harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0 = \sum_{k = 0}^{l}a^{( k )}( ( s )_{k}( k + h_{j}( m - 1 ) ) ( s )_{k}- a^{( k )}( s )_{k + 1}= ( h_{j}( m - 1 ) - s + l ) \sum_{k = 0}^{l}a^{( k )}( s )_{k}.
\begin{array}{rrll}\Pi_\gamma = \int_\gamma & \prod_{i=1}^N dY_i & \prod_{\beta = 1}^\ell \exp(-Y_{P_{\beta}}) dY_{P_\beta}& \prod_{a = 1}^k \delta(\sum_{i = 1}^N Q_{i a}Y_i - \sum_\beta d_{\beta a}Y_{P_{\beta}}- t_a) & \exp(-W')\end{array}
M^{\mu_n(i)}{}_{(j)}\left( p\right) f_{\left[ k\right]}^{\mu_n(j)}\left(p\right) =\lambda_{\left[ k\right]}^{\mu_n}\left( p\right) f_{\left[k\right]}^{\mu_n(i)}\left( p\right) \;\,,\;\,p\in V_{\mu_n}^{+}\;\,,
F_n\!=\!\frac{1-c_{1n}}{(1-c_{1n}+1-c_{2n}+1-c_{1n}c_{2n})^{1/2}},\,\,G_n\!=\!\frac{1-c_{2n}}{(1-c_{1n}+1-c_{2n}+1-c_{1n}c_{2n})^{1/2}},
Q^\delta_s(h,\widetilde{h}) = \iint_{(\mathbb{R}^+)^2}\delta^{2s-1}(x,y) h(x)\widetilde{h}(y) dx dy &=\int_{x\in I(h)}h(x)\left(\int_{y\in I(\widetilde{h})}\delta^{2s-1}(x,y)\widetilde{h}(y)dy\right)dx &=\int_{x\in I(h)}h(x)\delta^{2s-1}(x,I(\widetilde{h}))\left(\int_{y\in I(\widetilde{h})}\widetilde{h}(y)dy\right)dx = 0.
(-1)^N \overline u (r) \leqslant (-1)^N \overline u (0) + \sum_{l=1}^{N-1}\frac{(-1)^{N-l}(-\Delta)^l \overline u(0) r^{2l}}{\prod_{k=1}^{l}(2k) \prod_{k=1}^{l}[2N-1+2(k-1)]}.
\begin{aligned}\mathbb{E}(\phi(X(t))) = \int_{0}^{\infty}\int_{H}\phi(x)\mu_{\tau}^{Y}(\mathrm{d}x)f_{E_{t}}(\tau)\mathrm{d}\tau & = \int_{H}\phi(x)\int_{0}^{\infty}\mu_{\tau}^{Y}(\mathrm{d}x)f_{E_{t}}(\tau)\mathrm{d}\tau.\end{aligned}
\phi b^{m_0}xb^{m_k}\sim \phi b^{m_0}xb^{m_k-1}-\sum_{s\in S_b}\phi b^{m_0}xb^{m_k-1}s &\sim ... &\sim \phi b^{m_0}xb^{m_k-(m_k-n_k)}-\sum_{i=1}^{m_k-n_k}\sum_{s\in S_b}\phi b^{m_0}xb^{m_k-i}s &=\phi b^{m_0}xb^{n_k}-\sum_{i=n_k}^{m_k-1}\sum_{s\in S_b}\phi b^{m_0}xb^{i}s.
+ 3 c_{2}^{1}a_{1}^{2}a_{2}^{2}+ c_{1}^{0}a_{1}^{2}a_{1}^{1}+ c_{1}^{1}a_{2}^{1}a_{2}^{3}+ c_{1}^{0}a_{2}^{1}a_{2}^{4}- c_{2}^{0}a_{1}^{2}a_{2}^{3}+ c_{2}^{0}a_{1}^{1}a_{2}^{4}- 3 a_{1}^{0}c_{2}^{2}a_{2}^{3}
\displaystyle \mbox{$_4$F$_3$}(a,a,a,b; \,1+a,1+a,1+a;\,1)= \frac{{a}^{3}\Gamma \left( a \right) \Gamma \left( 1-b \right) \left( \Psi^{\prime}\left(a \right) -\Psi^{\prime}\left( 1+a-b \right) + \left( \Psi \left( a \right) -\Psi \left( 1+a-b \right) \right)^{2}\right) \mbox{}}{2\,\Gamma \left( 1+a-b \right)}
\begin{array}{c}^{out}<R_c (\beta_1') \bar R_{\underline d}( \beta_2') | R_a(\beta_1) \bar R_{\underline b}( \beta_2) >^{in}= \delta (\beta_1' - \beta_1) \delta (\beta_2' - \beta_2) F_{ab}^{cd}(\beta) - \delta (\beta_1' - \beta_2) \delta (\beta_2' - \beta_1) B_{ab}^{dc}(\beta),\end{array}
b_R(Rgx, Rgy) = \sum_{r \in R}b(rgx, rgy) = \sum_{r\in R}b(g^{-1}rgx, g^{-1}rgy) = \sum_{\underset{r\in R}{r' = g^{-1}rg}}b(r'x, r'y) = \sum_{r \in R}b(rx,ry) = b_R(Rx,Ry),
\left[ \left( \frac{H_{0}}{f}\right)^{2}- \left( \frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}\,^{2}}\right] + \frac{2 \alpha}{M^{2}}\left[ \left( \frac{H_{0}}{f}\right)^{2}- \left( \frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}\,^{2}}\right]^{2}= \frac{\Lambda_{b}}{6 M^{3}}.
\tilde{Y}(z) = \begin{pmatrix}1 0 0 0 0 1 0 1 0 \end{pmatrix}Y(z) \begin{pmatrix}1 0 0 0 0 1 0 1 0 \end{pmatrix}\begin{pmatrix}1 0 0 0 z^{-1/2}0 0 0 z^{1/2}\end{pmatrix}
max_{0<\theta<2\pi}2(1-\frac{sin(\theta)}{\theta})\le lim_{n\rightarrow\infty}\frac{k(n)^{2}}{n}=inf_{n\in\mathbb{N}}\frac{k(n)^{2}}{n}\le\frac{128^{2}}{6166}=2.6571...<\frac{8}{3}.
w\left( \{x : |T_{\alpha, m}f (x) | \geq \lambda \}\right) \leq C \lambda^{-\frac{n}{n - \alpha}}\sum_{i=1}^{m}\left(\int_{\mathbb{R}^{n}}|f(x)| [w_{A_i^{-1}}(x)]^{\frac{n- \alpha}{n}}dx \right)^{\frac{n}{n - \alpha}}.
f^{2}= 1 - \frac{1}{\kappa^{2}u_{b}^{2}}g ,{\cal E}\simeq - \frac{\pi \Delta u}{u_{b}}( \kappa^{2}u_{b}^{2}) \left[ 1 - \frac{1}{\kappa^{2}u_{b}^{2}}\left( L - \frac{1}{2}u_{b}^{2}b_{\mathrm{e x t}}\right)^{2}\right]^{2}.
\beta=\lim_m\beta^{(3)}_m,\beta^{(3)}_m=-\Big(\sum_{n=3}^m\frac{1-c_{1n}}{1+c_{1n}}\Big)^{-1}\sum_{n=3}^m\frac{(1-c_{1n})(1-c_{2n})}{1+c_{1n}}.
P^N\tilde{\tau}_t(G_{n}\times G_{n})=\bigcup_{\xi,\beta,\gamma}P^N\tilde{\tau}_t((\xi\times 1_{2m-2t})\beta\times\gamma)=\bigcup_{\xi,\beta,\gamma}P^N\tilde{\tau}_t(\beta\times(\xi\times 1_{2m-2t})\gamma).
m^{\star}(C_\mu \cap (\bigcup_{k \in \mathbb{N}}I_k)) = m^{\star}(\bigcup_{k \in \mathbb{N}}(C_\mu \cap I_k) \leq \sum_{k \in \mathbb{N}}m^{\star}(C_\mu \cap I_k) \leq \sum_{k \in \mathbb{N}}m(H \cap I_k) \leq (1-\mu)\sum_{k \in \mathbb{N}}m(I_k) < (1-\mu) m(G) < m^{\star}(C_{\mu})
\mathfrak{S}_{\delta}^{2^{\omega}}(x, D) = \sum_{d\le \sqrt{2D}}\mu(d) \sum_{D/d^2<\ell\le 2D/d^2}\tau(\ell) \psi\Big(\frac{x/d^2}{\ell+\delta/d^2}\Big) & = \sum_{d\le \sqrt{2D}}\mu(d) \mathfrak{S}_{\delta/d^2}^{\tau}(x/d^2, D/d^2).
N^{2}M \int_{0}^{M}{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{k^{2}}{k^{2}+ \sigma^{2}}}={\frac{N^{2}}{2 \pi^{2}}}M \left[{\frac{1}{3}}M^{3}- \sigma^{2}M + \sigma^{3}\arctan{\frac{M}{\sigma}}\right]
a_h^0(e_f,\psi_h) := -\sum_{i,j}\int_{T_{ij}}e_f v\,\partial_x\psi_h\,{\rm d}v\,{\rm d}x-\sum_{j = 1}^{N_v}\sum_{i = 0}^{N_x-1}\int_{J_j}\left(\widehat{v e_f}[\![\psi_h]\!]\right)_{i+1/2,v}\,{\rm d}v,
\sigma^{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \qquad \sigma^{2}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) , \qquad \sigma^{3}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) .
\mathcal{H}^1\left( J \cap B_{2s\rho}\right) \geq \mathcal{H}^1\left( J \cap B_{2 r_x}(x)\right) = \mathcal{H}^1\left( J \cap B_{\lambda_x}(x)\right) \geq \eta \lambda_x > \eta(1-s)\rho,
D_D(\zeta_0)= \exp\left[\zeta_0 \sqrt{2\lambda_0}\hat A^{\dagger}\right]\left( 1 -\vert \zeta_0 \vert^2\right)^{\lambda_0 g(\hat n;a,\lambda)}\exp\left[-\zeta_0^{*}\sqrt{2\lambda_0}\hat A\right].
\frac{(1+1/p)^{-1}\cdots (1+1/p^r)^{-1}}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}\cdot \frac{(1-1/p) \cdots (1-1/p^r)}{(1-1/p) \cdots (1-1/p^{r-l(\lambda)})}
\int_{0}^{L}\left(\psi'\right)^{2}=\int_{0}^{L}\left[\mu_{1}\left(\alpha- 2 v \right)\mathbf{1}_{v>0}+\mu_{2}\left(d + 2 v \right)\mathbf{1}_{v<0}\right]\psi^{2}+\lambda & =M_1\int_{\{\mu_1 > 0\}\cap\{v>0\}}\left(\alpha-2 v\right)\psi^{2}+M_2\int_{\{\mu_2 > 0\}\cap\{v<0\}}\left(d+2v\right)\psi^{2}+\lambda.
\left\|u_{(a_i')}- u_{(a_i)}\right\|_{C^{2m}(\overline B_{R})}\leq 1\mbox{and}\Delta^{m-1}u_{(a_i')}(R) \leq\frac{\Delta^{m-1}u_{(a_i)}(R)}{2}:= -M < 0.
\beta(\mathbf{f}_{11}(a),\mathbf{g}_{12}(b)) =-(-1)^{(1+|a|)(1+|b|)}\beta(\mathbf{g}_{12}(b),\mathbf{f}_{11}(a))=\boldsymbol{\pi}(a\bar{b}), \beta(\mathbf{f}_{21}(a),\mathbf{g}_{11}(b)) =-(-1)^{(1+|a|)(1+|b|)}\beta(\mathbf{g}_{11}(b),\mathbf{f}_{21}(a))=-\boldsymbol{\pi}(a\bar{b}),
3\sum_{\substack{i<j r \neq i,j}}d_T(i,j)x_ix_jx_r = 3 \sum_{i<j<k}[d_T(i,j) + d_T(i,k) + d_T(j,k)] x_ix_jx_k = 6 \sum_{i<j<k}d_T(i,j,k) x_ix_jx_k = \sum_{i,j,k}d_T(i,j,k) x_ix_jx_k
\frac{\left|{\sum}^* ~L(1,\chi,Y) |R(\chi)|^2 \right|}{{\sum}^* ~|R(\chi)|^2}\geq e^\gamma \log X \left(1 - \frac{1}{\log X}+ \mathcal{O}\left(\frac{1}{(\log X)^2}\right)\right) & = e^\gamma \left(\log_2 q + \log_3 q - \log B - 1 + \mathcal{O}\left(\frac{1}{\log_2 q}\right)\right),
\sigma_{2 3}^{2}= - 1 ~ , ~ ~ \sigma_{1 3}\sigma_{2 3}= \sigma_{2 1}~ , ~ ~ \sigma_{1 2 3}~ \left( \sigma_{2 3}, \sigma_{1 3}, \sigma_{1 2}\right) = - \left( \sigma_{1}, \sigma_{2}, \sigma_{3}\right) ~ ,
\begin{cases}-\Delta w^+ +\big[A_+(|w^+|^2-{t^+}^2)+B(|w^-|^2-{t^-}^2)\big]w^+=0, begin{align*}2mm]-\Delta w^- +\big[A_-(|w^-|^2-{t^-}^2)+B(|w^+|^2-{t^+}^2)\big]w^-=0,\end{cases}
\sum_{p}\frac{1}{p}= \sum_{n \geq 1}\frac{1}{n}(\pi_2(n+1) - \pi_2(n)) = \int_1^\infty \frac{1}{\lfloor x \rfloor}\mathrm{d}(\pi_2(x)) = = \left[ \frac{\pi_2(x)}{x}\right]_1^\infty - \int_1^\infty \pi_2(x) \mathrm{d}\left( \frac{1}{\lfloor x \rfloor}\right),
\begin{array}{ccc}b_{0,0}(z) = 1, b_{0,1}(z) = \overline{z}, b_{0,2}(z) = \frac{1}{\sqrt{2}}\overline{z}^2, begin{align*}1ex]b_{1,0}(z) = z, b_{1,1}(z) = |z|^2-1, b_{1,2}(z) = \frac{1}{\sqrt{2}}\,\overline{z}(|z|^2-2), begin{align*}1ex]b_{2,0}(z) = \frac{1}{\sqrt{2}}\,z^2, b_{2,1}(z) = \frac{1}{\sqrt{2}}\,z (|z|^2-2), b_{2,2}(z) = \frac{1}{2}(|z|^2 - 4|z|^2 + 2).\end{array}
\begin{cases}\partial_{t}G(t,x)=\nabla v\big(t,\Phi(t,0,x)\big)\cdot G(t,x) 0\leq t\leq T \qquad\qquad- (\nabla\times d)\big(t,\Phi(t,0,x)\big)\exp\left\{\int^t_0\nabla\cdot v\big(s,\Phi(s,0,x)\big){\rm d}s\right\}G(0,x)=B_0(x) x\in\mathbb{T}^3\end{cases}
\langle |F_k|^2 , \phi \rangle =\frac{3}{\pi}\cdot \langle 1, \phi \rangle + \sum_{j \geq 1}\langle |F_k|^2 , u_j \rangle \langle u_j , \phi \rangle + \frac{1}{4\pi}\int_{\mathbb{R}}\langle |F_k|^2, E(\cdot, \tfrac 12 + it)\rangle \langle E(\cdot , \tfrac 12 + it) , \phi \rangle dt.
f^{\lambda}= \left( f_{1}e_{a}^{\lambda}e^{b \nu}+ f_{2}e^{b \lambda}e_{a}^{\nu}+ f_{3}g^{\nu \lambda}\delta_{b}^{a}\right) \bar{\Omega}_{b \nu}^{a}+ N_{b}^{\mu \alpha \beta b \nu}\nabla_{\alpha}\nabla_{\beta}\bar{\Omega}_{b \nu}^{a}
\left( \prod_{i=1}^{n+1}(1+x_i) \right)h_{n-|I|}(\underline x_n^I) \prod_{i\in I}\frac{1}{1+x_i}=h_{n-|I|}(\underline x_n^I) \prod_{i\in [1,n+1]\setminus I}(1+x_i)
\Psi_{q^3}(1) \Psi_q(2) =\Psi_{q}(2) \Psi_{q^3}(12^3) \Psi_q(1 2^2) \Psi_{q^2}(1^2 2^3) \Psi_q(12) \Psi_{q^3}(1) \;.
(q;q)_{\infty}\sum\limits_{n = 0}^{\infty}d_{e}(n)q^{n}= \left( 1 + \sum\limits_{n = 1}^{\infty}(-1)^{n}q^{\frac{n(3n \pm 1)}{2}}\right)\sum\limits_{n = 0}^{\infty}d_{e}(n)q^{n}& = \sum_{n = 0}^{\infty}\sum_{k = 0}^{n}c_{k}d_{e}(n-k)q^{n}
f_j^{(n)}( \pmb{t}_{n-1}, \pmb{x}_{n-1},r,z,t,x)= f_{j}(\pmb{t}_{j - 1}, \pmb{x}_{j - 1},r,z) g_{n-j}(\pmb{t}_{j:n-1}, \pmb{x}_{j;n-1},r,z,t,x),
(\partial_{x}^{h}\phi)_{jk}=\sum_{p=-M/2}^{M/2-1}\sum_{q=-M/2}^{M/2-1}i\varrho^{x}_{p}\widehat{\phi}_{pq}\, e^{i \frac{2j p \pi}{M}}e^{i \frac{2k q \pi}{M}}, (\partial_{y}^{h}\phi)_{jk}=\sum_{p=-M/2}^{M/2-1}\sum_{q=-M/2}^{M/2-1}i\varrho^{y}_{q}\widehat{\phi}_{pq}\, e^{i \frac{2j p \pi}{M}}e^{i \frac{2k q \pi}{M}},
\langle \sigma_{-}( 0 ) \sigma_{+}( x ) \sigma_{-}( 1 ) \sigma_{+}( \infty ) \rangle \sim \sqrt{\sin ( \pi \theta )}[ x ( 1 - x ) ]^{- 2 \Delta_{\sigma}}[ F ( x ) F ( 1 - x ) ]^{- 1 / 2}\ ,
ds_{\rm can}^2 = \left(1+{4M\over r}-{4 g^2q^2 \over r^2}\right)^{-{1\over2}}dt^2 - \left(1+{4M\over r}-{4 g^2 q^2 \over r^2}\right)^{1\over2}d\vec x^2 \ .
\left(\beta\,M_X(s,t,z)-\alpha\,M_V(s,t,z)\right)\left(M_X(s,t,z)-M_U(t,s,z)\right) =0 \left(\beta\,M_U(t,s,z)-\alpha\,M_Y(t,s,z)\right)\left(M_V(s,t,z)-M_Y(t,s,z)\right) =0.
K_{N}^{1}(z,\zeta)=\sum_{k=n-r}^{n-1}C'_{k}\frac{\rho(\zeta)^{k+N}s\wedge\left(\partial_{\bar{\zeta}}Q\right)^{n-r}\wedge\left(\partial_{\bar{z}}Q\right)^{k+r-n}\wedge\left(\partial_{\bar{z}}s\right)^{n-k-1}}{\left|z-\zeta\right|^{2\left(n-k\right)}\left(\frac{1}{K_{0}}S(z,\zeta)+\rho(\zeta)\right)^{k+N}},
e_{k}^{\boldsymbol{q}}(x_1,\ldots,x_n)=\sum_{\ell}\sum_{1 \le i_{1}< \cdots < i_{\ell}\le n \atop j_{1}> i_{1}, \ldots ,j_{\ell}> i_{\ell}}e_{k-2\ell}(X_{\overline{I \cup J}}) \prod_{a=1}^{\ell}q_{i_{a},j_{a}},
\Phi ({\bf r}_{1}^{( 1 )}, \ldots ,{\bf r}_{n_{1}}^{( 1 )}, \ldots ,{\bf r}_{1}^{( M )}, \ldots ,{\bf r}_{n_{M}}^{( M )}, t ) = \left\langle 0 \left| \prod_{p = 1}^{M}\frac{1}{\sqrt{n_{p}!}}\Psi_{p}({\bf r}_{1}^{( p )}, t ) \cdots \Psi_{p}({\bf r}_{n_{p}}^{( p )}, t ) \right| \Phi \right\rangle
\varphi\circ D(ab)= \varphi\circ id_{A}(a) \varphi\circ Db+\varphi\circ id_{A}(b) \varphi\circ Da = \varphi(a)\varphi\circ Db+\varphi(b) \varphi\circ Da = \varphi(a.Db+b.Da)
\int_{\Omega}\delta ({\cal L}) \sqrt{g}d^{4}x = \int_{\Omega}J^{\mu \nu}{}_{\lambda \rho}\left\{- 2 \nabla_{\mu}\nabla^{\lambda}( \delta g )_{\nu}{}^{\rho}+ R_{\mu \nu}{}^{\sigma \lambda}( \delta g )_{\sigma}{}^{\rho}\right\}\sqrt{g}d^{4}x .
{\cal V}^{I}( \Phi ) ={\frac{m^{2}}{\beta^{2}}}+{\frac{\mu^{2}-{\cal M}^{2}}{8 \pi}}-{\frac{m^{2}}{\beta^{2}}}({\frac{\mu^{2}}{{\cal M}^{2}}})^{{\frac{\beta^{2}}{8 \pi}}}\cos ( \beta \Phi ) .
w^{(m)}=\left(\prod_{s=1}^{q-1}\prod_{k=1}^{\alpha_s}\ell_0^{(i_s)}\cdots \ell_{g-1}^{(i_s)}(\lceil \beta_{i_{s,k}+g}\rceil-1)\cdots (\lceil \beta_{i_{s,k+1}+p(2N+1)-1}\rceil-1)\right) \ell_0^{(j)}\cdots \ell_{m-1}^{(j)}.
H_0 = \frac{1}{l_0}\sum_{i}{\rm Tr}(\dot{\bf x}^i \dot{\bf x}^i)+ \frac{1}{l_0^5}\sum_{i,j}{\rm Tr}([{\bf x}^i,{\bf x}^j]^\dagger[{\bf x}^i,{\bf x}^j] )\ .
\int_{|y|>t^{\beta/\alpha}}|I_{B}|dy \le C t^{2\beta - 1}\int_{|r| > t^{\beta/\alpha}}|r|^{-d-\alpha + d - 1}dr +C t^{2\beta - 1}\int_{|r| > t^{\beta/\alpha}}|r|^{d-1-d-\alpha}|r|^{2\alpha}t^{-2\beta}dr =Ct^{2\beta - 1}(t^{\beta/\alpha})^{-\alpha}+ C t^{2\beta - 1 - 2\beta}(t^{\beta/\alpha})^{\alpha}= Ct^{\beta - 1}.
\left(f+\bar{\psi}\circ L\right)^{*}_\Phi\left(\phi\right)=\sup_{x\in x}\phi\left(x\right)-\bar{\psi}\left(Lx\right)-f\left(x\right)\geq \sup_{x\in x}\phi_1 \left(x\right)-f\left(x\right)=f^*_\Phi (\phi_1).
a^{\frac 1 2}_A = \left(\tau^{\frac 1 2}\right)^{1,2}+ \left(\tau^{\frac 1 2}\right)^{12,3}a^{\frac 1 2}_B = \left(\tau^{\frac 1 2}\right)^{2,3}+ \left(\tau^{\frac 1 2}\right)^{1,23},
\begin{array}{l}f_{(+)}(\theta_{+k})=Ce^{\chi_{R}(\theta_{+k})-i\chi_{J}(\theta_{+k})}, f_{(-)}(\theta_{-k})=\left.f_{(+)}^{*}(\theta_{+k}) \right|_{\theta_{+k}=\theta_{-k}}=Ce^{\chi_{R}(\theta_{-k})+i\chi_{J}(\theta_{-k})}.\end{array}
z_{i_1}(z_{i_2}\cdots z_{i_K})z_{i_{K+1}}=\sum_j z_{i_1}z_{i_2}\cdots z_{i_K}z_jz_j^*z_{i_{K+1}}=\sum_jz_{i_1}z_j^*z_{i_{K+1}}z_{i_2}\cdots z_{i_K}z_j &= \sum_j z_{i_{K+1}}z_{i_2}\cdots z_{i_K}z_{i_1}z_j^*z_j = z_{i_{K+1}}(z_{i_2}\cdots z_{i_K})z_{i_1}
\begin{aligned}\Vert \Lambda^{\alpha}(uv)- u\Lambda^{\alpha}v \Vert_{L^p}\lesssim_{\alpha,N}\Vert\nabla u \Vert_{L^{p_1}}\Vert \Lambda^{\alpha-1}v \Vert_{L^{p_2}}+\Vert \Lambda^{\alpha}u \Vert_{L^{p_1}}\Vert v \Vert_{L^{p_2}}.\end{aligned}
\mathcal{Z}(\widehat{\mu}_{\{b_k\},\{D_k\}})=\bigcup_{k=1}^{\infty}\left(b_1b_2 \cdots b_k \frac{\mathbb{Z}\setminus p_{k}\mathbb{Z}}{p_{k}t_{k}}\right) \subset \bigcup_{k=1}^{\infty}\frac{b_1 b_2\cdots b_k\mathbb{Z}}{p_{k}t_{k}}.
\gamma^{\mu}\Psi_{, \mu}= - i \gamma^{\mu}\frac{e^{\gamma^{5}\alpha}}{\rho}\{I m ( \overline{{\Psi}}_{, \mu}\Psi ) - j_{\mu}- \gamma^{5}[ I m ( \overline{{\Psi}}_{, \mu}\gamma^{5}\Psi ) - g_{\mu}] \}\Psi
\prod_{j=l_1}^{l_{m}}\dddot{f}_j\cdot\sigma_{r-m}(\ddot{f}_{k+1},\ldots,\widehat{\ddot{f}_l},\ldots,\widehat{\ddot{f}}_{l_1},\ldots,\widehat{\ddot{f}}_{l_{m}},\ldots,\ddot{f}_n)=0
\operatorname{vol}(M) = \int_M \frac{\omega_M^n}{n!}= \operatorname{vol}(B)\int_0^1 P(x)^{n-1}dx = \frac{(p+c)^n - c^n}{pn}\int_M s^H\frac{\omega_M^n}{n!}\overset{\eqref{ScalCalabi}}{=}\int_0^1 \frac{s^H_S - (P\varphi)''(x)}{P(x)}\operatorname{vol}(\mu^{-1}x) dx &= \operatorname{vol}(B)\int_0^1 \left[s^H_S - (P\varphi)''(x) \right] P(x)^{n-2}dx
X^H(t) = v_{i_k}^H t+\sum_{\substack{h=0 h\not=k}}^H (v_{i_h}^H-v_{i_k}^H) T_{(i_h)}(t) = g_k^H\Bigl(T_{(-i_k)}^H(t)\Bigr), \ \ \t\ge0,\, k=0,\dots, H.
\int \dfrac{x\sin(x;q)}{(\frac{-x^2}{q}(1-q)^2;q^2)_\infty}\,_2\phi_1(\frac{-x^2}{q}(1-q)^2,q^2;0;q^2,q)d_qx & =\frac{1}{(\frac{-x^2}{q}(1-q)^2;q^2)_\infty}\left( x\cos(q^{\frac{-1}{2}}x;q) \,_2\phi_1(\frac{-x^2}{q}(1-q)^2,q^2;0;q^2,q)-\dfrac{q \sin(x/q;q)}{1-q}\right),
\Delta_{0}^{( 1 )}= \frac 1{2 m^{2}}i p^{( 1 )}\left( i p^{( 1 )}- \varepsilon m \right) \left( 1 - \sigma_{p}^{( 1 ) 2}\right) = \Psi_{0}^{( 1 )}\cdot \overline{{{\Psi}}}_{0}^{( 1 )}
\xi_t(z) = \xi_0(z) + \sum_{w\sim z}\int_0^t (\xi_{s-}(w) - \xi_{s-}(z)) \, dP^{z,w}_s & + \sum_{w\sim z}\int_0^t \xi_{s-}(w)(1 - \xi_{s-}(z)) \, d\tilde P^{z,w}_s.
g\left(x_1,x_2\right) =\left(\tilde{f}*f\right)\left(x_1,x_2\right) &=\int_{G^2}{\overline{f\left(-y_1,-y_2\right)}f\left(x_1-y_1,x_2-y_2\right)}\{d^2}_{{\mu}_{G^2}}\left(y_1,y_2\right) &=\int_{G^2}{\overline{f\left(y_1,y_2\right)}f\left(x_1+y_1,x_2+y_2\right)}\{d^2}_{{\mu}_{G^2}}\left(y_1,y_2\right).
V^*V = \left((U^*U)^{-1}U^*+P(0\oplus G)R^*\right)\left(U(U^*U)^{-1}+R(0\oplus G)P\right) &= (U^*U)^{-1}+(0\oplus G^2)=(I\oplus T^{-1})+(0\oplus (I-T^{-1})) = I.
f_{s,k}(p):=-\Big(\sum_{m=1}^{s}\frac{1}{m!}\sum_{\substack{n_0+\ldots+n_m=s n_0\ge0 n_1,\ldots,n_m\ge1}}\{\{\ldots\{\{f_{n_0}+f_{n_0}^*,G^0_{n_1}\},G^0_{n_2}\}\ldots\},G^0_{n_m}\}\Big) + (ip\cdot k)g_{s,k}(p)\,.
f(y)-f(x)-\langle \nabla f(x), y-x\rangle = \int_0^1 \langle \nabla f(x+\tau(y-x))-\nabla f(x), y-x\rangle d\tau = \int_0^1 \langle \nabla g(Ex+\tau\cdot E(y-x))-\nabla g(Ex), Ey-Ex\rangle d\tau.
q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda n_s^{-1}(\mu))T_{n_s^{-1}}=q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(n_s^{-1}(\mu))\theta(\lambda)T_{n_s^{-1}}&=E(n_s^{-1}(\mu))E(\lambda n_s^{-1}).
\sum_{t\in\mathbb{F}_p^\times}\varphi(x-t)~{_2F_1}\left(\begin{array}{cc}\chi_4, \chi_4^3 \varepsilon\end{array}\mid t\right)=\frac{\varphi(x)}{p}+\varphi(x)\chi_4(-1)(\varphi(1+a)+\varphi(1-a)).
Z=\int{\cal D}g_{ij}\,{\cal D}\pi^{ij}\,{\cal D}\theta^a\,{\cal D}\chi^a\delta((M^{-1})^a_b[\theta]\chi^b) exp\int(-{\cal H}^{'}+i\pi^{ij}\partial_0g_{ij}+i\chi^a\partial_0\theta^a).
\tau_1 (t_f) - \tau_1 (t_i) \equiv \int_{t_i}^{t_f}\frac{ds}{a_1^2 (s)}= \frac{1}{\alpha ( \alpha t_i + \beta )}- \frac{1}{\alpha ( \alpha t_f + \beta )}= \frac{T}{a(t_f) a(t_i)},
(\Phi(t,L)-Id)u_0 = \Psi(t,L) L u_0 = \int_0^t k(t,s) \Phi(s,L) L u_0 ds = \int_0^t k(t,s) L \Phi(s,L) u_0 ds \Leftrightarrow \ \Phi(t,L)u_0 = u_0 + \int_0^t k(t,s) L\Phi(s,L)u_0 ds.
S=\sum_{r_{1}=1}^{R_{1}}\sum_{r_{2}=1}^{R_{2}}\cdot\cdot\cdot\sum_{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},...,r_{M}}e_{r_{1}}\otimes e_{r_{2}}\otimes\cdot\cdot\cdot\otimes e_{r_{M}},
c_1(t) = x_1 - \frac1\lambda(e^{\lambda t y}\cos(\lambda t \xi)-1)\sin\theta + \frac1\lambda e^{\lambda t y}\sin(\lambda t \xi) \, \cos\theta & c_2(t) = x_2 + \frac1\lambda e^{\lambda t y}\sin(\lambda t \xi)\sin\theta + \frac1\lambda (e^{\lambda t y}\cos(\lambda t \xi)-1) \, \cos\theta \,.
F_2 :=E(2C_R+1)\int_{t_0}^{u \wedge \tau_{nR}}|e^n_s|^2I_{B(R)}ds &\leq (2f(R)+1)\int_{t_0}^{u}E\sup_{t_0 \leq r \leq s}|e_{r \wedge \tau_{nR}}^n|^2 I_{B(R)}ds
P\left(\rho_1\geq 2c_y n^{1+\xi_y}\right) =P\left(\frac{1}{n}||Y||^2\geq 2c_y n^{\xi_y}\right) &\leq P\left(\frac{1}{n}||Y||^2\geq 2\sigma_y^2\right)\leq O\left(\exp\left(-c_Yn^{\xi_Y}\right)\right),
p_{}^{\prime}\left( x_{1},\ldots,x_{l}\right) =p_{l}\left(x_{l}|x_{1},\ldots,x_{l-1}\right) p_{l-1}\left( x_{l-1}|x_{1},\ldots,x_{l-2}\right) \cdots p_{2}\left( x_{2}|x_{1}\right) p_{1}\left(x_{1}\right)
\|\boldsymbol{\vartheta}_0\|_\infty\leq R,\qquad\|\boldsymbol{\theta}_{\boldsymbol{j}}^0\|_\infty\leq R2^{-\sum_{l=1}^d\alpha_lj_l\left(\frac{1}{d}+\frac{1}{2\alpha^{*}}\right)},
\mathcal{A}_{FT2}=\begin{pmatrix}I_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}I_{d\times d}0_{d\times d}0_{d\times d}I_{d\times d}0_{d\times d}0_{d\times d}-I_{d\times d}0_{d\times d}0_{d\times d}\end{pmatrix}.
0 \ge \left\langle \nabla_xd(y,x)-\nabla_xd(y,z),x-z \right\rangle-2C|x-z|^2 &=\left\langle \nabla_x\left(d(y,x)-C|x|^2\right)-\nabla_x\left(d(y,z)-C|z|^2 \right),x-z\right\rangle &=\left\langle \nabla_xd_{C}(y,x)-\nabla_xd_{C}(y,z),x-z\right\rangle.
1-\theta_{n,p}(x)=\frac{(1-v^{-1})^3(\log\log n)^{2}}{2\log n}-\frac{(1-v^{-1})^{2}(1+x-\log\{2\Gamma(\frac{1}{v})\})\log\log n}{\log n}+o\big(\frac{\log\log n}{\log n}\big)
\sigma_\mu^A \sigma_\nu^B \sum_{j=1}^{N_v}\langle g_\mu^A \vert j \rangle \langle j \vert f_\nu^B \rangle \langle j \vert \upsilon \rangle = \tilde{\it \Phi}_{\mu\nu}\ , 1\leq \mu \leq N_d \ , \ 1 \leq \nu \leq N_s \ .
S = \left( \begin{array}{c c c c}{{\tilde{a}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\tilde{d}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\tilde{e}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\tilde{e}}}\\ \end{array}\right) .
n =2\#\alpha + \#(\gamma^{-1}\alpha) +{\bf 1}_{f \equiv 1}+ \#(P_f^{-1}\alpha) &={\bf 1}_{f \equiv 1}+ 4p-(|\alpha| + |\gamma^{-1}\alpha|) - (|\alpha| + |P_f^{-1}\alpha|) &\leq{\bf 1}_{f \equiv 1}+ 4p - |\gamma| - |P_f| &\leq 1+3p+1 = 3p+2,
\times\exp \left(\sum_{b, b^\prime=1}^N \sum_{j=1}^{n_b}\sum_{j^\prime=1}^{m_{b^\prime}}2 \frac{\pi}{\pi+gN}\frac{1}{N}\; \mbox{K}_0\Big( e \sqrt{\frac{N}{\pi+gN}}|x_j^{(b)}- y_{j^\prime}^{(b^\prime)}|\Big) \right)
y_1 =\sum_{k=1}^{K}\bigl(\frac{d(u_{2k-2},u_{2k-1})}{d(a_1,b_1)}\,m_{u_{2k-2},u_{2k-1}}-\frac{d(u_{2k-1},u_{2k})}{d(a_1,b_1)}\,m_{u_{2k-1},u_{2k}}\bigr).
G_a(z) = \sum_{n \ge 0}z^{-n-1}\varphi(a^n), h_a(t) = \sum_{k \ge 0}t^k\varphi(a^k), G_{a,b}(z,w) = \sum_{m \ge 0,n \ge 0}z^{-m-1}w^{-n-1}\varphi(a^mb^n), H_{a,b}(t,s) = \sum_{p \ge 0,q \ge 0}t^ps^q\varphi(a^pb^q),
\lim_{n\rightarrow\infty}\left|\frac{n!}{(n+1)!}\frac{\left|\gamma(n+1,\lambda( \mathrm{\mathsf{x}}))\right|}{\left|\gamma(n,\lambda( \mathrm{\mathsf{x}}))\right|}\right|=\lim_{n\rightarrow\infty}\left|\frac{1}{(n+1)}\right|\lambda( \mathrm{\mathsf{x}})=0\;.
\vec{\psi}_{\mp}( x ) \left( \frac{d \vec{\phi}}{d x}\pm \vec{\nabla}W \right) | \vec{\phi}( x ) , \vec{\psi}( x ) \rangle = 0 \qquad , \vec{\psi}_{\pm}( x ) = \vec{\psi}_{1}( x ) \pm \vec{\psi}_{2}( x )
p(a,b;D):= \begin{cases}\frac{1}{2}, ~~{\rm if}~a\in D,~b\in D, \frac{1}{2}, ~~{\rm if}~a\in D^c,~b\in D^c, 1, ~~{\rm if}~a\in D^c,~b\in D, 0, ~~{\rm if}~a\in D,~b\in D^c.\end{cases}
\begin{array}{ccl}\left\vert \int_\Omega Ru^2dV_g\right\vert = \left\vert \int_\Omega R((\xi_{\tilde K}+\xi_E)u)^2dV_g\right\vert \leq \left\vert \int_\Omega R\xi_{\tilde K}^2u^2dV_g\right\vert+\left\vert 2\int_\Omega R\xi_E\xi_{\tilde K}u^2dV_g\right\vert+\left\vert \int_\Omega R\xi_E^2u^2dV_g\right\vert.\end{array}
\langle \psi_{1}^{I_{1}}( x_{1}) \cdots \psi_{n}^{I_{n}}( x_{n}) \rangle = e^{i ( \kappa_{1}+ \cdots + \kappa_{n}) \Omega}\langle \psi_{1}^{I_{1}}( x_{1}) \cdots \psi_{n}^{I_{n}}( x_{n}) \rangle ,
0=\mathcal{S}(V_k) =s_k\mathcal{L}_{0}(U_k)+t_k\mathcal{L}_0(W_k)+\frac{s_k^2}{2}\mathcal{S}_{0,2}(U_k,U_k)+s_kt_k\mathcal{S}_{0,2}(U_k,W_k) &+\frac{t_k^2}{2}\mathcal{S}_{0,2}(W_k,W_k) +O(s_k^3,s_kt_k^2,s_k^2t_k,t_k^3).
\frac{s^{2}}{4}= k^{\prime}+ \frac{1}{k^{\prime}}\ , \ \, e p s i l o n = 2 \frac{1 + k^{2}}{1 - k^{2}}= \frac{2 s^{2}}{\sqrt [ [object Object] ] ]{s^{4}- 6 4}}\ .
\psi_{\alpha,0}(x) = \frac{1}{2\alpha}+ \frac{1}{2\alpha\pi \sqrt{1-x^2}}\left(\int_{-\alpha}^{-1}+ \int_1^{\alpha}\right) \frac{\sqrt{t^2-1}}{|t-x|}dt, & = \frac{1}{2\alpha}+ \frac{1}{\alpha\pi \sqrt{1-x^2}}\int_1^\alpha \frac{t \sqrt{t^2-1}}{t^2-x^2}\, dt, x \in [-1,1].
n(PP^{- 1})_{ij}= \sum\limits_{r = 1}^n{P_{ir}n(P^{- 1})_{rj}}= \sum\limits_{r = 1}^n{(\delta_{r1}- \delta_{i1}+ \delta_{r1}\delta_{i1}+ \delta_{i,n + 2 - r})}&\qquad\qquad\qquad\times (2\delta_{r1}- 1 + n\delta_{r,n + 2 - j})
\varphi_{h , 0 , 0}( z_{1 0}, z_{2 0}, z_{3 0}) = \varphi_{h}( z_{2}, z_{3}; z_{0}) \, , \; \varphi_{1 1 1}( z_{i 0}) = \frac{1}{3}( \varphi_{0 0 3}+ \varphi_{0 3 0}+ \varphi_{3 0 0}) ( z_{i 0}) \, .
z_{i,\alpha}^{(0)}=[\begin{matrix}z_{i,\alpha0}^{(0)}\\z_{i,\alpha1}^{(0)}\\ \vdots\\z_{i,\alpha J-1}^{(0)}\end{matrix}]=A_{i,\alpha}^{(0)}[\begin{matrix}\eta_{\alpha}^{(0)}\\ \eta_{\beta}^{(0)}\end{matrix}]+v_{i,\alpha}^{(0)},
f\in C^\infty (\Gamma) \cap C^0 (\overline{\Gamma}) \mbox{is homogeneous of degree one and symmetric in}\lambda_i, &f>0\ \mbox{in}\ \Gamma,f = 0 \mbox{on}\partial\Gamma, &f_{\lambda_i}> 0 \ \mbox{in}\Gamma \ \forall 1 \leq i \leq n, &f.
\mathcal{M}_\Sigma (u)^A= \left\langle \vec{H}+\frac{X^\perp(u)}{2},\frac{\partial}{\partial y^B}\right\rangle(h_0)^{BA}e^{-|X(u)|^2/4+|X(0)|^2/4}\sqrt{\frac{\det g_u}{\det g_0}}.
D(z) = \{d_1,d_2,d_3,d_4,d_8,d_9,d_{10},d_{12},d_{13}\}, V_1(z) = \{1_{\partial'},2_{\partial'}, 1_{\partial''}\}, V_2(z) = \{1_{\partial},2_{\partial}, 1_{\partial'''}, 3_{\partial}\}.
\begin{array}{cccc}R_{i}^{~j}=\left(\begin{array}{cc}0 -\zeta^{ab}\bar{\zeta}_{ab}0 \end{array}\right)~~~~ ~~~~~\zeta^{ab}\bar{\zeta}_{bc}=\delta^{a}_{~c}~~ ~~\zeta^{ab}=\zeta^{ba}~~ ~~\bar{\zeta}_{ab}=(\zeta^{ab})^{\ast}\end{array}
\Delta(P)\star \Delta(H_1H_2)= \Delta(P)\star \Delta(H_1)\star \Delta(H_2)=\Delta(P\star H_1)\star \Delta(H_2) = \Delta(H_1 \star H_1(P))\star \Delta(H_2)=\Delta(H_1) \star \Delta(H_1(P))\star \Delta(H_2) = \Delta(H_1) \star \Delta(H_1(P)\star H_2)=\Delta(H_1) \star \Delta(H_2 \star H_2H_1(P)) = \Delta(H_1H_2) \star \Delta((H_1H_2)(P)).
\int dq_1^{\,3}\,dq_2^{\,3}\left(\frac{\left|\vec{\nabla}\psi\right|^2}{q^2}+ \frac{\mbox{i}}{q^2}\left(\vec{\nabla}_2\,\psi\,\vec{\nabla}_1\,\psi^{\ast}- \vec{\nabla}_1\,\psi\,\vec{\nabla}_2\,\psi^{\ast}\right)\right) < \infty \hspace{1cm}.
\frac{d}{dt}X_1(a_1,t) = \sin (X_1(a_1,t)), X_1(a_1,0) = a_1, &\frac{d}{dt}Y_{22}(a_1,t) = \cos (X_1(a_1,t)) Y_{22}(a_1,t), Y_{22}(a_1,0) = 1.