harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{r c l}{{x_{1}+ i x_{2}}}&{{=}}&{{y ( \partial_{\sigma}\bar{y})^{1 / 3}/ \mathrm{R e}\; ( \partial_{\sigma}y )^{1 / 3}}}\\{{x_{-}}}&{{=}}&{{- \mathrm{I m}\; ( \partial_{\sigma}y )^{1 / 3}/ \mathrm{R e}\; ( \partial_{\sigma}y )^{1 / 3}}}\\{{x_{+}}}&{{=}}&{{x_{-}| y |^{2}+ i \int d \sigma \; ( y \partial_{\sigma}\bar{y}- \bar{y}\partial_{\sigma}y ) + C}}\\ \end{array}
\tilde{W}: = \frac q{1 + q}\tilde{K}^{- 1}+ \frac 1{1 + q}\tilde{K}+ q^{- \frac 1 2}( q \! - \! 1 )^{2}\tilde{K}^{- 1}\tilde{L}^{-}\tilde{L}^{+}.
{\bar{T}}_{c}^{L}= \frac{1}{6 ( 2 \pi )^{4}}\left( \frac{g}{\pi m}\right)^{2}\cos \left( m \Theta_{0 j}p^{j}\right) \left\{\cos \left[ m \Theta_{0 j}\left( p^{j}- k^{j}\right) \right] + \cos \left[ m \Theta_{0 j}\left( p^{j}- k^{\prime}\,^{j}\right) \right] \right\}.
T'_2 \ll M^2Q^{\varepsilon}(M'_1M'_2M'_3)^{-\frac{1}{3}}\sum_{\stackrel{m_j\sim M'_j}{G(\mathbf{m})=0}}\sum_{d\leq Q}d^{-\frac{1}{2}+\varepsilon}&\ll M^2Q^{\frac{1}{2}+\varepsilon}.
\pi(k,\alpha,\beta;\mathbf{v}):=\begin{cases}\pi(k,\alpha,\beta), \mbox{if}\ \ \mathbf{v}\in\{\tilde{x}_1,\cdots,\tilde{x}_n,\tilde{y}_1,\cdots, \tilde{y}_n\}, \pi(k,\alpha,\beta)- j, \mbox{if}\ \ \mathbf{v}=\tilde{z}_j, \pi(k,\alpha,\beta)+j, \mbox{if}\ \ \mathbf{v}=\bar{\tilde{z}}_j.\end{cases}
D_{U}\big(\langle F_{t}(W),G_{t}(W)\rangle\big) =\langle D_{U}F_{t}(W),G_{t}(W)\rangle +\langle F_{t}(W),D_{U}G_{t}(W) \rangle,\intertext{and}D_{U}\nabla\phi(F_{t}(W)) =\nabla_{D_{U}F_{t}(W)}\nabla\phi(F_{t}(W)).
\mathring E\{\bar W(\psi)\}=E\{W(\psi)\}+\lambda^{ar}\lambda^{st}\beta_a(2\lambda_{rs,t}+\lambda_{rst})-2\lambda^{ar}\beta_{a/r}+\lambda^{ab}(\beta_{ab}-\beta_a\beta_b)+O_p(n^{-3/2}) &=E\{\bar W(\psi)\}+O_p(n^{-3/2}),
\begin{aligned}\sum_{\substack{p\leq T \,(p,N)=1}}\log D(p^{-1},{\rm sym}^m[g(\theta^+_{m,p})])+o(1) \geq \log L(1,{\rm sym}^m f) &\geq \sum_{\substack{p\leq T \,(p,N)=1}}\log D(p^{-1},{\rm sym}^m[g(\theta^-_{m,p})])+o(1).\end{aligned}
\begin{array}{l l}{{d e t_{q}L ( u ) =}}&{{a ( u + i \kappa / 2 ) d ( u - i \kappa / 2 ) - b ( u + i \kappa / 2 ) c ( u - i \kappa / 2 )}}\\{{\,}}&{{= Q_{1}u^{2}+ Q_{2}u - Q_{3}+ \kappa^{2}/ 4 Q_{1}\quad ,}}\\ \end{array}
{(-1)^{\ell(\hat\sigma)}x_1^{o(\hat\sigma)}x_2^{e(\hat\sigma)}y^{o(\hat\sigma)}z^{e(\hat\sigma)}}= (-1)^{n-1}y^{\lceil \frac{n-1}{2}\rceil}{(-1)^{\ell(\sigma)}x_1^{e(\sigma)}x_2^{o(\sigma)}y^{o(\sigma)}z^{e(\sigma)}}
A(\frac{e^{i\theta}}{\alpha})B(re^{-i\theta})=\frac{\alpha^{n+1}e^{i(n+1-q)\theta}}{(\alpha-e^{i\theta})^{n+1}(e^{i\theta}-r)^{n+1}}-\alpha^q\frac{\sum_{k=0}^{q-1}\binom{k+n}{k}e^{i(n+1+k-q)\theta}}{(e^{i\theta}-r)^{n+1}},
\mathfrak{p}= \{\mathfrak{p}_+,\mathfrak{p}_-,\{q_i,r_i\}_{i \in I}\}[1,n] = \mathfrak{p}_+ +\mathfrak{p}_- + \sum_{i \in I}\{q_i,r_i\}\mathfrak{p}_+ \subset \;\omega_+,\; \mathfrak{p}_- \subset \omega_-,\;q_i\in \omega_-,\;r_i\in \omega_+,\;q_i > r_i
\left\langle P\left( a_{\hbar},a_{\hbar}^{\dag}\right) \right\rangle_{U_{\hbar}\left( \alpha\right) \psi}=P\left( \alpha,\bar{\alpha}\right) +O\left( \sqrt{\hbar}\right) \operatorname*{Var}\nolimits_{U_{\hbar}\left( \alpha\right) \psi}\left(P\left( a_{\hbar},a_{\hbar}^{\dag}\right) \right) =O\left( \sqrt{\hbar}\right) ,
E_I^{\nu}=\Psi_{x_1}^{\nu_1}...\Psi_{x_n}^{\nu_n}\mathcal{L}, \Psi_{x_i}^{\nu_i}=\begin{cases}\psi_{x_i}^{\nu_i}\mathcal{L}[-1] i \notin I \phi_{x_i}^{0}\mathcal{L}[-1] .\end{cases}
\|\sum_{n \in A}\pm e_n \| \le 2\beta \|\lambda e_1 + \sum_{n \in A}\pm e_n\| &\le 2\beta \overline{K}\sigma^w_{w_{1}}(x) \le 2\beta \overline{K}(1+ \varepsilon) \| e_{1}\| \intertext{(since $w(A) \le w_1$)}= 2\beta \overline{K}(1+ \varepsilon).
T_{z_0}(\boldsymbol{x},\boldsymbol{x}')=\int d^2 p \, e^{-ipz_0-i\bar p\bar z_0}\int \mathcal{D}_{N-1}\boldsymbol{u}\,\boldsymbol{\mu}^{(B)}_N(\boldsymbol{u})\overline{\langle \Psi_B(p,\boldsymbol{u})|\Psi_A(\boldsymbol{x}')\rangle}\langle \Psi_B(p,\boldsymbol{u})|\Psi_A(\boldsymbol{x})\rangle\,,
\sigma(x) = - 2\sum_{i=0}^1 \sum_{k \in I_i}(-1)^{i}\left(\sum_{j=0}^{m_{ki}- 1}\alpha_{k+j,i}\sum_{p=0}^{j}\tilde \varphi_{k+p, i}(x)\varphi_{k+(j-p), i}(x) - \frac{1}{2}\alpha_{ki}\right).
g(\xi_0)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}\xi_0^{-\frac{\alpha}{\beta}}\quad\mbox{and}g'(\xi_0)=-\frac{\alpha}{\beta}\lambda^{-\frac{\rho_1}{(1-m)\beta}}\xi_0^{-\frac{\alpha}{\beta}-1}.
F_i(\mathbf u^{(l)})=\frac{(\alpha_i-\alpha_1)(\alpha_i-\alpha_2)\cdots(\alpha_i-\alpha_{\check i})\cdots(\alpha_i-\alpha_{n})}{(\alpha_i-\lambda_1)(\alpha_i-\lambda_2)\cdots(\alpha_i-\lambda_{n+1})}\cdot(\alpha_i)^{l}.
\omega = t \bar{z}_1 \xi+ t^2 \bar{z}_1^2+ r(tz_1,t\bar{z}_1,\xi,tz_3,\ldots,tz_n) , \omega = t (-z_2-\bar{z}_1) \xi+ t^2 (-z_2-\bar{z}_1)^2+ r(tz_1,t(-z_2-\bar{z}_1),\xi,t z_3,\ldots, t z_n) .
\Vert (\sigma,u)^\ell(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha+s\alpha}_{2,1}}+\Vert \sigma^h(t)\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}+\Vert u^h(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\le C(1+t)^{-s}.
p_i = G \sqrt{\Delta t}w_i , i \geq 1 , q_i = \sum_{j=1}^i p_j , x_i = \begin{cases}q_i - \left\lfloor q_i \right\rfloor , \left\lfloor q_i \right\rfloor 1 - q_i + \left\lfloor q_i \right\rfloor , \left\lfloor q_i \right\rfloor \end{cases}
{\times}\int{d^{3}x \: \mathrm{t r}\Bigl [ ( \partial_{j}A_{k}+ A_{j}A_{k}) ( \partial_{i}f \, g - \partial_{i}g \, f ) + \partial_{i}( A_{j}A_{k}) \Bigl ( \frac{1}{1 + \mu}\, f g - \frac{\mu}{1 + \mu}\, g f \Bigr ) \Bigr ]}.
(1)\epsilon_{r}=\epsilon[\begin{matrix}1&0&iQm_{y}\\ 0&1&-iQm_{x}\\ -iQm_{y}&iQm_{x}&1\end{matrix}]+[\begin{matrix}B_{1}m_{x}^{2}&B_{2}m_{x}m_{y}&0\\B_{2}m_{x}m_{y}&B_{1}m_{y}^{2}&0\\ 0&0&B_{1}m_{z}^{2}\end{matrix}]
I'_{h,m,n}(x) =\frac{1}{2 \pi n^{1/3}}\,\int_{- \frac{\pi}{m \delta}}^{\frac{\pi}{m \delta}}\,\exp\left[n^{2/3}\left(\frac{L_m(x)^3}{2 ( \Re L_m(x) - iz)^2 - iz)^2}+ ( \Re L_m(x) - iz)\right)\right] \, dz &+O\left(\exp\left(n^{2/3}[ \tfrac{3}{2}\Re L_m(x) - K ]\right)\right) .
\frac{\beta_{s}^{2}}{\beta_{r}^{2}}= \frac{( 1 + \operatorname{c o s}\theta_{s})^{2}\operatorname{s i n}^{2}\theta_{r}}{( 1 + \operatorname{c o s}\theta_{r})^{2}\operatorname{s i n}^{2}\theta_{s}}e^{2 i ( \varphi_{r}- \varphi_{s})}\; .
\mathfrak{V}(n) = \{(x_1,y_1,\hdots,x_n,y_n) \,|\,x_i\in\mathbb{R}-\{0\}\ \mbox{and}\y_i\in\mathbb{R}\ \mbox{for all}\i, &s_1\cdots s_n = 1,\ \mbox{and}\q_n(s_1,t_1,\hdots,s_n,t_n) = 0\ \}
G \otimes^{\mathbf{L}}\tilde{G}= ([\check{T_1}\rightarrow \check{T_2}] \otimes^{\mathbf{L}}\mathbb{Z}(a)[1]) \otimes^{\mathbf{L}}([\hat{T_2}\rightarrow \hat{T_1}] \otimes^{\mathbf{L}}\mathbb{Z}(d+1-a)) \rightarrow \mathbb{Z}[1] \otimes^{\mathbf{L}}\mathbb{Z}(a)[1] \otimes^{\mathbf{L}}\mathbb{Z}(d+1-a) \rightarrow \mathbb{Z}(d+1)[2].
\varphi(z) = \begin{cases}-\pi i + 3\pi i c_{0,V}z^{\frac{1}{3}}+ \mathcal{O}\left(z^\frac{2}{3}\right), , begin{align*}5pt] \pi i - 3\pi i c_{0,V}z^{\frac{1}{3}}+ \mathcal{O}\left(z^\frac{2}{3}\right), ,\end{cases}
F_{N, M}=\frac{Z_{N-M, \beta}^{\frac{N}{N-M}V;\boldsymbol{A}}}{Z_{N-M, \beta}^{V;\boldsymbol{A}}}\,,G_{N, M}=\frac{Z_{N-M, \beta}^{V;\boldsymbol{A}}}{Z_{N, \beta}^{V;\boldsymbol{A}}}\,.
\begin{gathered}\mathfrak{S}(M)^+=\mathfrak{S}(M)^+_+\sqcup\mathfrak{S}(M)^+_- \mathfrak{S}(M)^-=\mathfrak{S}(M)^-_+\sqcup\mathfrak{S}(M)^-_- \mathfrak{S}(M)_+=\mathfrak{S}(M)^+_+\sqcup\mathfrak{S}(M)^-_+ \mathfrak{S}(M)_-=\mathfrak{S}(M)^+_-\sqcup\mathfrak{S}(M)^-_- \end{gathered}
(1-\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{S^-}+r_2\delta_{\alpha_2 y}^{S^+}\}) S_{i,j}^{n+1}+(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{S^-}+r_2\delta_{\alpha_2 y}^{S^+}\}) S_{i,j}^{n}=2 S_{i,j}^*
\L \Theta = -\frac{\gamma}{m}v^2-\frac{1}{m}\sum_{k=1}^N\lambda_k z_k^2-\sum_{k>N}\lambda_k k^{-2s}z_k^2-\frac{1}{m}\sum_{k>N}\sqrt{c_k}z_k v &\qquad+\sum_{k>N}\sqrt{c_k}k^{-2s}z_k v+ \frac{\gamma}{m^2}+\frac{1}{m}\sum_{k=1}^N\lambda_k+\sum_{k>N}\lambda_k k^{-2s}.
\frac{X_{t_-}^{-\infty}}{t_-}\frac{t_-}{t_+}= \frac{X_{t_-}^{-\infty}}{t_+}\leq \frac{X_t^{-\infty}}t \leq \frac{X_{t_+}^{-\infty}}{t_-}= \frac{X_{t_+}^{-\infty}}{t_+}\frac{t_+}{t_-}.
\sum_{k = 1}^K \min\big\{S^{\mbox{\tiny \upshape (\itshape k\upshape)}}+ \sum_{a \in A}\max\{0, x_a - x^{\mbox{\tiny \upshape (\itshape k\upshape)}}_a\}; h \big \}\geq \sum_{v = 1}^V \widetilde{\beta}_v \Big(n_v \overline{S}_v + (K - n_v)h\Big) = \sum_{v = 1}^V \widetilde{\beta}_v R_v \geq \min_{v \in \{1, \ldots v\}}R_v,
FT(h)(y_0,g)= \frac{1}{2\pi}\int\displaylimits_{-\infty}^{+\infty}\int\displaylimits_{-\infty}^{+\infty}e^{-\frac{i}{2c}(x_0y_0+rg)}h(x_0,r)\ \mathrm{d}x_0\ \mathrm{d}r.
{\cal E}[ h ] ={\cal E}_{\mathrm{c l}}[ h ] + \Gamma_{2}[ h ] - \frac{1}{\pi}\sum_{\ell}( 2 \ell + 1 ) \int_{0}^{\infty}d k \: \bar{\delta}_{\ell}( k ) \frac{k}{E ( k )}+ \sum_{j}( E_{j}- M ) \ .
(\xi_1+\xi_2)I_1(c_1,c_2,\xi_1,\xi_2,\xi_3,\xi_4)=\int_0^1w^{\xi_1c_2/c_1+\xi_3-1}(1-w)^{\xi_4}\left(1-w^{c_2/c_1}\right)^{\xi_2}dw+\xi_2I_1(c_1,c_2,\xi_1,\xi_2-1,\xi_3,\xi_4).
{\cal D}_{g_{\mu \nu}( w )}\nabla_{\sigma ( x )}^{g_{\kappa \lambda}( z )}\equiv \delta ( z - w ) \left[ t_{\kappa \lambda \ , \sigma \tau}^{\ \mu \nu}\nabla^{\tau ( x )}\right] \delta ( w - x ) +{\cal O}[ l^{- 1}] ,
\mathrm{Proof.}\displaystyle \sum_{m=1}^{k-r}(-1)^{k-m-r}\binom{k}{m}r! \genfrac{\{}{\}}{0pt}{}{k-m}{r}=\displaystyle \sum_{m=r+1}^{k}(-1)^{k-m}\binom{k}{m-r}r! \genfrac{\{}{\}}{0pt}{}{k-m+r}{r}&=\displaystyle \sum_{l=r+1}^{k}(-1)^{k-l}l! \genfrac{\{}{\}}{0pt}{}{k}{l}.
\sum_{k=1}^{n}\sum_{\ell=1}^{d}\sum_{i=1}^{n}\sum_{j=1}^{d}a_{k\ell ij}^2(x)\left(\sum_{k=1}^{n}\sum_{\ell=1}^{d}a_{k\ell k\ell}(x)\right)^{\!-2}\leq \frac{1}{nd-1+\varepsilon}
(\alpha \otimes \Delta) \Delta(l) =(\Delta \otimes \alpha)\Delta(l) (\alpha \otimes \Delta)(l_1 \otimes l_2) =(\Delta \otimes \alpha)(l_1 \otimes l_2) \alpha(l_1) \otimes l_{21}\otimes l_{22}=l_{11}\otimes l_{12}\otimes \alpha(l_2)
\begin{aligned}\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\exp(\sum h(\lambda_i)))\geq \exp(\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))).\end{aligned}
\delta_{{\hat{\xi}}_{2}}L [{\hat{\xi}}_{1}] = \left\{L [{\hat{\xi}}_{2}] , L [{\hat{\xi}}_{1}] \right\}= L [ \{{\hat{\xi}}_{1},{\hat{\xi}}_{2}\}_{\mathrm{\scriptsize ~ S D}}] + K [{\hat{\xi}}_{1},{\hat{\xi}}_{2}]
\frac{1}{G}= \frac{2^{[ \frac{D + 1}{2}]}N}{( 4 \pi )^{D / 2}}\int_{1 / \Lambda^{2}}^{\infty}\frac{d s}{s^{D / 2 - [ ( D - 1 ) / 2 ]}}e^{- s \rho^{2}}\prod_{k = 1}^{[ ( D - 1 ) / 2 ]}e H_{k}\coth ( e H_{k}s ) .
A\left( Vz,0;\chi_{1},G_{2}\right) -\chi_{1}\left( c\right) \chi_{2}\left( b\right) A\left( z,0;\chi_{2},G_{1}\right) =\pi i\chi_{1}\left( c\right) \chi_{2}\left( b\right) s\left( d,c;\chi_{2},\chi_{1}\right) .
\mathcal{K}_{\mu}(\Xi_1, \Xi_2; U_1, V_1; U_2, V_2):= \int_{\Bbb{R}^4}K^{(\Xi_1), (\Xi_2)}_{w_6}( \xi_1 \eta_1 \Xi_1, \xi_2 \eta_2 \Xi_2; \mu) &\times e(\xi_1 U_1+\eta_1 V_1+\xi_2 U_2+\eta_2 V_2) \W(\xi_1) W(\eta_1)\overline{W}(\xi_2) \overline{W}(\eta_2) d\xi_1\, d\xi_2\, d\eta_1\, d\eta_2.
dg_t = dL_{g_t}(\xi_{t})dt, d\xi_t^{\vee}= b(t,g_t,\xi_t) dt + \sigma_W dW_t, dY_t = h(t,g_t,\xi_t)dt + \sigma_B dB_t, (g_0,\xi_0^{\vee}) \sim \nu_0, Y_0 = 0.
\sum\limits_{k = 1}^m x^{( 2k - 1)( m + 1)}- \sum\limits_{k = 1}^m x^{2km}= \sum_{l=1}^{\lfloor m/2 \rfloor}\{x^{a_l}(1-x^{b_l}) +x^{c_l}(x^{b_l}-1)\},
_{n^{\prime}}\langle a a | \hat{H}| a a \rangle_{n}= \Big [ \frac{I_{n}( \zeta )}{n}+ \frac{I_{K - n}( \zeta )}{K - n}\Big ] \Big ( \delta_{n}^{n^{\prime}}+ \delta_{K - n^{\prime}}^{n}\Big ) ,
A = \left\lbrace\left|\langle B_1,B_2\rangle\sum_{i=1}^n \int_{(i-1)\Delta_n}^{i\Delta_n}\Delta_n^{-\frac 12}\langle (\mathcal S(i\Delta_n-s)-I) B_1,B_2\rangle ds\right|>c\right\rbrace.
\binom{n}{t-k-3}+\binom{n}{t-k}+\binom{n}{t+k+1}+\binom{n}{t+k+4}= \binom{n}{t-k-1}+\binom{n}{t-k-2}+\binom{n}{t+k+2}+\binom{n}{t+k+3},
S ={\frac{1}{4 \pi}}\int_{0}^{2 \pi}d \sigma \int d \tau \left[ \sqrt{g}g^{\alpha \beta}G_{i j}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}+ \epsilon^{\alpha \beta}B_{i j}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}-{\frac{1}{2}}\sqrt{g}\phi{\cal R}^{( 2 )}\right] ,
x^{a'}= x^{a'}(x), y^{I'}_{w}= \sum \frac{1}{n!}y^{J_{1}}_{w_{1}}y^{J_{2}}_{w_{2}}\cdots y^{J_{n}}_{w_{n}}T_{J_{n}\cdots J_{2}J_{1}}^{\:\:\:\:\:\:\:\:\:\:\:\: \:\:\:\: I'}(x),
\Xi_{k+1}= \mathcal{A}\Xi_k + \mathcal{B}\mathcal{G}(\hat{\Psi}_k(\mathbf{u},{\tilde{\Xi}})+\hat{E}_k(\mathbf{u},{\tilde{\Xi}}) + \hat{D}_k(\omega)),{\mathbf{\tilde{y}}_k}{= \mathcal{C}\Xi_k + \mathbf{w}_k},
x_t \overset{\mathrm{Def.}}{=}x_0+\int_0^t\vartheta_s\,\mathrm{d}{s}+ 2c_H^{B,R}\int_0^t\varphi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}}+\int_0^t\psi_s\delta{R}^H_s
( H - \epsilon_{a}) J_{a}\propto \left| \begin{array}{c c c}{{( \epsilon_{c}- \epsilon_{a}) m_{c}}}&{{( \epsilon_{b}- \epsilon_{a}) m_{b}+ v_{b c}m_{c}}}&{{( \epsilon_{a}- \epsilon_{a}) m_{a}+ v_{a b}m_{b}+ v_{a c}m_{c}}}\\{{\epsilon_{c}- \epsilon_{a}}}&{{v_{b c}}}&{{v_{a c}}}\\{{0}}&{{\epsilon_{b}- \epsilon_{a}}}&{{v_{a b}}}\\ \end{array}\right| \, .
q^0_-(\tau)=-\frac{\Omega\vartheta'_1(0)}{2\pi}~\frac{e^{\Omega\tau\frac{\vartheta'_1}{\vartheta_1}(\Omega y)}\vartheta_1(\Omega(y-\tau))+e^{-\Omega\tau\frac{\vartheta'_1}{\vartheta_1}(\Omega y)}\vartheta_1(\Omega(y+\tau))}{\vartheta_1(\Omega\tau)\vartheta_1(\Omega y)},
\frac{\lambda_{n-1}\left( n-1\right)^{2/p-1}}{\lambda_{n}n^{2/p-1}}\geq\left( 1-\frac{\left( 1-\xi\right) /n}{1-\left( 1-\xi\right)/n}\right) \cdot\left( \frac{1-1/n}{1-\left( 1-\xi\right) /n}\right)^{2/p-1}& =\left( 1-\frac{1-\xi}{n-1+\xi}\right) \cdot\left( 1-\frac{\xi}{n-1+\xi}\right)^{2/p-1}.
{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}u_{\alpha}(x,t)=\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}u_{\alpha}(x,t)-(2H)^{\alpha}\frac{t^{-2H\alpha}}{\Gamma(1-\alpha)}u_{\alpha}(x,0),
\Sigma_{\psi}= - i ( \not \! \! p - M ) \int \frac{d^{D}k}{( 2 \pi )^{D}}\frac{\cos^{2}( k \wedge p )}{[ ( k + p )^{2}- M^{2}] [ k^{2}- 4 M^{2}]}\Delta_{\lambda}( k ) .
I_j(q,z):=\frac{i_j^*i_*J(\tau,z)}{e(N_{p_j/\P^4})}=z\sum_{d\geq 0}q^d\frac{\prod\limits_{k=0}^{5d}(5\alpha_j+kz)}{\prod\limits_{l=0}^4\prod\limits_{k=0 \atop (l,k)\neq (j,0)}^d(\alpha_j-\alpha_l+kz)}
(a x;q)_{m-k}(a /x;q)_{m-k}={(ax;q)_\infty\over (qx/a;q)_\infty}{(q^{-m+k+1}x/a;q)_\infty\over (q^{n-k}ax;q)_\infty}(-a/x)^{m-k}q^{(m-k)(m-k-1)/2}&={(ax;q)_\infty\over (qx/a;q)_\infty}(-a/x)^{m-k}q^{(m-k)(m-k-1)/2}\sum_{l\geq 0}{(q^{-2m+2k+1}/a^2;q)_l\over (q;q)_l}(q^{m-k}ax)^l.
a_n = \frac{2C}{(n\log n)^2}+ O\left(\frac{1}{n^2(\log n)^3}\right), b_n = \frac{1}{2}+ \frac{1}{16n^2}+ \frac{C}{(n \log n)^2}+ O\left(\frac{1}{n^2(\log n)^3}\right).
M_{3,1}=6\int_{-\infty}^{+\infty}\Phi(x)\varphi^{2}(x)dx=6\int_{-\infty}^{+\infty}(\Phi(x)-\frac{1}{2}+\frac{1}{2})\varphi^{2}(x)dx=6\int_{-\infty}^{+\infty}\frac{1}{2}\varphi^{2}(x)dx
(\begin{matrix}X&-\overline{Y}\\Y&\overline{X}\end{matrix})^{*}=(\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix})(\begin{matrix}X&-\overline{Y}\\Y&\overline{X}\end{matrix})(\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix})\Leftrightarrow X^{T}=-X,\overline{Y}^{T}=Y.
\frac{d^3 x / d \mu^3}{d x / d \mu}= \frac{3}{2}\frac{d^2 x / d \mu^2}{(d x / d \mu)^2}- \frac{1}{2}\frac{dx}{d \mu}\left ( \frac{1}{x^2}+ \frac{1}{x(1-x)}+ \frac{1}{(1-x)^2}\right ).
\mu\otimes\mu =\left(a\lambda+(1-a)\varepsilon_{\frac{1}{2}}\right)\otimes \left(a\lambda+(1-a)\varepsilon_{\frac{1}{2}}\right) &=a^2(\lambda\otimes\lambda)+a(1-a)(\lambda\otimes\varepsilon_{\frac{1}{2}})+(1-a)a(\varepsilon_{\frac{1}{2}}\otimes\lambda)+(1-a)^2(\varepsilon_{\frac{1}{2}}\otimes\varepsilon_{\frac{1}{2}}).
\Psi_0^{(1)}=Ne^{-\pi x_4^2/V}\theta_2(\tau|\bar{z})\quad ;\quad \Psi_0^{(2)}=Ne^{-\pi x_4^2/V}\theta_3(\tau|\bar{z})\quad ;\quad \tau\equiv 2i{R_4\over R_5}\ ,
\frac 12\Sigma_{\mu \nu}^{(+)}\widetilde{F}_{\mu \nu}=\mathbf{S}\widetilde{\mathbf{X}},\hspace{1.0in}\frac 12\Sigma_{\mu \nu}^{(-)}\widetilde{F}_{\mu \nu}=\mathbf{S}\widetilde{\mathbf{X}}^{*},
Z_{Kup}=\sum\limits_{(\Lambda)}\lambda\left(S(\Lambda'_{(3)})\Lambda_{(1)}\Lambda'_{(5)}S^{-1}(\Lambda_{(3)})\Lambda'_{(1)}S^{-1}(\Lambda_{(5)})\right) &\cdot\lambda\left(\Lambda_{(4)}S(\Lambda'_{(4)})S(\Lambda'_{(2)})\Lambda_{(2)}\Lambda'_{(6)}\Lambda'_{(7)}\Lambda'_{(8)}\right)
\mathbf{\Psi}=(\mathbf{u}_{\lambda_{11}})^{\dag}\mathbf{u}_{\lambda_{12}}=({\mathbf{u}^{H}_{\lambda_{11}}}\mathbf{u}_{\lambda_{11}})^{-1}{\mathbf{u}^{H}_{\lambda_{11}}}\mathbf{u}_{\lambda_{12}},
a^{3412}=a^{3142},a_{1}^{3412}=a_{1}^{3142},a_{2}^{3412}=a_{2}^{3142},a_{3}^{3412}=a_{3}^{3142},a_{4}^{3412}=a_{4}^{3142}, a_{12}^{3412}=a_{12}^{3142},a_{13}^{3412}=a_{13}^{3142},a_{14}^{3412}=a_{14}^{3142},a_{24}^{3412}=a_{24}^{3142},a_{34}^{3412}=a_{34}^{3142}, a_{124}^{3412}=a_{124}^{3142},a_{134}^{3412}=a_{134}^{3142}
R^{(\lambda)}_n \leq R(1^n,1^n_*)+R(1^n_*,2^n_*)+R(2^n_*,2^n) &= \frac{3\lambda}{2}R^{(\lambda)}_{n-1}+\frac 23(3\lambda R^{(\lambda)}_{n-1}+(3\lambda)^n)+\frac{3\lambda}{2}R^{(\lambda)}_{n-1}&= 5\lambda R^{(\lambda)}_{n-1}+ 2\cdot3^{n-1}\lambda^n.
\Vert \mathcal{K}(\varepsilon ;\vartheta)\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant C_K^{(1)}, &\Vert \varepsilon \mathbf{D}\mathcal{K}(\varepsilon ;\vartheta)\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant C_K^{(2)}\varepsilon +C_K^{(3)}.
N_{n}^{(2)}=\begin{cases}\emptyset, n=2,3, \prod_{k=0}^{\frac{n-5}{2}}(C_{2k+2}, C_{2k+1}, \ldots, C_{1}), n \equiv 1 \pmod2, n>3, \prod_{k=0}^{\frac{n-4}{2}}(C_{2k+1}, C_{2k},\ldots, C_{1}), n\equiv 0 \pmod 2, n>2, \end{cases}
\int_{S^2}J(u_j) \, dA_{S^2}= \int_{D_j}J(u_j) \, dA_{S^2}+ \int_{S^2 \setminus D_j}J(u_j) \, dA_{S^2}\geqslant (4 \pi - \tfrac14 \eta) - \frac12 \int_{S^2 \setminus D_j}|\nabla u_j|^2 \, dA_{S^2}\geqslant (4 \pi - \tfrac14 \eta) - \big((8 \pi - \eta) - (4 \pi - \tfrac14 \eta)\big) = \tfrac12 \eta > 0.
E_q ={_\mathbb{Q}}\{\varphi_n, n \in \mathbb{N}\}+{_\mathbb{Q}}\{h_m, m \in \mathbb{N}\}&= \bigcup_{n,m \in \mathbb{N}}\Big\{\sum_{i=1}^nq_i\varphi_i + \sum_{j=1}^m r_j h_j, (q_1,...,q_n,r_1,...,r_m)\in \mathbb{Q}^{n+m}\Big\}
A = (a_{ij}) : (M^i,N^i,P^i) \mapsto (a_{11}M^i+a_{12}N^{i}+a_{13}P^{i}, a_{21}M^i+a_{22}N^{i}+a_{23}P^{i},a_{31}M^i+a_{32}N^{i}+a_{33}P^{i}).
V_1(\rm x) = \frac{1}{32}\rm x^6 - \frac{\omega}{16\rm g}\rm x^4 +(\frac{\omega^2}{32\rm g^2}- \frac{3j}{4}- \frac{3}{8}) \rm x^2 -\frac{\omega}{4\rm g}(3j-\frac{1}{2})
~~\left(\frac{k-1}{2k}+\left(\frac{k-1}{2k}\right)^2+\cdots\right) \binom{n}{(k+r)/2}+ \frac{1}{2}\binom{n}{(k+r)/2}+\frac{k-1}{2k}\binom{n}{(k+r)/2}&<\frac{k-1}{k+1}\binom{n}{(k+r)/2}+ \frac{2k-1}{2k}\binom{n}{(k+r)/2}=\frac{4k^2-k-1}{2k^2+2k}\binom{n}{(k+r)/2}\le \binom{n}{k}.
\sum_{\substack{\alpha\in U^{r}\alpha\not=\beta_{0}}}\sum_{\substack{x<n\le x+H \gcd(n,M)=1}}\chi_{\alpha}(n)\ge -2^{\omega(M)}(2^{r}-1)\left(\dfrac{1}{3\log 3}\sqrt{\Gamma}\log \Gamma+\dfrac{13}{2}\sqrt{\Gamma}\right).
f\left(m_{u}^{(l)},{\sigma^{2}_{u}}^{(l)}\right) =\left( f\left(m_{v}^{(l)},\,{\sigma_{v}^{2}}^{(l)}+\sigma^2_d\right)\right)^{d_c-1}, &g\left(m_{u}^{(l)},{\sigma^{2}_{u}}^{(l)}\right)=\left( g\left(m_{v}^{(l)},\,{\sigma_{v}^{2}}^{(l)}+\sigma^2_d\right)\right)^{d_c-1}.
(\delta f)(x_0,x_1,\ldots,x_n)=\sum_{i=0}^{k}(-1)^{i}\rho(x_i)f(x_0,\ldots,\hat{x_i},\ldots,x_k) +\sum_{0\leq i<j\leq n}(-1)^{i+j}f([x_i,x_j],x_0,\ldots,\hat{x_i},\ldots,\hat{x_j},\ldots,x_k),
N =N_L+N_R+1, Y =Y_L+Y_R+N_L, X_{123}=X_{123,L}+X_{123,R}+Y_L, X_{213}=X_{213,L}+X_{213,R}+\binom{N_L}2-Y_L, X_{231}=X_{231,L}+X_{231,R}+Y_LN_R+N_LN_R, X_{312}=X_{312,L}+X_{312,R}+N_LY_R+Y_R.
\psi([p,q,r][a,b,c]) = \psi[pq_1, r_1raa_1, b_1c] &= pq_1(r_1raa_1)^{-1}b_1c &= pq_1a_1^{-1}a^{-1}r^{-1}r_1^{-1}b_1c &= p(raa_1q^{-1})^{-1}r_1^{-1}b_1c &= pq^{-1}r_1^{-1}b_1c raa_1q^{-1}= q\eqref{eq:cal1}&= pq^{-1}rab^{-1}c r_1^{-1}b = rab^{-1}\eqref{eq:cal2}&=\psi[p,q,r]\psi[a,b,c]
\mathcal{E}_{n}^{\left( \alpha+m+1\right)}\left( m+1;\lambda\right)=\frac{2m}{\left( \alpha+m\right) \lambda}\mathcal{E}_{n}^{\left(\alpha+m\right)}\left( m;\lambda\right) -\frac{2}{\left( \alpha+m\right)\lambda}\mathcal{E}_{n+1}^{\left( \alpha+m\right)}\left( m;\lambda\right).
\begin{array}{rcl}\left(- \zeta\psi'\right)' \sim{G \over 2 \zeta\left[ (4Y_0)^2 \ln^2 [ (4Y_0)^2/\zeta] \right]^{1/2}}\!, \noalign{\vspace{0.2ex}}\phi'' \sim{2G \over\zeta \left[(4Y_0)^2 \ln^2 [ (4Y_0)^2/\zeta] \right]^{3/2}}\!.\end{array}
<{S_{\mathrm{\scriptsize ~ B}}}> ={\frac{g}{2 \sqrt{2 \alpha^{\prime}}}}{\int_{- \infty}^{+ \infty}}d \tau \int{d^{2 6}}k \int{\mathcal{D}_{\tilde{g}}}( \bar{X}, Y ) \exp \left\{-{S_{0}}+ i k \cdot \left[{X_{0}}( \tau ) + \bar{X}( \tau ) \right] \right\}{L_{\mathrm{\scriptsize ~ B}}}( k , \tau ) ,
\mathcal{U}(t) = \begin{pmatrix}u(t) v(t) \end{pmatrix}= U(t) \begin{pmatrix}u_0 u_1 \end{pmatrix}+ \int_0^t U(t-s) \begin{pmatrix}0 -|u(s)|^{p-1}u(s) \end{pmatrix}\,ds
X:=ax+at\sum_{m\ge1}\binom{m+1}{1}(2b)^mb_m, &T_1:=\left(b_1+\sum_{m\ge2}\binom{m+1}{2}(2b)^{m-1}b_m\right)a^2t, &T_2:=\left(b_2+\sum_{m\ge3}\binom{m+1}{3}(2b)^{m-2}b_m\right)a^3t, &\quad\ldots
\sum_{i,j=1}^{n,\nu}\gamma_{ijk}x_{i}^{\:\left(t\right)}y_{j}^{\:\left(t\right)}\quad\Bigl(\mbox{resp.}\sum_{i,j=1}^{n,\nu}\widetilde{\gamma}_{ijr}x_{i}^{\:\left(t\right)}y_{j}^{\:\left(t\right)}\Bigr).
\alpha=\frac{b_1\left( d_N-b_N\right) \,m\,N+\left( b_N+b_1\right) \,{m}^{2}+b_N\left( b_1-d_1\right) \,m}{(b_N-d_N)(b_1-d_1)N+\left( d_N+d_1-b_N-b_1\right) \,m+\left( b_1-d_1\right)(d_N-b_N)}, \beta=\frac{\left( b_N\,d_1-b_1\,d_N\right) \,m}{(b_N-d_N)(b_1-d_1)N+\left( d_N+d_1-b_N-b_1\right) \,m+\left( b_1-d_1\right)(d_N-b_N)}.
L^{'}={\frac{1}{2}}\{{\frac{{\dot{x}}^{2}}{e}}+ e m^{2}- i ( \xi_{\mu}{\dot{\xi}}^{\mu}- \xi_{5}{\dot{\xi}}_{5}) - i \varphi ({\frac{\xi_{\mu}{\dot{x}}^{\mu}}{e}}- m \xi_{5}) \},
{\cal A}_{\mathrm{e f f}}^{\mathrm{m f}}\, = \, \frac{R \beta M^{2}}{2}\, \sqrt{\rho_{0}\rho_{1}}\, \left[ \left( \frac{\lambda_{0}\, + \, 1}{\rho_{0}}\, + \, \frac{\lambda_{1}\, + \, 1}{\rho_{1}}\right) \, - \, \left( \lambda_{0}\, + \lambda_{1}\right) \right] ,
d(ks - \ell)(G^{ks-\ell-1}F^{\ell+1})_{kv + w-t}= ((F\partial - \ell \partial F)(G^{ks-\ell}F^\ell))_{kv + w - t'}= \sum_{y \in B_d}(kv_i + w_i - t'_i - (\ell+1)y_i) F_y (G^{ks-\ell}F^\ell)_{kv + w-t'-y}.
\vert J_2(s)\vert\leq \int_{x}^{x+2s}\frac{\vert x+s-t\vert f(t)dt}{(x+s-t)^2+v_{\xi}(s)^2}\leq 2s\int_{x}^{x+2s}\frac{f(t)dt}{(x+s-t)^2+v_{\xi}(s)^2}=2s\xi,
E_{\alpha,i}\cdot (\mathfrak{A}|_{X_{\alpha}})^{n-1}\le \mathfrak{D}\cdot X_{\alpha}\cdot \mathfrak{A}^{n-1}= (-m K_{\mathfrak{X}}+m\mathfrak{h}^*\mathfrak{L}-\mathfrak{A}) \cdot X_{\alpha}\cdot \mathfrak{A}^{n-1}.
\phi: \bigoplus_{\substack{I \subseteq [n] |I| = t}}\Gamma(D(x_{I}), \Omega_{X}^{j}(\log f)) \to \bigoplus_{\substack{I \subseteq [n] |I| = t}}\Gamma(D(x_{I}), \Omega_{X}^{j}(\log f))
\begin{array}{l}{\omega_{1 \beta}= \displaystyle \frac 1{( \Lambda \Pi )^{2}}( \Lambda \Pi_{\sigma}) P^{+ \sigma \gamma}\Lambda_{n}\Gamma^{n}\tilde{\Gamma}^{m}\Pi_{m \gamma}k_{\beta},}\\{\omega_{2 \beta}= \displaystyle \frac 1{( \Lambda \Pi )^{2}}( \Lambda \Pi_{\sigma}) P^{+ \sigma \gamma}\Pi_{n \gamma}\Gamma^{n}\tilde{\Gamma}^{m}\Lambda_{m}s_{\beta}.}\\ \end{array}
x\in H^0(P,{\mathcal O}_{P}(1)\otimes p^{*}{\mathcal L}^2),\;\;\;\;y\in H^0(P,{\mathcal O}_{P}(1)\otimes p^{*}{\mathcal L}^3),\;\;\;\; z\in H^0(P,{\mathcal O}_{P}(1)).
\mathbb{C}=(\mathbb{C}_{ijklmn}):\mathbb{R}^{3\times3\times 3}\rightarrow\mathbb{R}^{3\times3\times 3}\ \ \ \ \mathbb{L}=(\mathbb{L}_{ijkl}):\mathbb{R}^{3\times3}\rightarrow\mathbb{R}^{3\times3}.
L = \frac{6 4 k^{2}}{a^{8}\xi}M = \frac{D - 1}{4 \xi}N = \frac{4}{3 a^{2}\xi}\left( 1 + \frac{D - 1}{2}- \frac{k^{2}}{a^{2}}\right) - \frac{4}{3 a^{4}}
w_{ij,k}- w\left( R^\varphi_{ij,k}- \frac{S^\varphi_k}{m-1}\delta_{ij}\right) - \left( R^\varphi_{ij}- \frac{S^\varphi}{m-1}\delta_{ij}\right) w_k - R^\varphi_{ij,k}+ \frac{S^\varphi_k}{m}\delta_{ij}= 0 \, .
E_{1}= \begin{bmatrix}1 0 0 1 \end{bmatrix},\,A_{1}= \begin{bmatrix}-1 4\pi -4\pi -4 \end{bmatrix}&E_2=\begin{bmatrix}1 0 1 0\end{bmatrix},A_2= \begin{bmatrix}-1 4\pi -\pi -1\end{bmatrix}
\sigma_{n}^{2}=\frac{(n-1)\sigma_{n-1}^{2}+(x_{n}-\overline{x}_{n-1})(x_{n}-\overline{x}_{n})}{n}=\sigma_{n-1}^{2}+\frac{(x_{n}-\overline{x}_{n-1})(x_{n}-\overline{x}_{n})-\sigma_{n-1}^{2}}{n}.
\hat{\mu}_{_{2}}=\frac{1}{n}\sum\limits_{r=1}^{(n-1)/2}(Y_{[1]2r-1}+Y_{[n]2r})+\frac{Y_{[1]n}+Y_{[n]n}}{2n}, &\hat{\mu}_{_{3}}=\frac{1}{n}\sum\limits_{r=1}^{(n-1)/2}(Y_{[1]2r-1}+Y_{[n]2r})+\frac{Y_{[\frac{n+1}{2}]n}}{n},