harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal T}_{\mu \nu}^{\alpha \beta}= 2{\it g_{1}}^{2}\epsilon_{\mu}^{\alpha \rho \sigma}\epsilon_{\nu}^{\beta \lambda \gamma}\bigg ( F_{\rho \sigma}F_{\lambda \gamma}+ \tilde{F}_{\rho \sigma}\tilde{F}_{\lambda \gamma}+{\frac{1}{\sqrt 2}}\hat{\theta}_{5}\big ( F_{\rho \sigma}\tilde{F}_{\lambda \gamma}+ \tilde{F}_{\rho \sigma}F_{\lambda \gamma}\big ) \bigg ) ,
{\frac{1}{1 6 \pi G_{\mathrm{\scriptsize ~ i n d}}}}\simeq 8 . 2 \cdot 1 0^{- 4}\mu^{2}, \hspace{1 c m}{\frac{2 \Lambda_{\mathrm{\scriptsize ~ i n d}}}{1 6 \pi G_{\mathrm{\scriptsize ~ i n d}}}}\simeq 4 . 2 \cdot 1 0^{- 8}\mu^{4},
\begin{aligned}J_2(\beta,N,T,\hat{\mathbf{R}}_2) \leq N^2 |\hat{\mathbf{R}}_2 | &\leq C N^2 \beta^{-1/3}N^{-1/3}T \leq C \beta^{-1/3}N^{5/3}T. \end{aligned}
R_k^j=\int_{k-1}^{k}\frac{d [q(s)-Q(s)]}{\Gamma(1-\alpha)(j-s)^{\alpha}}=-\frac{\alpha}{\Gamma(1-\alpha)}\int_{k-1}^{k}\frac{q(s)-Q(s)}{(j-s)^{\alpha+1}}d s,1\leq k\leq j.
0=\partial_T[\mathcal{N}(\psi^T)]=\mathcal{N}'(\psi^T)\partial_T \psi^T =\mathcal{N}'(\psi^T)\left[-\frac{r}{(T-t)^2}f'\left(\frac{r}{T-t}\right)\right] &=\mathcal{N}'(\psi^T)[-e^{\tau}\rho f'(\rho)].
\displaystyle \lim\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\mathbb{E}[| \sum_{\substack{j\in J_{K}}}\Phi(\frac{B_{t_{j}}+B_{t_{j+1}}}{2},t_{j}) (B_{t_{j+1}}-B_{t_{j}}) |^2 ]=0
\begin{array}{ll}D=\begin{pmatrix}c e^{\int \frac{1}{A(t)}dt}+ e^{\int \frac{1}{A(t)}dt}\int A^3(t) e^{-\int \frac{1}{A(t)}dt}c e^{\int \frac{1}{A(t)}dt}+ e^{\int \frac{1}{A(t)}dt}\int A^3(t) e^{-\int \frac{1}{A(t)}dt}c e^{\int \frac{1}{A(t)}dt}+ e^{\int \frac{1}{A(t)}dt}\int A^3(t) e^{-\int \frac{1}{A(t)}dt}\end{pmatrix}.\end{array}
\begin{cases}B_3 (1+\tau)^{\frac{1}{1+\beta}}\le B^{-1}(\tau)\le B_4 (1+\tau)^{\frac{1}{1+\beta}}( \beta \in (-1,1]), \exp \left( B_3 (1+\tau) \right)\le B^{-1}(\tau)\le \exp \left( B_4 (1+\tau) \right) (\beta =-1)\end{cases}
\frac{\partial H_1}{\partial x^1}+ \frac{\partial H_2}{\partial x^2}+ \frac{\partial H_3}{\partial x^3}= 0 \frac{\partial E_1}{\partial x^1}+ \frac{\partial E_2}{\partial x^2}+ \frac{\partial E_3}{\partial x^3}= 4 \pi \rho
y_\ell^{\vphantom +}(r) = \sqrt{r}\left(C_\ell^-\left(\frac{R}{r}\right)^{\left((\ell + 1)^2 - M^2 N\right)^{1/2}}+C_\ell^+\left(\frac{r}{R}\right)^{\left((\ell + 1)^2 - M^2 N\right)^{1/2}}\right) \ , \ \M^2 \leq \frac{(\ell + 1)^2}{N}.
\Delta f=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial f}{\partial r})+\frac{1}{r^{2}sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial f}{\partial\theta})+\frac{1}{r^{2}sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}},
\begin{pmatrix}\eta_1 \eta_3 \vdots \eta_j\end{pmatrix}=\bigg\{\frac{1}{t_1^1},...,\frac{1}{t_1^j}\bigg\}\begin{pmatrix}1 1 ... 1 1^2 3^2 ... j^2 \vdots \vdots \vdots \vdots 1^{j}3^j ... j^j \end{pmatrix}^{-1}\begin{pmatrix}0 \vdots 0 (-1)^{\frac{j-1}{2}}j! \end{pmatrix}.
\Delta f^{i}- \frac{\partial}{\partial x^{i}}(\delta f) = 2 \frac{\partial}{\partial x^{j}}(R^{0}|_{S^{1}}f)_{i,j}= \frac{\partial}{\partial x^{j}}\left( \frac{\partial f_{i}}{\partial x^{j}}- \frac{\partial f_{j}}{\partial x^{i}}\right) = - \left({\rm curl}^{\intercal}\,{\rm curl}\, (f) \right)_{i},
m(\omega\cap (B^\delta_{h,h_0,M})^c) = m \{t' \in \omega \colon dist(t', \partial\omega) \leq \frac{M+1}{1-\epsilon}|h|\}\le 2\frac{M+1}{1-\epsilon}|h| &\le\frac{2(M+1)|h|}{(1-\epsilon)|\omega|}|\omega| &\le \frac{2C_6(M+1)}{\delta (1-\epsilon) N(t,h)^{\mathcal{K}\frac{\log \lambda}{2}}}|\omega|.
U_{\bf j}(v)=U_{\bf j}=\alpha(v)\cdot [u_0(v),\,u_{\bf j}(v)]-[u_{\bf i}(v),\,u_{\bf k}(v)]{=:}T(v)\begin{pmatrix}{\bf j}0 0{\bf j}\end{pmatrix}~\quad &U_{\bf k}(v)=U_{\bf k}=\alpha(v)\cdot [u_0(v),\,u_{\bf k}(v)]+[u_{\bf i}(v),\,u_{\bf j}(v)]=T(v)\begin{pmatrix}{\bf k}0 0{\bf k}\end{pmatrix}.
\int_R^\infty \exp_p(\mu|u|^{p'})r^\theta\mathrm dr =\sum_{j=0}^\infty\dfrac{\mu^{p-1+j}}{\Gamma(p+j)}\int_R^{\infty}|u|^{(p-1+j)p'}r^\theta\mathrm dr &=\dfrac{\mu^{p-1}}{\Gamma(p)}\|u\|_{L^{p}_\theta}^{p}+\dfrac{\mu^{p}}{\Gamma(p+1)}\|u\|_{L^{pp'}_\theta}^{pp'}+\sum_{j=2}^\infty\dfrac{\mu^{p-1+j}}{\Gamma(p+j)}\int_R^{\infty}|u|^{(p-1+j)p'}r^\theta\mathrm dr.
F_p \left( \phi_0, \phi_1, \cdots, \phi_p \right) = \sum_{m_0 \cdots m_p}\alpha_{m_0 \cdots m_p}\, \, a_{m_0}\left( \phi_0 \right) a_{m_1}\left( \phi_1 \right)\cdots a_{m_p}\left( \phi_p \right)
\omega_{4}^{3}= - i \{\frac{1}{2}d \ln ( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 - g_{2}\overline{{g}}_{2})}\},
V_{2}^{[\ell^{1},\ell^{2}]}(x) \coloneqq \sup\limits_{\tau_{2}\in \mathcal{T}}M^{x}_{2}(D_{[\ell^{1},\ell^{2}]},\tau_{2}) = M^{x}_{2}(D_{[\ell^{1},\ell^{2}]},D_{[r,1]}),\forall x \in [0,1].
\bar{p}(\mathbf{z})=\mathrm{Tr\,}\rho_{\mathbf{z}}\Phi_{c}[\rho ]=\int_{\mathbf{Z}}\underline{p}(\mathbf{w})\mathrm{Tr\,}\rho_{\mathbf{z}}\rho_{\mathbf{w}}\frac{d^{2s}\mathbf{w}}{\pi^{s}}.
||G_n(z)^{-1}P(z)-I|| = ||G_n(z)^{-1}\lim_{k\to\infty}G_k(z)-I|| = \lim_{k\to\infty}||G_n(z)^{-1}G_k(z)-I|| = \lim_{k\to\infty}||B_{n+1}\exp(\tilde{B}_{n+1})\ldots B_k\exp(\tilde{B}_k)-I|| \leq \lim_{k\to\infty}\exp\left[\sum_{s=n+1}^k \frac{|\frac{z-z_0}{z_s-z_0}|^{s+1}}{1-|\frac{z-z_0}{z_s-z_0}|}\right]-1,
J_{T_{*}}^{(2)}= - \frac{2 \pi \sigma^{3}}{n T_{*}^{3/n}}\sum_{k=0}^{\infty}\frac{1}{k!}\Gamma \left( \frac{km-3}{n}\right) \left( \frac{1}{T_{*}}\right)^{(n-m)k/n}.
(\alpha_{-1}- \beta_{-1}) \cdot a_0 = \left(\frac{2t}{3}- \frac{1}{2^3}\right) a_2 - \frac{1}{2^3}(a_{-2}- a_3 - a_{-3}) - \frac{1}{2^4}(v_{(1,2)}- v_{(1,3)}+ 4v_{(2,3)}) & + \frac{1}{2^3 \cdot 3}(7a_1 + 4a_{-1}) \cdot v_{(2,3)}- \frac{2}{3}a_2 \cdot v_{(1,3)}+ \frac{1}{2 \cdot 3}(a_3 + a_{-3}) \cdot v_{(1,2)}.
\dot{V}\left(\frac{\partial}{\partial L}\right) \langle W (L) \rangle -\int_{0}^{L}dJ \langle W(J) W( L - J ) \rangle \sim f(z) \exp \left \{-\frac{2}{5}\sqrt{6}z^{\frac{5}{2}}\right \}( 1 + \cdots )
\mathbf{G}( a ) = N_{G}^{2}( a ) \exp \left( \int_{0}^{\infty}\frac{d t}{t}[ a^{2}e^{- 2 t}- \sum_{\alpha > 0}\sinh ( ( \alpha^{2}b^{2}/ 2 + 1 ) t ) F_{\alpha}( a , t ) ] \right) .
\phi^{-1}(\rho t^{-1})=\phi^{-1}(t^{-1})\frac{\phi^{-1}\rho t^{-1})}{\phi^{-1}(t^{-1})}\geq C_L^{1/\gamma}\rho^{1/\gamma}\,\phi^{-1}(t^{-1}).
M \frac{d \Omega}{d y}+ ( 1 + q ) ( \frac{d \Omega}{d y})^{2}- \frac{1}{2}f_{0}^{2}= - \frac{\mu}{4}\Omega^{2 + 2 q}\exp ( M \int \frac{d y}{\Omega}+ 2 \rho_{0})
A_{k,\beta}(t,\xi)=\frac{1}{(\xi-kt)^2+\beta^2k^2},\,k\neq 0,A_{0,\beta}(t,\xi)=\frac{1}{\xi^2+\beta^2};A_{*,\beta}(t,\xi)=\sum_{k}e^{-2|k|}A_{k}(t,\xi),
F(\lambda) = \sum_{n=1}^N F(\lambda_n) \frac{[\lambda - \lambda_n + t + \sum_{m=1}^N \lambda_m]}{[t + \sum_{m=1}^N \lambda_m]}\prod_{\substack{m=1 m\neq n}}^N \frac{[\lambda-\lambda_m]}{[\lambda_n-\lambda_m]}\ .
F_{k,n}:=\left\{\sum_{m=k}^{\lfloor n/\ell \rfloor}\frac{1}{C_m}\in \left( \left(\frac{n}{\ell}\right)^\alpha\cdot \left(\frac{1}{\alpha}\pm \frac{5}{n^{1/4}}\right)\right)\right\}\bigcap \left\{\left\vert C_{k}-k^{\frac{1}{\ell+1}}\right\vert < 10 k^{\frac{1}{\ell+1}-\frac14}\right\}.
\Lambda(x, \beta, N)r_n(x) \equiv \left[-\mathcal{L}(x, \beta; N) +\left(\frac{\beta_2-\beta_0}{2}\right)\left(\frac{\beta_2-\beta_0}{2}-1\right)\right]r_n(x) \hphantom{\Lambda(x, \beta, N)r_n(x)}{}=\kappa\left(n, \frac{\beta_2-\beta_0}{2}\right)r_n(x),
\frac{\partial l}{\partial\sigma^{2}}=-\frac{n}{2\sigma^{2}}+\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{2\sigma^{4}}+\frac{2\mu}{\sigma^{4}}\sum_{i=1}^{n}\frac{x_{i}e^{-\frac{2\mu x_{i}}{\sigma^{2}}}}{1+e^{-\frac{2\mu x_{i}}{\sigma^{2}}}}
\sum_{j_o}c_{j_o}\,(1-s_1^2)^{\gamma_{j_o}/2}\,s_1^{\nu_{j_o}}=1 - \sum_{j_e}c_{j_e}\, (1-s_1^2)^{\gamma_{j_e}/2}\,s_1^{\nu_{j_e}}, s_1 \in (0, 1).
h_{\mu}(Z,\lambda) =h_{\mu}(Y,\nu)+\lim_{n\to\infty}\int_{X}\left(H\left(\xi_{n}^{x}\right)-H\left(\xi_{n-1}^{x}\right)\right)d\eta(x) =h_{\mu}(Y,\nu)+\lim_{n\to\infty}\frac{1}{n}H\left(\xi_{n}|X,\eta\right).
\begin{aligned}\|\nabla^l (\varrho(t), u(t),\nabla n(t))\|_{H^{m-l}}^2+\int_0^t ( \|\nabla^{l+1}\varrho\|_{H^{m-l-1}}^2+\|\nabla^{l+1}(u, \nabla n)\|_{H^{m-l}}^2 )d\tau &\le C\|\nabla^l (\varrho_0, u_0, \nabla n_0)\|_{H^{m-l}}^2.\end{aligned}
\lim_{z \downarrow 0}\int_0^\infty w_z(\lambda) \lambda^{4(1-s+w)}\frac{d\lambda}{\lambda}\int_0^1 du \int_0^\infty \frac{dt}{t^3}\phi(n_{u}^t a_{\frac{1}{t}}) \sum_{m = 1}^\infty \hat{F}_{\pm}(\lambda a_{\frac{1}{t}}\cdot (0,0,0,m))
\begin{aligned}D(1,1)=4t_\infty^{(1)}=1+12\cdot\sum_{n=2}^\infty \frac{(n+3-2)(-2)^{n-1}}{(n+3)!}& =1+12\cdot\sum_{n=2}^\infty \frac{(-2)^{n-1}}{(n+2)!}+12\cdot\sum_{n=2}^\infty \frac{(-2)^{n}}{(n+3)!}& =1-\frac{3}{2}\left(\sum_{n=2}^\infty \frac{(-2)^{n+2}}{(n+2)!}+\sum_{n=2}^\infty \frac{(-2)^{n+3}}{(n+3)!}\right)=1-3e^{-2}\end{aligned}
\sum\limits_{\ell=1}^q(-1)^{\ell-1}{a_2\brack p}_{v^b}{a_1-bp\brack q-\ell}_{v^c}{a_1-bp+\ell-1\brack \ell}_{v^c}\quad\quad\quad\quad=-{a_2\brack p}_{v^b}\sum\limits_{\ell=1}^q{a_1-bp\brack q-\ell}_{v^c}{bp-a_1\brack \ell}_{v^c}={a_2\brack p}_{v^b}{a_1-bp\brack q}_{v^c},
S_{0}= \int d^{2}x e ( 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})
-\partial_x\left(\overline{A_x}\partial_x\overline{w}\right)=\frac{1}{L_z}\partial_x\left(\int_{\Omega_z^1}A_x \partial_xw^\prime_1 \mathrm{d}z+\int_{\Omega_z^2}A_x \partial_xw_1^\prime\left(x,z_\iota\right) \mathrm{d}z+\int_{\Omega_z^2}A_x \partial_x\xi_2^\prime\mathrm{d}z\right).
(a;q)_{\infty}= \prod_{k=0}^\infty (1-aq^k), (a;q)_{n}= \frac{(a;q)_{\infty}}{(aq^n;q)_{\infty}}, (a_1,a_2,\dots,a_j;q)_{\infty}= (a_1;q)_{\infty}(a_2;q)_{\infty}\dots(a_j;q)_{\infty}, (a_1,a_2,\dots,a_j;q)_{n}=(a_1;q)_{n}(a_2;q)_{n}\dots(a_j;q)_{n}.
\Gamma_{kl}=\frac{1}{(4\pi)^2}\int{\rm d}^8z\frac{W^{\alpha kl}W^{kl}_\alpha\bar{W}^{kl}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}kl}}{(\Phi^{kl})^2(\bar{\Phi}^{kl})^2}
\mu_n C_{Sn}(i) - \mu_k C_{Sk}(i) = \alpha^{\rm R}_n(i) + \beta^{\rm R}_k(i),\; \forall k\neq n \mu_n C_{Sn}(i) - (1-\mu_m) C_{mD}(i) = \alpha^{\rm R}_n(i) + \beta^{\rm R}_m(i),\; \forall m.
P(n,z,p) = \sum_{j=0}^{p-1}\frac{(-2)^j}{n^{j+1}}\sum_{k = 1}^\infty \frac{(- 1)^k \zeta (2k) z^{2k}}{k^{p-j}}+ \left (- \frac{2}{n}\right )^p S_{2,n}(z).
I ( x ) = \mathrm{P}\int_{1}^{\infty}\frac{\coth ( \pi x z )}{( z^{2}- 1 )^{3 / 2}}d z = - \frac{1}{2}\oint_{| z - 1 | = \epsilon}\frac{\coth ( \pi x z )}{( z^{2}- 1 )^{3 / 2}}d z + \int_{1 + \epsilon}^{\infty}\frac{\coth ( \pi x z )}{( z^{2}- 1 )^{3 / 2}}d z .
\sum_{i = 1}^{p - 1}b_i^2 < \frac{p^3(p - 1)(2p - 1)}{6}- \frac{p - 2}{2}\cdot cp(p - 1) - p\left(\frac{1}{18}p - 1\right)\left(\frac{2}{5}p\sqrt{p}- 18p\right)\left(\frac{8}{9}p - 1\right) + \frac{p - 2}{2}\cdot c^2 < \frac{1}{3}p^5 - \frac{1}{2}p^3c - \frac{1}{60}p^4\sqrt{p}+ \frac{1}{2}pc^2 + \frac{1}{4}p^4 + p^2c + \frac{1}{6}p^3,
G_sf(x) =\bigg(\sum_{k\in \Bbb Z}\big|[(\delta_0-\phi_k)\ast\nu_{s+k}]\ast\sum_{l\in \Bbb Z}\Delta_{l-k}^2f(x)\big|^2\bigg)^{1/2}&\le \sum_{l\in \Bbb Z}\bigg(\sum_{k\in \Bbb Z}|\Delta_{l-k}[(\delta_0-\phi_k)\ast\nu_{s+k}]\ast\Delta_{l-k}f(x)|^2\bigg)^{1/2}:=\sum_{l\in \Bbb Z}G_s^{l}f(x).
p^u_i :=\sup\{t>0: Q_\epsilon(f^{i-n}(x))\geq e^{-{\epsilon}n}t\textrm{for all}n\geq 0\}, p^s_i :=\sup\{t>0: Q_\epsilon(f^{i+n}(x))\geq e^{-{\epsilon}n}t \textrm{for all}n\geq 0\}.
\begin{aligned}\Vert \sigma^n\Vert_{\widetilde{L}^\infty_T(\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}})}+\Vert u^n\Vert_{\widetilde{L}^\infty_T(\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1})}+\Vert \sigma^n\Vert_{L^1_T(\widetilde{B}^{\frac{N}{2}+1,\frac{N}{2}+2-\alpha})}+ \Vert u^n\Vert_{L^{1}_{T}(\dot{B}^{\frac{N}{2}+1}_{2,1})}\le C X_0,\end{aligned}
A(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0))\le 0 \u(x_0)>0 B(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0))\le 0 \u(x_0)<0 -\mathcal{Q}_\infty \phi(x_0)\le0 \u(x_0)=0
\left( - \partial_{N}- V^{\prime}( \bar{X}) \partial_{N}- \frac 1 2 V^{\prime \prime}( \bar{X}) ( \partial_{N}\bar{X}) + \frac 1 2 \hat{w}^{\prime \prime}( \bar{X}) \right) \xi |_{\partial{\cal M}}= 0 \; .
\|\tilde l_N(t) - l_N(t)\|_{H^{s}_{r}(\mathbb R)}= \| \sum_{j,m}\mu_{j,m}(t)(h_{j,m}- \tilde h_{j,m})\|_{H^{s}_r(\mathbb R)}& \leq \max_{t\in[0,T]}\sum_{j,m}|\mu_{j,m}(t)| \| (h_{j,m}- \tilde h_{j,m})\|_{H^{s}_{r}(\mathbb R)}&<\varepsilon.
C_{f_{\tau}}\ll\int_{\left[0,\infty\right)^{n+1}}\left(\left(\prod_{i=1}^{n}\left|u_{2i-1}\right|\right)^{p+q-3}\left|u_{2n}\right|\int_{\left(S_{p}\times S_{q}\right)^{n}}\mathbb{1}_{\left[1-3\tau,1+3\tau\right]}\left(\left\Vert g_{0}^{-1}\left(u_{1}\eta_{1},u_{1}\eta_{2}\dots,u_{2n}\eta_{2n}\right)\right\Vert \right)d\eta\right)du_{o}.
\tilde{Q}[\psi] + b = \frac{1}{W}\left( \sigma^{ij}- \frac{(h')^2 d^i d^j}{W^2}\right) \left( h'' d_i d_j + h' d_{i;j}\right) = \frac{1}{W}\left( h'' + h' \Delta d - \frac{(h')^2 h''}{W^2}\right) = \frac{1}{W}\left( \frac{h''}{W^2}- h' H(x) \right) = \frac{h''}{W^3}- \frac{h'}{W}H(x),
\int_{B_{1/2}}r^{2-n}|\nabla u|^2 \d x \leq C \int_{B_{1}}r^{2-n}u_r^2 \d x + C \varepsilon^2 \int_{B_{1}}r^{3-n}|\nabla u|^2 \d x \varepsilon \leq \varepsilon_0,
W_{i-1}= \sum_{m=l}^k A_m^{(i-2)}-\sum_{g=1}^n \Lambda_{l,g}^{(i-2)}= (A_l^{(i-2)}-\sum_{g=1}^n \Lambda_{l,g}^{(i-2)}) + \sum_{m=l+1}^k A_m^{(i-2)}
u(s,t)=\int_0^t\int_0^s\{b(\xi,\zeta,W_{\xi,\zeta}(\omega)+u(\xi,\zeta))-b(\xi,\zeta,W_{\xi,\zeta}(\omega))\}\mathrm{d}\xi\mathrm{d}\zeta,\forall\,(s,t)\in[0,1]^2.
\mathcal{F}^T :=\{f:T\to\mathbb{R}\ :\ \mathcal{E}^T(f,f)<\infty\}, \mathcal{E}^T(f,g) :=\frac{1}{2}\sum_{x,y\in T: x\sim y}(f(x)-f(y))(g(x)-g(y))\ \f,g\in{\cal F}^T.
c ( 1 ) ~ = ~ 4 [ ( p_{1}\! \cdot \! p_{2}) ( \epsilon_{1}\! \cdot \! \epsilon_{2}) ( p_{3}^{+}p_{4}^{-}p_{5}^{+}p_{6}^{-}) ( \epsilon_{3}^{-}\epsilon_{4}^{+}\epsilon_{5}^{-}\epsilon_{6}^{+}) ] .
TZ^{p}(X) := TZ^{M}_{p}(D^{\mathrm{perf}}(X) =\mathrm{Ker}\{Z^{M}_{p}(D^{\mathrm{perf}}(X[\varepsilon])) \xrightarrow{\varepsilon =0}Z^{M}_{p}(D^{\mathrm{perf}}(X)\}.
|f_{n-}^{\mu}(x_{2})-f_{n-}^{\mu}(x^{*})| =\left|\int_{\mathcal{X}}t(x_{2}|x_{1})-t(x^{*}|x_{1})d\pi_{n-}^{\mu}(dx_{1})\right| &\leq \int_{\mathcal{X}}|t(x_{2}|x_{1})-t(x^{*}|x_{1})|d\pi_{n-}^{\mu}(x_{1})\leq \epsilon
\gamma_R(X)=- \frac{|x'|^2}{2R}+ 2(n-1)\frac{x_n r}{R}+ 2 \beta \frac{r}{R}, \bar \gamma_R(X)=- \frac{|x'|^2}{2R}- 2(n-1)\frac{x_n r}{R}- 2 \beta \frac{r}{R}
~[x^k][y^n]\frac{1+y-x^2y^2}{(1-xy)(1-xy^2)-xy^3}= [x^k][y^n]f(x,y) +[x^k][y^{n-1}]f(x,y) -[x^{k-2}][y^{n-2}]f(x,y) = \sum_{j=0}^k \binom{n-2j}{k-j}\binom{j}{n-k-j}+\sum_{j=0}^k \binom{n-2j-1}{k-j}\binom{j}{n-k-j-1}~-\sum_{j=0}^k \binom{n-2j-2}{k-j-2}\binom{j}{n-k-j}.
\left|\frac{1}{|z-s|^2}-\frac{1}{s^2}\right| =\frac{1}{s^2}\frac{|2(s-x)x+x^2-y^2|}{(x-s)^2+y^2}&\leq\frac{1}{s^2}\left(\frac{2\delta|x-s|y}{(x-s)^2+y^2}+\frac{x^2+y^2}{y^2}\right)\leq\frac{1}{s^2}\left(\delta+\delta^2+1\right)\in L^{1}(\nu).
||h_\mu||_s =\sum_{i=1}^\infty|(h_\mu)_i|s^i<s+\sum_{i=2}^\infty\prod_{j=1}^{i-1}\frac{|\mu+\lambda_j|}{|\lambda_j|}s^{i}<s+\sum_{i=2}^\infty\prod_{j=1}^{i-1}\frac{|\mu|+|\lambda_j|}{|\lambda_j|}s^{i}&<s+\sum_{i=2}^\infty s \left(\frac{(\varepsilon+\beta)s}{\alpha}\right)^{i-1}<\infty.
[ \bar{\Delta}_{m}^{a}, - i \hbar \frac{\delta}{\delta \eta_{I}}+ \phi^{I}] = 0 , \qquad [ \bar{\Delta}_{A}, - i \hbar \frac{\delta}{\delta \eta_{I}}+ \phi^{I}] = 0 , \qquad [ Y , - i \hbar \frac{\delta}{\delta \eta_{I}}+ \phi^{I}] = 0 ,
\begin{aligned}\alpha = \frac{1-z+\sqrt{1-5z}\sqrt{1-z}}{2}= \frac{u+1}{u^2+3u+1}, \beta = \frac{1-z-\sqrt{1-5z}\sqrt{1-z}}{2}= \frac{u(u+1)}{u^2+3u+1}\end{aligned}
\frac{I_1}{|B_1(0)|}\le \exp\left((b_j(b^\lambda_j\lambda)^{\frac1{1-\lambda}})^\lambda\right) = \exp\left((b_j\lambda)^{\frac\lambda{1-\lambda}}\right) \le \exp\left(b^{\frac\lambda{1-\lambda}}_j\right)
\bar{\Phi}= - \lambda r , \bar{F}_{t r}= Q e^{- ( 2 - \epsilon ) \lambda r}, \bar{T}= 0 , \bar{G}_{\mu \nu}= \left( \begin{array}{c c}{{- f}}&{{0}}\\{{0}}&{{f^{- 1}}}\\ \end{array}\right)
V_i' =\begin{cases}V_i \cap N(v) \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right) i \in [t+4]\setminus I_{small}, V_i \cap \left(\bigcap_{u\in U_1\setminus V_i}N(u)\right) i \in I_{small}.\end{cases}
\mathbf{a}(\psi, \theta) = [e^{j\mathbf{b}(\psi, \theta)^{T}l_{1}},e^{j\mathbf{b}(\psi, \theta)^{T}l_{2}},\cdots,e^{j\mathbf{b}(\psi, \theta)^{T}l_{K}}]^T,
v_r^{\alpha}(Q_{l+1}-R_iT^i)\ge\inf{\{v_r^{\alpha}(Q_l-R_iT^i),v_r^{\alpha}(Y_l)\}}\ge \inf{\{c_0^{\alpha}+ri+v^{\alpha}(R_i),c_l^{\alpha}+ri+v^{\alpha}(R_i)\}}=c_0^{\alpha}+ri+v^{\alpha}(R_i).
c_i+a_i+b_i=k , (0\leq i\leq D) c_i u_{i-1}(\theta_D)+a_iu_i(\theta_D)+b_i u_{i+1}(\theta_D)=\theta_D u_i(\theta_D) , (1\leq i\leq D-1) \gamma_i u_i(\theta_D)+(a_1+2-\gamma_i) u_{i+1}(\theta_D)=0 , (0\leq i\leq D-1)
a(bc\cdot d) =d(a\cdot bc)=d(c\cdot ab)=ab\cdot dc=c(ab\cdot d)=c(db\cdot a) =a(c\cdot db)=db\cdot ac=da\cdot bc=c(da\cdot b)=c(ba\cdot d) =d(c\cdot ba)=d(a\cdot cb)=d(b\cdot ac)=ac\cdot db=b(ac\cdot d) =b(dc\cdot a)=a(b\cdot dc)=dc\cdot ab=(ab\cdot c)d \Rightarrow a(bc\cdot d) =(ab\cdot c)d.
P_s[f^\frac1p]^{-p}( Ax + c)\gamma(Ax + c)P_sf(A^*y) \gamma(A^*y- c) &\le\frac{1}{(2\pi)^n}e^{- \frac12 ( |Ax+c|^2 + |A^*y - c|^2 )}e^{\frac12 |Ax+c-A^*y|^2}&= \frac{1}{(2\pi)^n}e^{- \frac12 |c|^2}e^{-\langle x, y\rangle}
\psi^{\otimes n}(\xi)^* \left( \sum_{n_i = m_i}\psi^{\otimes n_i}(\xi_i) \psi^{\otimes m_i}(\eta_i)^* \right) \psi^{\otimes n}(\xi) = \psi^{\otimes n}(\xi)^* y_0 \psi^{\otimes n}(\xi) &= \psi^{\otimes n}(\xi)^* \kappa_n^{\psi, \pi}(T) \psi^{\otimes n}(\xi) = \psi^{\otimes n}(\xi)^* \psi^{\otimes n}(T\xi) = \pi ( \langle \xi, T\xi \rangle) = \pi (g).
\sigma(\tilde{t}, \tilde{x_1}, \tilde{x_2}, \tilde{x_3}, \tilde{m}) =\tilde{\sigma}(\tilde{t}, \tilde{x_1}, \tilde{x_2},\tilde{x_3},\tilde{m}) \forall (\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3},\tilde{m}) \in \Gamma_-,
J_\nu(z;q)=J_\nu^{(3)}(z;q): =\frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty}z^{\nu}{_1\phi_1}(0;q^{\nu+1};q,qz^2) =\frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty}\sum_{n=0}^\infty (-1)^n\frac{q^{n(n+1)/2}z^{2n+\nu}}{(q;q)_n(q^{\nu+1};q)_n}, z\in\mathbb{C},
S^{\left( \Sigma_{1+1}\right)}=-2\pi i\theta f\left( \overline{t_{*}}\right) g\left( t_{*}\right) \int_{\Sigma_{1+1}}\left[ i\overline{\psi}^{L}\sigma^{a}D_{a}\psi^{L}-\overline{\psi}^{L}m\psi^{L}\right] ,
T^{\alpha \beta \{a b \}}= \frac{1}{2}( - 1 )^{\varepsilon_{\beta}}C^{\alpha \{a}C^{\beta b \}}, \; \hat{T}^{\alpha \beta}\varepsilon^{a b}= - \frac{1}{2}( - 1 )^{\varepsilon_{\beta}}\varepsilon_{c d}C^{\alpha c}C^{\beta d}\varepsilon^{a b}
\sum_{s=1}^{n}\frac{b''_s}{2(s-1)!}\sum_{\ell=0}^{n-s}\sum_{j=0}^{\ell}\sum_{k=0}^{n-s-\ell}a_ja_{\ell-j}\frac{(n-s-1+k)!}{(n-s-\ell)!}B_{n-s-\ell, k}(-a_1, -2!a_2, \ldots)\ell(s+n-2)(j+1)(\ell-j+1)
\gamma(\lambda,z)=(1-F_{22}(\lambda)z)^{-1}F_{21}(\lambda)\index{$\gamma(\lambda,z)$}\; \;\eta(\lambda,z)=\begin{bmatrix}1 z(1-F_{22}(\lambda)z)^{-1}F_{21}(\lambda) \end{bmatrix}=\begin{bmatrix}1 z\gamma(\lambda,z) \end{bmatrix}
|I_i|\leq \sum_{N_{f_i}}\big|M_{f_if_i}(\sigma_{\rm HHJ}-\Pi_{\rm HHJ}\sigma)\tau_{f_i}:\int_{N_{f_i}}(\sigma-\Pi_{\rm HHJ}\sigma)dx\big| +\sum_{T_{f_i}}\big|\int_{T_{f_i}}M_{f_if_i}(\sigma_{\rm HHJ}-\Pi_{\rm HHJ}\sigma)\tau_{f_i}:(\sigma-\Pi_{\rm HHJ}\sigma)dx\big|.
|I_i \cap [j_i+1, j_{i+1}]|\ = \ |J_i \cap [j'_i+1, j'_{i+1}]| \mbox{and}|I_{i+1}\cap [j_i+1, j_{i+1}]|\ = \ |J_{i+1}\cap [j'_i+1, j'_{i+1}]|.
[\mathbb{E}^V-\mathbb{E}^G] \theta [\int_I x(E)q(y(E)+\tilde{y}(E))dE]=\sum_{\gamma=1}^{\gamma_{\max}}\mathbb{E}\left( \theta[(\int_I x^S q(y^S+\tilde{y}^S))]-\theta[(\int_I x^T q(y^T+\tilde{y}^T))] \right).
dY(t,x)=[A_{u(t,x,Z)}Y(t,x)+a(t,x,Y(t,x),u(t,x,Z),Z)]dt+b(t,x,Y(t,x),u(t,x,Z),Z)dB(t) &+\int_{\mathbb{R}}c(t,x,Y(t,x),u(t,x,Z),Z,\zeta)\tilde{N}(dt,d\zeta); (t,x)\in (0, T)\times D.
H^z\cap \langle g\rangle = H^z\cap G\langle g^s\rangle \cap \langle g\rangle= H^z\cap \langle g^s\rangle &= H^z\cap G_1\langle g^s\rangle \cap \langle g^s\rangle=(H\cap G_1\langle g^s\rangle )^z\cap \langle g^s\rangle &=H_1^z\cap \langle g^s\rangle \leq \langle g_0\rangle
|\lambda \cap S_\mu(m)| + |\lambda \cap T_\mu(m)| - \frac{|\lambda|}{2}&= |\sigma \cap (S_\mu(m) \times T_\mu)| + |\sigma \cap (S_\mu \times T_\mu(m))| - |\sigma| &= |\sigma \cap (S_\mu(m) \times T_\mu(m))| - |\sigma \cap (\overline{S_\mu(m)}\times \overline{T_\mu(m)})|.
X(t) =x+\int_0^t f(s, X_s,X(s))ds +\int_0^t g(s,X_s,X(s))d W(s)+\int_0^t \int_{\mathbb{R}_0}h(s,X_s,X(s))(z)\tilde N(d s,d z) (X_0,X(0)) =(\eta,x).
i\Theta_{E,h,x_0}=i\sum_{j,k}\Theta_{jk}dz_j\wedge d\overline{z}_k=i\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}dz_j\wedge d\overline{z}_k\otimes e^*_\lambda\otimes e_\mu,~\,\, \overline{c}_{jk\lambda\mu}=c_{kj\mu\lambda}.
\delta_{\xi}I_{( m )}= \int_{\cal M}\sqrt{- g}d^{4}x \left[ \frac 1 2 T^{\mu \nu}{\cal L}_{\xi}g_{\mu \nu}+ \nabla_{\mu}( \theta_{( m )}^{\mu}( \xi ) - \xi^{\mu}L_{( m )}) \right] .
\omega=\log|{q^2|c_2|\over c_3|\Lambda|}{\cosh^2\sqrt{c_3\over 2}(z-z_3)\over\{-{c_1\over2}\sin\sqrt{c_2\over 2}(z-z_2)-{|c_2|\over2}\cos\sqrt{c_2\over 2}(z-z_2)\}^2}|.
S_X(f) = \sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \sum_{\gamma_{\alpha}}f(\gamma_{\alpha}) &= \sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \frac{1}{2\pi i}\left(\int_{(c)}- \int_{(1-c)}\right) \frac{L'(\chi_{\alpha},s)}{L(\chi_{\alpha},s)}f\left(-i\left(s-\frac{1}{2}\right)\right) ds.
H_{\mu \nu}\rightarrow H_{\mu \nu}+ \frac{1}{m^{2}}\bigl ( \lambda_{1}R_{\mu}{}^{\alpha}{}_{\nu}{}^{\beta}H_{\alpha \beta}+ \lambda_{2}R^{\alpha}{}_{( \mu}H_{\nu ) \alpha}+ \lambda_{3}R H_{\mu \nu}+ \lambda_{4}G_{\mu \nu}R^{\alpha \beta}H_{\alpha \beta}\bigr ) + O \left( \frac{1}{m^{4}}\right)
J(p)=\int_{-\infty}^{\infty}p(x)lnp(x)dx-\eta_{0}(\int_{-\infty}^{\infty}p(x)dx-1)-\sum_{j=1}^{n}\lambda_{j}(\int_{-\infty}^{\infty}f_{j}(x)p(x)dx-a_{j})
f ( y ) = A e^{- y^{2}/ 4}M \left( \frac{a}{2}+ \frac{1}{4}, \frac{1}{2}, \frac{y^{2}}{2}\right) + B y e^{- y^{2}/ 4}M \left( \frac{a}{2}+ \frac{3}{4}, \frac{3}{2}, \frac{y^{2}}{2}\right) ,
\phi_3=-c\phi_1-\frac{\epsilon}{\kappa}\phi_4-\frac{c}{\kappa}\phi_2+\frac{c}{\kappa}\rightarrow 0,\ \phi_1\rightarrow\phi_1^*,\ \phi_2\rightarrow 0 \ \ \phi_4\rightarrow 0\ \z\rightarrow -\infty
h=-\sum_{I=1}^{q}(\omega^{I})^{2}+\sum_{A=q+1}^{q+p}(\omega^{A})^{2}, \;\; h=g_{ij}dx^{i}dx^{j}, \;\; \mathrm{rank}|g_{ij}|=p+q<m, g_{ij}=g(\partial_{i},\partial_{j})=-\sum_{I=1}^{q}\omega^{I}_{i}\omega^{I}_{j}+\sum_{A=q+1}^{q+p}\omega^{A}_{i}\omega^{A}_{j},
I(\overline{C}^{N};\overline{F}^{N}|\overline{S}^{N}=s^{N})=\frac{1}{2}\sum_{i=1}^{N}log_{2}(\frac{|g_{i}^{2}s_{i}^{2}C_{U}+(\sigma_{v}^{2}+\sigma_{n}^{2})I|}{|(\sigma_{v}^{2}+\sigma_{n}^{2})I|}).
\begin{array}{l}{{{\cal K}={\cal A}+ V S_{m}^{\dagger}+ S_{m}W^{\dagger}+ S_{m}\tilde{{\cal K}}S_{m}^{\dagger}= \left( \begin{array}{c c}{{{\cal A}}}&{{V}}\\{{W^{\dagger}}}&{{\tilde{{\cal K}}}}\\ \end{array}\right) \, ,}}\\{{\varphi = X S_{m}^{\dagger}+ S_{m}\tilde{h}S_{m}^{\dagger}= \left( \begin{array}{c c}{{0}}&{{X}}\\{{0}}&{{\tilde{h}}}\\ \end{array}\right) \, .}}\\ \end{array}
\mathcal{L}_{\mathcal{I}^{({\rm CCU})}_{\rm d \rightarrow u}}(s)=\mathcal{L}_{\mathcal{I}^{({\rm CEU})}_{\rm d \rightarrow u}}(s)= \exp\left( - \frac{2}{\eta}\pi^{2}\lambda \left( s P_{\rm{d}}\right)^{\frac{2}{\eta}}\csc \left( \frac{2 \pi}{\eta}\right) \right),
H_{| 2}= \frac 1{4 l_{p}^{2}}\int_{\Sigma}d^{3}x \sqrt{g}G^{i j k l}\left[ K^{- 1 \bot}\left( x , x \right)_{i j k l}+ \left( \triangle_{2}\right)_{j}^{a}K^{\bot}\left( x , x \right)_{i a k l}\right]
\int_{t-y}^t \frac{|N_{h}(u)|}{u^2}du \ll B \frac{y}{t}(\frac{1}{(1-\kappa)\log t}\int_{t^{\kappa}}^{t}\frac{|M_{h}(u)|\log u}{u^2}du+R_h(\lambda)\int_1^{t^{\kappa}}\frac{M_{|g|}(u)}{u^2\log^{\lambda}(3u)}du &+ |A|\mu \int_1^{t}\frac{M_{|g|}(u)}{u^2}du) ,
\xi^\prime=y_2 y_1^\prime=\frac{y_2(y_1+y_2+y_3-z_0)}{n(\bar v)}y_2^\prime=-y_2-\frac{y_2(\xi-\xi_0+2(y_1+y_2)+y_3-(x_0+z_0+1))}{n(\bar v)}y_3^\prime=\frac{y_2((y_2+y_3)(y_1+y_2+y_3-r_0-z_0)+r_0(z_0-y_1))}{n(\bar v)}
[\begin{matrix}T_{1}^{(n)}&T_{2}^{(n)}&T_{3}^{(n)}\end{matrix}]=[\begin{matrix}n_{1}&n_{2}&n_{3}\end{matrix}]\cdot[\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}].