harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
X_{t}^{(u)}\, =\, \int^{t}_{0}e^{- (t-s)}u \widetilde{X}^{(u)}_{s}{\mathrm d}s + \int^{t}_{0}e^{-(t-s)}{\mathrm d}B_{s}\, , \widetilde{X}_{t}^{(u)}\, =\, \int^{t}_{0}\sum_{k=0}^{\infty}\mathfrak p_{0, k}(t-s; u) \,{\mathrm d}W_{s, k}\,, \mathfrak p_{0, k}(t-s; u)\, :=\, \frac{u^{k}(t-s)^{k}}{k!}e^{-(t-s)}\, ,
H \left( \begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\ \end{array}\right) = \left( \begin{array}{c}{P_{-}P_{+}\qquad 0}\\{0 \qquad P_{+}P_{-}}\\ \end{array}\right) \left( \begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\ \end{array}\right) .
x(t)=\sum_{n=1}^{\infty}[lim_{r\rightarrow0^{+}}(\frac{t^{\frac{2}{3}n}}{n!}\frac{d^{n-1}}{dr^{n-1}}(r^{n}(\frac{3}{2}(sin^{-1}(\sqrt{r})-\sqrt{r-r^{2}}))^{-\frac{2}{3}n}))]
\widetilde{P}_{N,M,k}^{( n^{2})}( z; \lambda_1) =4( n-k) ( n+k+N+M+1) P_{k}^{(N,M)}(z) \big[ \lambda_{1}+Q_{n}^{(N,M)}(z) \big] \hphantom{\widetilde{P}_{N,M,k}^{( n^{2})}( z; \lambda_1) =}{}+(1-z)^{N+1}(1+z)^{M+1}P_{n,k}^{( N,M)}(z) P_{n}^{(N,M)}(z) .
pdf_{MJ}(\beta;\theta)d\beta_{1}\cdot\cdot\cdot d\beta_{d}=\pi^{\frac{1-d}{2}}2^{-\frac{d+1}{2}}\theta^{\frac{1-d}{2}}K_{\frac{d+1}{2}}(\frac{1}{\theta})^{-1}e^{-\frac{\gamma(\beta)}{\theta}}\gamma(\beta)^{d+2}d\beta_{1}\cdot\cdot\cdot d\beta_{d}
P(s) =\sum_{v_1,...,v_l=0}^\infty P_{v_1,...,v_l}s_1^{v_1}\cdots s_l^{v_l},\,\,\,\,\, P_{v_1,...,v_l}\in \mathbb{C}^n, a(s) =\sum_{v_1,...,v_l=0}^{\infty}a_{v_1,...,v_l}s_1^{v_1}\cdots s_l^{v_l},\,\,\,\,\,,a_{v_1,...,v_l}\geq 0,
||D^{\alpha}_{\ast}\mathcal{F}||^2_{L^2(\Omega_t)}\lesssim||J^T\mathbf{F}||^2_{s,\ast,t}&\leq C(K)\Big(||\mathbf{F}||^2_{s,\ast,t}+||\mathbf{F}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s,\ast,t}\Big).
\delta ( x^{1 1}) ={\frac{1}{2 \pi{\alpha^{\prime}}^{1 / 2}}}+{\frac{1}{\pi{\alpha^{\prime}}^{1 / 2}}}\sum_{n = 1}^{\infty}c o s{\frac{n x^{1 1}}{{\alpha^{\prime}}^{1 / 2}}}\ .
\Sigma \! \! \! \! \! \displaystyle{\int}d^{4}p E_{p}( x ) \bar{E}_{p}( y ) \equiv \sum_{k}\int d p_{0}d p_{2}d p_{3}E_{p}( x ) \bar{E}_{p}( y ) = ( 2 \pi )^{4}\delta^{( 4 )}( x - y ) ,
\tanh_{q}t =-i\tan_{q}\left( it\right) =\frac{e_{q}\left( t\right)-e_{q}\left( -t\right)}{e_{q}\left( t\right) +e_{q}\left( -t\right)}=\frac{\mathcal{E}_{q}\left( 2t\right) -1}{\mathcal{E}_{q}\left(2t\right) +1}& =\sum_{n=1}^{\infty}\mathfrak{T}_{2n+1,q}\frac{\left( -1\right)^{k}t^{2n+1}}{\left[ 2n+1\right]_{q}!}.
|c_{I,\chi}^{0}| =\Big|\sum_{a\in \overset{\circ}{\overline{E}}_{I}\cap\overline{h}^{-1}(0)}\Omega_{\chi}(a)\Big| \leq \sum_{a\in \overset{\circ}{\overline{E}}_{I}\cap\overline{h}^{-1}(0)}1 = \#\big(\overset{\circ}{\overline{E}}_{I}\cap\overline{h}^{-1}(0)\big) &\leq\#\big(\overset{\circ}{\overline{E}}_{I}\big) \leq D_{I}p^{n-\#(I)}.
\left( \hat{\Gamma}_{\hat{\mu}\hat{\nu}}\right)^{\hat{\alpha}\hat{\beta}}\left( \hat{\Gamma}^{\hat{\rho}\hat{\sigma}}\right)_{\hat{\alpha}\hat{\beta}}= 1 6 \delta_{[ \hat{\mu}}^{[ \hat{\rho}}\delta_{\hat{\nu}]}^{\hat{\sigma}]}\ .
\rho[\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial t}+\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})(\overline{u_{j}}+u_{j}^{\prime})}{\partial x_{j}}]=-\frac{\partial(\overline{p}+p^{\prime})}{\partial x_{i}}+\mu[\frac{\partial^{2}(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}\partial x_{j}}].
\Delta |U| =\sum_j \left[\frac{u_j\Delta u_j}{|U|}+\frac{|\nabla u_j|^2}{|U|}-\frac{u_j\nabla u_j\cdot\nabla|U|}{|U|^2}\right] & =-\frac{1}{|U|}\sum_j\lambda_j(\Omega) u_j^2+\frac1{|U|^3}\sum_{i,j}\Big(u_i^2 |\nabla u_j|^2-u_iu_j\nabla u_i\cdot\nabla u_j\Big)\ge -\lambda_k(\Omega)|U|,
S_{\mathrm{m a t t e r}}= \int d^{2}\sigma \, \left[ \frac{1}{4}\left( F_{r s}\right)^{2}+ \frac{1}{2}\left( D_{r}Y^{I}\right)^{2}+ \frac{g^{2}}{4}\left( \{Y^{I}, Y^{J}\}\right)^{2}\right] \, ,
\theta ( | \mathbf{p}+ \mathbf{q}| - 1 ) = \theta ( \mathbf{\hat{p}_{.}q}+ p / 2 ) = \theta ( \mathbf{q_{.}\hat{p}}) + \sum_{m = 1}^{\infty}\frac{( p / 2 )^{m}}{m !}\delta^{( m - 1 )}( \mathbf{q_{.}\hat{p}})
\begin{array}{lcll}h_1(0,0) = 1, h_1(i,j) = 0 \mbox{for}(i,j)\neq (0,0), h_{2n}(i,j) = h_n(i,j) \mbox{for}i,j\geq 0, h_{2n+1}(i,0) = h_n(i-1,0) \mbox{for}i\geq 1, h_{2n+1}(0,j) = h_{n+1}(0,j-1) \mbox{for}j\geq 1, h_{2n+1}(i,j) = h_n(i-1,j)+h_{n+1}(i,j-1) \mbox{for}i,j\geq 1.\end{array}
E_{M}^{( 1 1 ) A}= \left( \begin{matrix}{e^{-{\frac{\Phi}{3}}}e_{\breve{\mu}}^{( 1 0 ) \breve{\alpha}}}&{e^{{\frac{2}{3}}\Phi}B_{\breve{\mu}}}\\{0}&{e^{{\frac{2}{3}}\Phi}}\\ \end{matrix}\right) ,
g_2 ( g_1\cdot v) \, b \,g_2 ( g_1\cdot v)^{-1}=g_2 ( g_1\cdot v) h_2 ( h_1\cdot v)g_2 (g_1\cdot v)^{-1}= h_2 ( h_1g_1 \cdot v) =e_{i+1}
e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};M)=e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(s)^{[d_s]};M)> 0\mbox{if}d_{s+1}+\cdots+d_r=0
{\cal V}= \frac{P_{\mathrm{M A X}}- P_{\mathrm{m i n}}}{P_{\mathrm{M A X}}+ P_{\mathrm{m i n}}}=_{N}\! < 0 \vert S^{[ N ]}\vert 0 >_{N}= \left( 1 - q \right)^{N / 2}
V = \frac{{q^{2}}}{{2 \lambda}}\left({1 + \frac{{4 \pi m \Sigma}}{{\lambda^{2}}}}\right) \left({1 - e^{- \lambda | y - y{\prime}|}}\right) + \frac{{q^{2}}}{2}\left({1 - \frac{{\frac{{e^{2}}}{\pi}}}{{\lambda^{2}}}}\right) | y - y{\prime}| .
\left(\pi_u\circ (\iota \times_\mathcal{T}\phi)\right)\big(i_{\mathcal{T}C^*(\Lambda^i)}\big(t_\mu^{\Lambda^i}\big)\big) =\pi_u\big(\phi\big(t_\mu^{\Lambda^i}\big)\big)=\pi_u\big(t_\mu^\Lambda\big)=u_\mu &=i_X^{\otimes d(\lambda)_i}\big(\Omega_{d(\mu)_i}\big(t_\mu^\Lambda\big)\big)=i_{\mathcal{T}C^*(\Lambda^i)}\big(t_\mu^{\Lambda^i}\big).
\widetilde{V}_n(x) := \begin{cases}\displaystyle{\frac{2(-1)^{\frac{n+1}{2}}i}{x}T_n \left(\frac{ix}{2}\right)}n: , \displaystyle{\frac{(-1)^{\frac{n}{2}}i}{x}U_{n-1}\left(\frac{ix}{2}\right)}n: .\end{cases}
(A-A^2 S')^{p+1}=(A-A^2 S')(A-A^2 S')^p &=(A-A^2 S')(A^p-A^{p+1}S') &=A^{P+1}-A^{P+2}S'-A^{P+2}S'+A^{P+3}{S'}^2 &=A^{p+1}-A^{p+2}S'.
y_{{R_i}}^I = \sqrt{1 -{\rho_i}}{y_{{R_i}}}+ n_i^b = \sqrt{1 -{\rho_i}}\sqrt{{P_i}}{g_{ii}}{x_i}+ \sqrt{1 -{\rho_i}}\times \sum\nolimits_{j = 1,j \ne i}^N{\sqrt{{P_j}}{g_{ji}}{x_j}}+ n_i^b,
\langle k_{S_{\mathcal{H}_{\mathcal{S}}}}(\hat x, \cdot), k_{S_{\mathcal{H}_{\mathcal{S}}}}(\hat y,\cdot) \rangle_{\mathcal{H}_{\mathcal{S}}^2}= \langle S_{\mathcal{H}_{\mathcal{S}}}(\hat x), S_{\mathcal{H}_{\mathcal{S}}}( \hat y) \rangle_{\widetilde{\mathbb{R}\oplus \mathcal{H}_{\mathcal{S}}}}
F =\prod_{j=1}^{n_1}(y+a_{1,j}z)^{e_{1,j}}+g_1+xyzg_0 &=\prod_{j=1}^{n_2}(z+a_{2,j}x)^{e_{2,j}}+g_2+xyzg_0 &=\prod_{j=1}^{n_3}(x+a_{3,j}y)^{e_{3,j}}+g_3+xyzg_0,
(\alpha_{\frac n2-\lambda}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda}\varepsilon_\xi i_\xi) =(\alpha_{\frac n2-\lambda-1}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda-1}\varepsilon_\xi i_\xi) +(i_\xi\varepsilon_\xi-\varepsilon_\xi i_\xi)
\begin{array}{l l}{{\begin{array}{l}{{Z [ e , \pi , \lambda , \rho ] =}}\\ \end{array}\int}}&{{[ D e_{i}^{a}] [ D \pi_{a}^{i}] [ D \lambda_{m n}] [ D \rho^{m n}] \J \ \delta ( \rho^{m n}) \delta ( \lambda_{m n}- \lambda_{m n}^{0})}}\\{{}}&{{\times \, \, d e t \, M_{\alpha \beta}\ \delta ( H_{\perp}) \delta ( H_{i}) \delta ( J_{a b}) \e x p \frac{i}{\hbar}S .}}\\ \end{array}
(M_r{V}^x)_S = 2^{r-|S|}\sum_a \left [\binom{|S|-1}{2}\binom{|S|-3}{a-3}+(a-1)\binom{|S|-1}{a-1}\right ] +2^{|S|-1}\sum_b \binom{r-|S|}{2}\binom{r-|S|-2}{b-2}.
\left(-kH^2 + \frac{r}{2}H^2+s+r\right)^2 \leq\ \left(-kH^2 + \frac{r}{2}H^2+s+r\right)^2 + 2k^2H^2 -4rs +4(r-k)^2 =\ \left(-kH^2 + \frac{r}{2}H^2+s-r\right)^2 + (2H^2+4)(r-k)^2.
\Box^{\ast}\left[\tau \left(u|\nabla\log u|^2+u\SS-2\Delta u\right)-u\log u-\frac{nu}2\log\left(4\pi\tau\right)-nu\right] &=-2\tau u\left|\alpha-\nabla\nabla\log u-\frac g{2\tau}\right|^2-\tau u\mathcal D_\alpha(-\nabla\log u).
\phi_{- 2 , 1}( \tau , z ) = - \frac{2 E_{6}( \tau ) \theta_{0 , 1}( \tau , z )}{\Delta ( \tau )}= - \frac{2}{q}+ 9 6 0 - 2 \sum_{( \gamma , \gamma ) = 2}\zeta^{\gamma}+ O ( q ) ,
\mathcal{K}_{n}\left( \mathcal{E},0\right) =\frac{\left( \gamma +\frac{1}{2}+(-D+1)\right)_{n}}{n!\left( 2\gamma +1\right)_{n}}\frac{\Gamma \left(\gamma +\frac{3}{2}-D+n\right)}{\Gamma \left( \gamma +\frac{3}{2}-D\right)}
A_{\pm}A_{\mp}\psi_{n_{\pm}, m_{\pm}}T_{n_{\pm}, m_{\pm}}= \left( \begin{array}{c}{{\scriptstyle n_{+}}}\\{{\scriptstyle{n_{-}+ 1}}}\\ \end{array}\right) \psi_{n_{\pm}, m_{\pm}}T_{\kappa_{\pm}; n_{\pm}, m_{\pm}}.
\bar{S_{s}}^{a b}= \frac{2^{n - 2}}{( d - 2 ) !}\varepsilon^{a b}{}_{a_{1}a_{2}\ldots a_{d - 2}}S_{s}^{a_{1}a_{2}}S_{s}^{a_{3}a_{4}}\ldots = \frac{( - i )^{n - 1}}{2}a^{a}a^{b}( a^{0}a^{1}\ldots a^{2 n}) .
R_1 =\Big\{(x_1,x_2,...,x_{k}) \mid 0\leq x_k\leq m-1;\ \ \ 0\leq x_i\leq 2\ \ \ \ 1\leq i\leq k-1, &\ \ \ \ \ \ \ \ \ \ \ \x_i=2,\ \ \ \x_j=0\ \ \ \j< i\leq k-1\Big\}
\|(T^hT^v(f,g)_\Phi)\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}\lesssim \|f_\Phi\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g_\Phi\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\eqref{1cohomologyOct14}=B\Big(d^2f(x,y,z),g(t)\Big)-\circlearrowleft_{x,y,z}B\Big(\rho\left( \alpha(x)\right) f(y,z),g(t)\Big) + \circlearrowleft_{x,y,z}B\left(\beta( f([x,y],t)),g(z)\right) +\circlearrowleft_{x,y,z}B(f(\alpha(a),\alpha(z)),g([x,y])).
H\coloneqq \limsup_{r\to\infty}\Big\{\int_{r}^\infty \frac{ds}{f_a^{n-1}(s)}\int_0^{r}a_0(t) f_a^{n-1}(t) dt - \int_0^{r}\int_t^\infty \frac{ds}{f_a^{n-1}(s)}a_0(t) f_a^{n-1}(t) dt \Big\}\le 0;
\alpha \mapsto \int \hat \Psi_{\alpha}^2 \:d\mu_{\alpha}\in C^1([\alpha_-,\alpha_+]) &\alpha \mapsto \int \hat \Psi_{\alpha}\cdot \hat \Psi_{\alpha}\circ F_{\alpha}^k\: d\mu_{\alpha}\in C^1([\alpha_-,\alpha_+]), \: k \geq 1.
|c_{n_0}(t)|\leq\{c_0(t)^2+\sum_{n=1}^{\infty}(c_n(t)^2+\tilde c_n(t)^2)\}^{\frac{1}{2}}\leq \hat K_1\|\hat\psi(t,\cdot;w,\omega)\|_{L^2}\leq \hat K_1\|\hat\psi(t,\cdot;w,\omega)\|,
{\cal F}_{2}\left( a , \epsilon_{1}, \epsilon_{2}\right) = \frac{2 0 a^{2}+ 7 \epsilon_{1}^{2}+ 1 6 \epsilon_{1}\epsilon_{2}+ 7 \epsilon_{2}^{2}}{\left( 4 a^{2}- \epsilon^{2}\right)^{2}\left( 4 a^{2}- \left( 2 \epsilon_{1}+ \epsilon_{2}\right)^{2}\right) \left( 4 a^{2}- \left( \epsilon_{1}+ 2 \epsilon_{2}\right)^{2}\right)}.
\Delta ( \begin{matrix}{{A}}\\ \end{matrix}{i}{j}) = \begin{matrix}{{A}}\\ \end{matrix}{i}{k}\otimes \begin{matrix}{{A}}\\ \end{matrix}{k}{j}, \, \epsilon ( \begin{matrix}{{A}}\\ \end{matrix}{i}{j}) = \delta_{j}^{i}, \, S ( \begin{matrix}{{A}}\\ \end{matrix}{i}{j}) = \begin{matrix}{{( A^{- 1})}}\\ \end{matrix}{i}{j}.
\begin{matrix}a_{1}&=&\frac{1}{2}(24+6)&=&15\\g_{1}&=&\sqrt{24\cdot6}&=&12\\a_{2}&=&\frac{1}{2}(15+12)&=&13.5\\g_{2}&=&\sqrt{15\cdot12}&=&13.4164078649...\\ &&\vdots&&\end{matrix}
\times \exp \left( \frac{1}{2}\sum_{b = 1}^{N}\sum_{j = 1}^{m_{b}}\sum_{j^{\prime}= 1}^{m_{b}}( 1 - \delta_{j j^{\prime}}) \ln ( y_{j}^{( b )}- y_{j^{\prime}}^{( b )})^{2}\right) .
D_{k-2}\Phi = \frac{k(k^2-1)}{12}G''' + G' \left( - \frac{(k+1)D_{k-2}}2 + 2 C_{k-2}\right) + G \left( \frac{k-1}2 D_{k-2}' + c_{k-2}' - D_{k-3}\right) .
|\mathbb{E}h(W)-\chi_{(m-1)}^2h| \leq\frac{1}{\sqrt{n}}\{3\|f''\|+14\|f^{(3)}\|\}\bigg\{\sum_{j=1}^m\frac{1}{\sqrt{p_j}}+2m\sum_{j=1}^m\sqrt{p_j}&\quad+\sum_{j=1}^m\sum_{k\not=j}^m\sqrt{p_k}+2\sum_{j=1}^m\sum_{k\not=j}^m\sum_{l=1}^m\sqrt{p_jp_kp_l}\bigg\}&\leq\frac{6}{\sqrt{n}}\{3\|f''\|+14\|f^{(3)}\|\}\sum_{j=1}^m\frac{1}{\sqrt{p_j}}.
\left|\ \frac{(z+h)^n-z^n}{h}-nz^{n-1}\ \right|\ = \ \biggl|\ \sum_{k=1}^n \bigl(\,(z+h)^{n-k}-z^{n-k}\,\bigr)z^{k-1}\ \biggr| \ \leq\ 2nr^{n-1}.
d_{iu}^{(j)}={\widetilde{X}}_{i,(u-1)(p+q)+j}-{\mathbb{E}}({\widetilde{X}}_{i,(u-1)(p+q)+j}|{\mathcal{F}}_{i,u-1}^{X}):={\widetilde{X}}_{iu}^{(j)}-{\mathbb{E}}({\widetilde{X}}_{iu}^{(j)}|{\mathcal{F}}_{i,u-1}^{X})\,,
B^{rrst}\geq{}20\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s \right)\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^t \right)-12\sum_{\alpha=1}^K (q_\alpha^r)^2 q_\alpha^s q_\alpha^t \geq{}8\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s \right)\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^t \right)\geq 0.
H^2\big(p_{<0}, q_{<0}\big) \leq \sum_{i=1}^m \frac{1}{n \int_{x_{i-1}}^{x_i}f}+ 16\sum_{i=1}^m \Big( \frac{\int_{x_{i-1}}^{x_i}f(x)-f_0(x) dx}{\Delta_i f_0(x_{i-1})}\Big)^2 =: (I) + (II),
|\chi_{\lambda}(x/t)|^3|b|^2|a| =|\chi_{\lambda}(x/t)|^3\left|b\right|^2|y/(2t)\phi''(c_{x,y}/t)|\left|\frac{e^{\pm i\phi''(c_{x,y}/t)y^2/(2t)}-1}{\phi''(c_{x,y}/t)y^2/(2t)}\right| &\lesssim\lambda\frac{1}{|y|}|\chi_{\lambda}(x/t)||y/(2t)\phi''(c_{x,y}/t)|\lesssim \frac{\lambda^2}{t}
-\frac{1}{2}\sum_{s=1}^{2n-1}\frac{(-1)^sq^{s(2n-s)+2n}}{y^{s}z^{2n-s}}- \frac{(-1)^nq^{n^2+2n}}{2z^{n}y^{n}}\cdot \frac{J_1^4}{J_2^2}\cdot \frac{j(yz;q^2)}{j(y;q)j(z;q)}
({\cal B}_{V})_{c l}= \sum_{{\vec{n}},{\vec{m}}}\sum_{s , s^{\prime}}\left[ ( a_{\vec{n}}^{s \dagger}\bar{u}_{\vec{n}}^{s}+ b_{-{\vec{n}}}^{s}\bar{v}_{-{\vec{n}}}^{s}) ( a_{\vec{m}}^{s^{\prime}}\gamma^{0}u_{\vec{m}}^{s^{\prime}}+ b_{-{\vec{m}}}^{s^{\prime}\dagger}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}}) \right]_{c l}F ({\vec{n}}-{\vec{m}}, X ) .
\sum_{t=0}^{k-1}p^{-t}U_{mk+t}^{(k)}=U_{m}^{k}(pU_{m})^{1-k}\left( \frac{U_{m+1}^{k}-p^{k}U_{m}^{k}}{U_{m+1}-pU_{m}}\right) &=p^{1-k}U_{m}\left( \frac{U_{m+1}^{k}-p^{k}U_{m}^{k}}{-qU_{m-1}}\right) &=\left( -\frac{p}{q}\right) p^{-k}\left( \frac{U_{m}}{U_{m-1}}\right)\left( U_{(m+1)k}^{(k)}-p^{k}U_{mk}^{(k)}\right).
g^{*}h_{\alpha \beta \gamma}^{B}\: = \: \left( h_{\alpha \beta \gamma}^{B}\right) \left( \nu_{\alpha \beta \gamma}^{F g}\right)^{- 1}\left( \nu_{\alpha \beta \gamma}^{R g}\right) \left( \kappa_{\alpha \beta}^{g}\right) \left( \kappa_{\beta \gamma}^{g}\right) \left( \kappa_{\gamma \alpha}^{g}\right)
{{\cal A}^{\prime}}_{( 2 )}^{I}= \frac{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}{\sqrt{V}}\;{\cal A}_{( 2 )}^{I}, \qquad{b^{\prime}}^{I}= \frac{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}{\sqrt{2 \; V}}\;{b}^{I}
\sum_{y \in X}b(x,y)f_t(y) \leq \left(\sum_{y \in X}b(x,y)f_1(y)\right)^t\left(\sum_{y \in X}b(x,y)f_2(y)\right)^{(1-t)}&= (\deg(x) - \lambda_1)^tf_{\alpha_1}(x)^t(\deg(x) - \lambda_2)^{(1-t)}f_{\alpha_2}(x)^{(1-t)}&\leq (\deg(x) - \lambda_t)f_t(x)
\mathcal{X}_{T^{*}}:=\widetilde{L}^{\infty}(0,T^{*}; \dot{B}^{-\alpha+\frac{n}{p}}_{p,q}(\mathbb{R}^{n}))\cap\widetilde{L}^{\rho_{1}}(0,T^{*}; \dot{B}^{s_1}_{p,q}(\mathbb{R}^{n})) \cap\widetilde{L}^{\rho_{2}}(0,T^{*}; \dot{B}^{s_2}_{p,q}(\mathbb{R}^{n}))
\# \{1 \leq j \leq q; \B_q u_j=q u_j B_q (u_j+u_{-j^*})=q (u_j+u_{-j^*}) \}= \frac{\varphi(q)+\widetilde{\varphi}(q)}{2}, & \# \{1 \leq j \leq q; \B_q (u_j-u_{-j^*})=-q (u_j-u_{-j^*}) \}=\frac{\varphi(q)-\widetilde{\varphi}(q)}{2},
R_{\mu \nu}- 2 \nabla_{\mu}\nabla_{\nu}\phi +{\textstyle \frac{1}{4}}H_{\mu}{}^{\rho \sigma}H_{\nu \rho \sigma}-{\textstyle \frac{1}{4}}e^{2 \phi}\sum_{n}{\textstyle \frac{( - 1 )^{n}}{( n - 1 ) !}}T^{( n )}{}_{\mu \nu}\, ,
L_2= \ -\frac{4H}{H^2-4K}(t-t^2) +\frac{2H}{H^2-4K}(1-\frac1q)\frac{(t\lambda_2+\lambda_1)^2}{K}&+\left[-\frac{4K}{H^2-4K}(1-\frac1q)-\alpha^2-\frac{\alpha}q\right]\frac{\lambda_1(\lambda_2t+\lambda_1)^2}{K^2}+\frac{2 \lambda_1(t+1)^2}{H^2-4K}.
0 = F_{u}^{\prime \prime}+ \frac{2}{r}F_{u}^{\prime}\left[ 1 + r L_{+}^{\prime}( k r ) \right] - \frac{s^{\prime}}{s + \frac{\omega}{m}}\left[ F_{u}^{\prime}+ F_{u}L_{+}^{\prime}( k r ) \right] + V ( r , m ) F_{u}
E_{\exp}^{\left( 1 \right)}= \left. - \frac \mu 2 \frac d{d t}\left( \sum_{n = - \infty}^{\infty}e^{- t \left( \left( \frac{2 n \pi}{L \mu}\right)^{2}+ \left( \frac m \mu \right)^{2}\right)^{\frac 1 2}}\right) \right\rfloor_{t = 0}.
-p\sum_{t\in\mathbb{F}_p^{\times}}\varphi(t(t-1)){_2F_1}\left(\begin{array}{cc}\chi_4, \chi_4^3 & \varepsilon\end{array}\mid\frac{1}{t}\right)=-1+\frac{p\varphi(-1)}{\Gamma_p(\frac{1}{4})\Gamma_p(\frac{3}{4})}.
\langle N \rangle_{\mathrm{T}}= \frac{1}{2 \pi}\int_{- \infty}^{\infty}d x \theta^{\prime}- \sqrt{\frac{2 m T}{\pi}}e^{- m / T}\int_{- \infty}^{\infty}d x \mathrm{s i n h}\left( \frac{\theta^{\prime}}{2 T}\right) + \dots
M(\vec{X},Y)=[\begin{matrix}\begin{matrix}\overline{\mu}_{1}\\ \overline{\Sigma}_{11}\\ \overline{\Sigma}_{21}\\ \end{matrix}&\begin{matrix}\overline{\mu}_{2}\\ \overline{\Sigma}_{12}\\ \overline{\Sigma}_{22}\\ \end{matrix}\\ \end{matrix}]
\int_{t^{*}}^{t^{*}+T}(\phi(u'(\xi)))' v(\xi) \,d\xi =\int_{t^{*}}^{t^{*}+T}\phi(u'(\xi)) u'(\xi) \,d\xi = \int_{0}^{T}\phi(u'(\xi)) u'(\xi) \,d\xi &\geq \int_{0}^{T}\bigl{(}b|u'(\xi)|-K_{b}\bigr{)}\,d\xi= b\|u'\|_{L^{1}}-K_{b}T= b\|v'\|_{L^{1}}-K_{b}T.
\frac{\partial \psi(x, S, Q)}{\partial x}\ge \sqrt{\frac{\partial \psi(x, S, Q)}{\partial x}}& = \sqrt{2 \ln (2 x) + \frac{1}{2}S^2 + 2 Q}& \ge \sqrt{2 \ln (2x) + \frac{1}{2}}& = \sqrt{\ln(4\sqrt{e}x^2)}& \ge \sqrt{\ln (6x^2)}& \ge \sqrt{\ln (1 + 2x^2)}\: .
\Gamma_{\varphi}(\varphi,\sigma,\mu) = \Lambda_{\varphi}(\varphi,\sigma)-\theta_{\varphi}(\varphi,\sigma)\mu, \Gamma_{\sigma}(\varphi,\sigma,\mu) = \Lambda_{\sigma}(\varphi,\sigma)-\theta_{\sigma}(\varphi,\sigma)\mu,
F_0(x)=\sum_{i=1}^{M}e^{-\beta_{i,q}x}\prod_{k=1}^{M}\frac{\beta_{i,q}-\beta_{k,p+q}}{\beta_{k,p+q}}\prod_{k=1,k \neq i}^{M}\frac{\beta_{k,q}}{\beta_{i,q}-\beta_{k,q}}, \ \x \geq 0.
V^{q}(\psi\otimes\bar{w},z) \; (\chi\otimes\bar{v}) : =\sum_{\bar{w}', \bar{v}'}\left(\widetilde{F}^{-1}\right)_{\bar{w}', \bar{v}'}^{\bar{w}, \bar{v}}V(\psi\otimes\bar{w}',z) \; (\chi\otimes\bar{v}'),
||\omega||^2_g =\int_{\mathbb{H}^3}||\omega||^2_{x}g(x,y) dV\geq\int^\infty_{r}\int_{S_t}||\omega||^2_{x}g(x,y) dA dt\geq \int^\infty_{r}\int_{S_t}||i_t^\ast (\omega)||^2_{x}g(x,y) dA dt &=\int^\infty_{r}||i_t^\ast (\omega)||^2_{g}dt\geq \int^\infty_{r}\frac{\tanh(r)}{\tanh(t)}||[i_r^\ast(\omega)]||^2_g dt=\infty.
\left|U(m_1,m_2, a_r) \right|\leq \ 2\pi \sum_{x=0}^{\tau_{r,j}-1}|m_1| \lambda_j^{n_{r,j}+x}\frac{\theta_r}{2^{n_{r+1}}}\leq \ 4\pi |m_1| \lambda_j^{n_{r+1},j}\frac{1}{(\lambda_j-1) 2^{n_{r+1}}}\leq \ \frac{4\pi |m_1|}{\lambda_j-1}.
\Gamma_{\phi^-}^{ba}(p)=\frac{iG}{2\Delta_C(p)}\left(\left\{[1+GL^{22}(p)]\delta^{b1}-GL^{21}(p)\delta^{b2}\right\}\delta^{a1}-\left\{[1-GL^{11}(p)]\delta^{b2}+GL^{12}(p)\delta^{b1}\right\}\delta^{a2}\right),
\lim_{n\to\infty}\frac{\Lambda_n(\theta)}{v(n)}=\lim_{n\to\infty}\frac{\log(1+\pi(n)(\mathrm{e}^\theta-1))}{\pi(n)}\lim_{n\to\infty}\frac{n-a_n}{n}&=\lim_{n\to\infty}\frac{\mathrm{e}^\theta-1}{1+\pi(n)(\mathrm{e}^\theta-1)}&=\mathrm{e}^{\theta}-1,
d s^{2}={\frac{\tau_{3}}{2}}{\frac{\cosh^{2}\chi}{\rho^{2}}}e^{2 A}d r^{2}+{\frac{\tau_{3}}{2}}{L^{2}}{e^{2 A}}\rho^{2}\left({\cosh^{2}\! \chi}\sigma_{3}^{2}+ \sigma_{1}^{2}+ \sigma_{2}^{2}\right) \ ,
s\left( c,d;\chi_{0},c_{2}\right) =2s\left( c,2d\right) -3s\left(c,d\right) +s\left( 2c,d\right) , s\left( d,c;c_{2},\chi_{0}\right) =s_{2}\left( 2d,2c\right)-s_{2}\left( d,2c\right) ,
\sum\limits_{i,j =1}^{2}h_{\dot{x}^{i}\dot{x}^{j}}y^{i}y^{j}= \frac{\lambda}{((\dot{x}^1)^2+(\dot{x}^2)^2)^{\frac{3}{2}}}(\dot{x}^2y^1-\dot{x}^1y^2)^2 = \frac{\lambda}{a}(y^{1}\cos t+y^{2}\sin t)^{2}.
\lim_{\varepsilon \rightarrow 0}P_{n}\left[ F_{1}, \dots, F_{n}\right]\left( \varepsilon \tau_{i},\tau_{j}\right) =P_{n-m+1}\left[ O_{m}\left[F_{1}, \dots, F_{m}\right] \left( \tau_{i}\right), F_{m+1}, \dots, F_{n}\right] \left( 0,\tau_{j}\right) .
M=\begin{pmatrix}g_{1,p-1}Xg_{1,0}\cdots Xg_{1,p-3}Xg_{1,p-2}g_{2,p-2}g_{2,p-1}\cdots Xg_{2,p-4}Xg_{2,p-3}\vdots \vdots \ddots \vdots \vdots g_{p-1,1}g_{p-1,2}\cdots g_{p-1,p-1}Xg_{p-1,0}g_{0,0}g_{0,1}\cdots g_{0,p-2}g_{0,p-1}\end{pmatrix}
l\sum h_{k}\nabla_{k}\chi =l\sum h_{k}\left[\frac{-f_{k}}{f}+\frac{h_{k}}{h}+\frac{\varphi^{'}}{\varphi}h_{k}+\frac{(\nabla G\cdot e_{i})\cdot w_{ki}}{G}\right] &\leq C_{1}l+l\frac{\nabla G\cdot e_{k}}{G}h_{k}w_{kk}=C_{1}l+l\frac{\nabla G\cdot e_{k}}{G}\rho \rho_{k}\leq C_{2}l,
A_{0,t}(x) = \sum_{j\ge 1}j^{t-1}C(x^j), A_{1,t}(x) = \sum_{j\ge 1}j^{t-1}x^jC'(x^j), A_{2,t}(x) = \sum_{j\ge 1}j^{t-1}(x^{2j}C''(x^j)+x^jC'(x^j)) \quad A_{3,t}(x) = \sum_{j\ge 1}j^{t-1}(x^{3j}C'''(x^j) + 3x^{2j}C''(x^j) + x^jC'(x^j)).
P_{BC_r,\,s}^{\Delta_L}(k_1,\,k_2|x)=P_{BC_r,\,s2}^{\Delta_S}(k_1,\,k_2|x), \quad P_{BC_r,\,c}^{\Delta_L}(k_1,\,k_2|x)=P_{BC_r,\,c2}^{\Delta_S}(k_1,\,k_2|x).
y^{(1)}_{2,3}(\theta)+ y^{(2)}_{2,2}(\theta) = 9(d^{[0]}_0(E \omega^3) d^{[1]}_0(E \omega^7) + i d^{[0]}_0(E \omega^7) d^{[1]}_0(E \omega^3)) d^{[2]}_0(-E \omega^3), y^{(1)}_{2,2}(\theta)+y^{(2)}_{2,3}(\theta) = 9(i d^{[0]}_0(-E \omega) d^{[1]}_0(-E \omega^5) -d^{[0]}_0(-E \omega^5) d^{[1]}_0(-E \omega)) d^{[2]}_0(E \omega^5) .
\frac{1}{(11)_b}+\frac{1}{(22)_b}+\cdots+\frac{1}{((b-1)(b-1))_b}= \frac{1}{b+1}+ \frac{1}{2b+2}+ \cdots + \frac{1}{(b-1)b+b-1}= \frac{x}{b+1}.
\begin{aligned}\exp(-rT_k)G(P(T_k))= \exp(-rS_k)G(P(S_k))+\int_{S_k}^{T_k}\exp(-rt)\mathcal{A}G(P(t))\mathrm{d}t &+\sigma\int_{S_k}^{T_k}\exp(-rt)P(t)G'(P(t))\mathrm{d}B(t)\end{aligned}
H_{(X_-,D_-)}(y_1,y_2) = e^{\frac{\xi_-\log y_1+ h_-\log y_2}{2\pi i}}\sum_{d_1,d_2\geq 0}\frac{y_1^{d_1}y_2^{d_2}[\textbf{1}]_{(d_1-d_2,\cdots,d_1-d_2)}}{\Gamma(1+\frac{h_-}{2\pi i}+d_2)^{r^\prime+1}\Gamma(1+\frac{\xi_--h_-}{2\pi i}+d_1-d_2)^{r^\prime+1}\Gamma(1+\frac{\xi_-}{2\pi i}+d_1)}.
\bar{v}^S + f_i^S = \begin{cases}\displaystyle h_i^1 + \sum_{j \in N}p_{ij}^1 \, f^S_j i \in S \displaystyle h_i^0 + \sum_{j \in N}p_{ij}^0 \, f^S_j i \in N \setminus S.\end{cases}
(\bar{c}_{1,x_0})^2 = (R_{x_0}B_{x_0}^{t}\nu_{x_0}) \cdot (R_{x_0}B_{x_0}^{t}\nu_{x_0}) = \nu_{x_0}\cdot A(x_0) \nu_{x_0}, (\bar{c}_{2,x_0})^2 = (R_{x_0}B_{x_0}^{t}e_{n+1}) \cdot (R_{x_0}B_{x_0}^{t}e_{n+1}) = a^{n+1, n+1}(x_0).
\mathbb{P}\left\{\mathcal{M}_i^{1, j}\right\}\leq \mathbb{P}\left\{\sum_{t=0}^{T-1}f^{(i)}_t V^{(i)}_t \leq \bar{t}\right\}+\mathbb{P}\left\{\sum_{t=0}^{T-1}g^{(i)}_t V^{(i)}_t > \bar{t}\right\},
{\cal U}_{\Sigma}{}^{\Lambda}\equiv \left. \frac{3}{W}g^{\Lambda \Xi}\partial_{\Sigma}\partial_{\Xi}W \right|_{\partial W = 0}= \left. \frac{3}{2 f}g^{\Lambda \Xi}\partial_{\Sigma}\partial_{\Xi}f \right|_{\partial f = 0}\, ,
s_{i}=f'_{i}(\tilde{y}^{(k)})-f'_{i}(y_{1}^{(k)},y_{2}^{(k)},\dots,y_{i-1}^{(k)},y_{i}^{(k)},y_{i+1}^{(k-1)},\dots,y_{m}^{(k-1)}).
\begin{cases}\partial_t\mathbf{w}+(\mathbf{v}\cdot \nabla)\mathbf{w}=(\mathbf{w}\cdot \nabla)\mathbf{v},\ \ D\times (0,T), \nabla\cdot \mathbf{v}=0,\ \ D\times (0,T), \mathbf{v}\cdot \mathbf{n}=v_n,\ \ \partial D\times (0,T), \mathbf{w}(x, 0)=\nabla\times \mathbf{v}_0(x), \ \ D.\end{cases}
C_m = 2^m m! (m-1) \|V^{-1}\| = 2^m m! (m-1) \max_{1\leq \beta\leq m-1}\frac{1}{\lambda_{\beta}}\prod_{\substack{\nu=1 \nu\neq\beta}}^{m-1}\frac{1+\lambda_{\nu}}{|\lambda_{\nu}- \lambda_{\beta}|}.
\nabla^{F^{*}+ F}_{u}(\eta + Y) = \nabla^{F}_{X}\eta -\frac{1}{3}{\mathcal H}(X, Y,\cdot ) +\nabla^{F}_{X}Y, & \nabla^{\mathcal G}_{u}(s):=\nabla_{X}s +\frac{2}{3}\mathrm{ad}_{r}(s)
4\pi L_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})\ ,
V(\alpha^*,\alpha) = \frac{g}{3!}\left( V_{abc}\hat{a}^a \hat{a}^b\hat{a}^c + 3 \hat{a}^{\dagger}_a V^a_{\; \; bc}\hat{a}^b \hat{a}^c + 3 \hat{a}^{\dagger}_a\hat{a}^{\dagger}_b V^{ab}_{\quad c}\hat{a}^c + \hat{a}^{\dagger}_a \hat{a}^{\dagger}_b \hat{a}^{\dagger}_cV^{abc}\right)
\rho_{\frac{3}{2}- m}(s) = \frac{(-1)^{m-1}(4\pi)^k \zeta(1 - m) \zeta(2s + m - 2) \Gamma(2s + m - 2)}{\Gamma(m) \Gamma(s) \Gamma(s + k - 1) \zeta^*(4 - 2s - 2m)}\times \quad\langle \mathcal{V}_{f, \overline{g}}, \overline{E(\cdot, s + m - 1)}\rangle.
A_{j}(z)=\sum_{n\in\mathbb{Z}}\frac{A_{[n,j]}}{z^{n+1}}=\sum_{n\in\mathbb{Z}}\frac{1}{z^{n+1}}\sum_{k\in\mathbb{Z}}A_{[k,0]}A_{[n-k-1,j-1]}^{(1)}=A_{0}(z) A_{j-1}^{(1)}(z).
V(\Gamma^e) =\{v^g: v\in V(\Gamma),\g\in A(\Gamma)\}, E(\Gamma^e) =\{\, \{v_1^{g_1}, v_2^{g_2}\}\,:\,v_1^{g_1}v_2^{g_2}=v_2^{g_2}v_1^{g_1}\ \mbox{in $A(\Gamma)$}\,\}.
\le\frac{2^m}{C_m}\int_{S(0,b{\sqrt m})}\ \frac{dv_1dv_2\ldots dv_m}{\left(\displaystyle\sum_{t=1}^mv_t^2\right)^{2s}},\{\rm as}\ [-b,b]^m\subseteq S(0,b{\sqrt m}), &=\frac{2^{m+1}\pi^{m/2}}{C_m\Gamma(m/2)}\int_0^{b{\sqrt m}}r^{m-4s-1}\,dr,\ \hbox{by\ \cite[pages 394-395]{stromberg1},}=\frac{2^{m+1}\pi^{m/2}m^{(m-4s)/2}b^{m-4s}}{C_m\Gamma(m/2)(m-4s)}.
(T^{1/3})^{\alpha+\beta}[u]_\alpha[f]_\beta \gtrsim\begin{cases}\|\lceil u, (\cdot)_T\rceil f\| \textrm{if}\alpha \in (0,1], \| \big( \lceil u, (\cdot)_T\rceil-\partial_1 u \lceil x_1, (\cdot)_T\rceil \big) f\| \textrm{if}\alpha \in (1, \frac32).\end{cases}
\hat{\Phi}( x ) = \int \frac{d^{3}k}{( 2 \pi )^{\frac{3}{2}}}\frac{1}{\sqrt{2 \omega_{\bf k}}}\left[ \hat{a}({\bf k}) e^{- i k x}+ \hat{a}^{\dagger}({\bf k}) e^{i k x}\right] \
\langle (f_n)_{n\geq 0}, (g_n)_{n\geq 0}\rangle := \mathbb E \sum_{n=0}^{\infty}\langle f_n, g_n\rangle ((g_n)_{n\geq 0}\in Q_{q'}^{p'}, \ (f_n)_{n\geq 0}\in Q_{q}^{p}).