harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-\mathrm{i}\frac{\Delta t\Delta x}{\alpha}\sum_m\left\{\mathcal{F}(\mathbf{z}_n^*,\mathbf{z}_{n+1}^*)_m\mathcal{F}(\mathbf{z}_n,\mathbf{z}_{n+1})_m-\mathcal{F}(\mathbf{z}_n,\mathbf{z}_{n+1})_m \mathcal{F}(\mathbf{z}^*_n,\mathbf{z}^*_{n+1})_m\right\}=0
\langle 0 | e^{- J \int_{V}\phi ( \vec{x}) d^{d}x}| 0 \rangle_{d}= \langle e^{- J \sqrt{\alpha_{\phi \phi}^{d}}\int_{\partial V}\phi ( \vec{x}) d^{d - 1}x}\rangle_{d - 1}^{\beta \to 0}.
F ( x^{+}, x^{-}, 0 ) = \frac{1}{2 \pi}\sum_{K , N_{\perp}, \alpha}\frac{1}{2 L}\frac{1}{l}\left( \frac{\pi K}{L}\right)^{3}e^{- i P_{\alpha}^{-}x^{+}- i P^{+}x^{-}}\frac{| \langle u | \alpha \rangle |^{2}}{l K^{3}| N_{u}|^{2}}.
T_{\mu \nu}^{e f f}= \frac{1}{2}\left( 1 + \frac{b_{k}}{b_{g}}\right) \left( \phi_{, \mu}\phi_{, \nu}- K \tilde{g}_{\mu \nu}\right) - \frac{\Delta^{2}K e^{2 \alpha \phi / M_{p}}}{2 b_{g}M^{4}}\left( \phi_{, \mu}\phi_{, \nu}- \frac{1}{2}K \tilde{g}_{\mu \nu}\right) + \tilde{g}_{\mu \nu}\frac{s M^{4}}{4 b_{g}}e^{- 2 \alpha \phi / M_{p}}
\tau^2 = |\xi - \rho \omega|^2 + \rho^2 + 2 \rho |\xi-\rho \omega| &= |\xi|^2 - 2\rho \xi \cdot \omega + \rho^2 + \rho^2 + 2 \rho (\tau - \rho) &= |\xi|^2 + 2\rho (\tau - \xi \cdot \omega),
\begin{pmatrix}g^+ g^-\end{pmatrix}\;=\;\begin{pmatrix}f^+ f^-\end{pmatrix}+a\, S_D^{-1}\begin{pmatrix}\Phi^+ \Phi^-\end{pmatrix}+\frac{b}{\gamma}\begin{pmatrix}\Phi^+ \Phi^-\end{pmatrix}\gamma\,:=\,\textstyle{\frac{\Gamma(2B)}{\Gamma(B)}\,\frac{1+\nu+B}{1+\nu}}
K(x,y) = (Q^{\otimes 2}\widetilde{K})(x,y) = \widetilde{K}(x,y) + n\widehat{\omega}_{\ell,\lambda}(0) = - \sum_{p\in \pi \mathbb{N}_0^3\backslash\{0\}}n \widehat \omega_{\ell,\lambda}(p) u_p(x) u_p(y),
\mathcal{L}=\sum f_{abc}\left\{F_{a}^{\mu\nu}A_{b,\mu}A_{c,\nu}+m^{2}A_{a,\mu}^{\rm P}A_{b}^{\mu}\phi^{c}\right\}, V_{\mu}=\sum f_{abc}F_{a}^{\mu\nu}(A_{b,\nu}+A_{b,v}^{\rm P})\phi^{c}, d_{e}(L-\partial^{\mu}V_{\mu})=0 \f_{abc}\ .
({\bf p}; l_L, l_R) \to({\bf p}'; l^{\prime}_L, l^{\prime}_R) =({\bf p}+ w x{\bf A};u_L -{\bf p}\cdot{\bf A}-{{wx}\over{2}}{\bf A}\cdot{\bf A},u_R -{\bf p}\cdot{\bf A}-{{wx}\over{2}}{\bf A}\cdot{\bf A}) \quad .
\int_{\pi/2 - \delta}^{\pi/2 + \delta}|P_n(\cos t)|^2 w_\mathbb{T}(e^{it}) dt = \int_{\pi/2 - \delta}^{\pi/2 + \delta}|P_n^*(e^{it})|^2 w_\mathbb{T}(e^{it}) dt + O(q^n) \leq \int_{\pi/2 - \delta}^{\pi/2 + \delta}|S_{2n}(e^{it})|^2 w_\mathbb{T}(e^{it}) dt + O(q^n) \leq \lambda_{2n}(\mu_\mathbb{T},e^{i\pi/2})+ O(q^n)
\delta_{H}^{2}\simeq \frac{5 1 2 \pi}{7 5}\frac{A}{q^{3}{M_{P}}^{6}}\frac{\phi^{2 + q}}{\phi_{0}^{q}}\left[ 1 + \xi ( 3 q - \frac{3 \kappa^{2}\phi^{2}}{2}+ \frac{\kappa^{2}\phi^{2}}{q}) \right]^{2}
\begin{array}{c c}{\gamma^{a}= \left( \begin{array}{c c}{0}{\rho^{a}}{\bar{\rho}^{a}}{0}\end{array}\right) \, , ~ ~ ~}{~ ~ \rho^{a}\bar{\rho}^{b}+ \rho^{b}\bar{\rho}^{a}= 2 \delta^{a b}\, .}\end{array}
\Delta = \langle M e, e \rangle \langle A_1 \xi, \xi \rangle + \langle M \xi, \xi \rangle \langle A_1 e, e \rangle + \langle M e, e \rangle \langle M \xi, \xi \rangle - 2 \langle M e, \xi \rangle \langle A_1 e, \xi \rangle - \langle M e, \xi \rangle^2.
{e}_{\hat{0}}^{0}{\bar{e}}_{\alpha}^{i}\left[ - \frac{1}{2}\frac{\partial \Gamma^{( 2 )}}{\partial v^{i}}\right] ={e}_{\hat{0}}^{0}{\bar{e}}_{\alpha}^{i}\left[ B_{i}^{v ( 2 )}\right] +{e}_{\alpha}^{\mu}{e}_{\hat{0}}^{\nu}K_{\mu \nu}^{g},
X^i(\sigma, \tau)= X^i_u+ p^i \tau + \frac{w^i \sigma}{2 \pi}+ \frac{i}{2 \pi}\sum_{n \neq 0}\left( \frac{\alpha_n^i}{n}e^{-in(\tau - \sigma)}+\frac{{\tilde \alpha}_n^i}{n}e^{-in(\tau + \sigma)}\right),
\varepsilon:\widetilde{\mathfrak{osp}}_{m|2n}(R,{}^-)\rightarrow R_-,\begin{pmatrix}A B -\overline{\rho(C)}^t C D_{11}D_{12}\overline{\rho(B)}^t D_{21}-\overline{D}_{11}^t\end{pmatrix}\mapsto\mathrm{Tr}(A)-\mathrm{Tr}(\rho(D_{11})-\overline{\rho(D_{11})}).
R_{ijij}=\rho^2\tilde{R}_{ijij}+\rho\rho_{ii}+\rho\rho_{jj}-|\nabla\rho|^2, i\neq j &R_{ijik}=\rho^2\tilde{R}_{ijik}+\rho\rho_{jk},~~ when~ \{i,j,k\}~ are~ distinct,
h^{(M)}(t) = \sum C_{p,q}\frac{(1-2t)^p}{(t(1-t))^{p+q+\frac 12}}= \sum \sum_{r=0}^p C_{p,q,r}\frac{1}{t^{r+q+\frac 12}(1-t)^{p+q-r+\frac 12}}
r_{a\rightarrow i}^{t}=y_{a}-\sum_{j\in[n]\backslash i}A_{aj}x_{j\rightarrow a}^{t}, x_{i\rightarrow a}^{t+1}=\frac{1}{\sigma_{i}^{2}}\eta_{t}\left(\sum_{b\in[m]\backslash a}A_{bi}r_{b\rightarrow i}^{t}\right),
u_m(t) = K_{\varphi,f}u_{m-1}(t) = T_0(t-s)\varphi + j^{-1}\int_s^t T_0^{\odot \star}(t-\tau)[B(\tau)u_m(\tau) + f(\tau) + B(\tau)[u_{m-1}(\tau) - u_m(\tau)]] d\tau.
\left. \frac{s h{\omega}^{'}x_{0}}{{\omega}^{'}}\Theta \left({\mu}^{2}-{\vec{p}}^{2}\right) \right] = \frac{- 1}{\left( 2 \pi \right)^{\nu / 2}}\int d^{\nu - 1}p e^{i \vec{p}\cdot \vec{r}}\frac{s i n \Omega x_{0}}{\Omega}
\frac{\langle C_{1},s\rangle\longrightarrow s^{\prime}}{\langle C_{1};C_{2},s\rangle\longrightarrow\langle C_{2},s^{\prime}\rangle}\frac{\langle C_{1},s\rangle\longrightarrow\langle C_{1}^{\prime},s^{\prime}\rangle}{\langle C_{1};C_{2},s\rangle\longrightarrow\langle C_{1}^{\prime};C_{2},s^{\prime}\rangle}\frac{}{\langle skip,s\rangle\longrightarrow s}
\begin{array}{cc}\displaystyle c_j(\xi,x,t)=\exp \int_x^\xi\left(\frac{b_{jj}}{a_{j}}\right)(\eta,\omega_j(\eta))\,d\eta,d_j(\xi,x,t)=\frac{c_j(\xi,x,t)}{a_j(\xi,\omega_j(\xi))}.\end{array}
\sum_{j = 1}^{N}\alpha_{j}^{\prime}\frac{\partial{\cal{G}}}{\partial \alpha_{j}^{\prime}}= R \;{\cal{G}}\hspace{1 0 m m}\mathrm{a n d}\hspace{1 0 m m}\mathrm{J a c o b i a n}= \left[ 1 + R \right] \;{\cal{G}}^{N}.
C_{m,n}(i,k)=\frac{1}{W_{i,k}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Psi_{i}(x_1)\Psi_{k}(x_2)\Psi_{m}(\widetilde{x}_1)\Psi_{n}(\widetilde{x_2})\,\mathrm{d}x_1\mathrm{d}x_2,
\frac{\epsilon_{0}}{2}E^{2}-\frac{1}{2\mu_{0}}B^{2}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F_{\rho\sigma}\eta^{\mu\rho}\eta^{\nu\sigma}
Tr\left( \exp\left(\frac{1}{4}t\Delta_{\nu}\right)\right) =e^{\left( \frac{n^{2}}{4}+\nu^{2}\right) t}\sum\limits_{m=0}^{+\infty}\left( \dim_{\mathbb{C}}\mathcal{A}_{m}^{\nu}\right) e^{-\left( m+\frac{n}{2}+\nu \right)^{2}t},
\frac{dA}{dt}=\frac{B^{2}+2(1+a_{2}^{2})BC+(1-a_{2}^{2})^{2}C^{2}}{BC}, \frac{dB}{dt}= \frac{AD-B^{2}+(1-a_{2}^{2})^{2}C^{2}}{AC}, &\frac{dC}{dt}=\frac{AD+B^{2}-(1-a_{2}^{2})^{2}C^{2}}{AB}, \frac{dD}{dt}= -\frac{D^{2}}{BC}.
\partial_t M_1 - d_{M_1}\Delta_x M_1 = \lambda_{m M_1}\frac{m}{k_{M_1}+ m}+ \lambda_{L_{ox}M_1}\frac{L_{ox}}{K_{M_1}+ L_{ox}}\, M_1 - \underbrace{\lambda_{M_1 F}\frac{M_1}{K_F + M_1}}_{transformation \, into \, F}\ - \ \beta_{M_1}M_1,
t_{i-\ell_0-2}\le t_{i-\ell_0-2-\ell}\prod_{h=0}^{\ell-1}a_{i-\ell_0-2-h}+U\left(1+\sum_{h_2=0}^{\ell-2}\prod_{h=0}^{h_2}a_{i-\ell_0-2-h}\right).
\mathrm{sign}(P_{n,k-1}P_{n,k+1},\Delta_{k})=\begin{cases}1 \mbox{for}\,\,k\,\,\mbox{even}, (-1)^{Z(n,k-1)+Z(n,k+1)}\mbox{for}\,\,k\,\,\mbox{odd}.\end{cases}
A(x,S_p(x,0)) = (2x^3+8x^2-x)S_p(x,0)^4-(x^4+3x^3-58x^2+19x-1)S_p(x,0)^3 & +(3x^4-30x^3+130x^2-56x+7)S_p(x,0)^2 & -(x^4+3x^3-58x^2+19x-1)S_p(x,0) & +(2x^3+8x^2-x).
\int_{c}^{C}\frac{\partial\varkappa}{\partial u}(y,\alpha) \, dy = \int_{c}^{C}\frac{2u\varkappa(y,\alpha)}{\sqrt{1-4|\alpha|^2\varkappa^2(y,\alpha)}}\frac{\partial \varkappa(y,\alpha)}{\partial y}\, dy &= - \frac{u}{2|\alpha|^2}\left(\sqrt{1-4|\alpha|^2\varkappa^2(C,\alpha)}-\sqrt{1-4|\alpha|^2\varkappa^2(c,\alpha)}\,\right)
- \sum\limits_{j=0}^n \gamma_{2j}\sum\limits_{i=j}^n \frac{1}{i+k}{-2i-2k \choose -2n-2k}{-2k-2j\choose -2k-2i-1}\frac{4^{i-j+1}-1}{i-j+1}B_{2(n-i)}B_{2(i-j+1)}.
\frac{1}{f(\theta)}=\sum\limits_{p=-\infty}^{\infty}b_{p}e^{ip\theta}=\left|\sum\limits_{j=0}^{\infty}\psi_{j}e^{-ij\theta}\right|^2=\left|\sum\limits_{j=0}^{\infty}\varphi_{j}e^{-ij\theta}\right|^{-2}.
J(\rho,m) = E\left[\int_0^T\int_A c(t,X(t),a,m(t))\rho_t(da) dt + \psi(X(T),m(T))\right] &=E\left[\int_0^T\int_A c(t,X(t),a,m(t))[\widehat{\gamma}(s,X(s))](da) dt + \psi(X(T),m(T))\right]
V^\mathrm{sampling}(N_1,N_0,n_1,n_0) =E\left[\mbox{var}\big(\widehat{\theta}\,|\,{\mathbf{X}},N_1,N_0\big)\,\big|\, N_1,N_0\right] =\frac{S^2_1}{N_1}\left( 1-\frac{N_1}{n_1}\right) +\frac{S^2_0}{N_0}\left(1-\frac{N_0}{n_0}\right),
{\cal F}_{m_1m_2}^{\pm}(x,y,t) = \frac{i}{{\hbar}^2}\frac{1}{({\varepsilon}_{m_1,\pm}\cdot{\rm sign}({\varepsilon}_{m_1,\pm}) +{\varepsilon}_{m_2,\pm}\cdot{\rm sign}({\varepsilon}_{m_2,\pm}))^2}
s^+_{1, k}s^+_{1, l}= -\delta_{kl}, s^+_{1, j}s^+_{2, 1}= 1 (j = 1, 2, 3, 4), s^\pm_{0}s^{\pm}_{i, j}= 0 (i = 1,2,\, j =1, 2, 3, 4)
\widetilde{\mathfrak{L}}_\alpha f :=\Big(\sqrt{x}\Big( \frac{d}{dx}-\Big(\frac{\alpha-1/2}{x}-1\Big)\Big)\Big(\sqrt{x}\frac{df}{dx}\Big) &=x\frac{d^2f}{dx^2}+ (-\alpha +x+1) \frac{df}{dx}.
R_{\tilde{x}}= (2-2\tilde{g}(\tilde{x})) - d_{\tilde{x}}(\varphi) (2-2g(\varphi(\tilde{x}))) - \sum_{\tilde{v}\in T_{\tilde{x}}(\widetilde{\Gamma})}(d_{\tilde{v}}(\varphi) -1),
w_{0}:= \iota(\phi(\beta_{0})) = c_{1}= a_{1}w_{1}:= \iota(\phi(\beta_{1})) = b_{2}b_{1}a_{1}b_{1}w_{2}:= \iota(\phi(\beta_{2})) = b_{2}b_{1}a_{1}a_{2}^{-1}a_{1}b_{1}a_{2}w_{c}:= \iota(\phi(c)) = b_{2}b_{1}a_{1}a_{2}^{-1}b_{1}^{-1}a_{1}a_{2}^{-1}b_{2}^{-1}
\int e^{-\mathcal Q}\prod_{k=1}^m f_k^{tc_k+(1-t)d_k}\circ B_k \le \left(\int e^{-\mathcal Q}\prod_{k=1}^m f_k^{c_k}\circ B_k \right)^t \left(\int e^{-\mathcal Q}\prod_{k=1}^m f_k^{d_k}\circ B_k \right)^{1-t}\le (\sup J_{\mathcal Q,B,c})^t (\sup J_{\mathcal Q,B,d})^{1-t}\prod_{k=1}^m \left(\int_{H_k}f_k \right)^{tc_k+(1-t)d_k}.
e^{-1}\mathcal{L}=\frac{1}{2}R-{\frac{1}{4}}G_{IJ}F_{\mu \nu}{}^{I}F^{\mu\nu J}-\frac{1}{2}\mathcal{G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu \nu \rho \sigma \lambda}C_{IJK}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{K}.
\widehat{\mu}_k(y) = \frac{1}{k}\sum_{t=0}^{k-1}\prod_{j=1}^g J_0\left(2\left|\sum_{\ell=0}^{k-1}\frac{Z(\gamma_{j,\ell}^{-1})}{k\gamma_{j,\ell}^{1-k}Z'(\gamma_j^{-1})}\frac{\gamma_{j,\ell}}{\gamma_{j,\ell}-1}e^{2\pi i[\ell t]/k}\right|y\right),
\mathbb{E}\left[ \left( 1-c'\gamma_{n}\right)\left\| Z_{n}-m \right\|^{2}2\left( p-1 \right) \gamma_{n}\left\langle \xi_{n+1}, Z_{n}-m \right\rangle \left( \left\| Z_{n}-m \right\|^{2}+ \gamma_{n}^{2}\right)^{p-2}\right] = 0 .
\left( (Q_{1,\vec{v}})_1 M_{\bullet,1}(Q_{1,\vec{v}})_2 M_{\bullet,1}(Q_{1,\vec{v}})_3 M_{\bullet,1}\right)= \begin{pmatrix}1+ 0 + 0 0+0+2 0+2+0 0+1+ 0 1+0+0 0+0+2 0+0+1 0+1+0 1+0+0\end{pmatrix},
\overline{x}=\frac{\sum_{i=1}^{n}(\frac{x_{i}}{\sigma_{i}^{2}})}{\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}}=\frac{\sum_{i=1}^{n}(x_{i}\cdot w_{i})}{\sum_{i=1}^{n}w_{i}},
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = 0$, $u(L,t) = 0$, $u_{x}(L,t) = h_2(t)$ in $(0,T)$, $v(0,t) =g_0(t)$, $v(L,t) = g_1(t)$, $v_{x}(L,t) = g_2(t)$ in $(0,T)$.\end{tabular}\right.
\|(f_t-h_0)e^{\lambda_1 t}-h_1\|_k = \|h_2 e^{(\lambda_1-\lambda_2)t}+ \cdots\|_k = e^{(\lambda_1-\lambda_2)t}\left\|\sum_{j\geq 2}e^{(\lambda_2-\lambda_j)t}h_j \right\|_k \leq e^{(\lambda_1-\lambda_2)t}\sum_{j\geq 2}e^{(\lambda_2-\lambda_j)t}\|h_j\|_k \leq e^{(\lambda_1-\lambda_2)t}C\sum_{j\geq 2}(1+j^N) e^{(\lambda_2-\lambda_j)t}.
y_1=\frac{1}{\sqrt{\mu}}x, \, y_2=\frac{1}{\sqrt{2}}(y+z), \, y_3=\frac{1}{\sqrt{2}}(y-z); [y_1,y_2]=[y_3,y_1]=-\frac{1}{2\sqrt{\mu}}(y_2+y_3), \, [y_2,y_3]=-c\mu Y_1+\sqrt{2}(y_2+y_3).
d s^{2}={\frac{1}{\sqrt{H_{p + r}H_{p + s}}}}d s_{p + 1}^{2}- \sqrt{{\frac{H_{p + r}}{H_{p + s}}}}d s_{s}^{2}- \sqrt{{\frac{H_{p + s}}{H_{p + r}}}}d s_{r}^{2}- \sqrt{H_{p + r}H_{p + s}}d s_{9 - p - r - s}^{2}\ .
\ln\frac{\sigma}{\tau}\int_{\Omega}q^2(K_H(x)\nabla w^{\sigma, \tau}_\varepsilon|\nabla w^{\sigma, \tau}_\varepsilon)dx=\frac{1}{\varepsilon^2}\int_{\Omega}\left (u_\varepsilon-q\ln\frac{1}{\varepsilon}\right )_+^{p}qdx.
h : \mathcal{N}_{\mathrm{Sp}}(2m,\alpha,L) \longrightarrow \mathcal{A}\coloneqq \bigoplus_{i=1}^{m}H^0(X, K^{2i}(D^{2i-1})) (E_*,\varphi,\Phi) \longmapsto (s_2,s_4,\dots , s_{2m}).
\rho_- \left\|Q_{j,-}\right\|_{L^2}^2 \leq \operatorname{Re}\int_{\mathbb{R}}Q_{j,-}(I+J) Q_{j,-}^H dz=\operatorname{Re}\int_{\mathbb{R}}\mathcal{P}^-(D_j) Q_{j,-}^H dz \leq\|D_j\|_{L^2}\left\|Q_{j,+}\right\|_{L^2}.
\frac{1}{c_{0}^{2}}\left( \frac{\ddot{a}_{0}}{a_{0}}+ \frac{\dot{a}_{0}}{a_{0}}\left( \frac{n - 1}{2}\frac{\dot{a}_{0}}{a_{0}}- \frac{\dot{c}_{0}}{c_{0}}\right) \right) = \frac{1}{2{\cal R}^{n}}\frac{d}{d{\cal R}}\Big ( H^{2}{\cal R}^{n + 1}\Big ) ,
\beta_{\nu}^{( 1 )}\overline{{\Gamma}}_{5}= \overline{{\Gamma}}_{5}\beta_{\nu}^{( \widetilde{1})}, \beta_{\nu}^{( \widetilde{1})}\overline{{\Gamma}}_{5}= \overline{{\Gamma}}_{5}\beta_{\nu}^{( 1 )}.
a_{m+k}= \log(N(\star_{i=1}^{m+k}G,\alpha^{m+k+1}) ) \le \log( N(\star_{i=1}^{m}G,\star_{i=1}^{m}\alpha^{2}) N(\star_{i=1}^{k}G, \star_{i=1}^{k}\alpha^{2}) ) &=\log( N(\star_{i=1}^{m}G,\star_{i=1}^{m}\alpha^{2})+ \log (N( \star_{i=1}^{k}G,\star_{i=1}^{k}\alpha^{2}) )=a_{m}+a_{k}.
q_2(h)=\sum\limits_{r=1}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=0}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+1,n_2]_<^r \mathbf{d'}\in [n_1+2,n_2]_<^t d_1=n_1+1}}\det M_h.
\Lambda ( \beta ) = - 8 ( 2 \pi \alpha^{\prime})^{- D / 2}\int_{E}\frac{d^{2}\tau}{2 \pi \tau_{2}^{2}}\; K_{h}( \bar{\tau}, \tau ; D ) \sum_{\ell \in Z ; \mathrm{o d d}}\exp \left[ - \frac{\beta^{2}}{4 \pi \alpha^{\prime}\tau_{2}}\; \ell^{2}\right] \quad ,
R_{ij}(r,t)=u^{\prime2}\{[f(r,t)-g(r,t)]\frac{r_{i}r_{j}}{r^{2}}+g(r,t)\delta_{ij}\},f(r,t)=\frac{R_{11}}{u^{\prime2}},g(r,t)=\frac{R_{22}}{u^{\prime2}}
\begin{aligned}J_1= \frac{\mathrm{d}}{\mathrm{d}x}\int_0^\infty (h+h_e\phi)D^\alpha h\cdot \langle y\rangle^{2(l+k)}D^\alpha u \mathrm{d}y &-\int_0^\infty (l+k)gD^\alpha h\cdot \langle y \rangle^{2(l+k)-2}2yD^\alpha u\mathrm{d}y.\end{aligned}
t_5(5)= |\{(5), (4,1), (3,1,1), (2,1,1,1), (1,1,1,1,1)\}|=5, t_6(5)= |\{(4,2), (3,2,1), (2,2,1,1)\}|=3, t_7(5)= |\{(4,3), (3,3,1), (3,2,2), (2,2,2,1)\}|=4, t_8(5)= |\{(4,4), (3,3,2), (2,2,2,2)\}|=3, t_9(5)= |\{(3,3,3)\}|=1
B_{H}(t)=B_{H}(0)+\frac{1}{\Gamma(H+1/2)}\{\int_{-\infty}^{0}[(t-s)^{H-1/2}-(-s)^{H-1/2}]dB(s)+\int_{0}^{t}(t-s)^{H-1/2}dB(s)\}
f_p(n) =(2p)^{2H}\biggl[ \biggl( 1+ \frac{n+2}{2p}\biggr)^{2H}- 2 \biggl( 1 + \frac{n+1}{2p}\biggr)^{2H}+\biggl( 1 + \frac{n}{2p}\biggr)^{2H}\biggr] &= \bigl( 2^{2H-1}H(2H-1) \bigr) p^{2(H-1)}\bigl( 1+ o(1) \bigr).
Y_m(x) - \widetilde{Y}_m(x)+\widetilde{Z}_m(x)-Z_m(x) =\sum_{k=r_m}^{r_{m+1}-1}\frac{1}{\sqrt{p_k}}\left(p_k^{-ix}-p_{r_m}^{-ix}\right)\left(e^{2\pi i\theta_k}-\frac{1}{\sqrt{2}}\left[W_k^{(1)}+iW_k^{(2)}\right]\right) :=\sum_{k=r_m}^{r_{m+1}-1}f_k(x)\left(e^{2\pi i\theta_k}-\frac{1}{\sqrt{2}}\left[W_k^{(1)}+iW_k^{(2)}\right]\right),
\mathcal{W}(L\downarrow\mathfrak{k})=\mathcal{W}(L\downarrow\mathfrak{g})\downarrow\mathfrak{k}\subseteq\Theta_{n}\downarrow\mathfrak{k}=\Gamma_{\mathfrak{g}}\downarrow\mathfrak{k}=\Gamma_{\mathfrak{k}}=BC_{r}\cup\{0\}.
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] ,
\P(S_{t}< 2a\phi^{-1}(at^{-1})^{-1}) =1-\P(1-e^{-\phi^{-1}(at^{-1})S_{t}}\geq 1-e^{-2a}) &\geq 1-\frac{1-e^{-t\phi(\phi^{-1}(at^{-1}))}}{1-e^{-2a}}=\frac{e^{-a}-e^{-2a}}{1-e^{-2a}}=:c_1>0
\frac{m-2}{2m-2}H_1^n(\alpha)+\frac{m}{2m-2}H_m^n(\alpha)\geq \frac{H_1^n(\alpha)+H_m^n(\alpha)}{2}&\ \ \ \geq \frac{H_1^n(\alpha)+H_4^n(\alpha)}{2}=1+(n-1)\cdot h(\alpha)=H_2^n(\alpha).
g_{ii}= \frac{-3(u^j-u^k)^2}{4}, g_{ij}= \frac{1}{4}(-2{u^i}^2-2{u^j}^2-3{u^k}^2 +u^iu^j+3u^iu^k+3u^ju^k) = -\frac{1}{4}((u^i-u^j)^2 + (u^i-u^k)^2 + (u^j-u^k)^2 + (u^i-u^j)(u^i-u^k)),
W_i=S_{i,1}{O_0}^{\alpha_{i,1}}S_{i,2}{O_0}^{\alpha_{i,2}}\cdots S_{i,n_i}{O_0}^{\alpha_{i,n_i}}S_i^{*}x +t \left( \sum_{k=1}^{n_i}(-1)^{\beta_{i,k}}\left( \prod_{j=1}^{k-1}S_{i,j}{O_0}^{\alpha_{i,j}}\right) S_{i,k}{O_0}^{\beta_{i,k}}\right) \mbox{,}
{}_2F_1\bigg[\begin{matrix}\frac14 \frac14 &1\end{matrix}\bigg|\,\frac14\bigg]_{p-1}\equiv \begin{cases}\big(\frac{2}{p}\big)\cdot (a+b\sqrt{-3})\pmod{p^2}, p\equiv1\pmod{4}, \big(\frac{2}{p}\big)\cdot 2(a\sqrt{-3}-3b)\pmod{p^2}, p\equiv3\pmod{4},\end{cases}
\ell(e)\mu_{can}(e)=\sup_{0 \ne \omega\in \mathcal{H}_{L^2}(\Gamma)}\frac{|\omega(e)|^2}{\|\omega\|^2}=\max_{0 \ne \omega\in \mathcal{H}_{L^2}(\Gamma)}\frac{|\omega(e)|^2}{\|\omega\|^2}\, .
V_{m+2}(k;x) +\frac{-x q^{2m+3}+q+1}{q^{m+2}-1}V_{m+1}(k;x)+\frac{q(q^{m+1}+1)}{1-q^{m+2}}V_{m}(k;x) &=V_m(k;x)R(m,k;x)-V_m(k-1;x)R(m,k-1;x),
G'_m (q,t) = (q^2+q^4+\cdots+q^{2m}) \left( 1+q^{2m}+\cdots +\frac12((1+q^m+\cdots)^2-1-q^{2m}-\cdots) \right) &+ \frac t2 ((q+q^2+\cdots+q^m)^2-q^2-q^4-\cdots-q^{2m})(1+q^m+\cdots)^2
\forall\,x:n\forall\,y:n\forall\,z:n((x+_ny)+_nz=x+_n(y+_nz)) \forall\,x:n(x+_n0_n=x) \forall\,x:n\forall\,y:n(x+_ny=y+_nx) \forall\,x:n(x+_nx=0_n) \forall\,x:n(1\cdot_n x=x) \forall\,x:n\forall\,y:n\forall\,a:0(a\cdot_n(x+_ny)=a\cdot_n x+_na\cdot_n y) \forall\,x:n\forall\,a:0\forall\,b:0((a+_0b)\cdot_n x=a\cdot_n x+_nb\cdot_n x)
M_2 = \sum_{\substack{r_1,r_2 \geq 0 a}}\frac{1}{2^{r_1}3^{r_2/2}N(a)^{1/2}}V \left(\frac{\pi 2^{2r_1+1}3^{r_2}N(a)}{z}\right)\sum_{c \equiv 1 \bmod{36}}\ \frac{g_6(c)\overline{\chi_c}(a)}{N(c)^{1/2}}W\left( \frac{N(c)}{y}\right).
d\phi_{ij}(\partial_j,\partial_r)=\partial_j(\phi_{ij}(\partial_r))-\partial_r(\phi_{ij}(\partial_j))=-\partial_r\Big(\frac{\Gamma_{ji}^i}{\varphi_j}\Big)=\frac{-\partial_r\Gamma_{ji}^i+2\Gamma_{rj}^j\Gamma_{ji}^i}{\varphi_j},
\max_{g \in \mathcal{G}}\liminf_{N \to \infty}\mathbb{E}\left[ \frac{1}{N}\displaystyle\sum\limits_{t = 1}^N \frac{1}{2}\log\left( 1 + g( t) \right) \right] & \leq \frac{1}{2}\log(1+pB_{max})
e_i(g_\nu)=\left(P_{\xi_{\tau},\check{\xi}_{\tilde{\tau}}}\left(\pi_{\nu}(g_\nu)v_{\pi_{\nu}}\right),\phi_i\right)=\int\limits_{K_{n,\nu}}\left<\xi_\tau,\tilde{\tau}\left(x^*\right)\check{\xi}_{\tilde{\tau}}\right>\left(\pi_{\nu}(xg_\nu)v_{\pi_{\nu}},\phi_i\right)dx.
A{}=A_{j-1}[X_j;\sigma_j,\delta_j][X_{j+1};\sigma_{j+1}]\ldots [X_N;\sigma_N], \hat{A}{}=A_{j-1}[X_j;\sigma_j,\delta_j][X_{j+1}^{\pm 1};\sigma_{j+1}]\ldots [X_N^{\pm 1};\sigma_N],
S_{f e r m i o n i c}= \frac{1}{2}\sqrt{d e t ( B )}( 2 \pi )^{d / 2}\int d^{d}x \psi \ast \sigma^{j}[ \nabla_{j}, \psi ]_{\ast}+ \psi \ast \sigma^{I}[ \phi_{I}, \psi ]_{\ast}.
P_{l}T^{\,j}P_{l}\,x=\sum_{k=b_{l}}^{b_{l+1}-1}x_{k}\,P_lT^{\,j}\,e_{k}= \sum_{k\,\in\,[b_{l},b_{l+1})\backslash I_{j}}x_{k}\,P_lT^{\,j}\,e_{k}+ \sum_{k\,\in\, I_{j}}x_{k}\,P_lT^{\,j}\,e_{k}
q^{-13/60}\varphi_1^{-6}\varphi_\frac{1}{2}^2G_\frac{1}{2}+q^{17/60}\varphi_1^{-6}\varphi_2^2H_2 = q^{-13/60}\varphi_1^{-6}(\varphi_\frac{1}{2}^2G_\frac{1}{2}+q^{1/2}\varphi_2^2H_2).
\left. \begin{array}{l}{\partial_{\phi}F^{a}= 0 , \partial_{\pi_{\phi}}G^{a}= 0 ,}\\{F^{b}\partial_{b}G^{a}- G^{b}\partial_{b}F^{a}+ \pi_{\phi}\partial_{\phi}G^{a}- 2 ( e^{\phi}- e^{- 2 \phi}) \partial_{\pi_{\phi}}F^{a}= 0 ,}\\ \end{array}\right.
\delta{\cal P}_{i}^{\left( 1\right) a}=-\epsilon_{0ijk}\pi^{jka},\;\delta{\cal P}_{i}^{\left( 2\right) a}=-\frac{1}{2}\epsilon_{0ijk}F^{jka},\;\delta{\cal P}^{a}=-\partial^{i}{\cal P}_{i}^{\left(2\right) a},
Skew[X]=-\frac{2(15\kappa^{6}+6\kappa^{4}+2\kappa^{2}+1)}{\beta^{3}\sigma_{\kappa}^{3}(\kappa^{2}-1)^{3}(36\kappa^{4}-13\kappa^{2}+1)}for0\le\kappa<1/3
\frac{1}{2}\int dt \, d^3 x h_\mu^\nu T_\nu^\mu = \frac{1}{2}\int dt \, d^3 x \left( H_{tt}^0 T^{tt}+ \frac{1}{2}H_{aa}^0 T^{aa}+ H_{zz}^0 T^{zz}+ 2 H_{tz}^0 T^{tz}\right) \,.
nD(i+1,j) =n+k_{i+1}-k_j-k_{n+i+1-j}=n+k_{i+1}+r-k_{j+1}-k_{n+i+1-j}=r+k_{n+i-j}-(r+k_{n+i-j})+n+k_{i+1}+r-k_{j+1}-k_{n+i+1-j}=nD(n-j+i)+nD(i+1,j+1).
\| m\|_{\mathcal{S}^\infty}:=\|\mathcal{F}^{-1}(m)\|_{L^1}, \|m(\xi,\eta)\|_{\mathcal{S}^\infty_{k,k_1,k_2}}:=\|m(\xi, \eta)\psi_k(\xi)\psi_{k_1}(\xi-\eta)\psi_{k_2}(\eta)\|_{\mathcal{S}^\infty},
g_{\mu \nu}g^{\alpha \beta}Q_{\alpha \rho}^{\kappa}Q_{\kappa \beta}^{\rho}( - 2 + \frac{a ( a - 2 ) ( 5 a^{2}+ 4 5 a + 2 6 )}{3 0 ( 1 - a )^{2}}) + g^{\alpha \beta}Q_{\mu \alpha \kappa}Q_{\beta \nu}^{\kappa}( \frac{- a ( a - 2 )}{2}) ,
(e^{i t \Delta}g_{K,\epsilon})(x) = \frac{\beta_\epsilon(t)}{(2\pi)^2}e^{- \frac{\epsilon^2\beta_\epsilon(t)}{2}|x|^2 + i \beta_{\epsilon}(t) x \cdot K- i t \beta_{\epsilon}(t) |K|^2}
C_{2n}^{\lambda}(t)= a_n^{\lambda}\,\,{}_{2}F_{1}\Big(\!-n,n+\lambda;\lambda+\frac{1}{2};1-t^2\Big), begin{align*}4pt] C_{2n+1}^{\lambda}(t)= b_n^{\lambda}\, t \,{}_{2}F_{1}\Big(\!-n,n+\lambda+1;\lambda+\frac{1}{2};1-t^2\Big),
\left[ \, \psi_{n}^{( K )}\, , \, \Phi_{\chi_{k}\otimes \chi_{\bar{k}}}^{\beta}\, \right] \ = \ \Phi_{k \left( \delta_{1}\left( \psi^{( K )}\right)_{n}\right) \chi_{k}\otimes \chi_{\bar{k}}}^{\beta}
(a\nabla (\eta u^{p/2}),\nabla (\eta u^{p/2})) = \frac{p^2}{4}u^{p-2}\eta^2 (a\nabla u,\nabla u)+p\eta u^{p-1}(a\nabla u,\nabla \eta)+u^{p}(a\nabla \eta,\nabla \eta),
q^t+1=|ST|\geq \frac{\left|\sum\limits_{X\in ST}|X^*|\right|}{|V(t,q)^*|}=\frac{|V(n,q)^*|-|Y^*|}{|V(t,q)^*|}=\frac{(q^n-1)-(q^{d}-1)}{q^t-1}\geq q^{d},
\|B^{(\ell)}x\|\leq \|\underline{B}^{(\ell)}\|+\|HB^{(\ell-1)}\|+ \sum_{t=1}^{\ell-1}\sum_{i=1}^r\nu_i^2\|\underline{B}^{(t-1)}\chi_i\| \left| \langle \check{\chi}_i, B^{(\ell-t-1)}x\rangle \right| +\sum_{t=1}^{\ell-1}\|\underline{B}^{(t-1)}\tilde{H}B^{(\ell-t-1)}\|+L^2\|\underline{B}^{(\ell-1)}\|+\sum_{t=0}^{\ell}\|R_t^{(\ell)}\|.
\sum_{A=1}^{\infty}\frac{G_k(A^2\slash X)}{A}=\frac{1}{2\pi i}\sum_{A=1}^{\infty}\int_{\mathrm{Re}(t)=\frac{3}{4}}\frac{\Gamma(k+t)^2}{(2\pi)^{2t}}\frac{X^t}{A^{2t+1}}\frac{dt}{t}=\frac{1}{2\pi i}\int_{\mathrm{Re}(t)=\frac{3}{4}}\frac{\Gamma(k+t)^2}{(2\pi)^{2t}}X^t\, \zeta(2t+1)\frac{dt}{t}.
V=\frac{1}{(\sum_{i=1}^{N}q_{i}m_{i})^{2}}\sum_{i=1}^{N}q_{i}(1-q_{i})m_{i}^{2}(a_{i}-\frac{\sum_{j=1}^{N}q_{j}a_{j}m_{j}}{\sum_{j=1}^{N}q_{j}m_{j}})^{2},
(\begin{matrix}A_{+}\\A_{-}\\A_{0}\end{matrix})=U^{*}(\begin{matrix}A_{x}\\A_{y}\\A_{z}\end{matrix}),U^{*}=(\begin{matrix}-\frac{1}{\sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ +\frac{1}{\sqrt{2}}&+\frac{i}{\sqrt{2}}&0\\ 0&0&1\end{matrix}),
X^{T}=(\begin{matrix}1&x_{11}&...&x_{1p}\\ 1&x_{21}&\cdot\cdot\cdot&x_{2p}\\ \vdots&\vdots&&\vdots\\ 1&x_{n1}&\cdot\cdot\cdot&x_{np}\\ \end{matrix})\in\mathbb{R}^{n\times(p+1)}
\|(H_\omega-\lambda)\psi_n\| =\|(H_\omega-\lambda)\varphi_n^{(j_n)}\| &=|(H_{\omega,j_n}-\lambda)\varphi_n\| &\leq\|(H_{\omega,j_n}-H_{\bar\omega})\varphi_n\|+\|(H_{\bar\omega}-\lambda)\varphi_n\| &\leq \frac{1}{n}+\|(H_{\bar\omega}-\lambda)\varphi_n\| &\to 0 n\to \infty.
Q^{1}= 1 + \chi_{4 \lambda}- \chi_{2 \lambda}, \, \, \, \, \, Q^{2}= \chi_{2 \lambda}, \, \, \, \, \, Q^{3}= \chi_{4 \lambda}, \, \, \, \, \, Q^{4}= \chi_{4 \lambda}+ \chi_{2 \lambda},
\tilde{S}_{(\lambda,\mu),(\lambda',\mu')}= i^{-n-\lvert\Phi^+\rvert}\frac{\sum_{w\in W}(-1)^{l(w)}\xi^{\langle 2\rho+\lambda,(w\bullet(\lambda',\mu'))_2\rangle + \langle\mu,(w\bullet(\lambda',\mu'))_1+2\rho\rangle}}{d^n}
\log\Upsilon(x)=\int\limits_0^\infty\frac{dt}{t}\;\biggl[\biggl(\frac{Q}{2}-x\biggr)^2e^{-t}-\frac{\sinh^2\bigl(\frac{Q}{2}-x\bigr)\frac{t}{2}}{\sinh\frac{bt}{2}\sinh\frac{t}{2b}}\biggr]
\frac{h_{H}(f^{n}(P))}{\delta^{n}}= -\sum_{i=0}^{n-1}\delta^{-1-i}c_{1}h_{Z_{1}}(p^{-1}(f^{i}(P)))+\sum_{i=0}^{n-1}\delta^{-1-i}h_{E}(p^{-1}(f^{i}(P))) &+\sum_{i=0}^{n-1}\delta^{-1-i}c_{1}h_{E_{1}'}(f^{i}(P))+\sum_{i=0}^{n-1}\delta^{-i}h_{N}(f^{i}(P))+h_{H}(P).
K' = \left(\begin{array}{c|c}K_{11}\begin{array}{cc}\hspace{4pt}0 \hspace{4pt}\cline{1-1}\mbox{\hspace{2pt}\small $-g$ \hspace{1pt}}\vline\hfill\end{array}\hline\begin{array}{cc}\hfill\vline \mbox{\hspace{2pt}\small $-g$ \hspace{1pt}}\cline{2-2}\hspace{4pt}0 \hspace{4pt}\end{array}K'_{22}\end{array}\right)\,.
\hat{\phi}_N(t) = \hat{\phi}_{N^{(1)}- N^{(2)}}(t)^{\frac{1}{2}}= \bigg| \frac{1}{n}\sum_{i=1}^n \cos \Big[ t \big( N^{(1)}- N^{(2)}\big) \Big] \bigg|^{\frac{1}{2}}.
\|\hat\delta(x) - \hat\delta(y) \|_{Lip_0(M)^*}= \sup_{\|g\|_{Lip_0(M)}\leq 1}|\langle g, \hat\delta(x) - \hat\delta(y)\rangle| &= \sup_{\|g\|_{Lip_0(M)}\leq 1}|E(g)(x) - E(g)(y)| &\leq \|E\| d(x, y).
|\nabla u_\eta|^2 + |\nabla u_{1-\eta}|^2 \le (1+ 2 \eta d \tan \frac{d}{2}) (|\nabla u_0|^2 + | \nabla u_1|^2) &- 2\eta (1+\frac{1}{2}d \tan \frac{d}{2}) | \nabla d|^2 -2 d \nabla \eta \cdot \nabla d + \textup{Quad}(\eta, |\nabla \eta|).
\big\| \big\| \Pi_{I_0}^{F, G, \tilde E}(\vec f, \vec g) \big\|_{L^{R^n}}\big\|_{\tilde s, \infty}= \big\| \big\| \Pi_{I_0}^{F, G, \tilde E}(\vec f, \vec g) \big\|_{L^{R^n}}^{r^{j_0}}\big\|_{\frac{\tilde s}{r^{j^0}}, \infty}^{\frac{1}{r^{j_0}}}.
\left|E[\beta(t,q_t^m)V(t,q_t^m)]-E[\beta(t,q_t)V(t,q_t)]\right| \leq E[|\tilde C(1+\|q_t\|^{\tilde p}+\|q_t-q_t^m\|^{\tilde p})\|q_t-q_t^m\||] = O(m^{1/2})