harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Xi_{[ 1 / 2 ]}= \left( \begin{matrix}{e^{i \phi}}&{0}\\{0}&{e^{- i \phi}}\\ \end{matrix}\right) , \quad \Xi_{[ 1 ]}= \left( \begin{matrix}{e^{i 2 \phi}}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{e^{- i 2 \phi}}\\ \end{matrix}\right) \quad ,
\begin{array}{c c c}{{\delta W :}}&{{}}&{{\left\langle \mathrm{{\bf ~ F}}^{n - 1}B_{a}\right\rangle = 0}}\\{{\delta \Psi :}}&{{}}&{{\left\langle \mathrm{{\bf ~ F}}^{n - 1}Q_{\alpha}\right\rangle = 0}}\\ \end{array}
y[n+1] = \left( 1- w^2 - \frac{1}{\tau}\right)y[n] + w^2T + \frac{T}{\tau}+ w^2\frac{c[n]}{\tau}&= \left( 2-w^2-\frac{1}{\tau}\right)y[n] &- \left( 1+\frac{1-\tau}{\tau}w^2 - \frac{1}{\tau}\right)y[n-1] + \frac{w^2}{\tau}T
\nabla \alpha (e_k,e_j)+\nabla \alpha (e_j,e_k) =-(\frac{i}{2}e_j.e_k.\psi,\psi)-(e_j.\psi,\frac{i}{2}e_k.\psi)-(\frac{i}{2}e_k.e_j.\psi,\psi)-(e_k.\psi,\frac{i}{2}e_j.\psi) =-\frac{i}{2}([e_j,e_k].\psi,\psi)-\frac{i}{2}([e_k,e_j].\psi,\psi)=0.
M=\left(\begin{matrix}0 \alpha^{5}z^{2}+ \alpha^{5}z \alpha z^{3}+ \alpha^{5}z^{2}+ \alpha^{6}z \alpha^{2}z^{3}+ z^{2}+ \alpha z 0 \alpha^{4}z^{2}+ \alpha^{2}z + 1 z^{3}+ \alpha \alpha z^{3}+ \alpha z^{2}+ \alpha^{3}z 0 \alpha z \alpha^{4}z^{2}+\alpha^{2}z \alpha^{5}z^{2}+\alpha^{2}z 0 0 0 1\end{matrix}\right).
-\frac{1}{e}\partial_{\mu}A^{\nu}{\bigg|}_{\tau_{s}}=\bigg{\{}\frac{K_{\mu}a^{\nu}}{\rho}+\frac{u^{\nu}}{\rho^{2}}\partial_{\mu}\rho\bigg{\}}{\bigg|}_{\tau_{s}}=\frac{1}{\rho^{2}}{\bigg(}K_{\mu}W^{\nu}+u_{\mu}u^{\nu}{\bigg)}{\bigg|}_{\tau_{s}},
\big| R_{r,n}(w)\big| = \left|\sum\limits_{m\not= 0}\left( \frac{w}{w+2\pi im}\right)^{r+1}\, \frac{\int\limits_0^\infty e^{-t}t^r \Big( \Big( \frac{t}{w+2\pi im}+ 1\Big)^n + \Big( \frac{t}{w+2\pi im}- 1\Big)^n\Big)\, dt}{\int\limits_0^\infty e^{-t}t^r \Big( \Big( \frac{t}{w}+ 1\Big)^n + \Big( \frac{t}{w}- 1\Big)^n\Big)\, dt}\right|\, .
0=\mathfrak{q}\left(d\mathfrak{s}\right) = \mathfrak{q}\left(\left(\partial_{z}u\right)\theta\otimes\mathfrak{w}+ \left(\partial_{\overline{z}}u\right)\overline{\theta}\otimes\mathfrak{w}\right) = \left(\partial_{\overline{z}}u\right)\overline{\theta}\otimes\mathfrak{w}
\Pi ( p^{2}, \mu ) \simeq \frac{b_{0}}{8 \pi^{2}}\left[ \log \frac{z_{1}}{z_{0}}+ \log m z_{0}- \frac{1}{4}\log \mu z_{0}- \frac{1}{4}\log \mu z_{1}+ \frac{m^{2}z_{0}^{2}}{8}\left( \log \frac{z_{1}}{z_{0}}- \frac{1}{2}\right) \right] \; .
\frac{m_{Z}^{2}}{2}=\frac{(m_{H_{d}}^{2}+\Sigma_{d}^{d}(i))-tan^{2}\beta(m_{H_{u}}^{2}+\Sigma_{u}^{u}(j))}{tan^{2}\beta-1}-\mu^{2}\simeq-m_{H_{u}}^{2}-\Sigma_{u}^{u}(i)-\mu^{2}
\psi^{-1}(u_{h_g})=1, &\gamma_{\psi'}^{-1}(\det c_{h_g})=\gamma_{\psi'}^{-1}(1-z_1z_2)=1, &\delta^{1/2}_{B_n}(b_g)=|1-z_1z_2|^{-1}, &\chi(\mathfrak{s}(b_g))=|1-z_1z_2|^{-s_i-s_{n-i}}=|1-z_1z_2|^{-s_i+s_{i+1}}.
\begin{aligned}\frac{d}{dt}H_t^k\leq \frac{k}{(n-1)^2}\mathbb{E}\left[\left|\sum_{j=2}^k \left(K(X_t^1-X_t^j)-\langle \mu_t,K(X_t^1-\cdot)\right)\right|^2\right] &+\frac{k(n-k)^2}{(n-1)^2}\mathbb{E}\left[\left|\mathbb{E}[K(X_t^1-X_t^n)\mid X_t^1,\cdots,X_t^k]-\langle \mu_t,K(X_t^1-\cdot)\right|\right] \end{aligned}.
\frac{\widehat{\mathfrak{Q}}_n(z)-\mathfrak{P}_{n}(z)}{\mathfrak{L}_{n}(z)}= \sum_{k=0}^{m}\left(\frac{\lambda_n}{\lambda_{n-k}}-1\right)\frac{b_{n,n-k}}{c_n^k}\; \frac{\mathfrak{L}_{n,n-k}(z)}{\mathfrak{L}_{n}(z)}.
\bigl(Ax\bigr)(t) =\bigl(\varphi^{-1}\bigl(\{(Tz)_i\}\bigr)\bigr)(t)=\bigl[(Tz)_k\bigr](t-k) &=\sum\limits_{m\in\mathbb Z^c}\bigl(b_{km}x_{k-m}\bigr)(t-k),t\in[0,1)^c+k.
M_{(k_1, \ell_1, \ldots, k_r, \ell_r)}= \left(\begin{matrix}1 1 0 1\end{matrix}\right)^{k_1}\left(\begin{matrix}1 0 1 1\end{matrix}\right)^{\ell_1}\dots \left(\begin{matrix}1 1 0 1\end{matrix}\right)^{k_r}\left(\begin{matrix}1 0 1 1\end{matrix}\right)^{\ell_r}.
u_{i}= \mathrm{Im}(\mu_{i}(\sigma \upsilon)) = \mathrm{Im}(\mu_{i}(\sigma \nu)) \stackrel{\eqref{reverse_formula}}{=}\sum_{j=1}^{i}\mathrm{Im}(\sigma \nu)_j = \sum_{j=1}^{i}t_{\sigma^{-1}j}
\widetilde{U}= (\widetilde{u}_1, \widetilde{u}_2, \ldots,\widetilde{u}_l)\in \mathbb{R}^{m\times l},\widetilde{V}= (\widetilde{v}_1, \widetilde{v}_2,\ldots,\widetilde{v}_l)\in \mathbb{R}^{n\times l}
\lim_{b\uparrow\infty}\frac{K_a^{(q,r)}(b)}{J_a^{(q,r)}(b)}=\frac{\tilde{Z}^{(q,r)}(-a)- \psi'(0+){Z^{(q)}(-a, \Phi(q+r))}}{qZ^{(q)}(-a,\Phi(q+r))}.
\tilde{P}_{n}(t, S) = ~ p(t, S) + \sum_{i=0}^{n}rK e^{-r(t - t_{i})}\aleph \Big(-d_{2}(S, t - t_{i}, \mathcal{B}_{n}(t_{i}))\Big) - \sum_{i=0}^{n}\delta S e^{-\delta (t- t_{i})}\aleph \Big(-d_{1}(S, t - t_{i}, \mathcal{B}_{n}(t_{i}))\Big).
-\int_0^T(\theta, \phi_t)dt-\int_0^T(u\theta, \nabla\phi)dt+\nu\int_0^T(\Lambda^{\frac{\alpha}{2}}\theta, \Lambda^{\frac{\alpha}{2}}\phi)dt=(\theta_0, \phi(x,0))+\int_0^T(f, \phi)dt.
j^{\vee}_{K_1,K_2}(z_{K_2}(v))=U^{h_{K_2}\left(v^*-\Lambda_{\overline{K_2}}\right)-h_{K_1}\left(v^*-\Lambda_{\overline{K_2}}\right)}z_{K_1}\left(v-\Lambda_{K_2-K_1}\right).
i{\cal D}^{- 1}( p^{2}) = \mu^{2}- ( p^{2}+ \omega^{2}) + \frac{N \beta^{2}}{2 \pi}\left( f \biggl ( - \frac{p^{2}}{\beta^{2}\sigma^{2}}\biggr ) - f \biggl ( \frac{\omega^{2}}{\beta^{2}\sigma^{2}}\biggr ) \right)
\begin{cases}\eta(0,t)=0,\,\,\eta(L,t)=0,\,\,\eta_{x}(0,t)=f(t) \,\, (0,T), w(0,t)=0,\,\,w(L,t)=0,\,\, w_{x}(L,t)=g(t) \,\, (0,T)\end{cases}
\begin{array}{|c|c|c|}\hline{\rm multiplet}{isospin}{\rm energy}\hline SD\left(E_0,2,J_0\right) 0\le J_0< k E_0={1\over 4}\sqrt{H_0+36}\hline SD\left(E_0,3/2,J_0\right) 0\le J_0< k E_0={1\over 4}\sqrt{H_0+36}-{3\over 2}\hline SD\left(E_0,3/2,J_0\right) 0\le J_0\le k E_0={1\over 4}\sqrt{H_0+36}+{3\over 2}\hline \end{array}
\begin{aligned}\frac{1}{\tau}\big\|(v_n-v_{n-1})_{n=k}^N\big\|_{L^p(X)}+ \big\|(v_n )_{n=k}^N\big\|_{L^p(D)}&\le C\Big(\big\|(f_n)_{n=k}^N\big\|_{L^p(X)}+\frac{1}{\tau}\big\|(v_i-v_{i-1})_{i=1}^{k-1}\big\|_{L^p(X)}+ \big\|(v_i )_{i=1}^{k-1}\big\|_{L^p(D)}\Big) . \end{aligned}
\hat{E}^{A}{}_{M}= \left( \begin{array}{c c}{{\begin{matrix}{{e^{0}{}_{t}}}&{{0}}\\{{0}}&{{e^{i}{}_{\underline{{i}}}}}\\ \end{matrix}}}&{{0}}\\{{E^{a}{}_{{\beta}}A^{( 1 ){\beta}}{}_{\underline{{i}}}}}&{{E^{a}{}_{\alpha}}}\\ \end{array}\right)
\sum_{n=0}^\infty \frac{q^{n^2}}{(q;q)_n}= \frac{1}{(q, q^4; q^5)_\infty}, \sum_{n=0}^\infty \frac{q^{n^2+n}}{(q;q)_n}= \frac{1}{(q^2, q^3; q^5)_\infty},
P_h(z,s) =\sum_j \langle P_h(\cdot,s),\mu_j \rangle \mu_j(z) + \frac{1}{4\pi}\int_{-\infty}^\infty\langle P_h(\cdot,s),E(\cdot,\tfrac{1}{2}+it)\rangle E(z,\tfrac{1}{2}+it)\,dt.
\partial_{\tau}|A_1|^2 +c\cdot\nabla |A_1|^2 = i (A_1A_2A_3+\overline{A_1A_2A_3}), \partial_{\tau}|A_2|^2 +c\cdot\nabla |A_2|^2 = -i (A_1A_2A_3+\overline{A_1A_2A_3}), \partial_{\tau}|A_3|^2 +c\cdot\nabla |A_3|^2 = -i (A_1A_2A_3+\overline{A_1A_2A_3}).
{\widehat P}_{ab}(p)= [n+1-\min(a,b)]\;p_{a-1}\;p_{b -1}- \sum_{i=\max(1,a+b-n-1)}^{\min(a,b)}\,(a+ b - 2i)\;p_{i-1}\;p_{a+b - 1-i}\,,\forall\,a,b=1,\ldots,n\,,
\int_0^1 \, d\rho \frac{1}{[(\rho x - y)^2]^{\frac{N}{2}-1}}= \int_0^\infty d\rho \, \frac{1}{[(\rho x - y)^2]^{\frac{N}{2}-1}}- \int_0^1 d\rho \, \frac{\rho^{N-4}}{[(x - \rho y)^2]^{\frac{N}{2}-1}}
E_{P^W}\left[\sup_{f\in\mathcal M_{\tilde\delta_n}}\Bigl\vert\frac{1}{\sqrt n}\sum_{i=1}^nW_i f(X_i)\Bigr\vert\right] \le K'v^{1/2}\int_0^{\tilde\delta_n/\|F\|_{L^2_Q}}(-\ln\epsilon) d\epsilon =K'v^{1/2}(\tilde\delta_n-\tilde\delta_n\ln(\tilde\delta_n)).
{\cal U}_{\cal A}=\exp\left(-\frac{2\pi{\mathrm i}}{l}\frac{\delta}{2b}\left(\frac{l}{2\pi}\hat{p}\right)^2\right)\exp\left(-\frac{2\pi{\mathrm i}}{l}\frac{1}{2b}\left[(1-a)\frac{l}{2\pi}\hat{p}+b\frac{l}{2\pi}\hat{q}\right]^2\right)
\mathcal{L}= - \overline{{{\Psi}}}( x ) \left[ \Gamma_{\mu}\left( \partial_{\mu}- i e A_{\mu}- g A_{\mu}^{B}I^{B}\right) + m \right] \Psi ( x ) - \frac 1 4 \mathcal{F}_{\mu \nu}^{2}- \frac 1 4 \left( F_{\mu \nu}^{B}\right)^{2}
(-\nabla_{\tilde X,\tilde Y,\tilde t}\Psi(P_j), H) = (\nabla_{X,Y,t}\Psi(P_j), H) \in \overline{J}^{2,+}(-u)(X_j,Y_j,t_j) = -\overline{J}^{2,-}u(X_j,Y_j,t_j), (\nabla_{\tilde X, \tilde Y, \tilde t}\Psi(P_j), E) \in \overline{J}^{2,+}\phi(\tilde X_j, \tilde Y_j, \tilde t_j),
{\cal E}^{n+1}_{\boldsymbol{v}}-{\cal E}_{\boldsymbol{v}}^n={\cal E}^{n+1}-{\cal E}^n-\triangle t_n\displaystyle\sum\limits_{s(i,j)}l_{(i,j)}\mu_{(i,j)}\left|\boldsymbol{v}_i-\boldsymbol{v}_j\right|^2,
\widetilde{\mu_i^n}(p)=\inf_{\substack{W\subset H^1_0(A_n) dim W=i}}\max_{\substack{v\in W v\neq 0}}\frac{\int_{A_n}\left(|\nabla v(x)|^2-p|u_p(x)|^{p-1}v(x)^2\right) dx}{\int_{A_n}\frac{v(x)^2}{|x|^2}dx}
\int_{0}^{\infty}b^{s-1}_{1}F_{1}\left( \alpha ,\beta ,-b\right) db=\frac{\Gamma \left( \alpha -s\right) \Gamma \left( \beta \right) \Gamma\left( s\right)}{\Gamma \left( \alpha \right) \Gamma \left( \beta -s\right)}.
X_{0}^{\lambda}=\left\| \bigwedge_{i=1}^{p}\nabla D_i\wedge\bigwedge_{j=1}^{k}\nabla I_j \right\|_{k+p}^{-2}\cdot\sum_{i=1}^{p}(-1)^{n-i}(-\lambda) (D- d_i) \Theta_i,
{\mathbf{B}}_{Q,n-k+1}= \prod_{k+1\le m\le n}^{\gets}\left({\mathbf{I}}-({\mathbf{T}}_{k})^{2}\,{\mathbf{T}}_{m,k+1}\right)\cdot \prod_{k+1\le m\le n}^{\to}\left({\mathbf{I}}-{\mathbf{T}}_{m,k}\right)^{-1}.
\left|\int_{\Omega}f_\delta\,\varphi\,dx-C(\beta)\,\int_\Omega |x|^{-\beta\,(p-1)-s\,p}\,\varphi\,dx\right| \le \|\varphi\|_\infty\,R^{-\beta\,(p-1)-s\,p}\,|\mathrm{supp}(\varphi)|\,\Big|C(\beta;\delta)-C(\beta)\Big|.
\mathbf{II}= \int_{\partial B_R(0) \cap \Omega_+}A \cdot N \, dS + \int_{\partial B_R(0) \cap \Omega_-}A \cdot N \, dS \to \frac{\gamma_n}{2}\rho_+ p_+ \cdot c - \frac{\gamma_n}{2}\rho_- p_- \cdot c,
\begin{aligned}\|v\|_*\leq~ 2\left(\|u_0\|_{H^1}+\|u_1\|_{L^2}\right)+2\rho_1\int_0^T\left\|\left(\frac{e^u}{\int_{M}e^u}-\frac{1}{|M|}\right)\right\|_{L^2}\mathrm{d}s &+2\rho_2\int_0^T\left\|\left(\frac{e^{-u}}{\int_{M}e^{-u}}-\frac{1}{|M|}\right)\right\|_{L^2}\mathrm{d}s.\end{aligned}
\frac{d}{dt}\{(1+t) \tilde{G}(v(t)) \}= \tilde{G}(v(t)) + (1+t) \frac{d}{dt}\tilde{G}(v(t)) \leq \tilde{G}(v(t)) - \frac{b_0}{32}(1+t) Z_0 (v(t)).
\frac{T_{\mathrm{d}}}{M_{\mathrm{t o t}}( 0 )}\, = \, \frac{2}{( d - 2 )}\left[ b ( | \alpha | ) | \alpha | \left( 1 \, + \, \sqrt{\frac{( d - 2 ) | \alpha |}{8 \pi}}\right) \right]^{- 1 / 2}\,{.}
S = \int_{0}^{1}\left[ - \frac{1}{2 e}\left( \dot{x}^{\mu}- i \psi^{\mu}\chi + i \varepsilon^{\mu \nu \rho \zeta}b_{\nu}\psi_{\rho}\psi_{\zeta}+{\frac{\alpha}{2}}b^{\mu}\right)^{2}- i \psi_{\mu}\dot{\psi}^{\mu}\right] d \tau \; ,
1 = \int d^{2}{\bf r}\, | \Psi({\bf r})|^2 =A_{l}^{2}\, 2\pi \kappa^{-2}\, \left\{{\mathcal K}_{i \Theta}(\kappa a) + \left( \frac{\kappa}{\tilde{k}}\right)^{2}\,\left[\frac{K_{i \Theta}(\kappa a)}{J_{l}( \tilde{k}a )}\right]^{2}\,{\mathcal J_{l}}( \tilde{k}a ) \right\}\; ,
\Delta_{\alpha}( S^{\pm}) = S^{\pm}\otimes q^{( \pm \alpha - 1 ) S^{3}}+ q^{( \pm \alpha + 1 ) S^{3}}\otimes S^{\pm}\, , \qquad \Delta_{\alpha}( S^{3}) = S^{3}\otimes 1 + 1 \otimes S^{3}
{}|I_4| \le c \frac{\Delta_n}{T_n}\frac{1}{(\prod_{l = 1}^d h_l)^2}\sum_{j = j_D + 1}^{n - 1}e^{- \rho \Delta_n j}& \le c \frac{\Delta_n}{T_n}\frac{1}{(\prod_{l = 1}^d h_l)^2}e^{- \rho \Delta_n (j_D + 1)}& \le c \frac{\Delta_n}{T_n}\frac{1}{(\prod_{l = 1}^d h_l)^2}e^{- \rho D}.
H_{2,1}\!=\!\int_{0}^{\theta_0}\!\!\!\int_{\frac{1}{2}}^{\frac{3}{2}}\frac{dr d\theta}{|\sin(\theta+\varphi)|^{\tau}|re^{i\theta}-1|^{q}}+\int_{\theta_0}^{\frac{\pi}{2}}\!\!\!\int_{\frac{1}{2}}^{\frac{3}{2}}\frac{dr d\theta}{|\sin(\theta+\varphi)|^{\tau}|re^{i\theta}-1|^{q}}\!=\!H_{2,1}^1+H_{2,1}^2.
\hat{B}( C , t ) = 3 D \mathrm{e x p}[ i g ( \int d^{3}x [ \hat{\vec{E}}( \vec{x}, t ) \int_{C}= d s \,{\vec{A}}^{A}( \vec{y}( s ) ) +{\hat{\Pi}}_{1}^{*}( \vec{x}, t ) \int_{C}d s \, \phi_{1}^{A}( \vec{y}( s ) ) ] ) ]
\|\lambda(b_N)\eta_{N-1}\|_2^2+\|\rho(b_N)\eta_{N-1}\|_2^2 &\le\big\|\lambda(b_N)\eta_{N-1}-\rho(b_N)\eta_{N-1}+(\lambda(b_N)-\rho(b_N))F(x)\hat{\;}\big\|_2^2 &=\|(\lambda(b_N)-\rho(b_N))(\eta-\eta_{N-1})\|_2^2 &\le4\|b\|^2\;\|\eta-\eta_{N-1}\|_2^2.
(\nabla v_{\alpha q}^{\underline\alpha})v_{\underline\alpha}^{~\beta r}={1\over 4}\delta_q^r\hat\Omega^{ab}(\gamma_{ab})_{\alpha}^{~\beta}+{1\over 4}\delta_\alpha^{~\beta}\hat\Omega^{ij}(\gamma_{ij})_q^{~r},
\bar\lambda(\sigma)=\sum_{i=1}^{\epsilon^{-2}T/|I|}\sum_{x\in I_{i}}\bar c^i(x,\sigma),\;\;\;\;\bar p(\sigma,\sigma')=[\bar\lambda(\sigma)]^{-1}\sum_{i=1}^{\epsilon^{-2}T/|I|}\sum_{x\in I_i}\bar{c}^i(x,\sigma)\mathbf{1}_{\sigma'=\sigma^{x}}.
\alpha_i=\sum\limits_{j=1}^s a_{ij}I_{\Omega_{ij}\times (\phi_{u_i,\nu})^{-1}(C_{ij})}, \ \ \alpha_i|_{\Omega_i^{*}}=0, \ \ \Omega_i^{*}\in \mathcal F_i\otimes (\phi_{u_i,\nu})^{-1}(\mbox{Bor}(Z^\nu_{u_i}))
F_v(x,t) = g_1(t) * F_v(\cdot,0)+ \int_0^t e^{(1-v_0^2)(t-\tau)}g_1(t-\tau)* \left(((v_0^2-(v_0+V)^2) + \varphi_{1})F_v - F_w + \varphi_3\right) d\tau.
I_{int}=\int d^n x\{\sum_a \big({-1 \over 2(p_{a}+1)!}H^a_{\mu_1\ldots \mu_{p_{a}+1}}H^{a \mu_1\ldots \mu_{p_{a}+1}}+g_aS^{a\mu_1 \ldots\mu_{p_a}}B^a_{\mu_1 \ldots \mu_{p_a}}\big)\},
\left\{- \frac{\partial^{2}}{\partial \xi^{2}}+ \frac{\alpha}{\sin^{2}\xi}+ \Omega \left( \xi \right) + \Omega \left( \xi - \pi \right) \right\}f_{l}( \xi ) = B_{l}^{2}~ f_{l}( \xi ) , ~ \mathrm{w h e r e}~ ~ B_{l}= \frac{b_{l}}{3}~ ~ .
-\sin\psi\cot\eta\cdot\left(\frac{F[p,q](i\mu_{0}+1)w_{1}[p,q](i\mu_{0}+1)w_{3}[p,q](i\mu_{0}+1)}{w_{2}[p,q](i\mu_{0}+1)}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\psi}u_{n}(\mu)\mid_{\mu=\mu_{0}-i}
t_{0 n}^{\ell}= ( - 1 )^{n}\left( \frac{( n - 2 \ell - 1 ) !}{n !{( - 2 \ell - 1 ) !}}\right)^{1 / 2}e^{- i \ell \psi}\frac{t^{n}}{( 1 - | t |^{2})^{\ell}},
E(t):=\sum_{k=0}^s \Big\{\int \frac{|z_{\alpha}|^{-2k+1}}{a|z_{\alpha}|}|\partial_{\alpha}^k u_t|^2 d\alpha + Re\int \bold{n}|z_{\alpha}|D^{k+1}f \overline{D^k f}d\alpha\Big\}
\iota_{\xi_\mathcal{V}}d^c\bigl(\eta+c\cdot\pi^*\varphi_k\bigr) =\iota_{\xi_\mathcal{V}}d^c\eta + c\cdot \iota_{\xi_\mathcal{V}}d^c\pi^*\varphi_k &=\iota_{\xi_\mathcal{V}}d^c\eta + c\cdot\iota_{\xi_\mathcal{V}}\pi^*d^c\varphi_k & =\iota_{\xi_\mathcal{V}}d^c\eta
\widehat{(\partial_{\Sigma_0, \epsilon}^{*})^{\otimes 2}\overline{u}_n}(\xi, \eta) = -4 \xi_u \eta_u \delta(\xi_u + \eta_u)\widehat{\overline{u}}(\xi_u) = \frac{(-i)^{n}\xi_u^{n + 2}}{n!}[\alpha - i \pi \mathrm{sgn}(\xi_u)].
\eta( \left\langle \eta, \xi^H\right\rangle )=1+ \left\langle \nabla_\eta \eta, \xi^H\right\rangle=1+ \left\langle \nabla_\eta J\eta, J\xi^H\right\rangle =1+ \left\langle \nabla_\eta \nu^H, J\xi^H\right\rangle &=1 + l(\xi)\left\langle \eta , J\xi^H\right\rangle=1- l(\xi)\left\langle \nu^H , \xi^H\right\rangle.
\frac{g_1^2}{4\pi m_0^2}=\alpha\left(\sigma_1\right)\,\frac{\mu_0}{\sigma_1m_0}\sqrt{1-\frac{1}{4}\left(\frac{\mu_0}{\sigma_1m_0}\right)^2}=\alpha_0+\frac{\alpha_1}{\sqrt{2}}\frac{\mu_0}{m_0}+O\left(\left(\frac{\mu_0}{m_0}\right)^2\right)\,,
\begin{aligned}\|Z^{1/p}-{A}^{1/p}\|_2\leq \|{Z}^{1/p}-{(Z+cI)}^{1/p}\|_2+ &\|({Z+cI})^{1/p}-{(A+cI)}^{1/p}\|_2+\|({A+cI})^{1/p}-{A}^{1/p}\|_2.\end{aligned}
i_h(h_1...h_p) =\sum_{j}\langle h,h_j\rangle h_1...\hat{h}_j...h_p &-\sum_{j<k}(h_k\circ h^t\circ h_j) h_1...\hat{h}_i...\hat{h}_j...h_p &-\sum_{k<j}(h_k\circ h^t\circ h_j) h_1...\hat{h}_i...\hat{h}_j...h_p .
\| v^\ast \Phi(a) v - \psi(a)\| \leq \sum_{i=1}^n \| v_i^\ast \Phi(a) v_i - \psi_i(a)\| + \sum_{1\leq j < k \leq n}(\| v_j^\ast \Phi(a) v_k \| + \| v_j^\ast \Phi(a^\ast) v_k\|) < \epsilon
\frac{1}{q_r}\sum_{c=0}^{q_r-1}D_{r,j}(c) < 2\Big(\frac{\lambda_j}{\lambda_j-1}\Big)^{3/2}\sqrt{\tau_{r,j}}(3+2\ln A_{r,j})^2 &\leq 2 \Big(\frac{\lambda_j}{\lambda_j-1}\Big)^{3/2}\sqrt{\tau_{r,j}}(3+\ln \tau_{r,j})^2.
\frac{1}{2}(\theta^{2})^{\alpha}{}_{\beta}=-(\theta_{\rm s}^{2})^{\alpha}{}_{\beta}\ ,\quad (\theta)^{2}=-3(\theta_{\rm s})^{2}\ ,\quad \Big[\Big(1+\frac{1}{2}(\theta)^{2}\Big)^{-1}\Big]^{y}{}_{y}=[{\rm det}(1-(\theta_{\rm s})^{2})]^{-1/2}\ .
\|E^{(n)}(t)\|_{H^{N+1}}\leq\sum_{p=2}^{\infty}\sum_{n'\geq\max(n,p)}C^pn^pt^{4p}\cdot\eta_0^{N+5}\cdot\eta_0^{-n'N/2}\leq (Cnt^4)^n\eta_0^{-nN/2}\eta_0^{N+5}\leq \eta_0^{-nN/4},
T(z) = R(z)E_{-x_{j}}(z)\Phi_{\mathrm{HG}}(\sqrt{r}f_{-x_{j}}(z);\beta_{j})(s_{j}s_{j+1})^{-\frac{\sigma_{3}}{4}}e^{\frac{\pi i \alpha}{2}\theta(z) \sigma_{3}}e^{-\sqrt{r}g(z)\sigma_{3}}.
u(t,x)= \int_t^T R_{s-t}\left[e^{-(s-t){A}}G B(s,\cdot)\right](x)\,ds+\int_t^T R_{s-t}\left[e^{-(s-t){A}}\nabla^Gu(s,\cdot) B(s,\cdot) \right](x)\,ds,
c(\pi_*(\mathcal{K}^\vee_i(a_j-a_{i}))) =(1+h_i)^{d_i+a_j-a_i+1}c(\pi_*(\mathcal{K}^{\vee}_i \otimes\mathcal{K}_j(a_{j}-a_{i}))) = (1+(h_i-h_j))^{d_i-d_j+a_j-a_i+1}.
\begin{cases}\dot{z}(\phi)=\dot{f}(\phi) -c- \frac{D(\phi)g(\phi)}{z(\phi)}\ \mbox{in}\ (0,\alpha)\cup (\alpha,\beta)\cup (\beta, 1), z<0 \ \mbox{in}\ (0,\alpha)\cup(\beta, 1), z>0 \ \mbox{in}\ (\alpha,\beta), z(0)=z(\alpha)=z(\beta)=z(1)=0.\end{cases}
E[W^k_{\ell_2\Delta,t}(j)W^k_{\ell_1\Delta,t}(j); W^k_{\ell_2\Delta,t}(j)>1]+E[W^k_{\ell_2\Delta,t}(j)W^k_{\ell_1\Delta,t}(j); W^k_{\ell_1\Delta,t}(j)>1] &=O\left(e^{\lambda\Delta \ell_1}e^{2\lambda \Delta\ell_2}\Delta^3\right),
d_0\circ \frak m^{\epsilon}_{1;\alpha_+,\alpha_-}+ \frak m^{\epsilon}_{1;\alpha_+,\alpha_-}\circ d_0+\sum_{\alpha; E(\alpha_-) < E(\alpha)< E(\alpha_+)}\frak m^{\epsilon}_{1;\alpha_+,\alpha}\circ\frak m^{\epsilon}_{1;\alpha,\alpha_-}= 0.
\begin{cases}\begin{array}{ll}\dot{v}=\log\det (v_{\alpha\bar{\beta}})-Av+g(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), v=\psi \mbox{on}\;\partial\Omega\times [0,T), v=v_0 \mbox{on}\;\bar{\Omega}\times\{0\}, \end{array}\end{cases}
\zeta(\rho,k) :=\begin{cases}\mbox{RHS of}\ \eqref{eq:values:diffZnk:1}\mbox{if}\,\,\ell\in[0:p-1], \mbox{RHS of}\ \eqref{eq:values:diffZnk:2}\mbox{if}\,\,\ell=p,\end{cases}\theta(\rho,k) :=\mbox{RHS of}\ \eqref{def:thetank}.
J=\frac{1}{N^\gamma}\begin{pmatrix}(1-2\rho^{A})(E_{A}-E_{C})-\rho^{B}(E_{B}-E_{C}) -\rho^{A}(E_{B}-E_{C}) -\rho^{B}(E_{A}-E_{C}) -\rho^{A}(E_{A}-E_{C})+(1-2\rho^{B})(E_{B}-E_{C})\end{pmatrix}
\mathrm{P}{\int}_{0}^{\infty}d k_{+}\left( \frac{1}{( k_{+}- a )^{2}}+ \frac{1}{( k_{+}+ a )^{2}}\right) = \mathrm{P}{\int}_{- \infty}^{\infty}d k_{+}\frac{1}{( k_{+}- a )^{2}}.
{\frac{d S^{( 1 )}}{d \ln M^{2}}}={\frac{\lambda}{6 4 \pi^{2}}}\int d^{4}x \mu^{2}\phi_{c l}^{2}( x ) +{\frac{\lambda^{2}}{2 5 6 \pi^{2}}}\int d^{4}x \phi_{c l}^{4}( x ) \, \, .
\begin{cases}U_t^{k+1}+ \mathcal{E}_1(U^k+U_s)U^{k+1}_x + \mathcal{E}_2(U^k+U_s)U^{k+1}_y + \ell (U^{k+1}) = F^{U^k}, U^{k+1}(0)=U_0, U^{k+1}|_{\boldsymbol\Gamma}= 0. \end{cases}
\mathbb{E}\left[\exp\left(-r\sum_{i=1}^n \left( f_i(Z_i) - \mathbb{E}\left[f_i(Z_i)\right] \right)\right) \right] \overset{(a)}{\geq}\exp\left(-r\mathbb{E}\left[\sum_{i=1}^n \left( f_i(Z_i) - \mathbb{E}\left[f_i(Z_i)\right] \right)\right]\right) = 1,
R(i,j) - r_{i,j}\geq -\left(\frac{1}{{d}(i)^2+{d}(i)}+\frac{1}{{d}(j)^2+{d}(j)}\right)> -\left(\frac{1}{{d}(i)^2}+\frac{1}{{d}(j)^2}\right)=-f_{i,j}.
\partial_t \overline{v}+ \mathbb{P}_h \Big(\overline{v}\cdot \nabla \overline{v}\Big) + \mathbb{P}_h P_0 \Big((\nabla\cdot \widetilde{v}) \widetilde{v}+ \widetilde{v}\cdot \nabla \widetilde{v}\Big) + \Omega \int_{\mathbb{T}^2}\overline{v}^\perp (\boldsymbol{x}') d\boldsymbol{x}' = 0.
\phi_x(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots)^x = \sum_{n=1}^\infty \left (\phi(\alpha_n)^x + \phi(\beta_n)^x\right) & = \sum_{n=1}^\infty \phi(\alpha_n)^x + \sum_{n=1}^\infty \phi(\beta_n)^x & = \phi_x(\alpha_1,\ldots)^x + \phi_x(\beta_1,\ldots)^x.
E_N^{(K)}(t,[s]_K) = \sum_{i=K+1}^N \int_{s_{K+1}=0}^\infty \! ... \! \int_{s_{i}=0}^\infty \varphi_i([s]_i) n_N^{(i)}(t, [s]_i) \,d[s]_{K+1, i},
\frac{\partial L^{(2)}}{\partial q^{i}}-\frac{d}{dt}\left(\frac{\partial L^{(2)}}{\partial \dot{q}^{i}}\right)+\frac{d^{2}}{dt^{2}}\left(\frac{\partial L^{(2)}}{\partial\ddot{q}^{i}}\right)=0
Q_2 =\mathrm{P}\left( z_m>x_m, \right. \left. x_m< \frac{\frac{\epsilon_{1,m}}{\rho}}{1- \max\left\{0, \frac{\left(x_m-\frac{\epsilon_{1,m}}{\rho}\right)}{x_m(1+\epsilon_{1,m})}\right\}(1+\epsilon_{1,m})}\right) &=\mathrm{P}\left( z_m>x_m, x_m< \max\left\{\frac{\epsilon_{1,m}}{\rho}, x_m \right\}\right).
\begin{array}{c}{{D ( \beta , R ) - i \mu \gamma^{0}= e^{\mu \beta t}D ( \beta , R ) \e^{- \mu \beta t},}}\\{{K_{b c}( t , \vec{x}; t^{\prime}, \vec{x^{\prime}}) = e^{\mu \beta t}k ( t , \vec{x}; t^{\prime}, \vec{x^{\prime}}) \e^{- \mu \beta t \prime},}}\\ \end{array}
\left( \left[{\alpha}_{m - n}^{\mu},{\alpha}_{- s}^{\sigma}\right] \left[{\alpha}_{n}^{\nu},{\alpha}_{s - m}^{\rho}\right] + \left[{\alpha}_{m - n}^{\nu},{\alpha}_{s - m}^{\rho}\right] \left[{\alpha}_{n}^{\nu},{\alpha}_{- s}^{\sigma}\right] \right)
\begin{pmatrix}n_1-2 1 0 0 \ldots 0 0 0 -1 -n_2/2 1 0 \ldots 0 0 0 0 -1 n_3 1 \ldots 0 0 0 0 0 -1 -n_4/2 \ldots 0 0 0 \vdots \vdots \vdots \vdots \ldots \vdots \vdots \vdots 0 0 0 0 \ldots -n_{2k-2}/2 1 0 0 0 0 0 \ldots -1 n_{2k-1}1 0 0 0 0 \ldots 0 -1 (2-n_{2k})/2 \end{pmatrix},
\psi(g^{-1}xg) = (g^{-1}(ab)^{\frac{q-1}{2}}, g^{-1})(\omega(x),\rho(x))((ba)^{\frac{q-1}{2}}g, g) &=(g^{-1}(ab)^{\frac{q-1}{2}}\omega(x)(ba)^{\frac{q-1}{2}}g, g^{-1}\rho(x)g) &=(\omega(x), g^{-1}\rho(x)g).
\begin{array}{rcll}-\Delta_{\xi}\Z(\xi) = 0, \xi \in \Pi_1 , \partial_{\xi_1}Z(\xi) = -1, \xi \in \partial_{\parallel}\Pi_1, \partial_{\xi_2}Z(\xi_1,0) = 0, (\xi_1,0) \in \partial\Pi_1 \setminus \partial_{\parallel}\Pi_1.\end{array}
u(t) = u_0 - \int_0^t \left[ \mathrm{A}u(s) + B(u(s)) - |\nabla u(s)|_{L^2}^2\,u(s) \right]\,ds &~~~+ \frac12\int_0^t \sum_{j=1}^m C^2_j u(s)\,ds + \int_0^t \sum_{j=1}^m C_ju(s)\,d\hat{W}(s).
g_{-}^{- 1}( u ) E_{j j}^{( n )}g_{-}( u ) = g_{-}^{- 1}( u ) \left( \frac{\partial}{\partial u_{j}^{( n )}}g_{-}( u ) \right) + \left( \frac{\partial}{\partial u_{j}^{( n )}}g_{+}( u ) \right) g_{+}^{- 1}( u ) .
\sum_{j=2}^{i+1}E_{2}(V(d w), B_{j}) \leq \frac{1}{2}\sum_{j=1}^{i}E_{2}(V(d w), B_{j}) + \frac{2c_{4}\lambda^{\frac{p}{2}}}{\sigma^{n}}\sum_{j=1}^{i}\omega\left( R_{j}\right).
\|DT_F(u)h\|_{1,2}\;\; = \; \; \|DT_F(u)h\|_2 \; + \; \|(DT_F(u)h)'\|_2 \; \leq \; \; \|F\|_\infty \|h\|_2 \; + \; \|F''\|_\infty \|h\|_\infty \|u'\|_2 \; + \; \|F'\|_\infty\|h'\|_2.
1-\frac{1}{2}(2e)^d(\sqrt{5})^{3d}\cdot 4e^{-C_4d}-(2e)^d\sum_{h=1}^H(\sqrt{5})^{(h+3)d}\cdot 4e^{-C_3d}\geq 1-\frac{\varepsilon}{6\cdot 2^{2d}}.
\mathbf{k}_{\tilde{\rho},\epsilon}^{(c)}(t,\,x) =\frac{g_{x}^{(c)}(t,\,F_{\rho}(t,\,x))}{\lambda+1}\Biggl[\int_{\mathbb{T}}\tilde{b}(t,\,y)^{\dagger}\tilde{\rho}(t,\,y)dy+\lambda b_{c}(t,\,x)+\sum_{k=1}^{m}\gamma_{c,k}(t,\,x)\rho_{k}*\iota_{\epsilon}(x)\Biggl]
\frac{f(w+1; k \lambda)}{f(w; k \lambda)}= d^k \cdot k^{k(m-n)}\cdot \prod_{i=0}^{k-1}\dfrac{\left(w+ \frac{u_1 + i}{k}\right) \cdots \left(w+ \frac{u_m +i}{k}\right)}{\left(w+ \frac{v_1 + i}{k}\right) \cdots \left(w+ \frac{v_n +i}{k}\right)}.
\begin{cases}\displaystyle i\partial_t \Psi_j+\alpha_j\Gamma_j \partial_\sigma^2\Psi_j+\sum_{k\neq j}\Gamma_k \frac{\Psi_j-\Psi_k}{|\Psi_j-\Psi_k|^2}=0,1\leq j\leq N, \Psi_j(0,\sigma)=\Psi_{j,0}(\sigma). \end{cases}
z_{sym}(k,\chi,v_1,v_2,y) = (\frac{|k|^{-2}}{\chi \cdot v_1 -\Psi(y)}- \frac{|k|^{-2}}{\chi \cdot v_2 -\Psi(y)})= \frac{|k|^{-2}\chi(v_2-v_1)}{(\chi \cdot v_1 -\Psi(y))(\chi \cdot v_1 -\Psi(y))},
H_{c}= \int [ \frac{1}{2}( \pi_{i}^{a})^{2}+ \frac{1}{2}m^{2}( A_{i}^{a})^{2}+ \frac{1}{4}( F_{i j}^{a})^{2}+ \frac{m^{2}}{2}( A_{0}^{a})^{2}- A_{0}^{a}T_{2}^{a}]
(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(yb\otimes 1)\bigr) =S\bigl((\psi\otimes\operatorname{id})((a^*\otimes 1)(\Delta(yb)))\bigr) &=S(yz)=S(z)S(y)=(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(b\otimes1)\bigr)S(y) &=(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(b\otimes S(y))\bigr),
\begin{aligned}\mathbb{E}(Y) =(q^2+q+1)\mathbb{P}(\mbox{a given point is not determined}) &= (q^2+q+1)(1-p)^{(q^2+q+1)}\left(\frac{p}{1-p}+ \left(1+q\frac{p}{1-p}\right)^{q+1}\right).\end{aligned}
\tilde{\mathbf P}(\tilde V^{I,+\delta}_n(\theta'_n,c)=\emptyset\cap\tilde{\mathfrak{W}}(c)\ne\emptyset)\le \tilde{\mathbf P}(\tilde{\mathfrak{W}}(c)\not\subseteq\tilde V^{I,+\delta}_n(\theta'_n,c))=\tilde{\mathbf P}(L^c_n)\le \tilde{\mathbf P}(A_n)<\eta/2,
f_n({\bf x})-f_n({\bf x}') = -C_{n,p, \lambda}\left ( x_i(a+x_i)^{p-1}+x_{i+1}(a+x_i+x_{i+1})^{p-1}-x_{i+1}(a+x_{i+1})^{p-1}-x_{i}(a+x_i+x_{i+1})^{p-1}\right ),
\frac{dV}{dt}= \left[\frac{a_ix_i}{1+x_i}-d_i\right]y_i\rho_j-d_jy_j\rho_i <-(\delta y_i\rho_j+d_jy_j\rho_i) &< -\delta^* ( y_i\rho_j+d_jy_j\rho_i)=-\delta^* V(y_i,y_j)
(VHV^{-1}-H)f(e^{i\varphi})=\frac{1}{\pi}\int[\frac{g^{\prime}(\theta)e^{ig(\theta)}}{e^{ig(\theta)}-e^{ig(\varphi)}}-\frac{e^{i\theta}}{e^{i\theta}-e^{i\varphi}}]f(e^{i\theta})d\theta.
|f(y) - f(x) | \lesssim \| f- f^m_T \|+ \| \partial_1 f^m_T \|d(y,x) + \| \partial_2 f^m_T\| d^2(y,x) &\lesssim (T^{\frac{1}{4}})^{\alpha}+ (T^{\frac{1}{4}})^{\alpha-1}d(y,x) + (T^{\frac{1}{4}})^{\alpha -2}d^2(y,x).