harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
~ \left\{(x,y) \in X \times Q^{-1}(B) : (x,y) \in H(z) \cap \mathcal{T}^{-1}_{nb}\right\}\crcr ~ = \left( \left\{x\in X : \, \exists\, y \in Q^{-1}(B),\, (x,y) \in H(z) \cap \mathcal{T}^{-1}_{nb}\right\}\times X \right) \cap H(z) \cap \mathcal{T}^{-1}_{nb}.
u_{i+1/2}^{*}=u_{i+1/2}^{*}(u_{i+1/2}^{L},u_{i+1/2}^{R}),u_{i-1/2}^{*}=u_{i-1/2}^{*}(u_{i-1/2}^{L},u_{i-1/2}^{R}),
h_{x_{i_0}}(h_{x_1}^{d_1}\cdots h_{x_{i_0}}^{d_{i_0}}\cdots h_{x_{n}}^{d_n})=h_{x_1}^{d_1}\cdots h_{x_0}^{d_{i_0}+1}\cdots h_{x_{n}}^{d_n}+f
B_\varepsilon=\gamma_\varepsilon-\frac{t_\varepsilon}{\gamma_\varepsilon}+\frac{S_{0,\varepsilon}}{\gamma_\varepsilon^3}+\frac{S_{1,\varepsilon}}{\gamma_\varepsilon^5}+(A(\gamma_\varepsilon)-2\xi_\varepsilon)\frac{S_{2,\varepsilon}}{\gamma_\varepsilon}+\frac{\zeta_\varepsilon \bar{w}_\varepsilon}{\gamma_\varepsilon}\,.
\max_{t\in [\mathcal{T}_0 , \mathcal{T}_0 + \mathcal{T}]}||z_{\epsilon}^{N,M}(t) - \bar{\bar{z}}(t)||\ \leq \ \kappa_5 (\epsilon , \mathcal{T}), \ \ \ \ \{\rm where}\ \ \ \ \ \lim_{\epsilon\rightarrow 0}\kappa_5(\epsilon , \mathcal{T}) = 0,
\bar{a}_{\hbar}:=\frac{q_{\hbar}+ip_{\hbar}}{\sqrt{2}}=\sqrt{\frac{\hbar}{2}}\left( M_{x}+\frac{d}{dx}\right) a_{\hbar}^{\ast}:=\frac{q_{\hbar}-ip_{\hbar}}{\sqrt{2}}=\sqrt{\frac{\hbar}{2}}\left( M_{x}-\frac{d}{dx}\right) .
Volume=\|\vec{x}_{1}\|\|\vec{x}_{2}\|\sqrt{1-(\frac{\vec{x}_{1}\cdot\vec{x}_{2}}{\|\vec{x}_{1}\|\|\vec{x}_{2}\|})^{2}}=\sqrt{\|t\vec{x}_{1}\|^{2}\|\vec{x}_{2}\|^{2}-(\vec{x}_{1}\cdot\vec{x}_{2})^{2}}
\partial_{t}\big( \partial_{s}\psi^{s}_{t}- \pi(\delta^{R}_{s}\gamma^{s}_{t})\psi^{s}_{t}\big)=\pi(\delta^{R}_{t}\gamma^{s}_{t})\big(\partial_{s}\psi^{s}_{t}- \pi(\delta^{R}_{s}\gamma^{s}_{t})\psi^{s}_{t}\big)\,.
A_{1}=\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big)^{\ast}H_{1}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big)=O\big(\vert z_{1}\vert^{-2\vert \kappa\vert}\big), A_{2}=\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big)^{\ast}\sigma H_{1}\sigma\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) =O\big(\vert z_{2}\vert^{-2\vert \kappa\vert}\big),
D_m\Big\{\big[v(t),w(t)\big];(0,0)\Big\}=\Bigg[ \frac{16\varepsilon^2+3.2\varepsilon-0.09}{\varepsilon}w(t)^2 &+(0.4-16\varepsilon)v(t)w(t)+(4\varepsilon-0.1)v(t)^2\Bigg]^{1/2}.
q_{kj}^{-1}q_{bj}t_{ik}^{ab}=q_{ij}q_{aj}^{-1}t_{ik}^{ab}=q_{jk}q_{jb}^{-1}t_{ik}^{ab}=q_{ji}^{-1}q_{ja}t_{ik}^{ab}, q_{ji}^{-1}q_{ja}t_{ik}^{ab}t_{jb}^{uv}-q_{ui}^{-1}q_{ua}t_{ib}^{av}t_{jk}^{ub}=t_{jk}^{bv}t_{ib}^{au}-t_{ij}^{ab}t_{bk}^{uv}=q_{vu}q_{ku}^{-1}t_{bk}^{av}t_{ij}^{bu}-q_{vj}q_{kj}^{-1}t_{ik}^{bv}t_{bj}^{au}.
O_B^{(a^{-1})}=\widetilde\rho O_B^{(1)}+\frac{\widetilde\rho-\widetilde\rho^{-1}}{\big(Q^2-Q^{-2}\big)^2}O_A^{(1)}O_A^{(1)}O_B^{(1)},\qquad\textrm{where}\quad\widetilde\rho=\mathrm{pexp}\left(-\frac{Q^8-Q^{-8}}{(1+p^{-1})(1+s)(1+ps^{-1})}\right).
\begin{array}{l}({\rm{Bose}\oplus{Fermi\oscillators})_L^{\left( l_L\right)}}\times ({\rm{Bose}\oplus{Fermi\oscillators})_R^{\left( l_R\right)}}\times \,\,|vac,\,\,p^\mu ;\vec{m},\vec{n}>\end{array}
f\circledast g(x_1, y_0) =f\circledast g(x_0, y_1)=\delta(f\circledast g)(x_0, y_0)=0 f\circledast g(x_0, y_0) =2f(x_0)g(y_0), \delta(f\circledast g)(x_1, y_1) =2\sqrt{-1}f(x_1)g(y_1).
\left(\begin{array}{lll}w_1 w_4 w_2 w_5 w_3 w_6\end{array}\right)^*=\left(\begin{array}{lll}-\overline w_3^t \overline w_2^tF' \overline w_1^t -\overline w_6^t \overline w_5^tF' \overline w_4^t\end{array}\right).
\phi^{(1)}_{d \, (-)}(t, \xi) = - \frac{\sqrt{3}}{2}\frac{\sinh \xi}{\cosh^{2}\xi}\int_{R}d \xi' \int_{0}^{t}dt' j_{d}(t',\xi' ) \sin [ \sqrt{3}(t-t') ]\frac{\sinh \xi'}{\cosh^{2}\xi'}.
\sum_{j_1=N_1}^{J_{n,1}}\sum_{k_1=0}^{2^{j_1}-1}\cdots\sum_{j_d=N_d}^{J_{n,d}}\sum_{k_d=0}^{2^{j_d}-1}2^{\frac{\sigma_0^2}{\sigma^2}\sum_{l=1}^dj_l}2\omega_{\boldsymbol{j},n}\lesssim n^{\frac{1}{2\sigma^2}\kappa(\underline{\gamma})-\frac{1}{2}}\sigma\prod_{l=1}^d\sum_{j_l=N_l}^{J_{n,l}}2^{j_l\left(\frac{\sigma_0^2}{\sigma^2}-\mu_l\right)}.
\chi^{}_\mathrm{min}(k) = \log\sqrt{L}-\underset{n\rightarrow\infty}{\lim}\dfrac{1}{n}\log\left\Vert \big(B^{\left(n\right)}(k)\big)^{-1}\right\Vert \chi^{}_\mathrm{max}(k) = \log\sqrt{L}+\underset{n\rightarrow\infty}{\lim}\dfrac{1}{n}\log\left\Vert B^{\left(n\right)}(k)\right\Vert.
A_k=\bigcap_{i=0}^{J_k}\left\{\max_{\substack{0\le l\le L^k_i 0\le |n| \le N^k_i}}Z_{y_i^k}(s_{i,l}^k,\tau_{i,n}^k)\le m(y_i^k)-\frac{(\theta_i^k)^{\frac{\hat\alpha}{2}}}{m(y_i^k)}\right\}.
\int \frac{y d y}{c_{1}y^{2}+ c_{2}y + c_{3}}= \frac{1}{2 c_{1}}\ln \bigl ( c_{1}y^{2}+ c_{2}y + c_{3}\bigr ) - \frac{c_{2}}{2 c_{1}}\int \frac{d y}{c_{1}y^{2}+ c_{2}y + c_{3}}
J_1^N(m_0^{\otimes N},{\gamma}^N, \iota_{\gamma^N_1})= \int_{\mathcal{P}(\mathcal{X})^{T+1}}J_1^N(m_0^{\otimes N},{\gamma}^N_m,\iota_{\gamma_{m,1}^N}) \rho_2 (dm) ,
\left(1 - \frac{Rt}{p_n}\right) \left(1 - \frac{Rt}{p_{-n}}\right)= \left(1 - \frac{R^2t^2}{p_{-n}^2}\right)\left(1 + \frac{Rt}{1+ \frac{Rt}{p_{-n}}}\left(-\frac{1}{p_n}- \frac{1}{p_{-n}}\right) \right).
(\partial_t - \partial_x^2) u = u\xi+ (c_{e_{(\xi,0)}+ e_{(\xi,e_0)}}+ c_{e_{(\xi,0)}+ 2e_{(\xi,e_0)}}+ c_{e_{(\xi,0)}+ 3e_{(\xi,e_0)}}) u &+ c_{2e_{(\xi,e_0)}+ e_{e_1}}\partial_x u.
\psi_{1}^{( \pm )}= \hat{\psi_{1}}\pm \varepsilon \Biggl ( \frac{1}{2}t a n \frac{\chi_{0}}{2}- \frac{\mid a_{0}\mid^{2}s i n \chi_{0}}{1 + \mid a_{0}\mid^{4}+ 2 \mid a_{0}\mid^{2}s i n \chi_{0}}\Biggr )
R(m,W)=\frac{\int_{\Gamma_k \backslash V_{m,k}}W d\mu}{(\Gamma_k \backslash V_{m,k})}\sum_{\Gamma_k \vec{h}\in H_k(m)}\frac{1}{|\Gamma_{k,\vec{h}}|}+\sum_{f_{\lambda}}\langle W,f_{\lambda}\rangle R(m,f_{\lambda})+R(m,W_).
\Psi( E_{j+1}(x, d-1)) - \Psi(E_j(x, d-1)) = = 2 \sqrt{d-1}\left( \csc\left( \dfrac{\pi}{2j+4}\right) - 1 \right) - 2 \sqrt{d-1}\left( \cot\left( \dfrac{\pi}{2j+2}\right) - 1 \right) = 2 \sqrt{d-1}\left( \csc\left( \dfrac{\pi}{2j+4}\right) - \cot\left( \dfrac{\pi}{2j+2}\right) \right) = f_j \, \sqrt{d-1}\, .
\Big|\sum_{k=m(n)+1}^n\langle Ae_k,e_k\rangle\mu(k,V)\Big| \leq \|A\|_{\infty}\sum_{k=m(n)+1}^n\mu(k,V) \leq \frac{\|A\|_{\infty}\|V\|_{1,\infty}}{n}\sum_{k=m(n)+1}^n1 = O(1).
H(z, q, N)=z \frac{\partial}{\partial z}\frac{(q^{N}z)_{\infty}}{(z)_{\infty}}=z \frac{\partial}{\partial z}\frac{1}{(z)_N}= \frac{z}{(z)_N}\sum_{r=0}^{N-1}\frac{q^r}{1-zq^r}
\dfrac{g_{i-1}(u - \dfrac{1}{2}\hbar(i-1))}{g_i(u - \dfrac{1}{2}\hbar(i-1))}\prod_{a=1}^n \dfrac{u + tm_a - \hbar(a - \dfrac{3}{2}+ \delta( j_a \leq i-1 )) - \dfrac{1}{2}\hbar(i-1)}{u + tm_a - \hbar(a - \dfrac{3}{2}+ \delta( j_a \leq i)) - \dfrac{1}{2}\hbar(i-1)}.
g_5(C_1,C_2,q,\alpha) =g_0(C_1,C_2,q)+I(X_1;X_2|U)+1-h_2\left(\alpha\star\left(\frac{q}{2}+(1-q)h_2^{-1}\left(\min\left(1,\frac{\left(R_{\max}^{(l)}-h_2(q)\right)^+}{1-q}\right)\right)\right)\right)
\begin{cases}\psi_s=-\tfrac{\rho-1}{\rho}\psi-\left(\ell(H_a^1,\psi)+\ell(H_a^2,\psi)+\ell^*(H_a^1,\psi)+\ell^*(H_a^2,\psi)\right), \psi(t,\cdot)=\psi^*,\end{cases}
C_{T}\frac{\Gamma ( c ) \Gamma ( a - b )}{\Gamma ( c - b ) \Gamma ( a )}( \alpha - 1 )^{b - a}2^{a - b}= \frac{\Gamma ( c ) \Gamma ( b - a )}{\Gamma ( c - a ) \Gamma ( b )}\left( C_{R}+ \frac{\Gamma ( 2 - c ) \Gamma ( c - a )}{\Gamma ( c ) \Gamma ( 2 - c - a )}\right) .
\mathrm{P}\left(U_1\in \mathcal{S}_1, E\right) = \mathrm{P}\left( [\mathbf{C}_{1}]_{q,q}> \phi_1, [\mathbf{C}_{2}]_{q,q}<\tilde{\tau}\right) +\frac{1}{2}\sum^2_{i=1}\mathrm{P}\left( [\mathbf{C}_{1}]_{q,q}>\phi_i, [\mathbf{C}_{2}]_{q,q}>\tilde{\tau}\right),
D_{0}'(m \otimes [a_{1}| \dots | a_{n}]) = d_{M}(m) \otimes [a_{1}| \dots | a_{n}] - \sum_{i=1}^{n}(-1)^{\tilde{\epsilon}_{i}}m \otimes [a_{1}| \dots | d_{A}(a_{i}) | \dots | a_{n}] + \sum_{i=2}^{n}(-1)^{\tilde{\epsilon}_{i}}m \otimes [a_{1}| \dots | a_{i-1}a_{i}| \dots | a_{n}],
dX_t^{(n)}=X_{j-1}^{(n)}+ f(X_{j-1}^{(n)}) B_{t^{(n)}_{j-1},t}+\frac12 Df(X_{j-1}^{(n)})f(X_{j-1}^{(n)})\langle B\rangle_{t_{j-1}^{(n)},t}, \ \t\in [t_{j-1}^{(n)},t_{j}^{(n)}],
\mathcal{E}\left\{\log \left(1+x_n\phi \right) \right\}= - \int^{\infty}_{0}\log \left(1+x\phi \right) d (1-F_{x_n}(x)) = \frac{\phi}{\ln 2}\int^{\infty}_{0}\frac{1-F_{x_n}(x)}{1+x\phi}dx.
s[\begin{matrix}u\\v\\ 1\end{matrix}]=[\begin{matrix}f_{x}&\gamma&u_{0}\\ 0&f_{y}&v_{0}\\ 0&0&1\end{matrix}][\begin{matrix}r_{11}&r_{12}&r_{13}&t_{1}\\r_{21}&r_{22}&r_{23}&t_{2}\\r_{31}&r_{32}&r_{33}&t_{3}\\ \end{matrix}][\begin{matrix}x\\y\\z\\ 1\end{matrix}]
S_{\mathrm{{S u g r a}}}^{( D )}( \tilde{\hat{g}}, \tilde{\hat{B}}, \tilde{\hat{\phi}}) = S_{\mathrm{S u g r a}}^{( D )}( \hat{g}, \hat{B}, \hat{\phi}) + \int d^{D}x \A ( \hat{g}, \hat{B}, \hat{\phi}) \partial_{\underline{{x}}}B ( \hat{g}, \hat{B}, \hat{\phi}) ,
\frac{e^{-t_0|\xi|^2}-e^{-t_0(|\xi-\eta|^2+|\eta|^2)}}{|\xi-\eta|^2+|\eta|^2-|\xi|^2}=t_0e^{-t_0|\xi|^2}\sum_{k=1}^\infty(-1)^{k+1}\frac{\big(t_0(|\xi-\eta|^2+|\eta|^2-|\xi|^2)\big)^{k-1}}{k!}= t_0e^{-t_0|\xi|^2}+t_0e^{-t_0|\xi|^2}\sum_{k=2}^\infty(-1)^{k+1}\frac{\big(t_0(|\xi-\eta|^2+|\eta|^2-|\xi|^2)\big)^{k-1}}{k!}:=t_0e^{-t_0|\xi|^2}+G(\xi-\eta,\eta).
\lim_{t\rightarrow\infty}\mathbb{E}\big[e^{-\beta t}p_ty_{(1,t)}\big]=0,\lim_{t\rightarrow\infty}\mathbb{E}\big[e^{-\beta t}Q_t\mathcal{Z}_{(1,t)}\big]=0,\lim_{t\rightarrow\infty}\mathbb{E}\big[e^{-\beta t}q_tx_{(1,t)}\big]=0.
\varphi_{n}(x+y)= \sum_{\substack{k_{1},k_2, \ldots, k_{l}\geq 0 k_{1}+k_2+\cdots+k_{l}=n}}\binom{n}{k_{1}, k_2,\ldots, k_{l}}\varphi_{k_{1}}(x)\varphi_{k_{2}}(y) \cdot \prod_{t=3}^{l}\varphi_{k_{t}}(0)
{\cal O}^{( 6 )}( p ) e^{F}= \sum_{n \geq 0}\frac{1}{p^{n + 1}}\left[ W_{n - 3}^{6}+ 3 n ( n - 1 ) W_{n - 4}^{4}+ 2 n ( n - 1 ) ( n - 2 ) ( n - 3 ) W_{n - 5}^{2}\right] e^{F}.
\psi (s_{\lambda,r}(x_0,x_1,\dots,x_{m-1})) =\left (v_-^{\otimes r}\otimes v_+^{\otimes m-r}\otimes \sum_{\mu \preceq \lambda}K_{\lambda\mu}M_{\mu}(t_1,t_2,\dots, t_r)\right ) &=\left (v_-^{\otimes r}\otimes v_+^{\otimes m-r}\otimes s_\lambda(t_1,t_2,\dots, t_r)\right ).
L_{0}= - \left( \dot{z}- A_{\mu}\dot{x}^{\mu}\right)^{2}- \left( \dot{w}- A_{\mu}\dot{u}^{\mu}\right)^{2}+ \eta_{\mu \nu}\left( \dot{x}^{\mu}\dot{x}^{\nu}+ \dot{u}^{\mu}\dot{u}^{\nu}\right) ,
S_{q f}( \mathrm{e x}) \sim N_{e f f}~ H_{m a x}^{3}V \sim e^{- \phi_{e x i t}}M_{s}^{3}~ V \sim \left( l_{s}^{2}/ l_{P}^{2}\right)_{e x i t}~ V l_{s}^{- 3}\sim S_{H B}( \mathrm{e x}) ~ ,
\sum_{k=0\atop k\neq m}^{n}\left[\begin{matrix}n k\end{matrix}\right]\frac{(q/x)_k(x)_{n-k}}{1-q^{k-m}}x^k &=(-1)^mq^{\frac{m(m+1)}{2}}\left[\begin{matrix}n m\end{matrix}\right](xq^{-m})_n\left(\sum_{k=0}^{n-1}\frac{xq^{k-m}}{1-xq^{k-m}}-\sum_{k=0\atop k\neq m}^{n}\frac{q^{k-m}}{1-q^{k-m}}\right).
\m^{2}(\beta)=m^{2}_{0}+\frac{\lambda_{0}}{(2\pi)^{D/2}}\sum^{\infty}_{n=1}\biggl(\frac{m(\beta)}{\beta n}\biggr)^{\frac{D}{2}-1}K_{\frac{D}{2}-1}\left(m(\beta)n\beta\right).
Z(B_{m,n}) = Z(C(a_d,a_{d-1},\ldots,a_3,a_2,2a_1,a_2,a_3,\ldots,a_{d-1},a_d)) = Z(C(a_d,a_{d-1},\ldots,a_3,a_2,2a_1))\cdot Z(C(a_2,a_3,\ldots,a_{d-1},a_d)) &+Z(C(a_d,a_{d-1},\ldots,a_3,a_2))\cdot Z(C(a_3,\ldots,a_{d-1},a_d)) = Z(C(2a_1,a_2,\ldots,a_d))\cdot n+n\cdot Z(C(a_3,\ldots,a_d)) = n(Z(C(2a_1,a_2,\ldots,a_d))+ Z(C(a_3,\ldots,a_d))) = 2mn.
\Vert \partial_j \Delta_qb_i + \partial_i \Delta_qb_j \Vert^2_{L^2_x}= \Vert \partial_j \Delta_qb_i \Vert^2_{L^2_x}+ \Vert \partial_i \Delta_qb_j \Vert^2_{L^2_x}+2(\partial_j \Delta_qb_i,\partial_i \Delta_qb_j) = \Vert \partial_j \Delta_qb_i \Vert^2_{L^2_x}+ \Vert \partial_i \Delta_qb_j \Vert^2_{L^2_x}+2(\partial_i \Delta_qb_i,\partial_j \Delta_qb_j).
\theta^{AA^{\prime}}=(\mathbf{L}^{-1})_{b}^{a}d\mathbf{x}^{b}=(\mathbf{L}^{-1})_{B^{\prime}}^{A^{\prime}}(\mathbf{L}^{-1})_{B}^{A}d\mathbf{x}^{BB^{\prime}},
\sum_{i=0}^{d-1}x_{l_{i+1}}\cdots x_{l_{d}}x_{l_{1}}\cdots x_{l_{i}}\equiv\sum_{i=0}^{d-1}f_{l_{d}}(S(x_{l_{i+1}}\cdots x_{l_{d-1}}),x_{l_{1}}\cdots x_{l_{i}})=\begin{cases}x_{l_{1}+\cdots+l_{d}}d:{\rm odd}0 d:{\rm even},\end{cases}
\frac{\mathrm{d^2}}{\mathrm{d}x^2}\cos_p(x) =(2-p)\sin_p(x)^{2p-2}\cos_p(x)^{3-2p}-(p-1)\sin_p(x)^{p-2}\cos_p(x)^{3-p}&=\sin_p(x)^{p-2}\cos_p(x)^{3-2p}\left[(2-p)\sin_p(x)^p-(p-1)\cos_p(x)^p\right],
\gamma_2 =\sigma_2 (C^+_K)^{\alpha\beta}1_{\{C^+_K>0\}}+\sigma_1 (C^-_K)^{\alpha\beta}1_{\{C^-_K>0\}}\;\; \;\; \gamma_1=\sigma_2 |C^+_K|^{\alpha\beta}1_{\{C^+_K<0\}}+\sigma_1 |C^-_K|^{\alpha\beta}1_{\{C^-_K<0\}}
R_{j_1i_1}+ R_{j_2i_2}= C\left(\frac{\rho^{\alpha_{j_1i_1}}}{1+\rho^{\alpha_{j_1i_2}}}\right)+ C\left(\frac{\rho^{\alpha_{j_2i_2}}}{1+\rho^{\alpha_{j_2i_1}}}\right) .
\sum_{\ell=1}^\infty\frac{|B(x_0,r)|}{\Phi(r)}\cdot\frac{\Phi(2^{\ell+1}r)}{|B(x_0,2^{\ell+1}r)|}\le\sum_{\ell=1}^\infty\left(\frac{D(\Phi)}{2^n}\right)^{\ell+1}&\le C.
g_{I(G)}(k+1) =\min\{g_{I(G_1)}(k+1)+1,g_{I(G_2)}(k),g_{I(G_3)}(k),g_{I(G_3)}(k+1)+1\}, g_{I(G)}(k) =\min\{g_{I(G_1)}(k)+1,g_{I(G_2)}(k-1),g_{I(G_3)}(k-1),g_{I(G_3)}(k)+1\}.
\inf_{x\in I_i}\left|\frac{d^{r_1}f}{d x^{r_1}}(x)\right| \geq\left|\frac{d^{r_1}f}{d x^{r_1}}(x_0)\right| -\sup_{x\in I_i}\left|\frac{d^{r_1}f}{d x^{r_1}}(x)-\frac{d^{r_1}f}{d x^{r_1}}(x_0)\right| &\geq \tau-\sup_{x\in I_i}\left|\frac{d^{r_1+1}f}{d x^{r_1+1}}(x)\right|\cdot |I_i| &\geq \tau-A\cdot\frac{\tau}{2A}=\frac\tau2.
\begin{aligned}| \b y-\b w| = | \b y-\b q - \frac{\varepsilon}{2}(\b y-\b q) | = | (\b y-\b q)(1-\frac{\varepsilon}{2})| & \leq | \b y-\b q| \underbrace{(1-\frac{\varepsilon}{2})}_{<\;1}<| \b y-\b q|.\end{aligned}
h(X,Y) = \frac{\langle D_XY,\xi\rangle}{\langle \eta,\xi\rangle}= -\frac{\langle Y,d\xi_pX\rangle}{\langle\eta,\xi\rangle}= -\frac{\langle du^{-1}_{\eta(p)}Y,d\eta_pX\rangle}{\langle\eta,\xi\rangle},
\mathfrak{c}_{N}=\left\vert \left\{m_{i}^{+}\right\}_{i=1}^{\mathfrak{m}_{N}^{+}}\cap\left\{m_{i}^{-}\right\}_{i=1}^{\mathfrak{m}_{N}^{-}}\right\vert \geq\left( 1-2\frac{\left( 1+\delta\right) c_{+}}{M}\right)\frac{\left\Vert x\right\Vert}{N}\ .
\frac{d}{dt}\large|_{t=d_{\mathbb{K}min}}\Lambda (t,u) \equiv \,\frac{d}{dt}\large|_{t=d_{\mathbb{K}min}}\left(\Omega\,\circ \Phi_\mathcal{E}(t, E (u))\,-\Omega\,\circ \Phi^c_\mathcal{E}(t,\, E (u))\right) & =\Xi_\Omega(E(u)).
\begin{cases}\frac{\partial^{\gamma}u}{\partial t^{\gamma}}= \epsilon \triangle u - \vec{\beta}\cdot\nabla u + r u, (x,y,z)\in \Omega = (0,1)^3,\t \in [0,T], u(x,y,z,t) = 0, (x,y,z)\in \partial\Omega,\t \in [0,T], u(x,y,z,0) = x(1 - x)y(1-y)z(1 - z), (x,y,z)\in \bar{\Omega}.\end{cases}
p(\theta|\mu,\Sigma)=\frac{e^{-\frac{1}{2}\mu^{\top}\Sigma^{-1}\mu}}{2\pi\sqrt{|\Sigma|}v^{\top}\Sigma^{-1}v}(1+T(\theta)\frac{\Phi(T(\theta))}{\phi(T(\theta))})I_{[0,2\pi)}(\theta)
\pmb{F}_\ell = \sqrt{\frac{P_t}{d}}\zeta_{\min}^d\left\{\sum\limits_{k=1,k \neq \ell}^{K+1}\pmb{H}_{k,\ell}^H(\pmb{I}- \pmb{U}_k\pmb{U}_k^H)\pmb{H}_{k,\ell}\right\}
K_{H}(t,s)=c_{H}\left[ \left( \frac{t}{s}\right)^{H-\frac{1}{2}}(t-s)^{H-\frac{1}{2}}+\left( \frac{1}{2}-H\right) s^{\frac{1}{2}-H}\int_{s}^{t}u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}du\right] .
F_k(\theta, E)=\left(\begin{array}{cc}\tilde{P}_k(\theta,E) -\tilde{P}_{k-1}(\theta+\alpha,E)\cos{\pi\theta}\tilde{P}_{k-1}(\theta,E)\cos{\pi (\theta+(k-1)\alpha)}-\tilde{P}_{k-2}(\theta+\alpha,E)\cos{\pi\theta}\cos{\pi(\theta+(k-1)\alpha)}\end{array}\right).
\frac{A}{2 \pi}\int d^{2}x \epsilon^{\mu \nu}\partial_{\mu}\omega_{\nu}+ \frac{B}{2 \pi}\int d^{2}x \epsilon^{\mu \nu}\partial_{\mu}a_{\nu}= ( 2 - 2 g ) A + \frac{B V}{2 \pi}= N ,
\begin{aligned}(\gamma_2\theta_{|a|})^{1/2}|x| >|x^a|\ge|x_{ij}|\ge |z_{ck}|-2|x^c| &>((\gamma_1^{-1}\rho_{|c|})^{1/2}-2(\gamma_2\theta_{|c|})^{1/2})|x|.\end{aligned}
\delta S \vert_{\hat{E}^{I}= 0 = \tilde{E}^{i}}={\frac{1}{2}}\int_{{\cal M}^{1 + 1}}\left( \hat{M}_{2}^{I}U_{\underline{{{m}}}}^{I}+ j_{1}\wedge \left( \hat{E}^{+ +}U_{\underline{{{m}}}}^{--}- \hat{E}^{--}U_{\underline{{{m}}}}^{+ +}- 2 m f^{i}u_{\underline{{{m}}}}^{i}\right) \right) \delta \hat{X}^{\underline{{{m}}}}
w^{\xi^+_i}_{\eta^+_i}(\rho_\tau) =\sum_{[\{\alpha,\beta;\epsilon\}]\in\vec{\triangle}^{\xi^+_i}_{\eta^+_i}}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho_\tau)+\ell_{\beta}(\rho_\tau))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
x_{+}= \frac{c}{( q - q^{- 1})}, \quad x_{-}= \frac{b}{( q - q^{- 1})}, \quad h = \frac{a - d}{( q - q^{- 1})}, \quad t = \frac{q d + q^{- 1}a}{c ( q + q^{- 1})},
\begin{aligned}\partial_{k}\left(\frac{\pi}{8}m(k)\right) = \frac{(\mathcal{E}+ (k^2 - 1) K)^2}{k (1 - k^2)}+ k K^2 = \frac{k^4 \Theta^2 + k^2(1 - k^2) K^2}{k (1 -k^2)}.\end{aligned}
\Lambda=(q^{-1/2}-q^{3/2}) I_1I_2I_3+q I_1^2+q^{-1}I_2^2+q I_3^2-(1-q)\,\iota_1\, I_1-(1-q^{-1})\,\iota_2\,I_2-(1-q)\,\iota_3\, I_3,
\langle\!\langle C_{n}^{\lambda,\beta},Q_{m}^{\pi_{\lambda,\beta}}\rangle\!\rangle_{\pi_{\lambda,\beta}}=\int_{\mathbb{R}}m!\sum_{k=m}^{\infty}\frac{S(k,m)}{k!}\left(\nabla^{k}C_{n}^{\lambda,\beta}(x)\right)\,\mathrm{d}\pi_{\lambda,\beta}(x) =m!\sum_{k=m}^{\infty}\frac{S(k,m)}{k!}k!s(n,k) =m!\sum_{k=m}^{n}s(n,k)S(k,m) =n!\delta_{n,m}.
\partial x_{n+1}= \left( \partial c(\theta_n) \right) \cdot p(x_n) + \partial x_0 \prod \limits_{j = 0}^n c(\theta_j) \cdot p'(x_j)+ \sum \limits_{k = 1}^n \left( \partial \theta_{k-1}p(x_{k-1}) \prod \limits_{j = k}^n c(\theta_j) \cdot p'(x_j) \right).
{\cal F}( p ) = - \frac{i N}{2 g}- N \int \frac{d^{D}k}{( 2 \pi )^{D}}\cos^{2}( k \wedge p ) \frac{k \cdot ( k + p ) + M^{2}}{( k^{2}- M^{2}) [ ( k + p )^{2}- M^{2}]}
~~\| u_{(\beta,w)}\|_{L^\infty(\Omega_{5 R_0})}+ \| u_{(\beta,w)}\|_{W^{2,2}(\Omega_{5 R_0})}+ \| q_{(\beta,w)}\|_{W^{1,2}(\Omega_{5 R_0})}& \leq C \big ( M (\alpha,\beta,F, w) + \| f \|_{L^2 (\Omega_{6 R_0})}\big )\,.
W_1(\mu_n^{j+1},\nu_n^{j+1})=W_1(\Theta^{s_n^{j+1}}{}_\#(\pi_n^j *\beta_n^{j+1}),\nu_n^{j+1}) \leq W_1(\Theta^{s_n^{j+1}}{}_\#( \pi_n^j*\beta_n^{j+1}),\Theta^{s_n^{j+1}}{}_\#\beta_n^{j+1}) &+ W_1(\Theta^{s_n^{j+1}}{}_\#\beta_n^{j+1},\nu_n^{j+1})).
{\rm Im}{\cal L}^{(1)}_{\rm scalar}= \frac{(qE)(qB)}{4 (2 \pi)^2}\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}}{n}{\rm csch}\Bigl( \frac{n \pi B}{E}\Bigr) \exp \Bigl(- \frac{n \pi m^2}{qE}\Bigr).
\sum_{\lambda = 1}^{i-1}= \sum_{\lambda = 1}^{i-1}(-1)^\lambda ~ h ( a^{m+n-1}_{1,\lambda -1}, ~ \alpha^{m+n-2}(a_\lambda \cdot a_{\lambda+1}), ~a^{m+n-1}_{\lambda +2 , i}, ~ f^n, ~ a_{i+m +1, j}^{m+n-1}, ~ g^m, ~ a^{m+n-1}_{j+n+1, m+n+p-1}),
\Psi(z)=\left(I+\sum^\infty_{j=1}C_j z^{-j}\right)z^{-\frac{1}{4}\sigma_3}Ne^{z^{1/2}\sigma_3}, C_j = \begin{pmatrix}q_j(s) -i r_j(s) i p_j(s) -q_j(s)\end{pmatrix}.
\Big ( a_{\mu}^{( b )}, T_{\mu \nu}a_{\nu}^{( b )}\Big ) = \Big ( \varepsilon_{\mu \nu}\partial_{\mu}a_{\nu}^{( b )}, \frac{- 1}{\triangle}\; \varepsilon_{\rho \sigma}\partial_{\rho}a_{\sigma}^{( b )}\Big ) \; ,
(u-c) \partial_q E = \left(u-c, v \right) \cdot \nabla E & = \left( (\sqrt{\varrho}(u-c))_y - (\sqrt{\varrho}v)_x \right) \left\{-\sqrt{\varrho}(u-c)v + \sqrt{\varrho}v(u-c)\right\}& +{g}y(u-c,v) \cdot \nabla \varrho ={g}y(u-c) \partial_q \varrho = 0.
\frac{(Tn-1)(n-\tanh{T})-\frac{T}{\cosh^2(T)}}{(Tn+1)(n+\tanh{T})-\frac{T}{\cosh^2(T)}}= \frac{Tn^2-n(1+T\tanh{T})+\tanh{T}-\frac{T}{\cosh^2(T)}}{Tn^2+n(1+T\tanh{T})+\tanh{T}-\frac{T}{\cosh^2(T)}}&=\frac{an^2-bn+c}{an^2+bn+c}.
R_{k}^{a}(p) = \int\frac{d^{4}l}{(2\pi)^4}\Bigg(\frac{2\pi\delta((p+l)^2-m^2)}{e^{\beta |p_0+l_0|}-1}\Bigg)\Bigg(\frac{-ig^k}{(32\pi^2)^{k-1}}(\log (-l^2))^{k-1}\Bigg).
\Delta^{(n,\sigma)}_{2}(P)= \left(\begin{array}{cc}\Delta^{(n,\sigma)}(xx^{*}) \Delta^{(n,\sigma)}(xy^{*}) \Delta^{(n,\sigma)}(yx^{*}) \Delta^{(n,\sigma)}(yy^{*}) \end{array}\right)=\sum_{i=1}^{n}A_{i}\otimes E_{ii},
i(\begin{bmatrix}ad-bc &a b &c d & 1\end{bmatrix}\begin{bmatrix}1 \,-y x z &1 x & 1 y & 1\end{bmatrix})=\begin{bmatrix}a b c d & 1\end{bmatrix}\begin{bmatrix}1 x &1 y & 1\end{bmatrix}
B_1 :=\left(-\frac{1}{4}\Box\omega^X-\frac{1}{2}\nabla_X V_1-\frac{M}{r^2}fV_1\right), B_2 :=\left(-\frac{1}{4}\Box\omega^X-\frac{1}{2}\nabla_X V_2-\frac{M}{r^2}fV_2\right),
\frac{3}{2}\left(\prod_{\substack{\{k,\ell\}\in S \{k,\ell\}\neq\{i,j\}}}\frac{\alpha_S}{d_{k\ell}}\right) \lceil n/t\rceil^{r-2}\le2(\alpha_S/\delta)^{\binom{r}{2}-1}(n/t)^{r-2}
\langle x,l_1,y,l_2 | m_{12}, l_{12}\rangle = \displaystyle\sum\limits_{m_1,m_2}\langle x,l_1,y,l_2 |m_1,l_1,m_2,l_2\rangle \langle m_1,l_1,m_2,l_2 | m_{12}, l_{12}\rangle &= \langle x,l_1,y,l_2 |0,l_1,0,l_2\rangle \displaystyle\sum\limits_{m_1,m_2}\langle m_1,l_1,m_2,l_2 | m_{12}, l_{12}\rangle P_{m_1}^{l_1}(x) P_{m_2}^{l_2}(y).
\frac{\partial \mathcal{L}}{\partial \mathbf{S}}=\frac{\mu\,\left(1-\rho\right)}{\sigma^2\,\ln2}\;\mathbf{H}^{\rm H}\left(\mathbf{I}_{N_R}+\left(1-\rho\right)\sigma^{-2}\mathbf{H}\mathbf{S}\mathbf{H}^{\rm H}\right)^{-1}\mathbf{H}+\rho\,\mathbf{H}^{\rm H}\mathbf{H}-\nu\mathbf{I}_{N_T}=0,
\left[ - \frac{d^{2}}{d w^{2}}- \frac{2 4}{1 + e^{\sqrt{8}w}}+ \frac{3 0}{\left( 1 + e^{\sqrt{8}w}\right)^{2}}\right] \chi_{n}( w ) = \nu_{n}^{2}\chi_{n}( w ) ,
\frac{1}{2t}B = (\lambda_2 + \lambda_3 - \lambda_1 - \beta) e_1 \otimes e_1 + (\lambda_1 + \lambda_3 - \lambda_2 - \beta) e_e \otimes e_2 begin{align*}6pt] + (\lambda_1 + \lambda_2 - \lambda_2 - \beta) e_3 \otimes e_3 + (1 + \beta) (\lambda_1 + \lambda_2 + \lambda_3) e_4 \otimes e_4 + O(t).
A d ( \tilde{g})^{T}= \left( \begin{matrix}{1}&{0}&{0}&{- \beta^{- 1}\rho e^{- \sigma}}\\{- \rho}&{e^{\sigma}}&{{\beta}^{- 1}{\rho}}&{\beta^{- 1}\rho^{2}e^{- \sigma}}\\{0}&{0}&{1}&{{\rho}{e^{- \sigma}}}\\{0}&{0}&{0}&{e^{- \sigma}}\\ \end{matrix}\right)
\pi_{2}\circ \pi_{1}(x^{(f)}_i) = x^{(f)}_{\pi_{1}( \pi_2(i))}=x^{(T_1(f))}_{\pi_2(i)}= x^{(g)}_{\pi_2(i)}= &= x_i^{(T_2(g))}= x_i^{(T_2 \circ T_1(f))}.
\pi = \tau_{r_{1}}^{-(v_{1}+1)}\cdot \tau_{r_{1}+1}^{v_{1}+1}\cdot \tau_{r_{2}}^{-(v_{2}+1)}\cdot \tau_{r_{2}+1}^{v_{2}+1}\cdots \tau_{r_{z}}^{-(v_{z}+1)}\cdot \tau_{r_{z}+1}^{v_{z}+1}.
a\left( x\right) :=\begin{cases}-\left( r-1\right) r^{q}x\in\left[ 0,1\right] , -\frac{p^{\prime\prime}\left( x\right)}{\left[ p\left( x\right) \right]^{q}}x\in\left[ 1,2\right) , a\left( -x\right) x\in\left( -2,0\right] .\end{cases}
\forall \gamma^{\mathbb{N}^\mathbb{N}},\lambda^{\mathbb{N}^\mathbb{N}},x^X\Big(\gamma>_\mathbb{R}0\land\lambda>_\mathbb{R}0\land x\in\mathrm{dom}(J^A_\lambda) &\qquad\qquad\qquad\rightarrow J^A_\lambda x=_X J^A_{\gamma}\left(\frac{\gamma}{\lambda}x+_X\left(1-\frac{\gamma}{\lambda}\right)J^A_\gamma x\right)\Big).
0<\kappa_\nu(u) =\det(f_u(u,0),f_{vv}(u,0),f_{uu}(u,0)) =\det(T \circ \tilde f_u(u,0),T \circ \tilde f_{vv}(u,0), T \circ \tilde f_{uu}(u,0)) =\det(\tilde f_u(u,0),\tilde f_{vv}(u,0),\tilde f_{uu}(u,0)) =\xi_u(u,0) \det(\check f_\xi(\xi(u,0),0), \check f_{\eta\eta}(\xi(u,0),0), \check f_{\xi\xi}(\xi(u,0),0)).
[\begin{matrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta}&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta}\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta}&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta}\end{matrix}].
M_{p, 1}(\delta, \nu) &\leq (C\nu^{-1/p})^{\sum_{j = 0}^{N}(1 - \alpha)^j}\prod_{j = 0}^{N}(\log\frac{1}{\nu^{2^j}})^{\frac{1}{2}(1 - \alpha)^j}M_{p, 2^{N + 1}}(\delta, \nu)^{(1 - \alpha)^{N+1}}\prod_{j = 0}^{N}D_{p}(\frac{\delta}{\nu^{2^j}})^{\alpha(1 - \alpha)^j}
d(g_{i}^{n}\circ h_{i}(x),g_{i}^{n}\circ h_{i}(y)) \leq d(g_{i}^{n}\circ h_{i}(x),f_{i}^{n}(x))+d(f_{i}^{n}(x),f_{i}^{n}(y))+d(f_{i}^{n}(y),g_{i}^{n}\circ h_{i}(y)) &=d(h_{i+n}\circ f_{i}^{n}(x),f_{i}^{n}(x))+d(f_{i}^{n}(x),f_{i}^{n}(y))+d(f_{i}^{n}(y),h_{i+n}\circ f_{i}^{n}(y)) \leq \tilde{r}/3+\tilde{r}/3 +\tilde{r}/3= \tilde{r}.
\left| \det \begin{pmatrix}b_1 (x,\xi' + \zeta', i |\xi'|_x+ \delta) b_2 (x,\xi' + \zeta', i |\xi'|_x+ \delta) begin{align*}2pt] \partial_{\xi_d}^2 b_1 (x,\xi' + \zeta', i |\xi'|_x+\delta_s ) \partial_{\xi_d}^2 b_2 (x,\xi' + \zeta', i |\xi'|_x+\delta_s ) \end{pmatrix}\right| \lesssim |\xi'|_x^{k_1 + k_2 -2},
\epsilon=0 \hspace{0.7cm},\hspace{0.7cm}x = \sqrt{\frac{7-p}{8(8-p)}}\hspace{0,7cm},\hspace{0.7cm}{\lambda}= \frac{7-p}{16x}\hspace{0.7cm},\hspace{0.7cm}\nu = \frac{p-3}{8 \sqrt{2}x}\quad .
Y_\infty(\lambda)\underset{\lambda\to\infty}=\left\{I+\sum_{k=1}^\infty\Phi_k(\infty)\left(\frac1{\lambda}\right)^k\right\}\begin{pmatrix}\lambda^{-\theta_\infty/2}0 0 \lambda^{\theta_\infty/2}\end{pmatrix},
\begin{aligned}\bigwedge_i a^L_i \wedge a^L \le a^R \vee \bigvee_j a^R_j \in A, b^L_i \le f_i(a^L_i) \vee b^R_i \in B_i \;\forall i, c^L_j \wedge g_j(a^R_j) \le c^R_j \in C_j \;\forall j.\end{aligned}
z ( 1 - z ){\frac{d^{2}\tilde{\phi}}{d z^{2}}}+ ( 1 - z ){\frac{d \tilde{\phi}}{d z}}+ \left({\frac{A_{1}}{z}}+{\frac{s / 4}{1 - z}}+ B_{1}\right) \tilde{\phi}= 0 ,