harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\partial_t u + \mathcal{L}u= \F, (0,T] \times \Omega u(0,\cdot) = \u_0, \, \Omega u= \g_D^{}, (0,T] \times \Gamma_D^{}, \mathbf{n}\cdot \nabla u= \g_N^{}, (0,T] \times \Gamma_N^{},\end{aligned}
\zeta^1 \delta^{n}(T)(a)(a_1,...,a_n) =\delta^{n}(T)(a)(a_1,...,a_n) &=a\cdot T(a_1,...,a_n) &- T(aa_1,...,a_n)+... &+(-1)^nT(a,a_1,...,a_{n-1})\cdot a_n &=(a\cdot T)(a_1,...,a_n)-(T\cdot a)(a_1,...,a_n) &=\delta_{n}^0(T)(a)(a_1,...,a_n),
z^{2}\! = \! R \sqrt{\frac{r^{2}- r_{-}^{2}}{r_{+}^{2}- r_{-}^{2}}}\sinh \left( \frac{r_{+}}{R}\phi \! - \! \frac{r_{-}}{R^{2}}t \right) z^{3}\! = \! R \sqrt{\frac{r^{2}- r_{-}^{2}}{r_{+}^{2}- r_{-}^{2}}}\cosh \left( \frac{r_{+}}{R}\phi \! - \! \frac{r_{-}}{R^{2}}t \right) ,
<R(X_{i},X_{j})X_{k},X_{l}> =<\alpha(X_{i},X_{l}),\alpha(X_{j},X_{k})>-<\alpha(X_{i},X_{k}),\alpha(X_{j},X_{l})> &=\|\eta\|^{2}\{\delta^{i}_{l}\delta^{j}_{k}-\delta^{i}_{k}\delta^{j}_{l}\}
{\bf x}^{(l)}\geq{\bf 0},~~{\bf w}^{(l)}:={\bf q}^{(l)}+\mathcal{A}^{l}({\bf x}^{(l)})^{m-1}\geq{\bf 0}~~{\rm and}~~({\bf x}^{(l)})^{\top}{\bf w}^{(l)}=0.
\sum_{n\leqslant\frac{(\lambda_2-\alpha\mu_2)(k-1)K}{N\det(\Lambda_d)}B}\mathtt{1}_{1-\{n\theta\}\leqslant\frac{k\varepsilon K}{N(\lambda_2-\mu_2\alpha)}B^{1-\frac{1}{r}}}=\frac{k(k-1)\varepsilon K^2}{N^2\det(\Lambda_d)}B^{2-\frac{1}{r}}+O\left(b\det(\Lambda_d) \log\left(\frac{(k-1)K(\lambda_2-\alpha\mu_2)}{N\det(\Lambda_d)}B\right)\right).
\mathcal{B}_\varepsilon =b(\mathbf{D})^*g^\varepsilon (\mathbf{x})b(\mathbf{D})+\sum \limits_{j=1}^d\left( a_j^\varepsilon (\mathbf{x})D_j +D_ja_j^\varepsilon (\mathbf{x})^*\right) +Q^\varepsilon (\mathbf{x}).
\mathfrak g=g-[g]_{Q}u-\sum_{P\in \P\setminus \P_4}g_P +\sum_{P\in \P\setminus \P_4}\frac{[g]_P}{[u]_P}u_P-\sum_{P\in \P\cap \P_4}g_P +\sum_{P\in \overline{\P}_4}\frac{[g]_P}{[u]_P}u_P-\sum_{P\in{\P'_4}}\frac{[g]_P}{[u]_P}u_P
\phi_k(x) \leq \Big(1 -\frac{A_0}{A_k}\Big) f(x) + \frac{A_0}{A_k}\phi_0(x) + \frac{\tilde \varepsilon_{k}}{A_k}= \Big(1 - \frac{1}{A_k}\Big)f(x) + \frac{1}{A_k}\phi_0(x)+ \frac{\tilde \varepsilon_{k}}{A_k}.
\alpha_1 \lesssim \int_0^\infty e^{\mu_1s_1}\{(\lambda\dot\lambda-\sigma) + \lambda^\frac{n-2}{2}\eta^\frac{2}{1-q}(T-t)^{{\sf d}_1}+ (T-t)^{{\sf d}_1}\sigma \}ds_1 \lesssim (T-t)^{{\sf d}_1}\sigma |_{t=0}\int_0^\infty e^{\mu_1s_1}ds_1 \lesssim (T-t)^{{\sf d}_1}\sigma |_{t=0}.
u_{ij}^*(x_1,\ldots,x_k;y_1,\ldots,y_k) &= \sum_{\substack{M\in\mathcal{E}: M\neq 0}}\frac{\rho(0,M)}{1 - \rho(0,0)}\left[u_{ij}^*(x_1-m_{1,\cdot},\ldots,x_k-m_{k,\cdot};y_1-m_{\cdot,1},\ldots,y_k-m_{\cdot,k}) + m_{ij}\right].
\left| (\rho-1)a\alpha^n-\left(d_1\cdot \rho^{\ell+m}-(d_1-d_2)\cdot \rho^m-(d_2-d_3)\right)\cdot \rho^k\right| = |-(\rho-1)\Pi(n)-d_3| &\leqslant (\rho-1)\cdot \alpha^{-n/2}+(\rho-1)<2(\rho-1).
||h(\tau,m)||_{(\nu',\beta,\mu,k_{2},h')}=\sup_{\tau \in \bar{S}_{d,\kappa,\delta,h'}, m \in \mathbb{R}}(1+|m|)^{\mu}\frac{1 + |\tau|^{2k_{2}}}{|\tau|}\exp( \beta|m| - \nu'|\tau|^{k_2}) |h(\tau,m)|
\prescript L{}{}\ell{}^R(x,\alpha) = \langle \mu,\alpha\rangle + \chi_R(\alpha) -\chi_L(w\alpha) = \langle \mu,\alpha\rangle + \Phi^+(\alpha)-\Phi^+_R(\alpha) -\Phi^+(w\alpha)+\Phi_L^+(w\alpha) = \ell(x,\alpha) - \Phi^+_R(\alpha) + \Phi_L^+(w\alpha)
\Psi(k,n,m)=\frac{1}{\sqrt{c_{0}^{n}}}\cdot\Psi[\frac{k-mc_{0}^{n}}{c_{0}^{n}}T]=\frac{1}{\sqrt{c_{0}^{n}}}\cdot\Psi[(\frac{k}{c_{0}^{n}}-m)T]
|\int_{X}Tr(i_{\nu}F_{A}\wedge F_{A}\wedge d(\ast Q))| \leq\int_{X}|\nu|\cdot|F_{A}|^{2}\cdot\max_{x\in X}|d(\ast{Q})| &\leq\max_{x\in X}|d(\ast{Q})|\int_{B_{\tau(1+\epsilon)}(p)}\tau(1+\epsilon)|F_{A}|^{2}dV_{g}
0<y_{0}^{(m)}= \langle y_0, \phi_{m}\rangle, 0<y_{0}^{(M)}= \langle y_0, \phi_{M}\rangle &0\le y_{0}^{(k)}= \langle y_0 , \phi_{k}\rangle m< k < M,\quad0=y_{0}^{(k)}\quad \quad0\leq k < m.
\mathcal{E}_{\zeta}^{({\tau_1},{\tau_2})}(c):=\begin{cases}-g_{{\zeta},{\tau_1},{\tau_2},c}+(\chi{}(\cdot{},{\zeta})-1)\cdot{}c A_{\zeta}({\tau_1})\cap{}C_{\zeta}({\tau_2}) -g_{{\zeta},{\tau_1},{\tau_2},c}A_{\zeta}({\tau_1})\setminus{}C_{\zeta}({\tau_2})\end{cases}
M'_I :=\left\{z=(x,y) \in S_{n,m}: x_i \in (0,1)\hbox{for all}i \in I,\hbox{and}x_j \in (1/2,\infty)\hbox{for all}j \in I^c\right\}, M''_I :=\left\{z=(x,y) \in S_{n,m}: x_i \in (0,2)\hbox{for all}i \in I,\hbox{and}x_j \in (1/4,\infty)\hbox{for all}j \in I^c\right\},
g_{++}(z)=\lim_{t_1,t_2\rightarrow0^+}G(z_1+it_1,z_2+it_2), g_{+-}(z)=\lim_{t_1,t_2\rightarrow0^+}G(z_1+it_1,z_2-it_2), &g_{-+}(z)=\lim_{t_1,t_2\rightarrow0^+}G(z_1-it_1,z_2+it_2), g_{--}(z)=\lim_{t_1,t_2\rightarrow0^+}G(z_1-it_1,z_2-it_2).
\delta(u,y,s)>0 \Longrightarrow p_{Y|X,S}(y|\psi(u,s),s)+\epsilon\delta(u,y,s) &\qquad=b(y,s)+\epsilon(a(y,s)-b(y,s)), \delta(u,y,s)<0 \Longrightarrow p_{Y|X,S}(y|\psi(u,s),s)+\epsilon\delta(u,y,s) &\qquad=a(y,s)+\epsilon(b(y,s)-a(y,s)),
\begin{cases}u_{tt}- \Delta u +b_{1}(x) u_{t}-{\rm div}\left(b_{2}(x)\nabla u_{t}\right) = 0 \mbox{in}(0, \infty) \times \Omega, u(t,\sigma) = 0 \mbox{on}(0, \infty) \times \partial\Omega, u(0,x) = u_{0}(x) \mbox{in}\Omega u_{t}(0,x) = u_{1}(x) \mbox{in}\Omega.\end{cases}
f_3 (r) = 2 \sqrt{\frac{\bar{s}^2 - Qc}{Qc}}e^{- \tilde{c}\sqrt{\bar{s}^2 - Qc}}\frac{\left( 1 - \frac{r_0^2}{r^2}\right)^{(1 + \sqrt{(\bar{s}^2 - Qc)/D_2}\ ) /2}}{1- e^{-2 \tilde{c}\sqrt{\bar{s}^2 - Qc}}\left( 1 - \frac{r_0^2}{r^2}\right)^{\sqrt{(\bar{s}^2 - Qc)/D_2}}}.
k_{1}^{T E}= 3^{1 / 3}\frac{3 \pi^{2 / 3}}{4 \ 2^{1 / 3}}, \qquad k_{2}^{T E}={3}^{{2}/{3}}\frac{3{\pi}^{{4}/{3}}}{1 6 0 \ 2^{2 / 3}}.
v_{\mathbf i, b, \mathbf j}=v_{i_r}\otimes v_{i_{r-1}}\otimes \cdots\otimes v_{i_1}\otimes b\otimes\bar v_{j_{1}}\otimes \bar v_{j_{2}}\otimes \cdots\otimes \bar v_{j_{t}}\in M_{pq}^{rt}.
{\rm mult}_{(x,y)}(N(\lambda))=\begin{cases}\displaystyle\Big\lfloor\frac{\min(x-k,y-\ell)+2}2\Big\rfloor y\leq k, begin{align*}2ex] \displaystyle\Big\lfloor\frac{\min(x-k,y-\ell)}2\Big\rfloor-\Big\lfloor\frac{y-k-1}2\Big\rfloor y>k. \end{cases}
\hat W_r(q+k)-\hat W_r(2q) \le \frac{2}{\pi}\left( \frac{1}{q+k}+\frac{1}{2q}\right) \le \frac{4}{\pi q}\le 4r-\frac{2}{\pi}\sin(2\pi r) =\hat W_r(0)-\hat W_r(1).
{{\bf{H}}_{k,{\rm{R}}}}= \sum\nolimits_{l \in{\mathcal L}_{{\rm{R}}}}{{\alpha_{{k,{\rm{R}}},l}}{{\bf{a}}_{\rm{R}}}\left({\Phi_{{k},l}^{\rm{A}}}\right){\bf{a}}_{{\rm{S,}}k}^H\left({\Phi_{{k},l}^{\rm{D}}}\right)},
\sideset{}{^*}\sum_{d (Nb)}\chi(d) e \left(\frac{m^2d}{Nb}\right) \ = \ - \chi(N)\sideset{}{^*}\sum_{u_1 (b)}\chi(u_1) e \left(\frac{m^2u_1}{b}\right) = \ - \chi(N) G_{\chi}(m^2).
\F_0(P(v_1)\otimes E(\alpha_1))= P(v_1)\otimes E(\alpha_1)=\pi_*(S/p \wedge E_0^*\mathbb{J}), F_{s}(P(v_1)\otimes E(\alpha_1))= (\alpha_1,v_1)^{k(s)}(2p-2)(k(s)-1)\le s \le(2p-2)k(s)-1
\adjustlimits\sup_{x \in X,\,y \geq 0,\,z\in \mathcal{U}\}\inf_{\tau \in \Theta^n}\ \sum_{i=1}^n\theta^i\left[u_i(x^i) + \delta^i z^i\right], begin{align*}5pt] \sum_{i=1}^n x^i + y \leq f(k), &\sum_{i=1}^n \tau^i z^i - V(y,\tau) \leq 0.
R_{ijkl}=q^{2}[g_{jl}\widetilde{M_{ik}}-g_{jk}\widetilde{M_{il}}+g_{ik}\widetilde{M_{jl}}-g_{il}\widetilde{M_{kj}}]+{{q^{2}(c_{0}^{2}-1)}\over{c_{0}^{2}}}F_{ij}F_{ml}
I_\varepsilon(u_\varepsilon,\Omega)=\int_\Omega|y|^\beta|\nabla u_\varepsilon|^2+f_\varepsilon(\mathcal{L}^n(\{u_\varepsilon>0\}\cap\mathbb{R}^n\cap\Omega))\leq I_\varepsilon(u_*,\Omega_*)=C,
\|\hat g_m - \hat g_n\|_S \leq \sum_{k=n}^{m-1}\| \hat g_{k+1}- \hat g_k\|_S< \sum_{k=n}^\infty \left(\frac{1}{2}\right)^k \|\hat g_1 - \hat g_0\|_S= \left(\frac{1}{2}\right)^{n-1}\|\hat g_1 - \hat g_0\|_S.
|(L+L) \cap [0,k]|+|(R+R) \cap [0,k]| \ >\ 2|(L+R) \cap [0,k]|, |(L^c+L^c) \cap [0,k]|+|(R^c+R^c) \cap [0,k]| \ >\ 2|(L^c+R^c) \cap [0,k]|,
B^{(l, \sigma)}= \Phi_{\left[ - \left( \frac{1}{2}- 4\varepsilon \right) 8L , \left( \frac{1}{2}- 4\varepsilon \right) 8L \right]}( \Delta^{(l, \sigma)}_1 \cap K' ) \cap \left( \xi^{(l, \sigma)}\right)^{-1}( V )
\begin{aligned}c(\mathbb{M}_{g},\mathbb{M}_{b}):=\sup\bigg\{\big|\langle\boldsymbol{g}_k,\boldsymbol{b}_k\rangle\big|:\boldsymbol{g}_k\in\mathbb{M}_{g}\cap(\mathbb{M}_{g}\cap\mathbb{M}_b)^\bot, \boldsymbol{b}_k\in\mathbb{M}_{b}\cap(\mathbb{M}_{g}\cap\mathbb{M}_b)^\bot,\|\boldsymbol{g}_k\|^2=\|\boldsymbol{b}_k\|^2=1\bigg\}\end{aligned}
(1-\gamma_{n+3})\left(\displaystyle\frac{\gamma_{n+3}}{b}-1\right)=\displaystyle\frac{m\gamma_{n+2}}{\gamma_{n+1}+a\gamma_{n+2}}<\displaystyle\frac{m\gamma_{n}}{\gamma_{n-1}+a\gamma_{n}}=(1-\gamma_{n+1})\left(\displaystyle\frac{\gamma_{n+1}}{b}-1\right),
\mathrm{R e s}_{z^{\prime}= z}^{\Psi}\partial^{\Psi}\! : \! \psi ( z^{\prime}) \! : \bar{\partial}^{\Psi}\! : \! \phi ( z ) \! : = - : \! ( \phi_{; \mu \eta}- i C_{\eta \nu \mu}\phi_{; \nu}) \psi^{; \eta}\bar{\partial}^{\Psi}\! X^{\mu}\! : + O ( \epsilon^{5})
\sum_{k=0}^{b-d}\binom{a}{k}\binom{b}{d+k}\binom{k}{c}\binom{d+k}{d}H_k H_{a-k}={}\binom{a+b-c-d}{a}\binom{a}{c}\binom{b}{d}[H_a^{(2)}-H_{a+b-c-d}^{(2)}+ (H_a-H_{a+b-c-d}+H_{a-c})(H_a-H_{a+b-c-d}+H_{b-d})]
{\cal V}=-{f_\pi^2\over 2}{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}-{1\over{4 e^2}}\biggl [{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}\;{\cal R}_{\mathrm{j}}\cdot{\cal R}_{\mathrm{j}}-{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{j}}\;{\cal R}_{\mathrm{j}}\cdot{\cal R}_{\mathrm{i}}\biggr ].
c_2 \left ({\mathcal E}\left ( \frac{d_1+d_2+d_3-1}{2}\right ) \right ) = c_2({\mathcal E})+c_1 ({\mathcal E}) \left ( \frac{d_1+d_2+d_3-1}{2}\right )+ \left ( \frac{d_1+d_2+d_3-1}{2}\right )^2,
\left( \left[ \omega_{1},\omega_{2},f\right] ,\left[ \tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{f}\right] \right)_{\mathbf{H}_{\rho}}=(\Delta\omega_{1},\Delta \tilde{\omega}_{1})_{\Omega}+(P_{\rho}^{\frac{1}{2}}\omega_{2},P_{\rho}^{\frac{1}{2}}\tilde{\omega}_{2})_{\Omega}+(f,\tilde{f})_{\mathcal{O}}.
\lim_{n \rightarrow \infty}\sup_{m \geq n}\sup_{\tau \in \mathcal{T}_{m,n}^{M,T}}\mathbb{E}[\sup_{t \in [0,\tau]}\lVert u_{\phi_m}- u_{\phi_n}\rVert_V^2 + \int_0^{\tau}\lVert A(u_{\phi_m}- u_{\phi_n}) \rVert_H^2dt ] = 0.
\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x_0}}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t)) \, \Bigl \vert \,\varphi(t) \in D_{+\delta}\cup \partial D_{+\delta}, \, \, \forall t \in [0, T] \biggr \}> T\bigl(r({\mathcal{K}}) - 2 \varkappa \bigr).
\sum_{n \geq 0}(-1)^n(q)_{n}q^{\binom{n+1}{2}}\beta_n = (1-q)\sum_{n \geq 0}(-1)^nq^{\binom{n+1}{2}}\alpha_n, \sum_{n \geq 0}\big(q^2;q^2\big)_{n}(-1)^n\beta_n = \frac{1-q}{2}\sum_{n \geq 0}(-1)^n\alpha_n.
\frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p}\rangle_{Np}}{\langle f, f \rangle_{Np}}= \frac{1}{p+1}\left(p+ 1+\frac{p+1}{p}- p^{1-\kappa}(\alpha_p^2+\beta_p^2 + 2p^{\kappa-1})\right) &= \frac{p}{p+1}\left(1-\frac{\alpha_p^2}{p^{\kappa}}\right)\left(1-\frac{\beta_p^2}{p^{\kappa}}\right).
S_N(f) = \sum_{1\leq j_1, j_2, \ldots, j_{n+1}\leq N}f(N\*(\theta_{j_2}-\theta_{j_1})_c, \ldots, N\*(\theta_{j_{n+1}}-\theta_{j_1})_c) = \frac{1}{(2\pi)^{n/2}\*N^{n}}\*\sum_{k\in \mathbb{Z}^n}\hat{f}(k_1\*N^{-1}, \ldots, k_n\*N^{-1})\*\prod_{j=1}^{n+1}T_{N, k_j},
f_{\mathbf{x}}(\mathbf{x}) = \sum_{s_1=0}^{\infty}\sum_{s_2=0}^{\infty}\ldots \sum_{s_d=0}^{\infty}\left[ \mathbf{C}_{s_1s_2\cdots s_d}\cdot (-1)^{\sum_{i=1}^{d}s_i}\prod_{p=1}^{d}H_{s_p}(x_p)G(x_p) \right] & \mbox{with,}\mathbf{C}_{s_1s_2\cdots s_d}= \frac{E\left\{\prod_{i=1}^{d}H_{s_i}(X_i) \right\}}{(-1)^{\sum_{i=1}^{d}s_i}\prod_{j=1}^{d}s_j!}
W^{+}_{ijkl}W^{+}_{ijpq,t}W^{+}_{klpq,t}= \,\, 4\lambda|\lambda_{t}\omega_{pq}+(\lambda-\mu)c_{t}\eta_{pq}+(\nu-\lambda)b_{t}\theta_{pq}|^{2}&+4\mu|\mu_{t}\eta_{pq}+(\lambda-\mu)c_{t}\omega_{pq}+(\mu-\nu)a_{t}\theta_{pq}|^{2}&+4\nu|\nu_{t}\theta_{pq}+(\nu-\lambda)b_{t}\omega_{pq}+(\mu-\nu)a_{t}\eta_{pq}|^{2}
D=t H -\frac{1}{2}\int^{\infty}_{-\infty}d x (x- v t){\cal P}(x- v t)-\frac{v}{2}t \int^{\infty}_{\infty}dx{\cal P}(x-v t)= t (H- \frac{v}{2}P) -\frac{1}{2}D_0,
[ \Lambda \frac{\partial \phantom{a}}{\partial \Lambda}+ \mu \frac{\partial \phantom{a}}{\partial \mu}+ \beta_{1}\frac{\partial \phantom{a}}{\partial g_{1}}+ \beta_{2}\frac{\partial \phantom{a}}{\partial g_{2}}- \gamma N_{F}] \Gamma^{( N )}( p_{1}, \ldots p_{N}) \approx 0 ,
{\cal H}_{-}={\frac{1}{2}}g_{i}^{2}+{\frac{1}{2 m_{-}^{2}}}( \epsilon_{i j}\partial_{i}g_{j})^{2}={\frac{1}{2}}( \pi_{g}^{2}+ m_{-}^{2}g^{2}) +{\frac{1}{2 m_{-}^{2}}}( \epsilon_{i j}\partial_{i}g_{j})^{2}.
\Xi_j \wedge \omega_{\mathbb{B}^n}^{p-1}\leq \Xi_j \wedge \big( \sum_{t=0}^{p-1}\omega^t_{\mathbb{B}^n}\wedge (F_j^*\omega_{\mathbb{B}^{N_j}})^{p-1-t}\big) \leq p \Xi_j \wedge \omega_{\mathbb{B}^n}^{p-1}.
\begin{array}{c c}{{\hat{M}_{0}( z ) Z_{0}= Z_{0}H ( z ) ~}}&{{~ H ( z ) = \left( \begin{array}{c c}{{\tilde{\hat{\omega}}( z ) - \frac{1}{2}\hat{\lambda}( z )}}&{{2 \hat{\rho}^{j}( z )}}\\{{0}}&{{\hat{T}_{i}^{~ j}( z )}}\\ \end{array}\right)}}\\ \end{array}
n' = 30h^2\alpha n^{\frac{1}{s+1}}+160 h^3\alpha n^{\frac{3}{2}-\frac{s}{s+1}}+3\cdot(6h)^{s+3}\alpha^{\frac{s+1}{2}}n begin{align*}5pt] \leq \left(30h^2+160h^3+3\cdot(6h)^{s+3}\alpha^{\frac{s-1}{2}}\right)\alpha n begin{align*}5pt] \leq 4\cdot (6h)^{s+3}\alpha n begin{align*}5pt] \leq 4\cdot (12sk)^{s+3}\alpha n.
\langle \zeta^{a}\left( z \right) \zeta^{b}\left( w \right) \rangle = \frac{i}{\pi}\theta^{a b}\mathcal{A}\left( z , w \right) - \frac{1}{\pi}G^{a b}\mathcal{B}\left( z , w \right) + \frac{1}{2 \pi}g^{a b}\mathcal{C}\left( z , w \right) ,
(\mathtt{H}1) d(\xi)-1\neq 0\quad\mbox{at}\xi=\mathbb{M}^{-1}\overline{\omega}, begin{align*}2mm] (\mathtt{H}2)_{j, k}\mbox{Fixed}\,\, j, k\in S^c,\, j\neq k,\quad\det(\mathbb{M}+B(j, k))\neq 0,
J^{\mu j}= - N_{j b}^{\mu \nu}\bigl ( \varepsilon_{\nu \lambda}\partial^{\lambda}\Phi^{b}- \frac i 2 \varepsilon^{b c d}\overline{{\chi}}^{c}\gamma_{\nu}\chi^{d}\bigr ) ={\cal J}^{\mu j}+ N_{j k}^{\mu \nu}K_{\nu}^{k},
E[z]\approx z(\mu_{1}\mu_{2})+\frac{1}{2}\{\frac{\partial^{2}z}{\partial x_{1}^{2}}\sigma_{1}^{2}+\frac{\partial^{2}z}{\partial x_{2}^{2}}\sigma_{2}^{2}\}+\frac{\partial^{2}z}{\partial x_{1}\partial x_{2}}\sigma_{1,2}
=VC \left( \sum_{n\neq a^{*}}\mathbb{E}[S_n] + \mathbb{E}[S_{a^{*}}] \right) &\leq V C \left(2\sum_{n\neq a^{*}}\mathbb{E}[S_n] +1\right) \leq VC\left(2\sum_{n\neq a^{*}}\mathbb{E}[\theta_{n,K}] + 1 \right) &\leq VC \left(2\sum_{n\neq a^{*}}\left[\frac{8\ln K}{\delta^2_n}+ 1+ \frac{\pi^2}{3}\right]+1\right ).
\int_\Omega \sum_{j=1}^n \eta^2 \langle D^2 L(\nabla u) \nabla u_{x_j}, \nabla \gamma^j \rangle = - \int_\Omega \sum_{j=1}^n \gamma^j \langle D^2 L(\nabla u) \nabla u_{x_j}, \nabla \eta^2 \rangle & \phantom{=}- \int_\Omega \sum_{j=1}^n \eta^2 \gamma^j f_{x_j}.
A:= \left[ \begin{array}{ccccc}\alpha_1 \alpha_1^q \alpha_1^{q^2}\cdots \alpha_1^{q^{n}}\alpha_2 \alpha_2^q \alpha_2^{q^2}\cdots \alpha_2^{q^{n}}\vdots \vdots \vdots \ddots \vdots \alpha_n \alpha_n^q \alpha_n^{q^2}\cdots \alpha_n^{q^{n}}x x^q x^{q^2}\cdots x^{q^{n}}\end{array}\right].
\bar{\xi}_{1}=(\xi^{1}_{1},\xi^{1}_{2}), \,\,\,\,\,\, \bar{\xi}_{2}=(\xi^{2}_{1},\xi^{2}_{2}), \,\,\,\,\, \cdots, \,\,\,\,\, \bar{\xi}_{d}=(\xi^{d}_{1},\xi^{d}_{2}).
\begin{pmatrix}2\phi_0 0 0 0 \gamma^a{}_0 \psi^a{}_b 0 0 \gamma^n{}_0 \gamma^n{}_a 2\phi_{n}0 \pi_a \pi_{ab}0 2\phi_0 \delta^a_b-\psi^a{}_b\end{pmatrix},
-e^{4}(\frac{(\overline{v}_{k}\gamma^{\mu}v_{k^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\mu}u_{p})}{(k-k^{\prime})^{2}})^{*}(\frac{(\overline{v}_{k}\gamma^{\nu}u_{p})(\overline{u}_{p^{\prime}}\gamma_{\nu}v_{k^{\prime}})}{(k+p)^{2}})
\begin{aligned}|H(y)| \leq (n-1)\max_{\alpha=1,\dots,n-1}\max_{z\in\Gamma}|\kappa_\alpha(z)|, |W(y)| \leq (n-1)^{1/2}\max_{\alpha=1,\dots,n-1}\max_{z\in\Gamma}|\kappa_\alpha(z)| \end{aligned}
A=\frac{1}{\Delta}\begin{pmatrix}-\tfrac{3}{2}t_1\beta-\tfrac{1}{12}d\Delta \tfrac{3}{2}\beta \Delta dt_1-\tfrac{1}{6}t_1d-(\tfrac{3}{2}t_1^2+\tfrac{1}{8}t_2)\beta \tfrac{3}{2}t_1\beta\Delta+\tfrac{1}{12}d\Delta\end{pmatrix}, \beta= 3t_3dt_2-2t_2dt_3.
G(z,n)=C \frac{(z-\eta_{k_0})(z-\eta_{k_0-1})}{(z-\lambda_{k_0})(z-\lambda_{k_0-1})}\prod_{\substack{k\in M k\ne k_0,k_0-1}}\left(1-\frac{z}{\eta_k}\right) \left(1-\frac{z}{\lambda_k}\right)^{-1}\,,
0 = \int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r})f_{h_{1}}(X_{r})\,dr +\int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r})\,d(A_{r}^{\nu^{+}}+A_{r}^{\mu}-A_{r}^{\alpha}) &\,\,-\int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r})\,dA_{r}^{\nu^{-}}+\frac 1 2 L_{t}^{0}(Y)+J_{t}^{+}-\int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r-})\,d(M_{r}-M_{r}^{1}).
\begin{array}{l c}{R ( \alpha_{i}) = \alpha_{\sigma ( i )},}&{R ( \alpha_{i}\alpha_{j}) = \alpha_{\sigma ( i )}\alpha_{\sigma ( j )},}\\{R ( \alpha_{i}\alpha_{j}\alpha_{k}) = \alpha_{\sigma ( i )}\alpha_{\sigma ( j )}\alpha_{\sigma ( k )},}&{\cdots ;}\\ \end{array}
\begin{cases}y^{k+1}=y^k+\beta(\sum_{i=1}^s A_i x_i^k-b), x_i^{k+1}=\mathrm{prox}_{\frac{\alpha_i}{\beta}f_i}(x_i^k-\frac{\alpha_i}{\beta}A_i^\top (2y^{k+1}-y^k)),~~i\in\mathbb{N}_s. \end{cases}
\|u\|_{W^{\mathfrak{m},p}(\Omega,\rho)}:=\Big(\sum_{|\alpha|\leqslant \mathfrak{m}}\|D^\alpha u\|^p_{L^p(\Omega,\rho)}\Big)^{1/p},D^\alpha u=\partial^{|\alpha|}u/\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots \partial x_N^{\alpha_N},\ \ \alpha\in\mathbb{N}^N.
\vee_t=[\vee^{4}_{13},\wedge^{4}_{24}],\ \wedge_t=[\wedge^{4}_{13},\vee^{4}_{24}], \ \vee_k=[\vee^{4}_{13},\vee^{4}_{24}], wedge_k=[\wedge^{4}_{13},\wedge^{4}_{24}], \mbox{and}\neg=[x^{2}_2,x^{2}_1],
Z(A_{m,n}) = Z(C(a_d,a_{d-1},\ldots,a_3,2a_2+1,a_3,\ldots,a_{d-1},a_d)) = Z(C(a_d,a_{d-1},\ldots,a_3,1,a_3,\ldots,a_{d-1},a_d))+2a_2Z(C(a_3,\ldots,a_{d-1},a_d))^2 = Z(C(a_d,a_{d-1},\ldots,a_3,1,a_3,\ldots,a_{d-1},a_d))+2a_2(m-n)^2.
\partial_{\mu_{( D - d + 1 )}}\phi^{a}\partial_{\mu_{( D - d - 2 )}}\phi^{a_{2}}\cdots \partial_{\mu_{D}}\phi^{a_{d}}\frac \partial{\partial \phi^{a}}\frac \partial{\partial \phi^{a_{1}}}( \frac 1{| | \phi | |^{d - 2 \}}) .
E(T) = (\tau_{ij}- \tau_{jj})\sum_{n=1}^{\infty}\left( H_{jj}^{n-1}- H_{jj}^{n}\right) + \tau_{jj}\sum_{n=1}^{\infty}n\left(H_{jj}^{n-1}- H_{jj}^n\right) &= \tau_{ij}+ \tau_{jj}\left(\frac{H_{jj}}{1-H_{jj}}\right)
|(q \circ g)_J(x)| \leq q\bigl(g(x)\bigr)\sum_{i=1}^{s}2^{2i-1}i^i t^i S(s,i)a(f(x))^{i(m+1)}D_g^i &\leq q\bigl(g(x)\bigr)s^{s+1}2^{2s-1}\max(1,t)^s B_s a(f(x))^{s(m+1)}D_g^s,
\pi_{0}(x)=\sum_{n}\frac{1}{n}\mu(n)f(x^{1/n})=f(x)-\frac{1}{2}f(x^{1/2})-\frac{1}{3}f(x^{1/3})-\frac{1}{5}f(x^{1/5})+\frac{1}{6}f(x^{1/6})-\cdot\cdot\cdot,
{(v\circ \widehat F^{n+1})}_+(x) =\sup_z v\circ \widehat F^{n+1}(x,z)=\sup_z v\circ \widehat F^n(Fx,G(x,z)) \le \sup_z v\circ \widehat F^n(Fx,z)={(v\circ \widehat F^n)}_+(Fx).
z_{2n-1}^{*}z_{1}^m = q^mz_{1}^{m}z_{2n-1}^{*}+ mq^{m}(1-q^{2})\epsilon_{2n-1}\epsilon_{1}q^{\rho_{2n-1}+\rho_{1}}z_1^{m-1}z_{2}z_{2n}^{*}, z_{2n-1}^*z_{2n-1}^m = z_{2n-1}^mz_{2n-1}^*+ (1-q^{2m})z_{2n-1}^{m-1}\omega.
\Big(K \big( (p_1,\hat{0}), v \big) \Big)^2 = \frac{\tilde{\alpha}^2 \vert v_1 \vert^4}{\tilde{t}^2\Big( \vert v_1\vert^2 \tilde{\alpha}^2 - p_1^2 \tilde{\alpha}^{2m}\vert v_1 \vert^{2m}\Big)^2}.
\begin{aligned}(\Pi_{\theta_1}\phi)^\dagger\gamma^0\Pi_{\theta_2}\varphi = [(\Pi_{\theta_1}-\Pi_{\theta_1}(x))\phi]^\dagger\gamma^0\Pi_{\theta_2}\varphi+(\Pi_{\theta_1}\phi)^\dagger\gamma^0(\Pi_{\theta_2}-\Pi_{\theta_2}(y))\varphi &\qquad\qquad+\phi^\dagger\Pi_{\theta_1}(x)\gamma^0\Pi_{\theta_2}(y)\varphi,\end{aligned}
e^{-t\Lambda_g}\textbf{\textit{V}}(x^{\prime}) = \int_{\partial \Omega}\textbf{\textit{K}}(t,x^{\prime},y^{\prime})\textbf{\textit{V}}(x^{\prime}) \,dS, \textbf{\textit{V}}\in (L^{2}(\partial \Omega))^{n+1}.
\int_{\partial B(0,r)}(g_{i}(x))^{2}dx=n(n-1)\left(\int_{\partial B(0,r)}x_{1}^{4}dx-\int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx\right)=\frac{2(n-1)}{n+2}r^{n+3}\mathrm{Vol}(S^{n-1}).
N(T_{n+1})=N(T_n)+A_{S_{n+1}}-\left[1-\delta_{N(T_n)}(\{0\})\right], &T_{n+1}=[1-\delta_{N(T_n)}(\{0\})](T_n+S_{n+1})+\delta_{N(T_n)}(\{0\})(T_n+S_{n+1}+I_{n+1}),
lim_{n\rightarrow\infty}(L_{n}-\frac{4}{\pi^{2}}ln(2n+1))=\frac{4}{\pi^{2}}(\sum_{k=1}^{\infty}\frac{2lnk}{4k^{2}-1}-\frac{\Gamma^{\prime}(\frac{1}{2})}{\Gamma(\frac{1}{2})})
\frac{d \rho_{0}}{d \Lambda}= \frac{\Lambda T^{2}}{2 \pi}g \left( \frac{\Lambda}{T}\right) \left[{\frac{6 \lambda_{{4}}+ 1 2 \rho_{{0}}\lambda_{{6}}}{{\Lambda}^{2}+ 4 \rho_{{0}}\lambda_{{4}}}}\right] \frac{1}{2 \lambda_{4}},
\prod_{i=1}^{j-h}(H_a-i+1)^{-1}\,\cdot\,\sigma_a(x)\,\cdot\,\prod_{i=1}^{j-h}(H_a+i+1)\,= &\prod_{i=0}^{j+h}(H_a+i+1)\,\cdot\,\sigma_a(x)\,\cdot\, \prod_{i=0}^{j+h}(H_a-i+1)^{-1}
\sum_{\substack{d|n d \leq \left\lfloor \frac{n}{m+1}\right\rfloor}}\scriptstyle{\binom{\frac{n}{d}-1-m+k}{k}}a_d = \sum_{q=0}^n \sum_{i=1}^{\left\lfloor \frac{n-q}{m+1}\right\rfloor}\sum_{j=0}^{\left\lfloor \frac{n-q-m}{i}\right\rfloor}\binom{k-1+j}{k-1}s_{n-q-m-ji, i}\cdot a_i \cdot p(q).
R^{\rho}_s=\rho+\int_{\tau_i}^s \frac{d-1}{2R^{\rho}_z}z+W^{(\rho)}_s-W^{(\rho)}_{\tau_i}\leq \rho+\rho^{-1}(d-1)\Delta_i^s+W^{(\rho)}_s-W^{(\rho)}_{\tau_i},
\sup_{x\geq 1}\Vert (w,\Omega)x^{-\sigma_0}\Vert_{L_\eta^2}^2+\Vert (w,\Omega)x^{-\sigma_0-\frac{1}{2}}\Vert_{L_x^2L_\eta^2}^2 +\Vert (w_\eta,\Omega_\eta)x^{-\sigma_0}\Vert_{L_x^2L_\eta^2}^2 \lesssim R(\delta,\sigma)^3+R(\delta,\sigma)^4+R(\delta,\sigma)^6.
H(Y_2^n|W_1Y_2^{n_1}S^n\Theta_2U_C) = H(U_B^C|W_1Y_2^{n_1}S^n\Theta_2U_C) &=H({K'}_B^C|W_1Y_2^{n_1}S^n\Theta_2U_C)\geq H({K'}_B^C|1_B^CW_1Y_2^{n_1}S^n\Theta_2U_C) &= H({K'}_B^C|1_B^C) - I({K'}_B^C;W_1Y_2^{n_1}S^n\Theta_2U_C|1_B^C)
g^n_m := \sum_{I\in\{1,\ldots,d\}^j}p^I_{m+\left\lceil\frac{j}{2}\right\rceil-1}(\tfrac{x}{|x|}) p^{n+\left\lceil\frac{j}{2}\right\rceil-1}_I(\tfrac{x}{|x|}) \,, m,n=1,\ldots,\left\lfloor\frac{j}{2}\right\rfloor+1 \,,
u(r,\theta)-\bar u(r, \theta) &=p(r)-\bar p(r)+\sum_{k=0}^{\infty}(C_k-\bar C_k)e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\Big(1+g_k(\frac{r^2}{4})\Big)\varphi_k &=p(r)-\bar p(r).
\dot{R}^2+k=\frac{\mu}{R^2}+ \Bigl(\frac{\tilde{\rho}^2}{\ell^2 \sigma^2}-q^2\Bigr)\frac{1}{R^4}+ \frac{2\tilde{\rho}}{\ell^2\sigma}\,\frac{1}{R}+\frac{\ell^2(\Delta\mu)^2}{16\,(R^3 +\tilde{\rho}/\sigma)^2}\,.
R_{\mu_{1}\nu_{1}\cdots \mu_{s}\nu_{s}}=\partial_{\alpha_{1}\cdots\alpha_{s}}h_{s}^{\beta_{1}\cdots \beta_{s}}\prod_{i=1}^{s}(\delta_{\mu_{i}\alpha_{i}}^{{}}\delta_{\nu_{i}\beta_{i}-}\delta_{\mu_{i}\beta_{i}}^{{}}\delta_{\nu_{i}\alpha_{i}}).
\begin{array}{@{}lll@{}}\Delta =+1\quad : \qquad E_{n}^+=E_{n+1}^->0\quad,\qquad E_{0}^-=0\quad , \Delta =-1\quad : \qquad E_{n}^-=E_{n+1}^+>0\quad,\qquad E_{0}^+=0\quad , \Delta =0\quad : \qquad E_{n}^-=E_{n}^+>0 \quad ,\end{array}
C ( x ) = \frac{\lambda_{1}- ( \lambda_{1}^{2}/ 4 ) e^{- 2 x}}{[ e^{2 x}- ( \frac{\lambda_{1}^{2}}{1 6}) e^{- 2 x}- \frac{1}{2}( \lambda_{o}+ \lambda_{1}) - \lambda_{1}x ]},
\dot{q}_\textbf{j}=\sqrt{-1}\left(\lambda_\textbf{j}q_\textbf{j}+\varepsilon\frac{\partial G}{\partial \bar{q}_\textbf{j}}\right),G(q,\bar q):=\int_{\mathbb{T}^d}g\left(\sum_{\textbf{j}\in\mathbb{Z}^d}\frac{q_\textbf{j}\phi_\textbf{j}+\bar q_\textbf{j}\bar\phi_\textbf{j}}{\sqrt{2\lambda_\textbf{j}}}\right)dx
\| \phi_1 * \phi_2 \|_p^p \geq \int_{G}^{}\int_{G}^{}\phi_1 (g)^{p_1}\phi_2 (g^{-1}g')^{p_2}dg dg' = \| \phi_1 \|_{p_1}^{p_1}\| \phi_2 \|_{p_2}^{p_2}= 1
(l+1)(l+2+2\sigma_n)\bar{p}_{\sigma,l+1}+ 2s(\sigma_n + 1)\bar{p}_{\sigma+\bar{n},l}+ (\sigma_i + 1)(\sigma_i+2)\bar{p}_{\sigma + 2\bar{\imath},l-1}= 0 \forall (\sigma, l).
M^b(K) := \left\{\sum_{j = 0}^n a_j\delta_{c_j}~:~a_j \in \mathbb{C}~~(j = 0, 1, 2, \dots, n)\right\}, M^1(K) := \left\{\sum_{j = 0}^n a_j\delta_{c_j}~:~a_j \geq 0 ~~(j = 0,1, 2, \dots, n), ~\sum_{j=0}^n a_j = 1\right\},
\|A^{\alpha}(\lambda^3 I + A)^{-1}x\|_\mathcal{H}\leqslant C \|(\lambda^3 I + A)^{-1}x\|_\mathcal{H}^{1-\alpha}\|A(\lambda^3 I + A)^{-1}x\|_\mathcal{H}^{\alpha}&\leqslant \dfrac{2^{\alpha}C}{\lambda^{3(1-\alpha)}}\|x\|_\mathcal{H}
(\widehat{\lambda}_{k}I-B)\,\mathbf{y}_{k+1}=\mathbf{x}_{k}, \mathbf{x}_{k+1}=\mathbf{y}_{k+1}\,/\,\Vert \mathbf{y}_{k+1}\Vert , \widehat{\lambda}_{k+1}=\max \left( \frac{B\mathbf{x}_{k+1}}{\mathbf{x}_{k+1}}\right) .
\theta( (F(t_2) - F(t_1))\,t_1^{k_1}\cdots t_n^{k_n}) = \left(-a_{(p,1,0,\dots,0)}+ a_{(q,0,\dots, 0)}\right)(t_1,\dots, t_n) & = q_{n-2}(t_1, \dots, t_n) \cdot e_1(t_1,\dots, t_n)
\sum_{i=1}^n\frac{\eta_i}{q_1^i}=1, \sum_{i=1}^n\frac{\eta_i}{q_2^i}+\sum_{i=n+1}^{\infty}\frac{M}{q_2^i}\ge 1\intertext{and}\left( \sum_{i=1}^n\frac{\eta_i}{q_3^i}\right) +\frac{1}{q_3^{n+t}}=1.
A(y,t,K) = \frac{L}{(ty)^\frac{1}{2}}\left(\frac{a_1 K^2y}{t^3}\right)^{\frac{\sigma}{2}}\left(\frac{a_2 K}{t}\right)^{2it}b_1(y,t,K), B(y,t,K) = \frac{a_3 t^3}{K^2}+ \frac{y}{2}\left( \log\left(\frac{a_4 K^2 y}{t^3}\right) +b_2(y,t,K)\right),