+nk}+|\varpi|^{m+nk}