harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi_{U V}= \lgroup 1 -{\frac{1}{4}}y^{2}\log{\frac{Y}{y}}+ y^{4}[{\frac{1}{4 8}}\log^{3}{\frac{Y}{y}}+{\frac{1}{6 4}}\log^{2}{\frac{Y}{y}}+{\frac{1}{1 2 8}}\log{\frac{Y}{y}}+ C_{k}] + \ldots \rgroup
\dot{x}{}^{a}\rightarrow \dot{f}{}^{- 1}\dot{x}{}^{a}, \, \, \, \, \, \, \, \, \acute{x}^{a}\rightarrow \acute{\varphi}{}^{- 1}\acute{x}{}^{a}- ( \acute{\varphi}\dot{f}){}^{- 1}\acute{f}\dot{x}{}^{a}.
\int dt'G(0,t;0,t') =\int\frac{dk}{2\pi}\left(\frac{i}{-k^{2}-4}\right) \left(\frac{2ik\left(ik-2\coth 2x_{0}\right)}{(ik)^{2}-2ik\sqrt{1+\sigma_{0}}\sqrt{1+\sigma_{1}}+2(\sigma_{0}+\sigma_{1})}\right).
\left(\begin{array}{cc}1 0 0 z^{\alpha}\left(1+z^{\alpha}D^{-1}C\right)^{-1}D^{-1}\end{array}\right)=\left(\begin{array}{cc}1 0 0 z^{\alpha}D^{-1}\end{array}\right)+\sum_{k=1}^{\infty}(-1)^{k}z^{(k+1)\alpha}\left(\begin{array}{cc}0 0 0 \left(D^{-1}C\right)^{k}D^{-1}\end{array}\right).
\frac{e^{nt}(e^{t}-1)^{r-n}}{(r-n)!}= \sum_{m=0}^{\infty}\sum_{i=0}^{n}(-1)^{n-i}{n \brack i}{m+i \brace r}\frac{t^{m}}{m!}, & \sum_{i=0}^{n}{n \brace i}\frac{e^{it}(e^{t}-1)^{r-i}}{(r-i)!}=\sum_{m=0}^{\infty}{m+n \brace r}\frac{t^{m}}{m!}.
\begin{cases}b_{11}=b_{22}b_{16}=b_{24}\end{cases}\quad\mathrm{or}\frac{b_{11}}{b_{16}}+\frac{b_{16}}{b_{11}}+\frac{b_{22}}{b_{24}}=0 \mathrm{or}\frac{b_{16}}{b_{11}}+\frac{b_{24}}{b_{22}}+\frac{b_{16}b_{24}}{b_{11}b_{22}}=0 \, .
\frac 12\bigl( \left(A(x)^2 + A(x^2)\right) = \frac12 \bigl((x+x^2+x^3+x^4+x^5+x^6)^2 + (x^2+x^4+x^6+x^8+x^{10}+x^{12})\bigr) &= x^{2}+x^{3}+2 x^{4}+2 x^{5}+3 x^{6}+3 x^{7}+3 x^{8}+2 x^{9}+2 x^{10}+x^{11}+x^{12}.
\widetilde{W}= \left( \begin{array}{l l}{A - \widehat{\theta}\Phi A}&{A \theta - \widehat{\theta}( \Phi A \theta + ( A^{t})^{- 1})}\\{\Phi A}&{\Phi A \theta + ( A^{t})^{- 1}}\\ \end{array}\right) = \left( \begin{array}{l l}{M}&{N}\\{R}&{S}\\ \end{array}\right)
0 = \left[ \Gamma_{\underline{{{0 1}}}}- E \Gamma_{*}\right] ( P - Q ) \gamma_{2}\epsilon = - 2 \left[ \Gamma_{\underline{{{0 1}}}}- E \Gamma_{*}\right] \gamma_{2}Q \epsilon = - 2 \gamma_{2}Q \left[ \Gamma_{\underline{{{0 1}}}}+ E \Gamma_{*}\right] \epsilon \, .
T_{a}\Psi^{( 0 )}= \left( \begin{array}{c}{{\phi^{( 0 )}( s^{i}+ a^{i})}}\\{{\phi^{\ast ( 0 )}( s^{i}+ a^{i})}}\\{{A_{k}^{( 0 )}( s^{i}+ a^{i})}}\\ \end{array}\right) ,
\arctan\left( \frac{7 - 2\beta_0R}{2 \sqrt{\alpha_0 + 7\beta_0R - \beta_0^2R^2}}\right)\ = arccos\left( \frac{2\sqrt{7\beta(r-1) - \beta^2(r^2-1)}}{4\beta^2-28\beta+49}\right) \ = arccos\left( \frac{2\sqrt{\beta(r-1)(7 - \beta(r+1))}}{4\beta^2-28\beta+49}\right)
\delta{\tilde{e}}_{\mu}^{\ \underline{{\nu}}}= - \bar{\tilde{\epsilon}}_{+ i}\tilde{\Gamma}^{\underline{{\nu}}}\tilde{\psi}_{\mu +}^{i}- 2 \kappa^{D - 2}W \bar{\tilde{\epsilon}}_{+ i}\tilde{\Gamma}^{\underline{{{\nu \lambda}}}}\tilde{\eta}_{-}^{i}\tilde{e}_{\mu \underline{{\lambda}}}+ c . c . \ .
\frac{df(x)}{dx}=0\Rightarrow-\frac{(x-\mu)}{\sigma^{2}}e^{-\frac{1}{2}\frac{(x-\mu)^{2}}{\sigma^{2}}}-\frac{(x+\mu)}{\sigma^{2}}e^{-\frac{1}{2}\frac{(x+\mu)^{2}}{\sigma^{2}}}=0
\begin{aligned}Z_t - a^{kl}Z_{kl}+ b^k Z_k \geq \zeta^2 \Big(\frac{\theta^2}{2}|\nabla^2 v|^2 - C|\nabla v|^3 - C |\nabla v|^2\Big) - C_1' + \mu &\geq \zeta^2 \Big(\frac{\kappa^2 \theta^6}{8n}|\nabla v|^4 - C|\nabla v|^3 - C |\nabla v|^2\Big) - C_1' + \mu &\geq - C_1 - C_1' + \mu >0,\end{aligned}
Q_1^{k,m}=\sum_{i=0}^{m}{{m+1}\choose{i}}_qq^{i(2k+i-1)/2}\prod_{j=0}^{i-1}(q^{k+j}r^2-r) \prod_{j=1}^{m-i}(1-q^{2k+m-j}r^2) Q_2^{k,m}= \frac{q^{(2k+m)(m+1)/2}(-r)^{m+1}-1}{q^{2k+m}r^2-1}\prod_{i=0}^{m}(1-q^{k+i}r)
z_i f_{\alpha,T}=\left(\alpha_i+1-(d_{\beta(T^{-1}w_\alpha(i))}-d_{\beta(T^{-1}w_\alpha(i))-\alpha_i-1})-r \mathrm{ct}(T^{-1}w_\alpha(i))c_0 \right)f_{\alpha,T}\zeta_i f_{\alpha,T}=\zeta^{\beta( T^{-1}w_\alpha(i))}f_{\alpha,T}
d_{\mathcal M}^2( m)\;\;\ge\;\;\inf_{\bar \xi \in \Gamma_1\times..\times \Gamma_k}\;\; \|m-\bar m_{{\bar\xi}}\|^2_{L^2(d\nu_{\xi})}\;\;\ge\;\; d_{\mathcal M}^2( m) - c \sum_{i=1}^{k-1}e^{-\alpha\; (\Gamma_{i+1},\Gamma_i) /2},
f_1(x) = x_2x_3\frac{r^{\beta_0}\sin\left(r^{-\alpha_0}\right)}{(1+r^2)^2},~f_1(x) =-\frac{1}{2}x_1x_3\frac{r^{\beta_0}\sin\left(r^{-\alpha_0}\right)}{(1+r^2)^2}
\binom{n}{k}^4 \frac{1}{a^{2k}}\leq \mathbb{E}[T_{n,k}^2] \leq \binom{n}{k}^2 \frac{1}{a^{k}}\, \sum_{j=0}^k \binom{n-k}{j}^2 \binom{n -j}{k-j}^2 \frac{1}{a^{j}}.
h(u).\gamma_{k}(x_{+}(t)) = k\sum_{r = 1}^{p-1}\binom{p-1}{r}t^{r}h(u).e_{2r}&= k\sum_{r = 1}^{p-1}\binom{p-1}{r}(u^{2}t)^{r}e_{2r}&= \gamma_{k}(x_{+}(u^{2}t)) &= \gamma_{k}({}^{h(u)}x_{+}(t))
\begin{cases}\overline{Y}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\overline{Y}(s),\overline{Z}(t,s), \overline{Y}(s+\delta_{s}) ds - \int_t^T \overline{Z}(t,s) dW(s), \ \t\in [0,T]; \overline{Y}(t)=\overline{\psi}(t), \ \t\in [T,T+K]. \end{cases}
\beta_{\bar{\mathbf{g}}}\mathbf{g}(\partial_t) =0 \beta_{\bar{\mathbf{g}}}\mathbf{g}(\partial_a ) =\beta_{\bar g}g(\partial_a ) + \bar u^{-2}u \partial_a u -\bar u^{-1}g(\nabla_{\bar g}\bar u, \partial_a )\mbox{for $a=1,\dots, n$}.
\operatorname{l n}\mathrm{d e t}_{3}\lower 7 p t \mathrm{~ \scriptscriptstyle ~ m^2 ~ \to ~ \infty ~}\mkern - 2 5 m u = \ \frac{1}{9 0 \pi m^{6}}\int d^{2}x \, B^{4}+{\mathcal{R}}( m^{2}) ,
\iint (u^h_j\partial_t^{-h}G_h \ast u^h_i - u^h_i \partial_t^{-h}G_h \ast u^h_j) \zeta +( u^h_j\nabla G_h \ast u^h_i - u^h_i\nabla G_h \ast u^h_j)\cdot \nabla \zeta \,dx\,dt=o(1).
\begin{cases}A_{\varepsilon_k}\; \longrightarrow \; A_0, 0 \leq A_0(x) \leq F_A(0), \frac{d}{dx}A_0 \leq 0, begin{align*}5pt] B_{\varepsilon_k}\; \longrightarrow \; B_0, 0 \leq B_0(x) \leq F_B(1), \frac{d}{dx}B_0 \geq 0, . \end{cases}
\liminf_{n\to\infty}\frac{\ell(t_n) - \ell(0)}{t_n}\geq \liminf_{n\to\infty}\cos\big( \angle^k_{\gamma(t_n)}(\gamma(0),s_n) \big) = \cos \big( \limsup_{n\to\infty}\angle^k_{\gamma(t_n)}(\gamma(0),s_n) \big) .
\langle \sigma^{2}( x ) \, \sigma^{2}( y ) \, \sigma^{2}( 0 ) \rangle ={\frac{1}{\pi^{6}| x | | y | | x - y |}}\left( 4 u ( s ) ( u ( s ) + 2 ) \right)^{3 / 4}\left({\frac{\alpha ( \mu )}{( 1 - \alpha^{2}( \mu ) ) ( 1 + \alpha ( \mu ) )^{2}}}\right)^{3 / 2},
\binom{n}{k-1}=\sum_{i=2}^{k-1}{\left(\frac{j}{2}\right)}^{i-1}\binom{n}{k-i}+{\left(\frac{j}{2}\right)}^{k-2}\binom{n}{1}+\sum_{i=2}^{k-1}{\left(\frac{j}{2}\right)}^{i-2}\frac{1+{\left(j+1\right)}{\left(i-1\right)}}{k-{\left(i-1\right)}}\binom{n}{k-i}.
E_{_{a}}^{\; \mu}= \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{\frac{\beta}{\rho}\cos \phi}}&{{- \frac{1}{\rho}\frac{\alpha}{\sin \alpha \theta}\sin \phi}}\\{{0}}&{{\frac{\beta}{\rho}\sin \phi}}&{{\frac{1}{\rho}\frac{\alpha}{\sin \alpha \theta}\cos \phi}}\\ \end{array}\right) .
v_{1}(t,r) + v_{2}(t,r) = r \left(\int_{t}^{\infty}\frac{\lambda''(x)}{1+x-t}dx -\frac{b}{t^{2}\log^{b}(t)}\right) +(t,r) + E_{v_{2}}(t,r), 0 \leq r \leq \frac{t}{2}
\frac{\partial}{\partial s}\bar \psi =\log \frac{(\kappa_\delta+\sqrt{-1}\partial\bar\partial\bar\psi)^n}{\kappa_\delta^n}+\mu\bar \psi - \bar h_{\kappa_\delta}, & \bar \psi_{0,\delta}=\bar\psi_{\delta}(0,\cdot)= \varphi_{t,\delta}+\Psi_{\delta}-\Phi_{\delta}^\beta.
z_{a b}^{V}= \left( \begin{array}{c c c c}{{z_{W}}}&{{0}}&{{0}}\\{{0}}&{{z_{W}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{z_{Z}\cos \theta_{Z}}}&{{- z_{A}\sin \theta_{A}}}\\{{0}}&{{0}}&{{z_{Z}\sin \theta_{Z}}}&{{z_{A}\cos \theta_{A}}}\\ \end{array}\right)
\overline I_{\gamma}[U,\hat h]=\frac{d_{\gamma}^{*}\int_{X}\left({\rho}^a|\nabla U|_{\bar{g}}^2+E(\rho)U^2\right)\,dv_{\bar{g}}}{(\int_{M}U^{2^{*}}\,dv_{\hat{h}})^{2/{2^{*}}}}.
I = \int_{\mathbb{R}}\lvert h(t+\mathrm{i}y)\rvert^q\,\mathrm{d}t = \int_{\{t\colon \lvert t-x_0\rvert>\delta_0\}}\lvert h(t+\mathrm{i}y)\rvert^q\,\mathrm{d}t + \int_{\{t\colon \lvert t-x_0\rvert\leqslant\delta_0\}}\lvert h(t+\mathrm{i}y)\rvert^q\,\mathrm{d}t = I_1+ I_2.
c^*(x) =((f_1(x)+2f_2(x)+2uf_{14}(x))a(x))^* =(f_1^*(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x))a^*(x) = (f_1(x)+2f_2(x)+2uf_{14}(x))a^*(x) = (f_1(x)+2f_2(x)+2uf_{14}(x)).c.a^*(x),
\begin{aligned}\iint g_s \partial_y g^\epsilon \epsilon v^\epsilon \leq \epsilon \Vert g_s \Vert_{L^\infty}\Vert g^\epsilon_y \Vert_{L^2}\Vert v^\epsilon\Vert_{L^2}&\leq \epsilon L\Vert g_s \Vert_{L^\infty}\Vert g^\epsilon_y \Vert_{L^2}\Vert v^\epsilon_x\Vert_{L^2},\end{aligned}
0\leq i\leq m, i\leq j\leq i+1 : d_is_i( c_{a_1}\otimes \dots \otimes c_{a_{m}}) = c_{a_1}\otimes \dots \otimes c_{a_{m}}= d_{i+1}s_i( c_{a_1}\otimes \dots \otimes c_{a_{m}}) .
\sum_{i=0}^{m-1}\{\sum_{i'=0}^{l_1 (\sigma_B^i(x))-1}f(\sigma_B^{i'}(h(\sigma_A^i(x))))-\sum_{j'=0}^{k_1 (\sigma_A^i(x))-1}f(\sigma_B^{j'}(h(\sigma_A^{i}(x)))) \}= \sum_{i'=0}^{l_1^m (x)-1}f(\sigma_B^{i'}(h(x)))-\sum_{j'=0}^{k_1^m (x)-1}f(\sigma_B^{j'}(h(\sigma_A^{m}(x)))).
D^\phi_+ \eta = \nabla^g \eta + \frac{1}{2}\sum_{i < j}g((\chi')^{-+-}e_i,e_j) e^j e^i \cdot \eta, = \nabla^g \eta - \frac{1}{2}\sum_{i < j}H(e_i,e_j,\cdot) e^j e^i \cdot \eta + \sum_{i < j}c(F(e_i,e_j),\cdot) e^j e^i \cdot \eta = \nabla^- \eta - c(F \cdot \eta,\cdot),
V_{p\Delta}=\int d^{4}an_{+}^{-\Delta}(x+a,y)e^{ipa}=y^{\Delta}e^{ipx}\int d^{4}ae^{ipa}\frac{1}{(a^{2}+y^{2})^{\Delta}}=N(\Delta)p^{\Delta -2}y^{2}K_{\Delta -2}(py)e^{ipx}
\begin{array}{l}h_{k,1}^{m-n}=I^{(2)}[-2\pi\frac{(R_n^{lb})^2}{R_{mk}}\cos(\theta_{mk})] h_{k,2}^{m-n}\!\!=\!\!I^{(1)}[\!-\!2\frac{(R_n^{lb})^2}{R_{mk}}\cos(\theta_{mk})\!\arccos(\frac{R_n^{lb}}{R_{mk}}\cos(\theta_{mk}))] h_{k,3}^{m-n}=I^{(1)}[2\sqrt{(R_n^{lb})^2-(\frac{(R_n^{lb})^2}{R_{mk}}\cos(\theta_{mk}))^2}]\end{array}
\left[\begin{array}{c}(p_n)^r(q_n)^0 (p_n)^{r-1}(q_n)^1 \vdots (p_n)^1 (q_n)^{r-1}(p_n)^0(q_n)^r\end{array}\right]=E(a_1;r)\left[\begin{array}{c}(\hat{p}_{n-1})^r(\hat{q}_{n-1})^0 (\hat{p}_{n-1})^{r-1}(\hat{q}_{n-1})^1 \vdots (\hat{p}_{n-1})^1(\hat{q}_{n-1})^{r-1}(\hat{p}_n)^0(\hat{q}_{n-1})^r\end{array}\right]
\partial_1 Q_k^n = (n-k) \left( P_k^{n-1}+k a_{k-1}(w_2) P_{k-1}^{n-2}+ k b_{k-1}(w_2) P_{k-2}^{n-3}\right), \partial_2 Q_k^n = k \left(P_{k-1}^{n-1}+ (n-k) a_{n-k-1}(w_1) P_{k-1}^{n-2}+(n-k) b_{n-k-1}(w_1) P_{k-1}^{n-3}\right).
\Gamma_{\underline{\alpha}\underline{\beta}}^{i}= \left( \begin{array}{c c}{0}&{\delta_{\beta}^{\alpha}( \gamma^{i})_{q}^{r}}\\{- \delta_{\alpha}^{\beta}( \gamma^{i})_{r}^{q}}&{0}\\ \end{array}\right) , \quad i = 1 , . . . , 5 ,
L_{1}(x) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big)\sum_{n=0}^{\infty}\alpha_{n}(x(u,0)) z^{n},
\Delta^{I}n_{1}=\Delta^{I}\Big(-\frac{au}{\sqrt{\Phi}}\Big)=\lambda_{11}\Big(-\frac{au}{\sqrt{\Phi}}\Big)+\lambda_{12}\Big(-\frac{bv}{\sqrt{\Phi}}\Big)+\lambda_{13}\Big(\frac{\sqrt{\omega}}{\sqrt{\Phi}}\Big),
\nabla f(x^*)+\sum\limits_{i=1}^m\lambda_i^*\nabla g_i(x^*)+\sum\limits_{i=1}^p\mu_i^*\nabla h_i(x^*) + \sum\limits_{i=1}^n \gamma_i^*e_i = 0, &g_i(x^*)\leq0,\ \lambda_i^*\geq0,\ \lambda_i^*g_i(x^*)=0,\quad\forall i=1,\dots,m, &h_i(x^*)=0,\quad\forall i=1,\dots,p, &\gamma_i^*=0,\quad\forall i\in I_{\pm0}(x^*,y^*).
\mathsf{M}^n= \{\mathsf{m}^n_1,\cdots,\mathsf{m}^n_n\}= \left\{g\bigl(\xi_0\bigr),g\bigl(\xi_1^{\mathsf{M}^n}\bigr),\cdots,g\bigl(\xi_{n-1}^{\mathsf{M}^n}\bigr)\right\}\in\Theta_n,
\eta_b = (-xX_1w_{1b}-X_2w_{2b}-X_3w_{3b}) dx + (\partial_xw_{1b}+X_3w_{2b}-X_2w_{3b}) x^{-1}\theta^1 + (-X_3w_{1b}+\partial_xw_{2b}+xX_1w_{3b}) \theta^2 + (X_2w_{1b}-xX_1w_{2b}+\partial_xw_{3b}) \theta^3.
\langle H\rangle=8E_{1}+\langle V_{ee}\rangle=8E_{1}+(\frac{e^{2}}{4\pi\epsilon_{0}})(\frac{8}{\pi a^{3}})^{2}\int\frac{e^{-4(r_{1}+r_{2})/a}}{|r_{1}-r_{2}|}d^{3}r_{1}d^{3}r_{2}
\frac{d N ( \omega )}{d \sigma}= \frac{1}{2 \pi}\frac{e^{2}}{\hbar c^{3}}( 2 a_{0}\omega_{0})^{3}\langle x^{2}\rangle \int_{- \infty}^{+ \infty}d s \, e^{- i s \omega / a_{0}\omega_{0}}\mathrm{c s c h}^{4}( s - i \epsilon ) ~ ,
\tilde\varphi\big(\tilde x\circ a(x)\big)=(a_{n-1}+a_0\tilde x+\cdots+a_{n-2}\tilde x^{n-1})\varphi &=\big(\tilde x a(\tilde x)-a_{n-1}(\tilde x^n-1)\big)\varphi=\tilde x a(\tilde x)\varphi=\tilde x \tilde\varphi\big(a(x)\big).
\sum_{n \neq 0}n \left[ N_{n}+ \widetilde{N}_{n}+ \sum_{I = 3}^{8}N_{n}^{( I )}+ \sum_{b = 1}^{8}F_{n}^{( b )}\right] = \frac{\sqrt{f}}{l_{s}}m R_{T}( \tilde{a}_{0}^{\dagger}+ \tilde{a}_{0}) ~ .
D_{\mu}f = \partial_{\mu}f -{\frac{i g}{2}}d^{B C C}( A_{\mu}^{B}\star f - f \star A_{\mu}^{B}) = \partial_{\mu}f -{\frac{i g}{2}}d^{B C C}[ A_{\mu}^{B}, f ]_{\mathrm{M}}
V^{\tfrac{1}{m_1}}(x(\varepsilon))\le V^{\tfrac{1}{m_1}}(x^0){+}\frac{1}{m_1}V^{\tfrac{1}{m_1}{-}1}(x^0)\sum_{i=1}^n\frac{\partial V(x^0)}{\partial x_i}y_i+\frac{\bar L}{2}.
\bar{V}^{I}= \sum_{i=1}^{n^a}\bar{U}^{I}_i, \bar{U}^{I}= X^{(2)}_{n:n}\bar{V}^{II}= \sum_{i=1}^{n^{b-a}}\bar{U}^{II}_i, \bar{U}^{II}= (X^{(1)}_{1:n^{2a}})_{n^{1\!-\!a}:n^{1\!-\!a}}\bar{V}^{III'}= \sum_{i=1}^{n^{b-a}}\bar{U}^{III'}_i, \bar{U}^{III'}= X^{(2)}_{n:n}
v_\infty(\partial_q\vartheta[p,q])=\begin{cases}+\infty p=0, q \in \{0, \frac{1}{2}\}, +\infty p=\frac{1}{2}, q=0, \frac{1}{2}p=0, q \not\in \{0, \frac{1}{2}\}, \frac{\langle p\rangle^2}{2}\end{cases}
x_{j_1}-x_{k_1}=(x_{j_1}-x_{j_0})+(x_{j_0}-x_{k_0})+(x_{k_0}-x_{k_1}) &= (M_{ij_0}-M_{ij_1})+(M_{1k_0}-M_{ij_0}+y_i-y_1)+(M_{1k_1}-M_{1k_0}) &=M_{1k_1}-M_{ij_1}+y_i-y_1,
\sqrt{\lambda_k}\langle \hat u_j,u_k\rangle =\frac{\sqrt{\lambda_k}}{\hat \lambda_j-\lambda_k}\langle \hat u_j,Eu_k\rangle &=\frac{\lambda_k}{\hat \lambda_j-\lambda_k}\big(\bar\eta_{kj}\sqrt{\lambda_j}\langle\hat u_j,u_j\rangle+\sum_{l\neq j}\bar\eta_{kl}\sqrt{\lambda_l}\langle\hat u_j,u_l\rangle\big)
{\tilde{G}}[{\tilde{x}},{\tilde{x}^{\prime}}] = \sum_{i}\left[ u_{i}({\tilde{x}}) u_{i}^{*}({\tilde{x}^{\prime}}) \right] \; \sum_{j}\left[ v_{j}^{*}({\tilde{x}}) \, v_{j}({\tilde{x}^{\prime}}) \right] .
U(\phi_1,\phi_2)=\frac{1}{2}\left(\frac{\partial W}{\partial\phi_1}\right)^2+\frac{1}{2}\left(\frac{\partial W}{\partial\phi_2}\right)^2\qquad ; \qquad W(\phi) = 4 \sqrt{2}\left(\frac{1}{3}\phi_1^3 -\frac{1}{4}\phi_1 + \frac{\sigma}{2}\phi_1\phi_2^2 \right)
\langle I^{\prime}(u_{n}),\xi_{j,\delta}u_{n}\rangle= \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\Phi(u_{n}(x)-u_{n}(y))(\xi_{j,\delta}u_{n}(x)-\xi_{j,\delta}u_{n}(y))K(x,y)dx dy &-\int_{\Omega}|u_{n}|^{p_{s}^{\ast}-2}u_{n}\xi_{j,\delta}dx-\lambda\int_{\Omega}f(x,u_{n})\xi_{j,\delta}u_{n}dx
\overline{G}_M(u,\mathcal{O})\,:=\,\left\lbrace x\in \mathcal{O}\,|\,\exists \,P_M\;\mbox{s.t.}\;P_M(x)=u(x)\;\;\mbox{and}\;\;P_M(y)\geq u(y)\;\forall y\in \mathcal{O}\right\rbrace,
\Phi_2(a, b; c; x, y) = \sum_{m=0}^\infty \frac{(a)_m}{(c)_m}\,{}_2F_1\left[ \begin{array}{c}-m,\,b 1-a-m \end{array};\,\frac{y}{x}\right] \, \frac{x^m}{m!}.
\begin{cases}\mu_{0,0}= \beta_0,\ \mu_{1,0}= \alpha_0 \mu_{j-1,j}= \gamma_j,\ \mu_{j,j}= \beta_j,\ \mu_{j+1,j}= \alpha_j \mu_{i,j}= 0,\ |i-j|>1 \end{cases},\j=1,2,\ldots
\langle \varphi^{2}\rangle_{M}^{\mathrm{t h}}= \frac{1}{2 \pi^{2}}\int_{0}^{\infty}\frac{\left( \omega^{2}+ 2 m \omega \right)^{1 / 2}d \omega}{e^{( m + \omega ) / T}- 1}, \quad T = \frac{1}{2 \pi \xi}
\xi (x_{1},x_{2},x_{3},x_{4}) = J(x_{1},x_{2},x_{3}x_{4}) \zeta (x_{1},x_{2},x_{3},x_{4}) = J(x_{1},x_{2},x_{3})x_{4}\varsigma(x_{1},x_{2},x_{3},x_{4},x_{5}) = J(x_{1}x_{2},x_{3}x_{4},x_{5})
\Gamma ( \mu^{2}) = \Bigl ( 1 - \frac{4 m^{2}}{\mu^{2}}\Bigr )^{1 / 2}\Bigl ( 1 + \frac{2 m^{2}}{\mu^{2}}\Bigr ) + \mathrm{~ m u l t i - l o o p ~ c o n t r i b u t i o n s}
\frac{1}{\sqrt{g}e^{- 2 \phi}}\frac{\partial \Gamma^{( 2 )}}{\partial \phi}= 8{\tilde{B}}^{\phi ( 2 )}+ 2 B_{\rho \sigma}^{b ( 1 )}B^{b ( 1 ) \rho \sigma}+ 2 B_{\rho \sigma}^{g ( 1 )}B^{g ( 1 ) \rho \sigma}- 2 ({\tilde{B}}^{\phi ( 1 )})^{2},
< \int d^{4}z ~ \left[ \Theta ( z ) - \left( 1 -{\frac{3 N_{c}}{N_{f}}}- \gamma ( g ) \right) L_{m a t t e r}( z ) \right] > = u_{0}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{4}B ( k )^{\mu \nu}{B}_{\mu \nu}( - k ) .
s\geq \sum_{j=1}^\infty \|\mu(S^+_{k,j})\|+\sum_{j=1}^\infty \|\mu(S^-_{k,j})\|\geq q \sum_{j=1}^\infty\pi(S^+_{k,j})+\pi(S^-_{k,j})=q\pi(S_k)=qd>s,
\sum_{|j| \leq K}\frac{1}{|a-j | ( K - |j| +1 )^{1/2}}\leq \sum_{|j| \leq K, K - |a| +1 \geq 2 (K - |j| +1 )}\frac{1}{|a-j | ( K - |j| +1 )^{1/2}}&+\sum_{|j| \leq K, K - |a| +1 \leq 2 (K - |j| +1 ) , j \neq a}\frac{1}{|a-j | ( K - |j| +1 )^{1/2}}=:A_1 + A_2.
e_i^{(k)}= \begin{cases}\dfrac{e_i^k}{[k]_i!}\ \i \in I^{\mathrm{re}}, e_i^k \ \i \in I^{\mathrm{im}},\end{cases}f_i^{(k)}= \begin{cases}\dfrac{f_i^k}{[k]_i!}\ \i \in I^{\mathrm{re}}, f_i^k \ \i \in I^{\mathrm{im}}.\end{cases}
K_{0}\left[ 2\frac{f^{3}}{p\dot{p}}r\dot{r}-4\frac{fp}{\dot{p}}g+2\frac{p^{3}}{f\dot{p}}\left( \frac{g^{2}}{r}\right) \left\{\frac{d}{dg}\left( \frac{g^{2}}{r}\right) \right\}\right] =\frac{d}{dg}\left( \frac{g\dot{r}}{r}\right) .
\int_{H}^{t_0}\frac{x^{-\nu_1(t)}}{t}N(\sigma, t)\le \frac{x^{-\nu_1(t_0)}}{t_0}\int_H^{t_0}N(\sigma, t) = \frac{x^{-\nu_1(t_0)}}{t_0}N(\sigma, t_0)
\sum_{k=N}^M a_k \bigg(\sum_{i=k}^M a_i \bigg)^s b_k = \sum_{k=N+1}^M (b_k - b_{k-1}) \sum_{i= k}^M a_i \big(\sum_{j=i}^M a_j\big)^s + \bigg(\sum_{k=N}^M a_k \big(\sum_{i=k}^M a_i\big)^s\bigg) b_N & \approx \sum_{k=N+1}^M (b_k - b_{k-1}) \bigg(\sum_{i= k}^M a_i\bigg)^{s+1}+ \bigg(\sum_{k=N}^M a_k\bigg)^{s+1}b_N,
( \check{R}( u , q ) \otimes I ) ( I \otimes \check{R}( u + v , q ) ) ( \check{R}( v , q ) \otimes I ) = ( I \otimes \check{R}( v , q ) ) ( \check{R}( u + v , q ) \otimes I ) ( I \otimes \check{R}( u , q ) ) \, .
\mathrm{itd}(x_\alpha)\bullet \mathcal{C}_T(Y_\alpha \bullet z) =\mathrm{itd}(x_\alpha)\bullet \mathcal{C}_T\big( \sum_v \Delta_{v(\alpha)}(p_v) f_v\big) =\sum_v \tfrac{\gamma(x_{v(\alpha)})}{x_{v(\alpha)}}\mathcal{C}(\Delta_{v(\alpha)}(p_v))f_v =\sum_v \tfrac{\mathcal{C}(p_{s_{v(\alpha)}v})-\mathcal{C}(p_v)}{x_{v(\alpha)}}f_v.
{\vert O_{2}\rangle}_{\nu \bar{\nu}}= \epsilon^{l_{1}\cdots l_{k - 2}}\, \epsilon^{t_{1}\cdots t_{k - 2}}\, ( J_{A_{1}B_{1}}\, \nu_{l_{1}}^{A_{1}}\, \nu_{t_{1}}^{B_{1}}) \cdots ( J_{A_{k - 2}B_{k - 2}}\, \nu_{l_{k - 2}}^{A_{k - 2}}\, \nu_{t_{k - 2}}^{B_{k - 2}}) \, \vert \Phi \rangle
\sum_{{k=1, k\neq j}}^n\dfrac{1}{z_{\nu,k}-z_{\nu,j}}=\lim_{z\to z_{\nu,j}}\left(\dfrac{K_{\nu+1}(z)}{K_\nu(z)}-1-\dfrac{2\nu}{z}-\dfrac{1}{z_{\nu,j}-z}\right) =\dfrac{1}{z_{\nu,j}}\lim_{z\to z_{\nu,j}}\dfrac{z(z_{\nu,j}-z)K_{\nu+1}(z)-(z+2\nu)(z_{\nu,j}-z)K_\nu(z)-zK_\nu(z)}{(z_{\nu,j}-z)\,K_\nu(z)}.
\frac{\delta}{\delta A_{\mu}(x)}\log{\cal{Z}}_{reg}[A]=e\frac{d}{ds}\left\{s\, \mbox{Tr}\left[\gamma_{\mu}K_{-s-1}(x,x;D[A]) \right]\right\}\left|\rule[-0.4cm]{0cm}{.4cm}_{s=0}\right. .
\Big| \mu(\mathcal{C}) - \mu(\widehat{\mathcal{C}}_{N}) \Big| = \Big| \mu(\mathcal{C}) - \mu(\widehat{\mathcal{C}}_{N}) \Big| I \{N > 0 \}+ \Big| \mu(\mathcal{C}) - \kappa_{0}\Big| I \{N=0 \},
W' =w_2,w_3,\ldots, w_n \mbox{or}W'=\emptyset \mbox{if}|W|=1. W'' =w_1,w_2,\ldots, w_{n-1}\mbox{or}W''=\emptyset\mbox{if}|W|=1. W^* =w_n,w_{n-1},\ldots,w_1.
f ( x^{2}) ={\frac{8 i \left( m^{2}\right)^{4}}{\pi^{3}}}\, \int_{0}^{\infty}d \kappa_{1}\, \int_{0}^{\infty}d \kappa_{2}\, \kappa_{1}\kappa_{2}\, e^{- i ( \kappa_{1}+ \kappa_{2}) ( m^{2}x^{2}- i \delta )}e^{- i ( 1 - i \epsilon ) ( \kappa_{1}+ \kappa_{2}) / ( 4 \kappa_{1}\kappa_{2})}.
\frac{\hat\sigma^2_{n,j}(\theta)}{\sigma^2_{P,j}(\theta)}=n^{-1}\sum_{i=1}^n\left(\frac{m(X_i,\theta)}{\sigma_{P,j}(\theta)}\right)^2-\left(n^{-1}\sum_{i=1}^n\frac{m(X_i,\theta)}{\sigma_{P,j}(\theta)}\right)^2.
\times \exp \left\{- \sum_{- \infty}^{+ \infty}[{\bar{\chi}}_{n}^{( 1 )}( \not \! d + \omega_{n}+ e \varphi ) \chi_{n}^{( 1 )}+{\bar{\chi}}_{n}^{( 2 )}( \not \! d - \omega_{n}- e \varphi ) \chi_{n}^{( 2 )}] \right\}.
\frac{q}{f(q)}\sum_{n \ge 1 \atop (n, P_{\Omega})=1}\frac{f(n)}{n}~+~ \delta_{\Omega}\log q~-~ \frac{r}{g(r)}\sum_{m \ge 1 \atop (m, P_{\Omega})=1}\frac{g(m)}{m}~-~ \delta_{\Omega}\log r ~=~ 0.
Bi(z)\sim\frac{1}{\sqrt{\pi}z^{1/4}}exp(\frac{2}{3}z^{3/2})[\sum_{n=0}^{\infty}\frac{\Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})(\frac{3}{4})^{n}}{2\pi n!z^{3n/2}}].
C_{1,0,2}(h)=(-1)^{l+1}\underset{t^{l-1}}{Coef}\cdot\underset{x=0}{Res}\Bigg\{\!\frac{1}{4}\frac{1}{(1-te^{-x})^{2}}\left( \frac{1+e^{-x}}{1-e^{-x}}\right)e^{2lx}\frac{(1-te^{-2x})^{2}}{(1+te^{-2x})^{4}}dx\! \Bigg\},
C_n=\frac{e^{\tau(\iota^{-1}(n) )}}{z^{d_n}}\times &\left(\phi_n(i_A^*P_1, \dots, i_A^*P_{r+2s}, z)+\phi_{n-1}(i_A^*P_1-zP_1(i_A)_*\iota^{-1}(n-1), \dots, z)+\cdots+\right. &\left.\phi_1(i_A^*P_1-zP_1(i_A)_*\iota^{-1}(1), \dots, z)\right).
\left(\int_{\mathbb{R}^n}u_1^{1/2}u_2^{1/2}\right)^2-\int_{\mathbb{R}^n}u_1\int_{\mathbb{R}^n}u_2 =\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\big((u_1\otimes u_2)^{1/2}(u_2\otimes u_1)^{1/2}- u_1\otimes u_2\big)
\sum_{k=1}^{p-1}\frac{(-\phi^3_+)^k}{k^3}\binom{2k}{k}+\sum_{k=1}^{p-1}\frac{(-\phi^3_-)^k}{k^3}\binom{2k}{k}-\sum_{k=1}^{p-1}\frac{(-1)^k}{k^3}\binom{2k}{k}.
\begin{pmatrix}0 m_{12}\big(b\mu_{21}^{-1}-\big(b\mu_{21}^{-1}\big)^\vee\big)\mu_{21}-m_{12}\mu_{21}^\vee\big(b\mu_{21}^{-1}-\big(b\mu_{21}^{-1}\big)^\vee\big) 0 \end{pmatrix}\!,
\begin{aligned}\frac{\sum_{\boldsymbol\mu}S_{\boldsymbol\nu,\boldsymbol\mu}}{S_{N,M}}=O\left(\binom{N}{\boldsymbol\nu}u^{-k_1\nu_1}(1-\rho+u\rho^2)^M\right) &=O\left(N^t\binom{N-t}{\nu_1}u^{-k_1\nu_1}(1-\rho+u\rho^2)^M\right).\end{aligned}
A_t(g)(X,z) = t\frac{g(z+t_0(X))-g(t_0(X))}{|z|}+(1-t)\frac{\nabla g(t_0(X))z}{|z|}, B(g)(X,z) = \frac{g(z+t_0(X))-g(t_0(X))-\nabla g(t_0(X))z}{|z|}.
{1\over a+bz}\left( \begin{array}{l}1 z'{z'}^2 \vdots \end{array}\right) ={1\over a+bz}\left( \begin{array}{l}1 \left({c + dz\over a + bz}\right) \left({c + dz\over a + bz}\right)^2 \vdots \end{array}\right).
\mathcal{E}_1= \bigcup_{\substack{(m_0,\ldots,m_N) \in \mathbb{Z}^{N+1}\setminus\{\mathbf{0}\}\forall n,~ |m_n| \leq \Delta}}\left\{\alpha \in \mathcal{A}~\middle|~ \prod_{n=0}^{N}(n+\alpha)^{m_n}=1 \right\}.
\mathrm{E}_{\theta}\tilde{\pi}_I( \|\vartheta-\theta\|^2 \ge M^2 R^2(I,\theta)| X)\le \frac{\mathrm{E}_{\theta}\|X(I)-\theta\|^2+\frac{K_n(I)\sigma^2|I|}{K_n(I)+1}}{M^2 R^2(I,\theta)}\le \frac{2}{M^2}.
\partial_{\boldsymbol e_i}\eta^{t,\bar t}= \left( \frac{\partial_N \partial_{\boldsymbol e_i}(u^{t,\bar t}-h^{t,\bar t})}{(N\cdot \nu^{t,\bar t})^2 \,\Delta h^{t,\bar t}}\right) (z, \eta^{t,\bar t}(x)).
\mu \simeq \frac{1}{\sqrt{2 n_{\mathrm{e f f}}}}\exp \biggl [ \pm i \int^{\eta}n_{\mathrm{e f f}}( n , \tau ) \mathrm{d}\tau \biggr ] \rightarrow_{\eta / \eta_{0}\rightarrow \pm \infty}\frac{1}{\sqrt{2 n}}e^{\pm i n \eta}.
H^2(g,h)=\frac{\lambda_k^k}{\lambda_1\cdots\lambda_k}=\frac{\lambda_k^{k+2n_2+3n_3+\cdots +mn_m}}{\lambda_1\cdots\lambda_k \lambda_k^{2n_2+3n_3+\cdots + mn_m}}\leq \frac{\lambda_k^Q}{\lambda_1\cdots\lambda_k\mu_{k+1}\mu_{k+2}\cdots\mu_n}.
\begin{array}{ccl}C_{I,01}-C_{I,10}= C_{I,\bar{1}}A_{11}-kC_{I,}A_{11,\bar{1}}, C_{I,0\bar{1}}-C_{I,\bar{1}0}= C_{I,1}A_{\bar{1}\bar{1}}+kC_{I,}A_{\bar{1}\bar{1},1}, C_{I,1\bar{1}}-C_{I,\bar{1}1}= iC_{I,0}+kWC_{I},\end{array}
M_{2}(\mathcal{NC}(G)) = \dfrac{q(q-1)}{2}(q^{14}-3q^{13}-4q^{12}+19q^{11}-47q^{9}+28q^{8}+43q^{7}-50q^{6}+11q^{5}+4q^{4}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-12q^{3}+19q^{2}-11q+2).
f_{0}( a , t ) = 2 \sum_{\alpha \in \mathbf{C}}\frac{\sinh ( b a_{\alpha}t ) \sinh ( ( b a_{\alpha}- 2 b Q_{\alpha}) t )}{\sinh 2 t \sinh ( \alpha^{2}b^{2}t ) \sinh ( ( 1 + b^{2}) 2 h t )},
(c_{i+j})_{0\leq i,j\leq n}=\begin{pmatrix}c_{0}c_{1}c_{2}\cdots c_{n}c_{1}c_{2}c_{3}\cdots c_{n+1}c_{1}c_{2}c_{3}\cdots c_{n+2}\vdots \vdots \vdots \ddots \vdots c_{n}c_{n+1}c_{n+2}\cdots c_{2n}\end{pmatrix}.
\begin{array}{c c c c c}{}&{S O ( 2 n )}&{\subset}&{S O ( 2 n + 1 )}&{R_{M}={\bf ( 2 n + 1 )}}\\{}&{S U ( 3 )}&{\subset}&{G_{2}}&{R_{M}={\bf 7}}\\{\tilde{R}_{M}={\bf 9}}&{S O ( 9 )}&{\subset}&{F_{4}}&{R_{M}={\bf 2 6}}\\ \end{array}\quad .
4 \beta \int \rho^{-Q+2}Zv \; \Delta_{H}v =2 \beta \int \operatorname{div}(\rho^{-Q+2}Z) |\nabla_{H}v|^2 -4 \beta \sum_{i=1}^{m}\int X_i v [X_i, \rho^{-Q+2}Z] v.
\widehat{g}(x)= C \int_{|\mathbf v|\lesssim \lambda^{-1}}\int_{I_*}e^{i \lambda x \cdot (\gamma_{\ast}(s) + (0,\mathbf{v}))}\lambda^{d}\widetilde g(\lambda(\gamma_{\ast}(s) + (0,\mathbf{v}))) ds d\mathbf{v}
H_{2}= p_{x}^{2}/ 2 \gamma + \gamma ( \omega_{1}^{2}+ \omega_{2}^{2}) x^{2}/ 2 - \gamma \omega_{1}^{2}\omega_{2}^{2}q^{2}/ 2 + p_{q}x + \gamma \omega_{1}^{2}\omega_{2}^{2}q p_{\lambda}.
u_v=i_X^{\otimes d(v)_i}\big(\Omega_{d(v)_i}\big(t_v^\Lambda\big)\big)=i_X^{\otimes 0}\big(\Omega_0\big(t_v^\Lambda\big)\big)=i_{\mathcal{T}C^*(\Lambda^i)}\big(\phi^{-1}\big(t_v^\Lambda\big)\big)=i_{\mathcal{T}C^*(\Lambda^i)}\big(t_v^{\Lambda^i}\big).