harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\beta_\lambda (r):=\frac{1}{\lambda}\bigl( r-J_\lambda (r) \bigr) :=\frac{1}{\lambda}\bigl( r-(I+\lambda \beta)^{-1}(r)\bigr), \beta_{\Gamma,\lambda}(r):=\frac{1}{\lambda}\bigl( r-J_{\Gamma,\lambda}(r) \bigr) :=\frac{1}{\lambda}\bigl( r-(I+\lambda \beta_\Gamma)^{-1}(r)\bigr){\rm for~}r \in \mathbb{R}.
\left|\{0\leq k <n \mid g^\prime(T^{kN}x) \neq *\}\right| \leq \sum_{k=0}^{n-1}1_{Y\setminus (E_1\cup \dots \cup E_m)}(S^{kN}y) \leq \sum_{k=0}^{nN-1}1_{Y\setminus (E_1\cup \dots \cup E_m)}(S^{k}y) < \delta nN
H_K = \begin{pmatrix}0_{K-1\times 1}I_{K-1}0_{K-1\times K(K-1)}0_{K-2\times K+2}I_{K-2}0_{K-2\times K(K-2)}\vdots \vdots \vdots 0_{K-l\times(l-1)K+l}I_{K-l}0_{K-l\times K(K-l)}\vdots \vdots \vdots 0_{1\times(K-2)K+K-1}1 0_{1\times K}\end{pmatrix}
S(\boldsymbol{\ell})_t=S_0+ \sum_{|J|\leq n}\ell^J_W\langle (\epsilon_{J};\epsilon_0)^{\thicksim},\mathbb{X}_{t}\rangle + \sum_{|J|\leq n}\ell^J_\nu\langle (\epsilon_{J};\epsilon_{1})^{\thicksim}-\lambda(\epsilon_{J};\epsilon_{-1})^{\thicksim},\mathbb{X}_{t}\rangle.
- \vec{\sigma}\cdot \vec{p}\psi_{+ ( 2 )}- e \vec{\sigma}\cdot ( \vec{t}\wedge \hat{r}) A ( r ) \psi_{+ ( 2 )}+ i m \psi_{+ ( 2 )}- i 3 v \vec{t}\cdot \hat{r}\psi_{+ ( 2 )}\phi ( r ) = 0 ,
\phi-\overline{\phi}\in C([0,T_*];H^3), \phi_t \in C([0,T_*];H^2), & u\in C([0,T_*]; H^{s'})\cap L^\infty([0,T]; H^3), \phi \nabla^4 u\in L^2([0,T_*] ; L^2), & u_t \in C([0,T_*]; H^1)\cap L^2([0,T_*] ; D^2),
J_{DS}(x)=\left( \begin{array}{cccccc}0 k 0 0 \cdots 0 0 0 k 0 0 0 0 0 \ddots \ddots \vdots \vdots \vdots \ddots k 0 0 0 0 0 k u_N u_{N-1}u_{N-2}\cdots u_2 0\end{array}\right).
\eta(s)\begin{cases}=1 \s\in [0, 1/2], \textrm{is decreasing}\s\in (1/2,1), =0 \s\in [1,\infty), \end{cases}\eta^*(s)=\begin{cases}0 \s\in [0, 1/2), \eta(s) \s\in [1/2,\infty).\end{cases}
G \left( x , \bar{x}; y , \bar{y}\right) = \int_{0}^{+ \infty}d s \int_{0}^{+ \infty}d \bar{s}\mathrm{e}^{- m^{2}\left( s + \bar{s}\right)}\int D z \int D \bar{z}\exp \left( - \frac 1 4 \int_{0}^{s}\dot{z}^{2}d \lambda - \frac 1 4 \int_{0}^{\bar{s}}\dot{\bar{z}}^{2}d \lambda \right) \left< W ( C ) \right> .
2 \log T + N_2 - N_0 = 2 \log \left({n \brack 1}_q \right) +{n-2 \brack k-2}_q -{n-1 \brack k-1}_q &\le 2n \log q - \left( 1 - \frac{q^{k-1}- 1}{q^{n-1}- 1}\right){n-1 \brack k-1}_q \le 2n \log q - \frac12 q^{(k-1)(n-k)}\rightarrow - \infty,
\left( N^{q}\cos^{q}\left( at\right) \right) \left( u;v\right) =\frac{1}{u}\sum\limits_{0}^{\infty}\left( -1\right)^{n}a^{2n}\frac{u^{2n}}{v^{2n}}q^{-2n\frac{\left( 2n-1\right)}{2}}.
m_{\theta,\geq |\lambda|^{\frac12}}(\xi,t) := \big (1-\chi (|\lambda|^{-\frac12}\xi) \big ) \frac{m_2(\xi)}{|\xi|^{2-\theta}}e^{-t|\xi|}, m_{2,\leq |\lambda|^\frac12}(\xi,t) := \chi (|\lambda|^{-\frac12}\xi) m_2 (\xi) e^{-t|\xi|}.
\begin{aligned}\|u\|_{Y_k(\Pi_{t_0})}&\leq c(T,k,b)\Bigl(\|u_0\|_{\widetilde H^k(\Sigma)}+t_0^{1/6}\|f\|_{M_k(\Pi_{t_0})}+\sum_{j=0}^{j_0-1}\|\partial_t^jf\big|_{t=0}\|_{\widetilde H^{k-3(j+1)}(\Sigma)}\Bigr).\end{aligned}
W_3=W_{31}+W_{32}= O \biggl( \max \biggl\{\frac{k^{1/2+2\beta/d}}{n^{1+2\beta/d}}\, , \, \frac{\log n}{nk^{1/2}}\, , \, \frac{k^{\frac{2 \alpha}{\alpha+d}- \epsilon}}{n^{\frac{2 \alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
\P\left[ \Phi_k (G_N)>(1+\varepsilon)\lambda \right]\leq \sum_{0\leq d_2\leq \dots \leq d_k}\ \sideset{}{^*}\sum_{s_1,\dots, s_k}\frac{\log N}{\left( \binom{SN}{k}-\binom{S(N-1)}{k}\right)^{(1+\gamma)}}
\frac{\partial}{\partial\hat{z}^1}=\left(\frac{1}{a}+\frac{\partial\frac{1}{a}}{\partial\hat{z}^1}\hat{z}^1\right)\frac{\partial}{\partial z^1}+\sum_{p=2}^n\frac{\partial z^p}{\partial\hat{z}^1}\frac{\partial}{\partial z^p},
W_n = \Omega_0 - S(L^{n-1}s) =\left( T_{n-1}- S(L^{n-1}s)\right) \cup\left( T_{n-1}^c - S(L^{n-1}s)\right) &=W_n \cup \left( T_{n-1}^c - S(L^{n-1}s)\right).
\frac{1}{\mu_{\alpha_i}^{n-\nu-2}}\left(\frac{2(n+1)}{n}c_2(X)-c_1(X)^2\right)\cdot[\tilde\omega_i]^{n-2}&\to\binom{n-2}{\nu}(2\pi)^\nu(3n)^{n-\nu-2}\left(\frac{2(n+1)}{n}c_2(X)-c_1(X)^2\right)\cdot(-c_1(X))^k\cdot\alpha_\infty^{n-k-2},
\mathbb{P}_{\lambda}^d(x\in L_n,y\in L_n) =EP_{\lambda}^{\omega}(\forall0\leq i\leq n-1, U_{x_i,x_{i+1}}<H_{x_i},U_{y_i,y_{i+1}}<H_{y_i}) &=E[\prod_{i=0}^{n-1}F(x_i,y_i;x_{i+1},y_{i+1})].
\phi(H)-\phi(H') = 2 (k - 1) - 2 \deg_H(v) - \big( k - 1 - \deg_H(v) - |\Gamma(v) \cap V_{\le k - 1}(H)| \big) = (k - 1) - \deg_H(v) +| V_{\le k - 1}(H) \cap \Gamma(v)| \ge 0.
\int_{\Omega}f(x)\, dx = \int_{B_H}f(\nabla \widetilde u (\xi)) \Phi(\xi)\, d\xi = \int_{B_{H_0}}f(\nabla \widetilde u (\nabla E_0(y)) \det(\nabla_{\xi}^2E(\nabla E_0(y)))\,{\rm det}(\nabla^2 E_0(y))\, dy = \int_{B_{H_0}}f(\nabla \widetilde u (\nabla E_0(y))) \, dy
\begin{tabular}[c]{cc}$-\Delta u\left( s,t,v\right) =f\left( s,t,v,u\left( s,t,v\right)\right) ,$ $\quad\left( s,t,v\right) \in\Omega$ $u\left( s,t,v\right) =0,$ $\quad\left( s,t,v\right) \in\partial\Omega$\end{tabular}
{\cal L}={\cal L_0}+\frac{1}{2}(D_\mu\pi)^2+\frac{g_4^{(1)}}{2}(D_\mu\pi\cdot D^\mu\pi)^2 +\frac{g_4^{(2)}}{2}(D_\mu\pi\cdot D_\nu\pi)(D_\mu\pi\cdot D_\nu\pi) + \cdots
\prescript{}{i_1}{(\prescript{}{i_0}{\hat{s}})}(i)= \begin{cases}\prescript{}{i_0}{\hat{s}}(i) i<i_1 \prescript{}{i_0}{\hat{s}}(i+1) i_1\leq i\end{cases}= \begin{cases}\hat{s}(i) i<i_1 \hat{s}(i+1) i_1\leq i\leq i_0-2 \hat{s}(i+2) i_0-1\leq i\end{cases}
l_{\mathcal{N}_{ij}}:= \begin{cases}\frac{1}{\mathcal{N}_i}\sum_{k=1}^N a_{ik}, \mbox{if}i=j -\frac{a_{ij}}{\mathcal{N}_i}, \mbox{if}i \neq j \mbox{and}(i,j) \mbox{are neighbors}0, \mbox{otherwise}\end{cases}
D^+_{z_k}(FG) =pI_q(g)I_{p-1}\bigl(f(z_k,\cdot)\bigr) +qI_p(f)I_{q-1}\bigl(g(z_k,\cdot)\bigr) \;+ pq I_{p-1}\bigl(f(z_k,\cdot)\bigr)I_{q-1}\bigl(g(z_k,\cdot)\bigr) =:p \tilde{F}_{z_k}G+q\tilde{G}_{z_k}F+pq \tilde{F}_{z_k}\tilde{G}_{z_k}
C_1= \ [\emptyset, 1,\emptyset, 3, \emptyset,\emptyset,\emptyset,\emptyset,\emptyset,10], C_2'= \ [6,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,12,15,\emptyset,\emptyset], C_2= \ [\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,\emptyset,12,15,17,\emptyset], C_3= \ [\emptyset,\emptyset,18,\emptyset,19,24,\emptyset,\emptyset,\emptyset,\emptyset].
\hat a^+\vert \hat{H}_2^+\vert+\hat a^-\vert\hat{H}_2^-\vert=\frac{\hat c^+\hat c^+_A}{((\hat c^+)^2+(\hat c^+_A)^2)^{1/2}}+\frac{\hat c^-\hat c^-_A}{((\hat c^-)^2+(\hat c^-_A)^2)^{1/2}}
\left\|\varphi|\widetilde{B}^{l}(\mathbb{R}^{n},\{\gamma_{k,m}\})\right\|:=\sum\limits_{k=1}^{\infty}\sum\limits_{m \in \mathbb{Z}^{n}}\gamma_{k,m}\delta^{l}(Q_{k,m})\varphi+\sum\limits_{m \in \mathbb{Z}^{n}}\gamma_{0,m}\|\varphi|L_{1}( Q_{0,m})\|.
t_q=[b_{\nu_q}s_{\nu_q}a_{\mu_q}][b_{\nu_{q+1}}s_{\nu_{q+1}}a_{\mu_{q+1}}]=[b_{l_q}^{-1}s_{l_q}a_{r_q}][b_{l_{q+1}}^{-1}s_{l_{q+1}}a_{r_{q+1}}]=b_{l_q}^{-1}s_{l_q}s_{l_{q+1}}a_{r_{q+1}}.
I_{\left( N,M,K\right)}^{}\left( f\right) =\frac{1}{2}{\displaystyle\sum_{k=0}^{K}}{\displaystyle\sum_{\ell=1}^{a_{k}}}{\displaystyle\sum_{j=1}^{N}}\lambda_{j,\left( k,\ell\right)}\times t_{j,\left( k,\ell\right)}^{-\frac{k}{2}}T_{M}\left( f_{\sqrt{t_{j,\left( k,\ell\right)}}}Y_{\left( k,\ell\right)}\right) .
\beta_q =\int_{C^q}\;\Big(\int_B 1_{Ag_1^{-1}}(x)...1_{Ag_q^{-1}}(x)\;x\Big)\;g_1...g_q =\int_B\;\Big(\int_{C^q}1_{x^{-1}A}(g_1)...1_{x^{-1}A}(g_q)\;g_1...g_q\Big)\;x =\int_{B}|x^{-1}A\cap C|^q\;x=\int_{B}|A\cap xC|^q\;x.
{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)=\frac{1}{2\pi i}\int_{-i\infty}^{+i\infty}\frac{{}^{\Psi}\Gamma^{(\alpha,\beta)}(s)B(b+s,c+s-b)}{B(b,c-b)}{}_{2}F_{1}(a,b+s;c+2s;z)p^{-s}ds.
\partial_{t}u_{i}^{\varepsilon}+\mathcal{A}_{d_{i}}^{\varepsilon}u_{i}^{\varepsilon}=\rho_{i}^{\varepsilon}\nabla^{\delta}\theta^{\varepsilon}\cdot\nabla u_{i}^{\varepsilon}+R_{i}\left(u^{\varepsilon}\right)\quad\mbox{in}\;Q_{T}^{\varepsilon},
\frac{1}{\sin \theta}\frac{\partial}{\partial \theta}\left( \sin \theta \frac{\partial Z}{\partial \theta}\right) +\frac{1}{\sin^2 \theta}\frac{\partial^2 Z}{\partial \varphi^2}-\left( \frac{c_1}{\cos^2 \frac{\theta}{2}}+\frac{c_2}{\sin^2 \frac{\theta}{2}}\right) Z = - A Z,
\mathbb{E}|\tilde{\mathcal{E}}_{1,2}|^{2}= N^{-3}\mathbb{E}\sum_{\substack{j_1,j_2,p_1 p_2,q_1,q_2}}W_{p_1j_1}\overline{W_{p_2j_2}}\mathbb{E}_{k}(G^{(1)}_{kp_1j_1})_{j_1q_1}\mathbb{E}_{k}(G^{(2)}_{kp_1j_1})_{q_1p_1}&\times\overline{\mathbb{E}_{k}(G^{(1)}_{kp_2j_2})_{j_2q_2}}\overline{\mathbb{E}_{k}(G^{(2)}_{kp_2j_2})_{q_2p_2}}
[\begin{matrix}N_{11}\\N_{22}\\N_{12}\end{matrix}]=\{\int_{-h}^{h}[\begin{matrix}C_{11}&C_{12}&C_{13}\\C_{12}&C_{22}&C_{23}\\C_{13}&C_{23}&C_{33}\end{matrix}]dx_{3}\}[\begin{matrix}u_{1,1}^{0}\\u_{2,2}^{0}\\ \frac{1}{2}(u_{1,2}^{0}+u_{2,1}^{0})\end{matrix}]
g_{X,\mathrm{hyp},s}(z,w)-\sum_{n=-\infty}^{\infty}g_{\mathbb{H},s}(\sigma_{p}^{-1}w,\gamma_{\infty}^{n}\sigma_{p}^{-1}z)= &\frac{4\pi\Im(\sigma_{p}^{-1}z)^{1-s}}{2s-1}\mathcal{E}_{X,\mathrm{par},p}(w,s)-\frac{4\pi}{2s-1}\Im(\sigma_{p}^{-1}w)^{s}\Im(\sigma_{p}^{-1}z)^{1-s}+ O\big(e^{-2\pi\Im(\sigma_{p}^{-1}z)}\big).
|E_{int,2}| \leq \frac{1}{2 b^{2}(\log(\log(t)))^{3/2}\log^{b}(t)}+ \frac{1}{\sqrt{\log(\log(t))}\log^{b+1}(t)}+\frac{1}{2(b+1) (\log(\log(t)))^{3/2}\log^{b+1}(t)}&\leq \frac{1}{100}\frac{1}{b \log^{b}(t) \sqrt{\log(\log(t))}}, t \geq T_{0}
{\cal{Z}}^{P F}( X_{1}, X_{2}, X_{3}; p ) ={\frac{\det ( X_{j}^{p + 7 - i}- X_{j}^{i})}{\det ( X_{j}^{2 M + 1 - i}- X_{j}^{i})}}~ ~ ; ~ ~ 1 \le i , j \le 3 ~ .
\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,z,T-t)u(z,t)~{\rm d}v_{\mathbb{H}^{n}}(z)=~\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,y,T) \, u_{0}(y)~{\rm d}v_{\mathbb{H}^{n}}(y) & +\int \limits_{0}^{t}\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,y,T-s)e^{\mu s}\, u(y,s)\big( e^{\beta u^{p}(y,s)}-1\big)~{\rm d}v_{\mathbb{H}^{n}}(y) \,{\rm d}s.
|\psi (t)-\psi (u)| =1-\psi (u)=1-\dfrac{u-a_{2n-1}}{b_{2n-1}-a_{2n-1}}=\dfrac{b_{2n-1}-u}{b_{2n-1}-a_{2n-1}}&\leq \dfrac{(b_{2n-1})^{1/k}}{b_{2n-1}-a_{2n-1}}<\dfrac{(b_{2n-1})^{1/k}}{2^{2n-1}(b_{2n-1})^{1/k}}<\dfrac{1}{2^{2n-1}}.
\int_{\mathbb{R}^{m}}\varphi(x)dx=\int_{q\in N}\mathrm{vol}_{N}(dq)\int_{x\in F^{-1}(q)}\frac{\varphi(x)}{\|F^*\mathrm{vol}_{N}\|}\mathrm{vol}_{F^{-1}(q)}(dx),
\Lambda(n)=\sum_{\genfrac{}{}{0pt}{}{kl=n}{k>v, l>w}}\Lambda(k)\sum_{\genfrac{}{}{0pt}{}{d|l}{d>w}}\mu(d)+\sum_{\genfrac{}{}{0pt}{}{kl=n}{l\le w}}\log k\ \mu(l)-\sum_{\genfrac{}{}{0pt}{}{klm=n}{l\le v, m\le w}}\Lambda(l)\mu(m),
C_{p,g+1}^{(\ell)}= (-1)^{p-1}\sum_{\gamma=0}^{g+1}(-2)^{\gamma}\binom{\ell}{\gamma}\binom{n-\gamma}{g+1-\gamma}\sum_{k=0}^{p-2g-3}(-1)^{k}\binom{n-\ell}{k}&= (-1)^{p-1}\sum_{\gamma=0}^{g+1}(-2)^{\gamma}\binom{\ell}{\gamma}\binom{n-\gamma}{g+1-\gamma}(-1)^{p-2g-3}\binom{n-\ell-1}{p-2g-3}.
H_{R}(NN) = N_F^{R}(NN) + \frac{1}{12}=\sum_{n=1}^{\infty}n ({d_{-n}}^{-j}{d_{n}}_{j}+{d_{-n}}^{j}{d_{n}}_{-j}) + \frac{1}{12}\;\;\;\;\;\;\; (j = 2)
\widehat{M}^{+}=\left\{(\vec{x}, \vec{u}) \in \mathbb{P}(V^{+})_{\widehat{U}}: \sum_{i=1}^a x_i w_{ij}(\vec{u})=0 \mbox{for all}1\le j\le b\right\}, &\widehat{M}^{-}=\left\{(\vec{y}, \vec{u}) \in \mathbb{P}(V^{-})_{\widehat{U}}: \sum_{j=1}^b y_j w_{ij}(\vec{u})=0 \mbox{for all}1\le i\le a\right\}.
s(g,\xi)=\hbox{Ad}^*_{g}.\xi,r(g,\xi)= \xi ,(g_1,\xi_1)(g_2,\hbox{Ad}^*_{g_1}.\xi_1)=({g_1}{g_2},\xi_1),(g,\xi)^{-1}=(g^{-1},\hbox{Ad}^*_{g}.\xi).
J_0(ax) J_0(bx) = \int\limits_0^{\infty}\!dt \,t J_0(tx)\!\int\limits_0^{\infty}\!dy y J_0(ay) J_0(by) J_0(ty) \;, J_0(cx) J_0(dx) = \int\limits_0^{\infty}\!dq \, q J_0(qx)\!\int\limits_0^{\infty}\!dz z J_0(cz) J_0(dz) J_0(qz) \;,
g^{i}=\frac{(\tilde{V}_R)_{1i}}{\det(\tilde{V}_R)}=\frac{\partial c_1}{\partial \lambda_i}\frac{1}{\prod\limits_{\stackrel{j=1}{j\neq k}}^n ( \lambda_i- \lambda_j)}=\frac{1}{\prod\limits_{\stackrel{j=1}{j\neq i}}^n ( \lambda_i- \lambda_j)}.
\begin{array}{l}{\nabla^{2}\phi_{e}= - \rho_{e}( \mathbf{x}) , ~ ~ ~ ~ \nabla^{2}\phi_{m}= \rho_{m}( \mathbf{x}) ,}{\nabla^{2}\mathbf{A_{m}}= - \mathbf{J_{e}}( \mathbf{x}) , ~ ~ ~ ~ \nabla^{2}\mathbf{A_{e}}= \mathbf{J_{m}}( \mathbf{x})}\end{array}
x_{0}(z)=\sum_{n\in 2\mathbb{Z}}x_{0}(n/2)z^{-\frac{n}{2}-1}=\sum_{n\in 2\mathbb{Z}}X(n)z^{-\frac{n}{2}-1}, &x_{1}(z)=\sum_{n\in 2\mathbb{Z}+1}x_{1}(n/2)z^{-\frac{n}{2}-1}=\sum_{n\in 2\mathbb{Z}+1}X(n)z^{-\frac{n}{2}-1}, &x(z)=\sum_{n\in \mathbb{Z}}x(n/2)z^{-\frac{n}{2}-1}=x_{0}(z)+x_{1}(z).
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{1-\nu^{2}}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{12}\end{matrix}]
\varphi_{k}(t,x) =\sum_{i=1}^{k}\alpha_{i}^{k}(t) w_{i}(x), \; \mu_{k}(t,x) = \sum_{i=1}^{k}\beta_{i}^{k}(t) w_{i}(x), \; \sigma_{k}(t,x) = \sum_{i=1}^{k}\gamma_{i}^{k}(t) w_{i}(x),
\begin{array}{c c c c}{\mathrm{l o n g ~ r o o t s :}}{\pm \, 2 \! e_{k}}{}{( 1 \leq k \leq 3 )}{\mathrm{s h o r t ~ r o o t s :}}{\pm \, e_{k}\pm e_{l}}{}{( 1 \leq k < l \leq 3 )}\end{array}~ .
\Lambda_{ij,\,lm}(\hat{k})=\delta_{il}\delta_{jm}-2\hat{k}_j \hat{k}_m \delta_{il}+\frac{1}{D-2}\left (-\delta_{ij}\delta_{lm}+ \hat{k}_l \hat{k}_m \delta_{ij}+\hat{k}_i \hat{k}_j \delta_{lm}\right ) +\frac{D-3}{D-2}\hat{k}_i \hat{k}_j \hat{k}_l \hat{k}_m \,.
\bigg|\int_{\Bbb R^n}\k_j(u,v)D_{k'}^\#(v,y)d\mu(v)\bigg| =\bigg|\int_{S(y,A_02^{3+k'})}\ [k_j(u,v)-k_j(u,y)]D_{k'}^\#(v,y) d\mu(v)\bigg| &\lesssim \int_{S(y,A_02^{3+k'})}\ \frac{1}{V_{-j}(u)}|T_j(v)-T_j(y)|^{\gamma}|D_{k'}^\#(v,y)|d\mu(v).
\max_{A\in\mathcal{P}_{x,n}}\left|1-\frac{\beta_{x}\left(A\right)}{q_{x,n}^{t}(A)}\right|\le\varepsilon_{x,n}(t)\quad\textrm{with}\quad\lim_{t\to\infty}\varepsilon_{x,n}(t)=0.
c_1^{\dagger}(k_{22},k_{21},k_{11})=q^{2n}c_2^{\dagger}(k_{22}-1,k_{21},k_{11})+q^{2n}c_1^{\dagger}(k_{22},k_{21}-1,k_{11})+c_1^{\dagger}(k_{22},k_{21},k_{11}-1), c_2^{\dagger}(k_{22},k_{21},k_{11})=c_2^{\dagger}(k_{22}-1,k_{21},k_{11})+c_1^{\dagger}(k_{22},k_{21}-1,k_{11})+c_1^{\dagger}(k_{22},k_{21},k_{11}-1),
D_{j}=\frac{C_0\,((pq)^{1/2}-1/2)\,(pq)^{-j+1}}{(pb_{j-1}+1)\,\ell_{j}^{pb_{j-1}+1}}Q_{j-1}^p\geqslant \frac{C_0M((pq)^{1/2}-1/2)}{B_0}(pq)^{-\frac{3j}{2}+1}\,Q_{j-1}^p, Q_{j}=\frac{C_0}{(q\beta_{j-1}+1)(q\beta_{j-1}+2)}D_{j-1}^q\geqslant\frac{C_0}{\widetilde{B}_0^2}(pq)^{-j}\,D_{j-1}^q.
\left\langle E^{ij}_0, ({m_{\Lambda}}_*f) \otimes ({m_{\Lambda}}_*g) \right\rangle = \Lambda^{2 - \mu_i + \mu_j}\left( \mathfrak{P}_{\ell}m_{\Lambda}\left\{\Psi^i(f), \Psi^j(g) \right\}_{\ell}^{\Sigma_0}\right)[0].
L_{r,m}\varphi(x) = \alpha(x,r,m)\frac{\partial \varphi}{\partial x}(x) + \frac{1}{2}\beta^2(x,r,m)\frac{\partial^{2}\varphi}{\partial x^2}&+\int_{\mathbb{R}}\{\varphi(x+\gamma(x,r,m,\zeta))-\varphi(x)-\nabla\varphi(x) \gamma(x,r,m,\zeta) \}\nu(d\zeta),
|1- \widehat{\rho}(\xi_n/\Lambda) \widehat{\rho}(\xi_j/\Lambda)| = |1-\widehat{\rho}(\xi_n/\Lambda) + \widehat{\rho}(\xi_n/\Lambda)(1-\widehat{\rho}(\xi_j/\Lambda))| &\leq 2(|\zeta_\Lambda(\xi_n)| + |\zeta_\Lambda(\xi_j)|),
\Vert f'_h - f' \Vert_1 = \int_{-\infty}^\infty |g'(t)| \, \mathrm{d}t = \sum_{k = 1}^M \int_{t_{k-1}}^{t_{k}}| g'(t)| \, \mathrm{d}t = \sum_{k=1}^M \big| g(t_k) - g(t_{k-1}) \big| &= |g(t_1) - g(-\infty)| + |g(\infty) - g(t_{M-1})| + \sum_{k=2}^{M-1}\big| g(t_k) - g(t_{k-1}) \big|,
\beta_{i,k}:= \begin{cases}0 \mbox{if}d_{i,k}-x_i\leq 0 c\P(\omega_k)\frac{x_i-d_{i,k}}{\bar{y}}\mbox{if}0\leq d_{i,k}-x_i\leq \bar{y}-c\P(\omega_k) \mbox{if}d_{i,k}-x_i \geq \bar{y}.\end{cases}
G_{n}(\omega)=\frac{1}{\sqrt{1+\frac{1}{\epsilon^{2}T_{n}^{2}(\omega_{0}/\omega)}}}=\sqrt{\frac{\epsilon^{2}T_{n}^{2}(\omega_{0}/\omega)}{1+\epsilon^{2}T_{n}^{2}(\omega_{0}/\omega)}}.
\left[\begin{array}{c}\dot{z}_1(t) \dot{z}_2(t)\end{array}\right]=\left[\begin{array}{cc}0 I_{N-1}-\bar{\Lambda}-\bar{\Lambda}\end{array}\right]\left[\begin{array}{c}z_1(t) z_2(t)\end{array}\right]+\left[\begin{array}{c}0 Q\end{array}\right]w(t)\,,
|x|^{1+\gamma}\, |\mathcal{R}[f](x)| \leq C \bigg( \min \big\{\frac{1}{|\alpha| |x|^{1-\gamma}}, \frac{|x|^\gamma}{|\alpha|^{\frac{1}{2}}}\big\}\| f\|_{L^1 (B_{R})}& + \min \big \{\frac{1}{|\alpha| |x|^{1-\gamma}}, |x|^{1+\gamma}\big \}\| f \|_{L^1 (B_{R})}+ \| |y|^{1+\gamma}f \|_{L^1 (B_{R})}\bigg ) \,.
\langle0|0\rangle_{\mathrm{ferm}}({\alpha_{\Re}}):=\langle0|(\mathrm{pf}\mathit{\Omega}_\lambda^{{\alpha_{\Re}}})_{W_a^-}|0\rangle=\mathrm{pf}({i\hbar \mathit{\Omega}})^{{\alpha_{\Re}}+1}\;,
P_{c,\mathrm{L}}(T) \approx\sum_{n=1}^{N_\mathrm{L}}(-1)^{n+1}{N_\mathrm{L}\choose{n}}\times \int_{0}^{\infty}\mathrm{e}^{-\frac{n \eta_\mathrm{L}x^{\alpha_\mathrm{L}}T\sigma^2}{C_\mathrm{L}M_\mathrm{r}M_\mathrm{t}}-Q_n(T,x)-V_n(T,x)}\hat{f}_\mathrm{L}(x)\mathrm{d}x,
\left\langle A(\bar{z}_n-x^*),\bar{w}_n-A\bar{z}_n\right\rangle=\frac{1}{2}\left\{||\bar{w}_n-Ax^*||^2-||A\bar{z}_n-Ax^*||^2-||\bar{w}_n-A\bar{z}_n||^2\right\}
\aligned\frac{1}{2}\mathcal{L}H^{2}= |\nabla^{\perp}{\vec H}|^{2}+H^{2}-\sum_{i,j,p,q}H^{p^{\ast}}h_{ij}^{p^{\ast}}\cdot H^{q^{\ast}}h_{ij}^{q^{\ast}}.\endaligned
S(X,\mathbf{d},\mathbf{D};L_1,L_2,L_3)=\sum_{\mathbf{x}\in\Lambda(\mathbf{D})\cap X\mathcal{R}}\tau\left(\frac{L_1(\mathbf{x})}{d_1}\right)\tau\left(\frac{L_2(\mathbf{x})}{d_2}\right)\tau\left(\frac{L_3(\mathbf{x})}{d_3}\right).
E_3^q=I_3-\sum_{j_3,j_2,j_1=0}^q C_{j_3j_2j_1}^2-\sum_{j_3,j_2,j_1=0}^q C_{j_3j_2j_1}C_{j_1j_2j_3}\ \ \ (i_1=i_3\ne i_2),
-\sum_{i=2}^{n-1}\langle x_i-\bar{x},B_{i-1}(x_{i-1})-B_{i-1}(\bar{x})\rangle \leq \sum_{i=2}^{n-1}\langle x_i-\bar{x},B_{i-1}(x_i)-B_{i-1}(x_{i-1})\rangle
\begin{aligned}\lim_{x \rightarrow 1^-}(1-x) F(x) = \lim_{n \rightarrow \infty}A_n &= \prod_{k=0}^{\infty}\left( 1 +q^{Nk+a(r)}\right) \frac{g(1)}{\prod_{n=1}^{\infty}\left(1-q^{nN}\right)}.\end{aligned}
\sup\limits_{B_{\frac{3\delta R}{4}}(\bar x^\prime)\cap \partial\Omega}\sup\limits_{|\tau|=1,\tau\cdot \nu=0}|u_{\tau\tau}| \le \epsilon \sup\limits_{B_{\frac{13\delta R}{16}}(\bar x^\prime)\cap\Omega}|D^2u| + C_\epsilon (1+ \frac{1}{(\delta R/16)^2}),
T_{\mu \nu}^{e f f}( \phi ) = \phi_{, \mu}\phi_{, \nu}- \frac{1}{2}\overline{{{g}}}_{\mu \nu}\phi_{, \alpha}\phi_{, \beta}\overline{{{g}}}^{\alpha \beta}+ \overline{{{g}}}_{\mu \nu}V_{e f f}( \phi ) , V_{e f f}( \phi ) = \frac{1}{4 U ( \phi )}( V + M )^{2}.
L_1 \widetilde{F}_{\eta}( (\widetilde{v}_{g,k},0),\omega_{k,1}\oplus \omega_{k,2}) L_1^*= d\Gamma(\omega_1)\otimes 1+1\otimes d\Gamma(\omega_{k_2}) &+ \eta W(\widetilde{v}_{g,k},-1)\otimes \Gamma(-1).
\int_{A_1}|u|\, d\mu_\omega \le C H_R^{1-2\beta}\, \left(\int_{A_1}g_{u,\rho}^2\, d\mu_\omega\right)^{1/2}H_R^\beta\, \mu_\omega(A_1)^{1/2}= C(\beta-1)\, R\, \left(\int_{A_1}g_{u,\rho}^2\, d\mu_\omega\right)^{1/2}\, \mu_\omega(A_1)^{1/2}.
J'_n = (x_1x_n, x_2x_n, \dots, x_{n-1}x_n, x_{n-1}x_{n+1}, \dots, x_{n-1}x_{2n-2}), J''_n = (x_i x_{n+j}- x_{n+i}x_j : 1 \leq i<j \leq n-2).
C_{\mu}= 2 c_{\mu,h}c' \mu(W)\big(1 + \mu_f^\star(W) \big) = \frac{2}{\mu_h(h^{-1})}\big( 1 + \mu_h(Wh^{-1})\big) \big( 1 + \mu_f^\star(W)\big) \frac{\mu(W)}{\mu(h)}.
\mathcal L_n(z_2)- \mathcal L_n(z_1) = \frac{z_1-z_2}{2\pi i}\oint_{|z|=2 r_n}\frac{\mathcal L_n(z)}{(z-z_1)(z-z_2)}dz = \mathcal O\left(n^{\frac{3}{2}}(z_1-z_2)\right)
g ( r ) = A ( r ) = C_{1}^{( 1 )}\sum_{n = 0}^{\infty}\frac{( \alpha )_{n}( \beta )_{n}}{n ! \left( \frac 5 2 \right)_{n}}\, r^{2 n}+ C_{2}^{( 1 )}r^{- 3}\sum_{n = 0}^{\infty}\frac{\left( \alpha - \frac 3 2 \right)_{n}\left( \beta - \frac 3 2 \right)_{n}}{n ! \left( - \frac 1 2 \right)_{n}}\, r^{2 n}, \qquad k = + 1 .
t_{o}\left( g , a , n \right) = \ln \left\{\frac{x_{o}^{2}\left( g , a , n \right)}{f \left[ x_{o}\left( g , a , n \right) , a , n \right]}\right\}= \quad ; \quad w_{o}\left( g , a , n \right) = \ln \left[ x_{o}\left( g , a , n \right) \right] =
\kappa_E^{-1}\left( \mathcal{Z}_E ( \mu \setminus F) \right) = \kappa_E^{-1}( \mathcal{Z}_E ( \mu ) ) \setminus \left( \bigcup_{e \in F}\kappa_E^{-1}( \mathcal{Z}_E(\mu e))\right) &= \mathcal{Z}_{E_\curlyvee}( \mu ) \setminus \left( \bigcup_{e \in F}\mathcal{Z}_{E_\curlyvee}(\mu e)\right)= \mathcal{Z}_{E_\curlyvee}( \mu \setminus F)
C_{011}(N) = \frac{(-1)^N}{N!}\sum_{j_0+j_1+j_2+ \cdots + j_N = N-1}B_{j_1}B_{j_2}\cdots B_{j_N}\frac{1^{j_1}2^{j_2}\cdots N^{j_N}}{j_0! j_1 !j_2 ! \cdots j_N!}
\left\|( u^s_{yy}+\partial_y\tilde{w}_1)\partial^{\alpha}\bar{v}\right\|_{L^2_{k+\ell+{\alpha_2}}}=\left\|( u^s_{yy}+\partial_y\tilde{w}_1)\partial^{\alpha_1}_x\partial^{\alpha_2-1}\bar{u}_x\right\|_{L^2_{k+\ell+{\alpha_2}}}&\qquad\qquad\le C (1+\| \tilde{w}_1\|_{H^{m-1}_{k+\ell}(\mathbb{R}^2_+)})\left\|\bar{w}\right\|_{H^{m-2}_{k+\ell}(\mathbb{R}^2_+)},
d^4([x_6])+d^2([x_2])\gamma_1^2 =2[y_1\gamma_1\gamma_2^2+y_2\gamma_1^2\gamma_2]+2[y_1\gamma_1^3+y_2\gamma_1^2\gamma_2] &=2[y_1\gamma_1^3-y_2\gamma_1^2\gamma_2]+2[y_1\gamma_1^3+y_2\gamma_1^2\gamma_2] &=4[y_1\gamma_1^3]
E_{\Delta'}(\lambda_1)E_{\Delta'}(\lambda_2) = q_{\lambda_1}^{1/2}q_{\lambda_1\lambda_2\lambda_1^{-1}}^{1/2}q_{\lambda_1\lambda_2}^{-1/2}E_{\Delta'}((\lambda_1\lambda_2\lambda_1^{-1})\lambda_1) = E_{\Delta'}(\lambda_1\lambda_2\lambda_1^{-1})E_{\Delta'}(\lambda_1).
A_{0}(\tau, s)( \varphi) \ = \v^{\frac{s-1}{2}}\, \varphi(0) \ + \W_{0,\infty}(\tau, s) \cdot \left( \prod_{\ell\in \Sigma}W_{0, \ell}(s, \varphi) \right) \cdot \frac{L^{\Sigma}(s, \chi_F)}{L^{\Sigma}(s+1, \chi_F)}.
\tilde{T}_1 := \bigvee_{i=1}^{M_1}X_i,\tilde{T}_2 := \bigvee_{i = M_1 + 1}^{M_1 + M_2}X_i,\dots, \tilde{T}_g := \bigvee_{i = M_1 + \cdots + M_{g-1}+ 1}^{M_1 + \cdots + M_g}X_i.
0 < \tfrac12\tfrac{d}{dt}|u(t^{**})|^2 = u(t^{**}) \cdot \dot{u}(t^{**}) , &= u(t^{**}) \cdot f\big(t^{**},u(t^{**}),u(t^{**}-\tau_1(t^{**},u(t^{**}))),\dotsc,u(t^{**}-\tau_N(t^{**},u(t^{**})))\big), &=x\cdot f\big(t^{**},x,u(t^{**}-\tau_1(t^{**},u(t^{**}))),\dotsc,u(t^{**}-\tau_N(t^{**},u(t^{**})))\big).
H_1=1+\sum_i{{Q_i}\over{[|\vec{x}-\vec{x}_{0\,i}|^2+{{4Q}\over{(p-3)^2}}|\vec{z}-\vec{z}_0|^{p-3}]^{2\over{p-3}}}},\ \ \H_2={Q\over{|\vec{z}-\vec{z}_0|^{5-p}}}.
Uniform_{\mathbb{GT}_c(N)}^{x^{(N)}}(dx^{(1)},\cdots,dx^{(N-1)})=\Lambda^N_{N-1}(x^{(N)},dx^{(N-1)})\cdots \Lambda_2^{3}(x^{(3)},dx^{(2)})\Lambda_1^{2}(x^{(2)},dx^{(1)}).
\alpha^a_i=\sqrt{\frac{\omega}{2\hbar g}}\left[\hat{q}^a_i+i\frac{g}{\omega}\hat{p}^a_i\right]\ \ ,\ \{\alpha^a_i}^\dagger=\sqrt{\frac{\omega}{2\hbar g}}\left[\hat{q}^a_i-i\frac{g}{\omega}\hat{p}^a_i\right]\ \ ,
N_2[\epsilon,v,w] = \{f_2(u) + f_2(\eta^\frac{2}{1-q}{\sf U}) - q\eta^{-2}{\sf U}^{q-1}(u+\eta^\frac{2}{1-q}{\sf U}) \}(1-\chi_1) \chi_2 = \tfrac{1}{2}f_2''(u_\kappa) (u+\eta^\frac{2}{1-q}{\sf U})^2 (1-\chi_1) \chi_2,
\begin{cases}-\Delta \psi_1''(c_0) = \lambda''(c_0) m \psi_1(c_0) + \lambda''(c_0) a (p-1) (w(c_0))^{p-2}\psi_1(c_0) + \gamma_1''(c_0) \psi_1(c_0) \mbox{in $\Omega$}, \partial_{\mathbf{n}}\psi_1''(c_0) = \lambda''(c_0) b (q-1) (w(c_0))^{q-2}\psi_1(c_0) + \gamma_1''(c_0) \psi_1(c_0) \mbox{on $\partial \Omega$}. \end{cases}
\frac{\partial}{\partial t}\frac{\delta\bar\ell^{\,\Gamma_0}}{\delta\sigma^1}+\operatorname{div}^{\Gamma_0}\frac{\delta\bar\ell^{\,\Gamma_0}}{\delta\sigma^2}=-\operatorname{ad}^*_{\sigma^1}\frac{\delta\bar\ell^{\,\Gamma_0}}{\delta\sigma^1}-\operatorname{Tr}\left(\operatorname{ad}^*_{\sigma^2}\frac{\delta\bar\ell^{\,\Gamma_0}}{\delta\sigma^2}\right).
G ( t - t^{\prime}, \tau ) = \frac{1}{a^{D}}\int_{1 / \Lambda^{2}}^{\infty}d s \frac{e^{- \frac{t - t^{\prime}}{4 s}}}{( 4 \pi s )^{\frac{1}{2}}}\int_{0}^{\infty}\phi_{r}( \tau ) | C ( r ) |^{2}e^{- \frac{s}{a^{2}}( r^{2}+ \rho_{D}^{2})}d r \, ,
p_i(w^{**}) =\cfrac{\overline{\overline{(a-1)a^\prime}b_i}+\overline{(a-1)a^\prime}a_i+\overline{\overline{(a+1)a^\prime}b_i}+\overline{(a+1)a^\prime}a_i}{m}-2 &=\cfrac{\overline{b_i-a^\prime b_i}+\overline{b_i+a^\prime b_i}+(m+2)a_i}{m}-2.
\psi''_{\mathrm{u}}( t_{1}) = ( p^{+}+ \gamma^{-}- 1 ) t_{1}^{p^{+}}\int_{M}| D \mathrm{u}( z ) |^{p( z )}\,\, dv_{g}( z ) &+ ( q^{+}+ \gamma^{-}- 1 ) t_{1}^{q^{+}}\int_{M}\mu ( z )\,| D \mathrm{u}( z ) |^{q( z )}\,\, dv_{g}( z ) &- \lambda ( r^{-}+ \gamma^{-}- 1 ) t_{1}^{r^{-}}\int_{M}| \mathrm{u}( z ) |^{r( z )}\,\, dv_{g}( z ).
\chi_c(U',{\cal F})+{\rm rank}\{\cal F}\cdot \deg(X\cap A_L) &=2\chi_c(U'\times_XY,{\cal F}|_{U'\times_XY})-{\rm dim\tot}\phi_u(j_!{\cal F},p_L)-\sum_{x\in X}{\rm dim\tot}\phi_x(j_!{\cal F},p_L).
4\Theta_{M_1 \cap T}(q) - 10\Theta_{M_2\cap T}(q) - 15\Theta_{M_3\cap T}(q) = \sum_{k = 0}^\infty \left(4N_{M_1 \cap T}\left(k/3\right) - 10N_{M_2\cap T}\left(k/3\right) - 15N_{M_3\cap T}\left(k/3\right)\right)q^{k/3}
\frac{d \hat{N}(t)}{dt}= \frac{1}{x_1}J^{}_{1/2}-\sum_{i=1}^{I-1}J^{}_{i+1/2}\left(\frac{1}{x_{i}}-\frac{1}{x_{i+1}}\right) - \frac{1}{x_I}J^{}_{I+1/2}+ \sum_{i=1}^I \sum_{k= i+1}^I \hat{N}_k(t) S(x_k) b(x_i,x_k) \Delta x_i - \sum_{i=1}^I \hat{N}_i(t) S(x_i) \sum_{j= 1}^{i-1}\frac{x_j}{x_i}b(x_j,x_i) \Delta x_j.