harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Theta=\left\{\theta\in(0,1)^{m^k\ell^k(m\ell-1)}\;:\;\sum_{(i',j')\in A\times B,(i',j')\neq (m,\ell)}\theta_{i_1^k,j_1^k,i',j'}<1,\;\mbox{for all}\;i_1^k\in A^k,j_1^k,\in B^k\right\},
\hat{\varphi}( \eta, \Tilde{\tau}) = \varphi^{\Lambda}( \eta, \Tilde{\tau}) := \varphi(\delta\eta + \mathrm{z}, \alpha\delta^2\Tilde{\tau}), \check{\psi}(\mathrm{x}, t) = \psi^\vee(\mathrm{x}, t) := \psi\Bigg(\frac{\mathrm{x}-\mathrm{z}}{\delta}, \frac{t}{\alpha\delta^2}\Bigg)
\Theta_0 (u) =~ \frac{u}{2}, ~ u \in C^0_\mathrm{RBA}((A,P), (M,Q)) = M, \Theta_n (f, g) =~ (f,~ 2^{n-2}g), ~ (f, g) \in C^{n \geq 1}_\mathrm{RBA}((A, P), (M, Q)).
(Y\triangleright c_{f,v}^{\Lambda})(X) = c_{f,v}^{\Lambda}(XY) = f(X\triangleright Y\triangleright v) = c_{f,Y\triangleright v}^{\Lambda}(X), (c_{f,v}^{\Lambda}\triangleleft Y)(X) = c_{f,v}^{\Lambda}(YX) = f(Y\triangleright X\triangleright v) = c_{f\triangleleft Y,v}^{\Lambda}(X).
\Sigma_{\mathrm{D}}(\delta) = \Pi \left(\mathcal{B}_1,\mathcal{B}_2 \right), \mathcal{B}_1 = B_{p_1}( \mu_1, \delta_1^{1/p_1}) \mathcal{B}_2 = B_{p_2}( \mu_2, \delta_2^{1/p_2}).
(f \cup_\alpha g) (a_1, \ldots, a_{m+n}) = ~ \mu ( f (\alpha^{n-1}a_1, \ldots, \alpha^{n-1}a_m) , g (\alpha^{m-1}a_{m+1}, \ldots, \alpha^{m-1}a_{m+n}) ) = ~ \mu ( \alpha^{n-1}f (a_1, \ldots, a_m), \alpha^{m-1}g (a_{m+1}, \ldots, a_{m+n})).
\nabla\cdot v=\frac{\partial v^{i}}{\partial q^{i}}+\frac{1}{2g}\frac{\partial g}{\partial g_{mi}}\frac{\partial g_{im}}{\partial q^{l}}v^{l}=\frac{\partial v^{i}}{\partial q^{i}}+\frac{1}{2g}\frac{\partial g}{\partial q^{l}}v^{l}
\left(\begin{array}{cc}\phi_{n+m}\psi_{n+m}\phi_{n+m}^* -\psi_{n+m}^*\end{array}\right)=\left(\begin{array}{cc}{\cal A}_m{\cal B}_m{\cal C}_m{\cal D}_m\end{array}\right)\left(\begin{array}{cc}\phi_{n}\psi_{n}\phi_{n}^* -\psi_{n}^*\end{array}\right)
\tau_{K,N}^{(t),0}(\theta)=\tau_{K,N}^{(t),1}(\theta) =\frac{1}{2}\left(\dfrac{2\sinh \left(\frac{\theta}{2}\sqrt{\frac{K}{1-N}}\right)}{\sinh \sqrt{\frac{K}{1-N}}\theta}\right)^{1-\frac{1}{N}}=\frac{1}{2}\left(\cosh \left(\frac{\theta}{2}\sqrt{\frac{K}{1-N}}\right)\right)^{\frac{1}{N}-1}
\sum_{M\geq0}F_M\,z^M =\frac{z(1-z-z^2)}{(1-z-z^2)^2}, \sum_{M\geq0}MF_M\,z^M =\frac{z+z^3}{(1-z-z^2)^2}, \sum_{M\geq0}\sum_{j=0}^MF_jF_{M-j}\,z^M =\frac{z^2}{(1-z-z^2)^2}.
y_r^{(m)}=\sum_{k=0}^{\min\left(J-1,m-1\right)}\rho^k\mathbf{a}_{r,m-k}\mathbf{x}&+\sum_{k=0}^{\min\left(\lceil\frac{J}{2}\rceil-1,m-1\right)}\rho^k z_r^{(m-k)}.
I(\phi)\;=\;\int_{M'\times M'}d^{d}{\bf x}d^{d}{\bf y}\,\phi_{0}({\bf x})\,G({\bf x},{\bf y})\,\phi_{0}({\bf y}),\;\;\;\;G({\bf x},{\bf y})\;\propto\;\sum_{\gamma\in\Gamma}\frac{|\gamma'({\bf y})|^{\Delta}}{|{\bf x}-\gamma{\bf y}|^{2\Delta}}.
F_k=\prod^{m_k}_{l=1}T^{\varepsilon_{l,k}}_{\alpha_{l,k}}\quad(k=1,2),W_{i,k}=\prod^i_{l=1}T^{\varepsilon_{l,k}}_{\alpha_{l,k}}\quad(i=1,2,\ldots,m_k;k=1,2)
{\cal L}= - \, \frac{1}{4}\, F_{\mu \nu}\, F^{\mu \nu}- \, \frac{1}{1 2}\, H_{\mu \nu \rho}\, H^{\mu \nu \rho}\, +{\frac{m}{4}}\, \epsilon^{\mu \nu \rho \lambda}\, B_{\mu \nu}F_{\rho \lambda}\, ,
\sum_{k=0}^{\frac{r+s}{2}-\frac{\mid r-s\mid}{2}}\frac{(C_{k+\mid \frac{r-s}{2}\mid}^{\frac{r+s}{2}})(C_k^{\frac{r+s}{2}})\mid 2k + \mid \frac{r-s}{2}\mid - \frac{r+s}{2}\mid}{C_{\frac{r+s}{2}}^{r+s}}.
\int_{B_r(x)}\nabla(\rho^2_\delta(|u|)\overline{u}) \cdot A \nabla u \,dy = \int_{B_r(x)}(\rho^2_\delta(|u|)\overline{u}) B\cdot \nabla u\,dy + \int_{\partial{B_r(x)}}(\rho^2_\delta(|u|)\overline{u})\nu \cdot A\nabla u \, d\sigma(y).
|U_i| =|U_i'|=\ell' +e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}') &=\ell+1+e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}')=\ell +e_C(U_{i+1}, U_i)+e_C(U_i, U_{i-1}).
x^2\mspace{1.5mu}\mathrm{K}_2(x) x>0, \mathrm{K}_2(x)=\sqrt{\frac{\pi}{2}}\,e^{-x}\left[\frac{1}{\!\sqrt{x}\:}+\mathcal{O}\!\left(\frac{1}{x^{3/2}\!}\right)\mspace{-1.5mu}\right]\!,x\rightarrow +\infty, \mathrm{K}_2(x)=\frac{2}{\,x^2\!}-\frac{1}{2}+\mathcal{O}\!\left(x^2\ln{x}\right)\!,x\rightarrow 0^+\!.
P_p(x,\zeta)=\sum_{m=0}^{\infty}Z^p_m(x,\zeta)=\frac{1-|x|^{2p}}{(x^2\overline{\zeta}^2-2x\cdot\overline{\zeta}+1)^{n/2}}\quad\textrm{for}x\in \widehat{B}_p,\,\, \zeta \in \widehat{S}_p.
U ( T ) \simeq \frac{\Delta n^{2}}{8}\, T{,}\; \; F ( T ) \simeq - \frac{\Delta n^{2}}{8}\, T \left[ \ln ( a T ) - c \right]{,}\; \; S ( T ) \simeq \frac{\Delta n^{2}}{8}\left[ \ln ( a T ) + c + 1 \right]{,}
C_{\{k \}}={\frac{1}{k !}}\sum_{a_{1}, \dots , a_{k}}T r_{F}\{t_{F}^{a_{1}}t_{F}^{a_{2}}\cdots t_{F}^{a_{k}}\}t_{R}^{( a_{1}}t_{R}^{a_{2}}\cdots t_{R}^{a_{k})}
\Delta S(\beta;{\bf m},{\bf n})=\frac{\pi}{2}NT_{II}\beta^2 [m_\ell\Omega_{\ell i}-n_i]((\Im \Omega)^{-1})_{ij}[\bar{\Omega}_{jk}m_k-n_j]={\rm g}^{\mu\nu}(\Omega){\cal N}_{\mu}{\cal N}_{\nu},
\left[\left({H_1' \over H_1}+{H_2' \over H_2}\right ) +{2 \over r}\right ] (h_1' -h_2') = -2 \left[\left({H_1'^2 \over H_1^2}+{H_2'^2 \over H_2^2}\right ) -{2 \over r^2}\right] h_2 - 4 \left({H_1'^2 \over H_1^2}-{H_2'^2 \over H_2^2}\right ) \varphi .
M= \begin{bmatrix}R_{1c}R_{1\theta}R_{1h}R_{2c}R_{2\theta}R_{2h}R_{3c}R_{3\theta}R_{3h}\end{bmatrix}= \begin{bmatrix}R_{1c}\lambda R_{1h}T'(c) -1 0 R_{3c}0 -1 \end{bmatrix}.
Z_{\mathrm{1 - l o o p}}^{( c )}( M ) ={\frac{( 2 \pi x )^{{\frac{1}{2}}( \mathrm{d i m}H_{c}^{0}- \mathrm{d i m}H_{c}^{1})}}{\mathrm{v o l}( H_{c})}}\mathrm{e}^{-{\frac{1}{x}}S_{\mathrm{C S}}( A^{( c )}) -{\frac{i \pi}{4}}\varphi}{\sqrt{| \tau_{R}^{( c )}|}},
V\left( \frac{g_2g_4v}{KX}\right) =\frac{1}{2\pi i}\Psi\Big( \frac{v}{XQ^{\vartheta}}\Big)\int_{(\varepsilon)}\left(\frac{XK}{g_2g_4 v}\right)^{s_4}\widetilde{V}(s_4) \,ds_4.
\left\langle (-\Delta_p)^s u, \varphi \right\rangle + \left\langle (-\Delta_p)^s v, \psi \right\rangle = \lambda \left[\alpha(p)\vert u(x_1)\vert^{\alpha(p)-2}u(x_1)\vert v(x_2) \vert^{\beta(p)}\varphi(x_1)\right. &\left.\ + \beta(p)\vert u(x_1)\vert^{\alpha(p)}\vert v(x_2) \vert^{\beta(p)-2}v(x_2) \psi(x_2) \right]
\begin{aligned}J_2(\beta,N,T) \leq C N^2\int_{0}^{\pi}\iint_{0\le s\le t\le T}\frac{1}{t+s}& \times \int_{\mathbf{B}_2(0)}\exp\left(-\frac{\big|z-D_t^{(k)}+D_s^{(\ell)}\big|^2}{2(t+s)}\right) dz dsdt d\theta. \end{aligned}
\psi_{0}=0,\psi_{1}=1,\psi_{2}=2y, \psi_{3}=3x^{4}+6sx^{2}+12tx-s^{2}, \psi_{4}=4y(x^{6}+5sx^{4}+20tx^{3}-5s^{2}x^{2}-4stx-(8t^{2}+s^{3})), \psi_{2k-1}=\psi_{k+1}\psi_{k-1}^{3}-\psi_{k-2}\psi_{k}^{3}, \psi_{2k}=\frac{\psi_{k}}{2y}\left(\psi_{k+2}\psi_{k-1}^{2}-\psi_{k-2}\psi_{k+1}^{2}\right),k\ge 3.
P_{Q}^{n}\left(I_{1},\dots,I_{n-1}\right)=\left\{v\in\mathbb{R}^{nd}:Q\left(v_{1}\right)-Q\left(v_{2}\right)\in I_{1},\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\in I_{n-1}\right\}.
\sum_{{s_{1}, s_{2}}={0 , 1}}( - 1 )^{s_{1}+ s_{2}}\theta (_{s_{2}}^{s_{1}})^{\prime}\theta^{3}(_{s_{2}}^{s_{1}}) \frac{d}{d \tau}[ \sum_{s_{1}, s_{2}}( - 1 )^{s_{1}+ s_{2}}\theta^{4}(_{s_{2}}^{s_{1}}) ] = 0 ,
|q_{t,\gamma}(x+h,y)-q_{t,\gamma}(x,y)| \leq |q_{t}(x+h,y)-q_{t}(x,y)| \leq \frac{C_N}{t^{n/2+1}}\Big(\frac{|h|}{\sqrt{t}}\Big)^\delta e^{-\frac{c|x-y|^2}{t}}\Big(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\Big)^{-N}.
\bar p^\nu_\xi = \begin{cases}\frac{p_\xi}{1-\alpha}\mbox{if}\xi \in \Xi_+(x^{\nu-1}) \frac{p_\xi(\sigma-\alpha)}{(1-\alpha)\tau}\mbox{if}\xi \in \Xi_0(x^{\nu-1}) 0 \mbox{otherwise},\end{cases}
f^{\prime \prime}={\frac{{f^{\prime}}^{2}}{2 f}}+{\frac{3}{2}}f^{3}+ 2 \omega x f^{2}+ ( \mathrm{~ \frac{1}{2}~}\omega^{2}x^{2}+ k_{3}- k_{1}) f -{\frac{k_{2}^{2}}{2 f}}.
F_1 := \int_{-\infty}^y G_{\epsilon}(v-z) \psi_{\epsilon}(s,v)dz, F_2 := \int_y^{\infty}G_{\epsilon}(v-z) \big[ \psi_{\epsilon}(s,v) - \psi(s,v) \big] dz, F_3 := \int_y^{\infty}G_{\epsilon}(v-z) \big[ \psi(s,v)-\psi(s,z)\big] dz.
\left(h^g\right)_{v}= \left(g^{-1}hg\right)_v = \left(g^{-1}\right)_{h(g(v))}h_{g(v)}g_v = \left(g^{-1}\right)_{g(v)}h_{g(v)}g_v = = \left(g_{g^{-1}g(v)}\right)^{-1}h_{g(v)}g_v = \left(g_{v}\right)^{-1}h_{g(v)}g_v,
(E_0=LK_X^{\frac{1}{2}}\oplus LK_X^{-\frac{1}{2}}\oplus L^{-1}K_X^{\frac{1}{2}}\oplus L^{-1}K_X^{-\frac{1}{2}},\theta_0=\begin{pmatrix}0 0 0 0 1 0 0 0 0 \mu 0 0 0 0 1 0\end{pmatrix}),
\frac{R_{12}}{i-\omega(2,2)}u_2+\frac{R_{14}}{i-\omega(4,4)}u_4=-i(-R_{12}+R_{14})=iR_{12}-iR_{14}=i=u_1; \frac{R_{32}}{i-\omega(2,2)}u_2+\frac{R_{34}}{i-\omega(4,4)}u_4=iR_{32}-iR_{34}=-i=u_3.
\overline D_n(\overline{\mu}_n | \overline{\nu}_{\widehat\theta, n}^N ) =\overline{\mathcal D}_n^0 (\overline{\mu}_n | \overline{\nu}_{\widehat \theta ,n}^N)\,+\, \overline D^{\widehat b}_n (\overline{\mu}_n | \overline{\nu}_{\widehat \theta ,n}^N) \, ,
A_1(t) \le C\sqrt{t}\int_0^\infty d\xi\int_{-\infty}^{\infty}dy\int_{-\infty}^{\infty}dz \int_0^t du\, \chi_t(\xi, y, z, u) \cdot \int_0^\infty dx_0 (a+b)(a+2b x_0)(1+\log^+(1/x_0))e^{-C'x_0^2}=C\sqrt{t}\int_0^\infty d\xi\int_{-\infty}^{\infty}dy\int_{-\infty}^{\infty}dz \int_0^t du\, \chi_t(\xi, y, z, u) (C_1a(a+b)+2C_2b(a+b))
\mu hK_1(b+c)(K+c)-\Gamma c(1+c)=0 \Rightarrow (\mu hK_1-\Gamma)c^2+(\mu hK_1(b+K)-\Gamma)c+\mu hK_1bK=0 \Rightarrow c=c_{-}(h)\,\,\textrm{and}\,\,c=c_{+}(h),
{\cal L}= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\hbar}{2}\bar{\psi}\gamma^{\mu}\left(i\partial_{\mu}- e A_{\mu}\right)\psi -\frac{\hbar}{2}\left[\left(i\partial_{\mu}+ eA_{\mu}\right)\bar{\psi}\right] \gamma^{\mu}\psi
({\bf P2}){\rm minimize}_{u_i\in L_{\cal F}^2(0,T;{\mathbb R}^{n_1})}\bar J_i(u_i,\hat f_{u_i}) =\{\mathbb E}\int_0^T \left\{| x_i-(\Gamma m_i +\eta)|^2_Q+u_i^T R u_i -\gamma | p_i(t)|^2 \right\}dt +{\mathbb E}x_i^T(T)Hx_i(T).
\prod_{p\nmid WB}\Bigl(1+\frac{\omega(p)(p-1)}{(p-\omega(p))^2}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^k=\prod_{p\nmid WB}\Bigl(1+\frac{\omega(p)}{p-\omega(p)}\Bigr)\Bigl(1+O\Bigl(\frac{k^2}{p^2}\Bigr)\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^k\ll \mathfrak{S}_{WB}(\mathcal{L})^{-1},
\begin{pmatrix}h[n+1] \Theta^\mathrm{pair}[n+1] \overline{\Theta^\mathrm{pair}[n+1]}h[n+1]\end{pmatrix}\circ \begin{pmatrix}\phi_{n+1}\overline{\phi_{n+1}}\end{pmatrix}= \mu_{n+1}\begin{pmatrix}\phi_{n+1}\overline{\phi_{n+1}}\end{pmatrix},
\Big\| \sum_{n=M+1}^{N}{\frac{a_{n}}{n^{\sigma + \varepsilon}}n^{-s}}\Big\|_{\mathfrak{X}(X)}\leq \frac{C}{N^{\varepsilon}}+ \frac{C}{M^{\varepsilon}}+ \sum_{n=M}^{N-1}{C \Big[ \frac{1}{n^{\varepsilon}}- \frac{1}{(n+1)^{\varepsilon}}\Big]}= \frac{2 C}{M^{\varepsilon}}
{\bf{H}}={c_0}\,{e^{- j2\pi{d_0}/{\lambda_c}}}\,{{\bf{u}}_{\mathrm r}}({{\theta_{0,a}}}) \,{\bf{u}}_{\mathrm t}^H ({{\phi_{0,d}}}) + \sum\limits_{l = 1}^L{{c_l}\,{e^{- j2\pi{d_l}/{\lambda_{\rm c}}}}\,{{\bf{u}}_{\mathrm r}}({{\theta_{l,a}}}) \,{\bf{u}}_{\mathrm t}^H ({{\phi_{l,d}}})}
S_0=\int d^2 xe(g^{\alpha\beta}\partial_\alpha\sigma\partial_\beta-i\bar\psi\gamma^\beta\gamma^\alpha\partial_\alpha\psi+2\bar\chi_\alpha\gamma^\beta\gamma^\alpha\psi\partial_\beta\sigma+\frac{1}{2}\bar\psi\psi\bar\chi_\alpha\gamma^\beta\gamma^\alpha\chi_\beta)
0 \mapsto \Omega^{0}( M ) \oplus \Omega^{0}( M ) \stackrel{T_{1}}{\mapsto}\Omega^{1}( M ) \oplus \Omega^{1}( M ) \stackrel{* d}{\mapsto}\Omega^{1}( M ) \oplus \Omega^{1}( M ) \mapsto 0 .
\begin{array}{rllll}\sigma^\star D_1 = -D_1 \sigma^\star D_2 = -D_2 \sigma^\star D_3 = D_1+D_3+\div(h_3) h_3=\frac{y}{x^3+a^{58}x^2+a^2x+a^{54}}\sigma^\star D_4 = D_2+D_4+\div(h_4) h_4=\frac{y}{x^3+a^{80}x^2+a^{103}x+a^{114}}\sigma^\star D_5 = D_5\end{array}
\underline{d}_{\mu}( x ) = \liminf_{N \rightarrow \infty}\frac{\log \mu \left( B_{K \beta_N}( x ) \right)}{\log \beta_N}\mbox{and}\overline{d}_{\mu}( x ) = \limsup_{N \rightarrow \infty}\frac{\log \mu \left( B_{K \beta_N}( x ) \right)}{\log \beta_N}
F_j(\lambda_1,\dots,\lambda_n)=(-1)^{\lambda_{j+1}+\cdots+\lambda_n}(\lambda_j+n+1-j+[\lambda_j+1]_2(p-n))^{\frac{1}{2}}\bigg(\prod_{\substack{\ell=1, \ell\neq i}}^n\frac{\lambda_j-\lambda_\ell-j+\ell}{\lambda_j-\lambda_\ell-j+\ell+[\lambda_j-\lambda_\ell]_2}\bigg)^{\frac{1}{2}}.
R \, = \, \left( \begin{array}{c c c c}{{Q^{- 1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\lambda^{- 1}}}&{{0}}&{{0}}\\{{0}}&{{\sigma}}&{{\lambda}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- Q}}\\ \end{array}\right) \, , \qquad \sigma \, = \, Q^{- 1}- Q \, .
A_{j i}^{- 1}\frac{| \alpha_{i}|^{2}}{2}= \left\langle \lambda_{j}, \lambda_{i}\right\rangle = \sum_{x = 1}^{h}\frac{x}{h}\left\langle \lambda_{j}, \sigma^{x + \frac{c_{i}- 1}{2}}\gamma_{i}\right\rangle ,
\sum_{K_1\ni k=0}^{\infty}\!\sqrt{\Phi(x^{\ell(k+1)})\!-\!\Phi(x^{k+1})}<\infty\ \{\rm when}\ \liminf_{K_1\ni k\to\infty}\frac{\Phi(x^{\ell(k)})-\Phi(x^{\ell(k+1)})}{\|x^{k+1}-x^k\|^2}=0,
\limsup_my_n\mathbf{d}x_m \leq\liminf_l\limsup_m(y_n\mathbf{d}y_l+y_l\mathbf{d}x_m) &\leq\liminf_l\limsup_m(y_n\mathbf{d}y_l+y_l\overline{\mathbf{d}}x_l+x_l\mathbf{d}x_m)=0
H(W_{n+u}) \leq~ \sum_{i \leq \ell}( H(W_i) - H(W_i | W_{[i+1,n+u-1]}) ) &=~ \sum_{i \leq \ell}H(W_i) + H( W_{[\ell+1,n+u-1]}) - H( W_{[1,n+u-1]}) &\leq~ \sum_{i \leq \ell}H(W_i) + H( W_{[\ell+1,n]}) +H(W_{[n+1,n+u-1}]) - H( W_{[1,n]})
( V_{1,r}, \left\| \cdot \right\|_{V_{1,r}}, \left< \cdot, \cdot \right>_{V_{1,r}}) = ( HS(U,H_{-r}), \left\| \cdot \right\|_{HS(U,H_{-r})}, \left< \cdot, \cdot \right>_{HS(U,H_{-r})}),
S_{B H}= \frac{2 \pi A_{9}}{\kappa^{2}}= \frac{2 \pi A_{3}}{\kappa_{5}^{2}}= \frac{8 \sqrt{2}\pi^{3}m^{3 / 2}}{\kappa^{2}}S_{1 2}S_{3 4}S_{5 6}\times \cosh \alpha_{1}\times \cosh \alpha_{2}\times \cosh \alpha_{3}
\int_0^\infty(CL_2^u(y)+DL_2^v(y))\mathrm{e}^{-\lambda y^2}\mathrm{d}y=\frac{\sqrt{mn}}{m(1+\frac{4t^\alpha}{\alpha}\lambda)^{\frac{c+1+\sqrt{mn}}{2}}}\mathrm{e}^{-\frac{\lambda x^2}{(1+\frac{4t^\alpha}{\alpha}\lambda)}},
\sum_{\nu =k+1}^{\infty}\dfrac{(1+\delta \nu - \delta ) \,\Gamma(\nu+\tau)\Gamma(\mu+1)}{\Gamma(\nu+\mu)\Gamma(\tau+1)}\, a_{\nu}\leqq 1-\gamma (a_{\nu}\geqq 0; \, 0 \leqq \gamma < 1).
\sum_{j= n+1}^{\infty}Cov(|X_1 - x| - E|X_1 -x| - B(X_1 -x) , |X_j - x| - E|X_j -x| - B(X_j -x)) & \le B\sum_{j=n+1}^{\infty}Cov(X_1 -x, X_j -x) = B\sum_{j=n+1}^{\infty}Cov(X_1, X_j) = O(n^{-(k-2)/2}).
\Phi_{\lambda,t}(z_\delta,w_\delta,t_\delta)= H(D_x \Phi_\lambda(z_\delta,w_\delta,t_\delta), z_\delta) \dot \xi_{t_\delta}- H(-D_y \Phi_\lambda(z_\delta,w_\delta,t_\delta), w_\delta) \dot \zeta_{t_\delta}.
\mathrm{Cov}(G(s), G(t))=\nu (f_t^{(0)}\cdot f^{(0)}_s) + \sum_{k>0}\nu (f_t^{(0)}\cdot f_s^{(0)}\circ T^k) +\sum_{k>0}\nu (f_s^{(0)}\cdot f_t^{(0)}\circ T^k) \, ,
\left(\eta'_a(\sigma) + i 4\pi G \, \epsilon_a^{~b}\, \eta_b(\sigma) \,{\delta \over \delta\eta_2(\sigma)}+ i 4\pi G \, \eta_3(\sigma) \,\epsilon_{ab}\,{\delta \over \delta \eta_b(\sigma)}\right) \, \Psi(\eta,q,X)\hspace*{\fill}
\langle N \rangle_{T}^{( d )}\sim - \frac{m T}{\pi}\frac{1}{d ! ( 2 T )^{d}}\, e^{- \frac{m}{T}}\, \int d^{2}x \, \frac{q ( \vec{x})}{| \vec{v}( \vec{x}) |}\left[ \left( \partial_{i}\hat{n}^{a}\, \partial_{i}\hat{n}^{a}+ | \vec{v}( \vec{x}) | \right)^{\frac{d}{2}}- \left( \partial_{i}\hat{n}^{a}\, \partial_{i}\hat{n}^{a}- | \vec{v}( \vec{x}) | \right)^{\frac{d}{2}}\right]
t^2\Lambda_5(t) \leq \frac{\mu}{2}t^2|u_{tt}|^2_{D^1_0}+C(c^6_3+c^2_3|v_t|^2_{D^1_0}) t^2|\sqrt{\rho}u_{tt}|^2_2+Cc^4_3t^2(|\rho_{tt}|^2_{2}+|P_{tt}|^2_{2}) &+Cc^4_3t^2|H_{tt}|^2_{2}+Ct^2(|v_{tt}|^2_{D^1_0}+|v_t|^2_{D^2})+Cc^{6}_3t^2|u_t|^2_{D^2}+Cc^{30}_3.
\inf_{x \in \mathcal{X}}\f(x), & \sup_{\mathcal{S}\subseteq [m]: |\mathcal{S}| \leq \Gamma}\left\{\sum_{i \in \mathcal{S}}\sup_{u^i \in \mathcal{U}_i}g_i(x,u^i) + \sum_{i \in [m] \setminus \mathcal{S}}g_i(x, \overline{u}^i) \right\}\leq 0.
[\mathbb{E}^V-\mathbb{E}^G] \theta(\frac{N^{2/3}}{\varpi}(\lambda_{\alpha^{\prime}}-a_{2k-1})) =-[\mathbb{E}^V-\mathbb{E}^G] \int_I \frac{N^{2/3}}{\varpi}\theta^{\prime}(\frac{N^{2/3}}{\varpi}(E-a_{2k-1})) \mathbf{1}(\lambda_{\alpha^{\prime}}\leq E)dE+O(N^{-D_1}),
\frac{f'}{f}(z)=\sum_{|\rho-s_0| \leq r-\delta}\frac{1}{s-\rho}+O\bigg( \frac{1}{\delta^2}\Big(\log M_{r}(s_0) +N_{r-\delta}(s_0) (\log 1/\delta+1) \Big)\bigg).
p'(\bar{x},\bar{y}) = m\big( \, w_1(\bar{x},\bar{y}), \dots, w_n(\bar{x},\bar{y})\, \big) &=m_1 \big( \, w_{i_1}(\bar{x},\bar{y}), \dots, w_{i_p}(\bar{x},\bar{y})\, \big) \cdot m_2 \big( \, w_{j_1}(\bar{x},\bar{y}), \dots, w_{j_q}(\bar{x},\bar{y})\, \big)
{R_{\alpha\beta\gamma}}^{\delta}=e_{\gamma}^{I}{R_{\alpha\beta I}}^{J}e_{J}^{\delta},R_{\alpha\beta}={R_{\alpha\gamma I}}^{J}e_{\beta}^{I}e_{J}^{\gamma},R={R_{\alpha\beta}}^{IJ}e_{I}^{\alpha}e_{J}^{\beta}
k_{e_m}=\limsup_{z\rightarrow0}\frac{\ln\|(z-T)^{-1}e_m\|_p}{\ln\|(z-T)^{-1}\|}&\leq\limsup_{z\rightarrow0}\frac{\ln\|(z-T)^{-1}e_m\|_p}{\ln\|(z-T)^{-1}e_N\|_p}=\frac{m+1}{N+1}.
SR^{-1}X_{\tt P}R =\left(\begin{array}{cc}Y_{\tt P}Z_{\tt P}O O\end{array}\right), SR^{-1}X_{\tt C}R =\left(\begin{array}{cc}Y_{\tt C}Z_{\tt C}O O\end{array}\right), SR^{-1}X_{\tt E}R =\left(\begin{array}{cc}Y_{\tt E}Z_{\tt E}O O\end{array}\right),
\sum_{1\leq j, k \leq n}f_j(\tau_k)f_k(\tau_j)= \sum_{j=1}^n\sum_{k=1}^nf_j(\tau_k)f_k(\tau_j)\geq \frac{1}{d}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2=\frac{1}{(\mathcal{X})}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2.
\lambda^2\|\partial_\nu e_\lambda\|_{L^1(\partial M)}\lesssim\int_{Z_\lambda}|\langle\nabla\triangle e_\lambda,N\rangle|ds\leq\|\nabla\triangle e_\lambda\|_{L^\infty(M)}|Z_\lambda| &\lesssim\lambda^{\frac{n+2}{2}}\|\partial_\nu e_\lambda\|_{L^1(\partial M)}|Z_\lambda|.
S_{\mathrm{C S}}= - \mathrm{i}\int \mathrm{d}^{3}x \epsilon^{\mu \rho \nu}\left( \frac{1}{2}A_{\mu}^{a}\partial_{\rho}A_{\nu}^{a}+ \frac{1}{3 !}g f^{a b c}A_{\mu}^{a}A_{\rho}^{b}A_{\nu}^{c}\right) ,
A_{s t}= \pi \alpha^{\prime}g_{s}I ( \theta_{1}, \theta_{2}, \theta_{3}) ={\frac{\alpha_{G U T}^{2 / 3}L ( Q )^{2 / 3}g_{s}^{1 / 3}I ( \theta_{1}, \theta_{2}, \theta_{3})}{4 \pi M_{G U T}^{2}}}\,
\begin{array}{l}{{\bar{c}( r^{1 1}) = c_{n_{r}}^{\prime}, \quad \bar{c}( r^{2 1}) = c_{q_{r}}^{\prime}, \quad N_{1 1}= N , \quad N_{2 1}= Q ,}}\\{{\bar{c}( r^{1 2}) = c_{m_{r}}^{\prime}, \quad \bar{c}( r^{2 2}) = c_{t_{r}}^{\prime}, \quad N_{1 2}= M , \quad N_{2 2}= T ,}}\\ \end{array}
\phi(u, v) = \int_{-v}^u \int_{-u_0}^v -F(u_0, v_0) dv_0 du_0+ \int_{-v}^u \phi_u(u_0, -u_0) du_0+ \phi(-v, v), = 2\int_{-v}^u \int_{-u_0}^v F(s, y) dy ds+ \int_{-v}^u \phi_u(u_0, -u_0) du_0+ \phi(-v, v), = 2\int_0^t \int_{x-t+s}^{x+t-s}F(s, y) dy ds+ \int_{-v}^u \phi_u(u_0, -u_0) du_0+ \phi(-v, v).
\left< u^{n+1},v \right> + \Delta t \left< \left(\frac{\left(u^{n+1/2}\right)^2}{2}\right)_{x}, v \right> + \Delta t \left< \mathcal{H}\left(u^{n+1/2}\right)_{x}, v_{x}\right> = \left< u^{n}, v \right>,
\widehat{y}_{i}^{[K,L]}(x) = (1-q) \sum_{n=K}^{L}s \frac{P_{\lambda}(x, s)}{s-b_{i}}\frac{(\alpha_1 s, \alpha_2 s, \alpha_3 s;q)_{\infty}}{(\beta_1 s, \beta_2 s, \beta_3 s;q)_{\infty}}\bigg|_{s= q^n \xi}, i=0,1,2,3, & b_0=0, b_1 = 1/\alpha_1, b_2 = 1/\alpha_2, b_3 = 1/\alpha_3
\mathbb{E}\Big(\Big\|\sum_{n=1}^m \epsilon_n x_n \Big\| \Big)=\frac{1}{2^m}\sum_{(\epsilon_n)\in\{-1,+1\}^m}\Big\|\sum_{n=1}^m \epsilon_n x_n \Big\|=\int_0^1\Big\|\sum_{n=1}^m r_n(t)x_n\Big\|_Xdt.
\sum_{I \subseteq \{2, 3, 4, 5\}}(-1)^{|I|}\Sigma_I = y^5 + y^2 - 4(y^4 - y^3 + y^2) + y^3 + 4(y^3 - y^2 + y) &\qquad+ (y-1)^3 + y^2 - 2y^2 - 2(y^2 - y + 1) + y begin{align*}2mm] = y^5 - 4y^4 + 10y^3 - 13y^2 + 10y - 3.
x(t,s_0,x_0) = X(t)\Bigg[X^{-1}(s_0)x_0 +\int_{s_0}^{t}X^{-1}(s){\rm d}[g(s)-g(s_0)]-\sum_{s_0< \tau \leq t}\Delta^{-}X^{-1}(\tau)\Delta^{-}g(\tau) & \ \ +\sum_{s_0\leq \tau < t}\Delta^{+}X^{-1}(\tau)\Delta^{+}g(\tau)\Bigg],
\langle T(z)\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\rangle=\sum_{i=1}^{N}(\frac{\Delta_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\frac{\partial}{\partial z_{i}})\langle\prod_{i=1}^{N}V_{\Delta_{i}}(z_{i})\rangle.
\delta\left({\sigma_{\tau}^{*}}(t)I\right)=\int^{t_{2}}_{t_{1}}\left[-\frac{d}{dt}\left(\frac{\partial L_{T}}{\partial \dot{Q}^{a}}\right)+\frac{\partial L_{T}}{\partial Q^{a}}\right]\delta Q^{a}dt+\left[\frac{\partial L_{T}}{\partial \dot{Q}^{a}}\delta Q^{a}\right]^{t_{2}}_{t_{1}},
-t_2 - t_1z + 2z^3=0, w(t_1)=\frac{2}{5},w(t_2)=\frac{3}{5},w(t_3)=1,w(z)=\frac{1}{5}, g_1 = \big(175t_1t_3 - 70t_1^3z + 70t_1^2z^3 + 378t_1z^5 - 540z^7\big)/175, g_2 = \big(10t_1^4 - 120t_1t_2^2 + 75t_2t_3 + 30t_1^2z^4 - 192t_1z^6 + 324z^8\big)/75, g_3 = \big(16t_1^5 + 80t_1^2t_2^2 + 25t_3^2 - 80t_1^3z^4 + 540t_1^2z^6 - 1080t_1z^8 + 432z^{10}\big)/50.
\partial_t u_{i}^{(n)}= \tilde{L}_k u_{i}^{(n)}- \delta_i\Bigl(\prod_{j=1}^q (u_j^{(n-1)})^{\alpha_j^{(i)}}\, u_{i}^{(n)}-\prod_{j=1}^q (u_j^{(n-1)})^{\beta_{j}^{(\nu_k(i))}}\, u_{\nu_k(i)}^{(n)}\Bigr).
V ( s , t ) = \frac{1}{8}\{e^{-{\sqrt{2}}\, ( 2 s +{\sqrt{3}}t )}}\ \left( 1 -{e^{2{\sqrt{2}}\, s}}\right)^{2}= \frac{1}{2}\e^{- \sqrt{6}t}\ \sinh^{2}\, ( \sqrt{2}\, s ) \ .
N(u) = \frac{\|\bar\Sigma^{1/2}u\|}{\theta_1}, \|u\|\le\frac{\|\bar\Sigma^{1/2}u\|}{\theta_1}\quad\frac{\|u_{T^c}\|_1}{\sqrt{s_1}}\le\frac{\|\bar\Sigma^{1/2}u\|}{\theta_1}
\theta_0(\pi^{n},\pi^{m},\pi^{k}) = \theta_0 \big( \operatorname{diag}(\pi^{n},\pi^{m},\pi^{k})^\rho \big) = \theta_0 \big( \operatorname{diag}(\pi^{n-k},\pi^{m-k}, \, \pi^{n}, \pi^{m}, \, \pi^{n+k}, \pi^{m+k}, \, \pi^{m+n-k}, \pi^{m+n}) \big),
a^{k}b^{l}[ f ] = \Delta_{1 1}( 0 ) \, \int_{C}d^{2}x \, F^{( k , l )}[ f ] ( x ) + \int_{C}d^{2}x d^{2}y \, \Delta_{C}( x ; y ) \, B^{( k , l )}[ f ] ( x , y ) = 0 \, .
\overline{G}=\varepsilon\frac{\sqrt{R}}{\sqrt{\theta}}\sum^{3}_{j=1}\frac{\partial\bar{\theta}}{\partial x_{j}}A_{j}(\frac{v-u}{\sqrt{R\theta}})+\varepsilon\sum^{3}_{j=1}\sum^{3}_{i=1}\frac{\partial\bar{u}_{j}}{\partial x_{i}}B_{ij}(\frac{v-u}{\sqrt{R\theta}}).
E\left[\exp\left(-\eta_{i_{j}}\min(W_{j}, t-T_{j}')\right)|W_{j}= w\right] = \int_{0}^{t-w}\exp(-\eta_{i_{j}}w)\frac{\lambda(x)}{m(t)}dx + \int_{t-w}^{t}\exp(-\eta_{i_{j}}(t-x))\frac{\lambda(x)}{m(t)}dx = \exp(-\eta_{i_{j}}w)\frac{m(t-w)}{m(t)}+ \exp(-\eta_{i_{j}}t)\int_{0}^{t}\exp(\eta_{i_{j}}x)\frac{\lambda(x)}{m(t)}dx
0\neq \sum_{i=1}^{\ell}h^{(i)}_{d_i}(s_i+\tilde{\lambda}_i(q)n_i)=\sum_{i=1}^{\ell}\frac{h^{(i)}(r_{d_i}^{(i)}+d_i(s_i+\tilde{\lambda}_i(q)n_i))}{\lambda_i(d_i)}=\Lambda(q)(a-a').
\{(x_1,y_1,z_1,w_1, \hdots,x_g,y_g,z_g,w_g,x,y)\,|\,x_i,z_i\in\mathbb{R}-\{0\}\ \mbox{and}\y_i,w_i\in\mathbb{R}\ \mbox{for all}\i,\ &x,y\in\mathbb{R},\x^2 =1,\ \mbox{and}\ \sum_{i=1}^g p(x_i,y_i,z_i,w_i) + y(x+1/x) = 0 \}
=\frac{1}{\sqrt{|D_{M}|}^{[K:M]}}\cdot\prod_{1\leq i_{1}<i_{2}\leq m}\;\;\prod_{j_{1}=1}^{k}\prod_{j_{2}=1}^{k}\left\vert \alpha^{(i_{1},j_{1})}-\alpha^{(i_{2},j_{2})}\right\vert .
d(u) + d(v) -1 + (r-1)(d(u)-1) = (d(u)+d(v)) + (r-1)d(u) -r \leq \omega + (r-1)\frac{\omega-1}{2}- r = \frac{(\omega-3)(r+1)}{2}+ 2 = \begin{cases}(g+5)(r+1)+2 (2g+2)(r+1)+2 \end{cases}= \ell-1.
x_{+}(t) = x_{\alpha_3}(3t)x_{001100}(2t^2)x_{001110}(t^3)x_{\alpha_4}(2t)x_{000110}(4t^2) x_{\alpha_5}(t) x_{010110}(2t^5) x_{011100}(t^5), x_{-}(t) = x_{-\alpha_3}(t)x_{-001100}(t^2)x_{-001110}(t^3)x_{-\alpha_4}(2t)x_{-000110}(3t^2) x_{-\alpha_5}(3t) x_{-010110}(3t^5) x_{-011100}(t^5).
(\partial_z u^z)(t,0) = C_2 \int \frac{z}{(r^2+z^2)^{\frac 32}}\frac 1 r \partial_r (r\omega^{\theta}) r dr dz &= 3C_2 \int \frac{r^2 z}{(r^2+z^2)^{\frac 52}}\frac{\omega^{\theta}(t,r,z)}r r dr dz,
R_{a}^{+}u_{a}^{+}= e^{+i\theta/2}u_{a}^{+}\hspace{0.5cm}R_{b}^{-}u_{b}^{-}= e^{+i\theta/2}u_{b}^{-}\hspace{0.5cm}R_{a}^{+}v_{a}^{+}= e^{+i\theta/2}v_{a}^{+}\hspace{0.5cm}R_{b}^{-}v_{b}^{-}= e^{+i\theta/2}v_{b}^{-},
\alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n}(x(u,0)) , \alpha_{2n-1}( x ( u,0 )) =\rho\left( \frac{\kappa}{2n-1-2\kappa},\frac{\kappa}{2n-1+2\kappa}\right) S_{2n-1}(x(u,0)) ,
\langle u(t)-u_\infty , \vartheta \rangle_{(H^1)',H^1}= \langle u(t)-u^{in}+ u^{in}-u_\infty , \vartheta \rangle_{(H^1)',H^1}& = \langle \Delta (A(t) - A_\infty) , \vartheta \rangle_{(H^1)',H^1}& = - \int_\Omega \nabla(A(t)-A_\infty)\cdot \nabla\vartheta\ \mathrm{d}x\,,
\sum\limits_{k=1}^{\infty}k^{2(\frac 1{p_1}+\frac 1{p_2})}s_{3k-2}(T)^2\leq \sum\limits_{k=1}^{\infty}k^{\frac 2{p_2}}s_{k}(E_2^{-1})^2s_{k}(A)^2k^{\frac 2{p_1}}s_{k}(E_1^{-1})^2 \leq (C_1C_2)^2 \sum\limits_{k=1}^{\infty}s_{k}(A)^2<\infty .
( ( 1 , 2 , 3 , 4 ) ) ( ( 5 , 3 , 2 , 7 ) ) = - ( ( 1 , 2 , 2 , 7 , 5 , 4 ) ) + ( ( 1 , 7 , 5 , 3 , 3 , 4 ) ) - ( ( 1 , 7 , 5 , 4 ) )
E_1: \quad I^{(s)}_{grav}+ (\frac{6}{\pi^2})^4 (x_{12}^2 x_{34}^2)^{-4}[64 u^{-4}G_{45}(\frac{1}{u}, \frac{v}{u}) -32 u^{-4}G_{44}(\frac{1}{u},\frac{v}{u})]
e^{I}{}_{i}( \Phi ) D_{\gamma c}D_{\alpha a}\Phi^{i}-{\frac{1}{4}}( \gamma^{I}{}_{J})_{a}{}^{b}e^{J}{}_{j}( \Phi ) D_{\gamma c}D_{\alpha b}\Phi^{j}= f^{I}{}_{\gamma \alpha c a},
\frac{i}{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}\, \frac{e^{2}}{k^{2}\Big ( 1 + ( - 1 )^{n}k^{2 n}/ \Lambda^{2 n}\Big ) ( k + p )^{2}}\to - \frac{1}{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{e^{2}}{k^{2}\Big ( 1 + k^{2 n}/ \Lambda^{2 n}\Big ) ( k + p )^{2}}.