harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V = \left({\it \rho_{1}}+{\frac{{{{\it x_{1}}}^{2}}}{1 0 \,{\it \rho_{1}}}}\right) \, \left({\it \rho_{2}}+{\frac{{{{\it x_{2}}}^{2}}}{1 0 \,{\it \rho_{2}}}}\right) \, \left( \frac{1}{1 6}+{\frac{x +{\it x_{1}}+{\it x_{2}}}{{{\left({{{\it \rho_{1}}}^{2}}+{{{\it \rho_{2}}}^{2}}+ 2 \,{\it \rho_{1}}\,{\it \rho_{2}}\, \left( 1 - x \right) \right)}^{{\frac{3}{4}}}}}}\right)
(\begin{matrix}1&1&1&0&0&0&0\\ 1&0&0&1&1&0&0\\ 1&0&0&0&0&1&1\\ 0&1&0&1&0&1&0\\ 0&1&0&0&1&0&1\\ 0&0&1&1&0&0&1\\ 0&0&1&0&1&1&0\end{matrix})
{h_0}^{\pm}=e^{i\theta}\Big[\,{h_{01}}^{\pm}\,+\,i{h_{02}}^{\pm}\,\Big], begin{align*}1mm]{h_j^1}^{\pm}=e^{i\theta}\Big[\,{h_{j1}^1}^{\pm}\sin{j\theta}\,+\,i{h_{j2}^1}^{\pm}\cos{j\theta}\,\Big], begin{align*}1mm]{h_j^2}^{\pm}=e^{i\theta}\Big[\,{h_{j1}^2}^{\pm}\cos{j\theta}\,+\,i{h_{j2}^2}^{\pm}\sin{j\theta}\,\Big].
W_{\delta}= -{\frac{\eta}{1 6 \pi^{3 / 2}}}\left[ m^{3}a_{0}\Gamma ( - 3 / 2 , m^{2}\delta ) + m a_{1}\Gamma ( - 1 / 2 , m^{2}\delta ) + \pi^{1 / 2}{\frac{1}{m}}a_{2}+ O ( m^{- 3}) \right] ,
\left[ Z_{ab},\bar{Z}_{c}\right] =\frac{1}{\sqrt{2}}\left[ Z_{ab},P_{c}\right] -\frac{1}{\sqrt{2}}\left[ Z_{ab},Z_{c}\right] , & =\frac{1}{\sqrt{2}}f_{ab,c}^{d}Z_{d}+\frac{1}{\sqrt{2}}f_{ab,c}^{d}P_{d}, & =f_{ab,c}^{d}\bar{P}_{d},
\begin{aligned}a = \frac 1{\vartheta_0}\, \left( \frac{A}{1 + A (1 +{\theta_r}) -B}\right)^{4/3}=: \frac 1{\vartheta_0}\, \left( \frac{A}{x_{A,B}}\right)^{4/3}\,, b = \frac 1{\pi_0}\, \frac{B\, A^{1/3}}{x_{A,B}^{4/3}}\,. \end{aligned}
\nabla_{Y(t,q)}Z(t,q)=\sum_{i,j}f_i(t,q)\frac{\partial g_j(t,q)}{\partial x_i}\frac{\partial\}{\partial x_j}\,+\,\sum_{i,j}f_i(t,q)g_j(t,q)\Gamma_{i,j}(q)\frac{\partial\}{\partial x_k}.
\Theta=\, (-2e^2+9e^3-3de^3+9de^4+d^3e^5)u^2\cr +(-4e+24e^2-2de^2v-36e^3+9de^3+2d^2e^3-6d^2e^4)uv\cr +(-2+15e+de-27e^2-6de^2+9de^3)v^2.
\begin{aligned}{\underline{u}_a}= \Big\lbrack \int_{\mathcal{A}_{k}^a}{u_a}\, \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}\mbox{and}{\underline{u}_b}= \Big\lbrack \int_{\mathcal{A}_{k}^b}{u_b}\, \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}\end{aligned}
\leq C \sum_{i=1}^{m}\int_{\mathbb{R}^{n}}\left\{\sum_{j=1}^{k}\left(\frac{\lambda_{j}|B_j|^{\frac{\alpha}{n}}\chi_{B_j}(x)}{\| \chi_{B_j}\|_{p(.)}}\right)^{p_0}\right\}^{\frac{q_0}{p_0}}w_{A_{i}^{-1}}(x) dx
\big|\mathcal S_L(t) \big|\le C \; \mathbb{E}_{\xi^{(\ell_1, \ldots, \ell_n)}}\left[ \prod_{m =1}^n q^{-4k\ell_m(t)+2[N_{\ell_m(t)+1}(\eta^{(L)})-N_{\ell_m+1}(\eta^{(L)})]}\right]
[{\cal Z}_{A}^{I}, \bar{\cal Z}_{B}^{J}]_{P . B .}= i \delta^{I J}\delta_{A B}\, , \quad [{\cal Z}_{A}^{I},{\cal Z}_{B}^{J}]_{P . B .}= [ \bar{\cal Z}_{A}^{I}, \bar{\cal Z}_{B}^{J}]_{P . B .}= 0 \, ,
\sum_{\alpha}\Vert x_{\alpha}\Vert^{q}\rho^{\vert \alpha \vert q}= \sum_{k=0}^{N}\rho^{qk}\sum_{\substack{\alpha \in F \alpha_{n}=k}}\Vert x_{\alpha}\Vert^{q}\rho^{(\vert \alpha \vert - \alpha_{n}) q}\,.
S|_V=\left(\begin{array}{cc}0 S_2 S_2 0 \end{array}\right),J|_V=\left(\begin{array}{cc}J_2 0 0 J_2 \end{array}\right), \quad\mbox{and}L|_V=\left(\begin{array}{cc}J_2 0 0 -J_2 \end{array}\right).
F_1=-\,\frac{1}{24}\log\Bigl\{\biggl(N_1+2\varphi\frac{1}{y_+^{3/2}\,y_-^{1/2}}\biggr)\biggl(K_1+2\varphi\frac{1}{y_+^{1/2}\,y_-^{3/2}}\biggr)(y_+-y_-)^4\Bigr\}.
R = \left( \begin{array}{l l l l}{{1}}&{{- h^{\prime}}}&{{h^{\prime}}}&{{h h^{\prime}}}\\{{0}}&{{\ 1}}&{{0}}&{{- h}}\\{{0}}&{{\ 0}}&{{1}}&{{\h}}\\{{0}}&{{\ 0}}&{{0}}&{{\ 1}}\\ \end{array}\right)
\int\frac{y-\frac{2b_{2}a-b_{1}}{2b_{2}}}{y^{2}+\alpha^{2}}dy=\frac{1}{2}ln(y^{2}+\alpha^{2})-\frac{2b_{2}a-b_{1}}{2b_{2}\alpha}arctan(\frac{y}{\alpha})+C_{0}
\begin{array}{c c c c c c c c c}{\cite{b h k}:}{x}{A}{N}{K}{H}{J}{\alpha}{\beta^{2}}{\cite{b f m 2}:}{\alpha r}{e^{\lambda + \nu}}{e^{- 2 \lambda}}{W}{\alpha H}{J}{\alpha}{\frac{\beta^{2}}{2}}\end{array}
\mathbb{E}\left[\boldsymbol{s}\boldsymbol{s}^{\rm H}\right] = \rho^2 \textbf{1}\cdot \textbf{1}^{\rm T}+ \rho(1-\rho)\mathbf{I}\mathbb{E}\left[\boldsymbol{s}^{\rm H}\right] = \rho \textbf{1},
K_T(\mathcal{DT}(d)) \otimes_{\mathbb{K}}\mathbb{F}= \bigoplus_{\begin{subarray}{c}d_1+\cdots+d_k=d 0\leq v_1/d_1 \leq \cdots \leq v_k/d_k <1 \end{subarray}}\mathbb{F}\cdot [\mathcal{F}_{d_1, w_1}] \ast \cdots \ast [\mathcal{F}_{d_k, w_k}]
X = \frac{3 \sigma{\log(1/\delta)}\pm \sqrt{(3 \sigma{\log(1/\delta)})^2 + 4 (\kappa +{L})\sqrt{\sum_{t=1}^T \|g_t\|^2}+ 12 (G^2 + G \tilde G){\log(1/\delta)}}}{2}
\sigma_{(p,q,n)}(0)=\begin{cases}(-1)^{\frac{(N-1)p(q-1)}{8}}\p, q, &(-1)^{\frac{(N-1)(p-1)q}{8}}\q, q, &(-1)^{\frac{(N-1)(p-1)(q-1)}{8}}\p,q, n\ , &(-1)^{\frac{N(p-1)(q-1)}{8}}\p,q, n \end{cases}
\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}\omega(x)|^{p}dx=\lambda^{p}\mu^{a_{1}p-Q}\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx,
V_{vdw}=-\frac{A_{H}}{6}[\frac{2a_{1}a_{2}}{R^{2}-(a_{1}+a_{2})^{2}}+\frac{2a_{1}a_{2}}{R^{2}-(a_{1}-a_{2})^{2}}+ln\frac{R^{2}-(a_{1}+a_{2})^{2}}{R^{2}-(a_{1}-a_{2})^{2}}]
\sum_{m_1}R_U(J,J';\varpi)_{m_{\phantom{2}}m'}^{m_2 m_1}\prod_{t=1}^{J'+m_2}(y'q^{2J}+yq^{\varpi +4J-2J'+2t-1})\prod_{s=1}^{J+m_1}(y'+yq^{-(\varpi -2J+2m_2 +2s -1)})
d(h(t,x,z)k(t,x,z)) =h(t,x,z)k(t,x,z)b(t,x,z) d G(t) +k(t,x,z)[dF(t,x,z)+xh(t,x,z)dG(t)] &+k(t,x,z) b(t,x,z)(xh(t,x,z)+y(t,x,z)q(t,x,z))dt
A_T= \begin{bmatrix}a_n a_{n+1}\dots a_{2n-2}a_{2n-1}a_{n-1}a_n \dots a_{2n-3}a_{2n-2}\vdots \vdots \ddots \vdots \vdots a_{2}a_3 \dots a_{n}a_{n+1}a_1 a_{2}\dots a_{n-1}a_{n}\end{bmatrix}.
qt_{ij}t_{jj}= qq^{-k_{i+1}}(f_{i}t_{i+1,j}-t_{i+1,j}f_i)q^{k_j}&= q^{k_j}q^{-k_{i+1}}(f_{i}t_{i+1,j}-t_{i+1,j}f_i) &= t_{jj}t_{ij}
W_{( 0 )}^{0}= \int \mathrm{\small{~ \frac ~ 1 2 ~}}[ \sqrt{g}g^{\alpha \beta}N_{\mu \nu}( X ) + \varepsilon^{\alpha \beta}F_{\mu \nu}( X ) ] \partial_{\alpha}X^{\mu}\cdot \partial_{\beta}X^{\nu}.
\bigl \langle \mathbb E \hat P_r , \mathbb E \tilde P_r \bigr\rangle = \bigl \langle \mathbb (1+b_r) P_r + T_r , (1+b_r) P_r + T_r\bigr\rangle = (1+b_r)^2 + \|T_r\|_2^2,
e(N(1)) =\sum_{t=n-m-1}^{n-1}\left[(-1)^{n-1-t}\cdot t\cdot f_{n-t-1}\right] + 1 & = (-1)^m(n-m-1)f_m+(-1)^{m-1}(n-m)f_{m-1}+(-1)^{m-2}(n-m+1)f_{m-2}+\cdots +(n-1)f_0 + 1.
\left[\phi_1,\phi_2\right]_{\Sigma}+\theta_{\Sigma}\left(\eta,\phi_2\right)=\theta_{\Sigma}\left(\phi_1+\eta,\phi_2\right),\, \forall \eta\in{A}_{\Sigma},\,\forall\phi_1,\phi_2\in{L}_{\Sigma}.
\left[\left(\prod_i x_i^{n_i}\right)\partial_{x_j},\left(\prod_i x_i^{n_i'}\right) \partial_{x_k}\right]=\left(n_j' \prod_{i=1}^n x_i^{n_i+n_i'-\delta_{ij}}\right)\partial_{x_k}- \left(n_k \prod_{i=1}^n x_i^{n_i+n_i'-\delta_{ik}}\right)\partial_{x_j}.
r_{b}(j)=C_{1}(j) &=\big[\underbrace{B\log_{2}\left(1+\zeta_{m,1}(j)\right)}_{r_{bm}(j)=C_{m1}(j)}-\underbrace{B\log_{2}\left(1+\zeta_{e,1}(j)\right)}_{r_{be}(j)=C_{e1}(j)}\big]^+,
\sum_{k=0}^{M}[4k-1]\frac{(q^{-1};q^2)_k^4(q^{-3};q^2)_k^2}{(q^2;q^2)_k^4(q^{4};q^2)_k^2}q^{11k}\equiv\frac{\Omega_q(n)[7][9]}{[2]^2[4]^2[6]^2}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3(q^{11};q^2)_k}{(q^2,q^6,q^8,q^8;q^2)_k}q^{2k}.
\mathrm{\fboxsep = . 1 i n \framebox{\displaystyle a_\mu^{\xi}( x ) : = \langle \Lambda |{\cal A}_\mu^\xi ( x ) | \Lambda \rangle , \quad{\cal A}_\mu^\xi ( x ) = \xi^\dagger ( x ){\cal A}_\mu ( x ) \xi ( x ) +{\frac{i}{g}}\xi^\dagger ( x ) \partial_\mu \xi ( x ) ,}}
\frac{V_{a}(P_{ra},T_{ra})}{V_{ca}}=\frac{V_{z}(P_{rz},T_{rz})}{V_{cz}}\iff V_{a}(P_{a},T_{a})=\frac{V_{ca}}{V_{cz}}\cdot V_{z}(P_{z}=\frac{P_{a}P_{cz}}{P_{ca}},T_{z}=\frac{T_{a}T_{cz}}{T_{ca}})
H_{1}\backslash A =\{(a,b,a,d):a,b>0,\d<0b\geq a,-d\geq a\}& \cup\{(a,b,b,d):a,b>0,\d<0a\geq b,-d\geq b\}& \cup\{(a,b,c,-c):a,b,c>0a\geq c,b\geq c\}.
d s^{2}= - g^{2}R^{2}d t^{2}- \frac{1}{R^{4}}( ( R^{2}- q ) d t - \alpha \sin^{2}\theta d \phi + \alpha \cos^{2}\theta d \psi )^{2}+ \frac{d R^{2}}{V ( R )}+ R^{2}d \Omega^{2},
\Gamma \left( p,P_{os}\right) = - \int \frac{d^{4}p^{\prime}}{\left( 2\pi\right)^{4}}K\left( p,p^{\prime},P_{os}\right) G_{1}\left( p_{1}^{\prime}\right) G_{2}\left( p_{2}^{\prime}\right) \Gamma \left( p^{\prime},P_{os}\right) .
\gamma_3 = 2\alpha_1\alpha_2 +2 \alpha_3-2 \beta_3 = 2qs-12q^2s+q^2s^2+28q^3s-4q^3s^2+q^4-32q^4s +4q^4s^2-4q^5+22q^5s+6q^6-12q^6s-4q^7+4q^7s \overset{(a)}{\le}2qs-12q^2s+q^2s^2+28q^3s \overset{(b)}{\le}4qs \, ,
d_X(F(x,y),x) = \lim_{j\rightarrow \infty}d_X(F(F^j(x_0,y_0),G^j(y_0,x_0)),F^j(x_0,y_0)) &=\lim_{j\rightarrow \infty}d_X(F^{j+1}(x_0,y_0),F^j(x_0,y_0)) &= 0
u_{1}= 2 \exp \left( i \frac{z}{2{\hbar}c}\right) \mathrm{R e}\left[ \frac{1}{{\Gamma}( 1 + i{\beta})}\exp \left( i \frac{z}{2{\hbar}c}\right) ( \frac{i z}{{\hbar}c})^{- 1 + i{\beta}}G ( 1 - i{\beta}, - i{\beta}; \frac{i z}{{\hbar}c}) \right] .
v([V_\lambda]) =\frac{v \left( \partial\sum\limits_{w\in W^\sharp}\sigma(w) e^{w(\lambda+\rho_0)}\right)}{v\left(\partial\, e^{\rho_0}\prod\limits_{\alpha\in\Delta_0^+}(1-e^{-\alpha})\right)}=\frac{\prod\limits_{\alpha\in\Delta_0^+}(\lambda+\rho_0|\alpha)}{\prod\limits_{\alpha\in\Delta_0^+}(\rho_0|\alpha)}.
\begin{array}{l}C_n(x)=\min\left\{\begin{array}{l}L_n(x)+\int_0^\infty C_{n-1}(x-\xi)f_n(\xi) \mathrm{d}\xi, \min_{x< y \leq x+B}\{K+v(y-x)+L_n(y)+\int_0^\infty C_{n-1}(y-\xi)f_n(\xi) \mathrm{d}\xi\}\end{array}\right\}, G_n(x)=vx + L_n(x)+\int_0^\infty C_{n-1}(x-\xi)f_n(\xi) \mathrm{d}\xi, C_n(x)=-vx + \min\{G_n(x),K + \min_{x\leq y \leq x+B}G_n(y)\}.\end{array}
\mathcal{L}(\boldsymbol{y},\boldsymbol{u},\boldsymbol{p}) := \mathcal{J}(\boldsymbol{y},\boldsymbol{u}) + \int_0^T \int_{\Omega}\big( \sigma \frac{\partial \boldsymbol{y}}{\partial t}+ \textbf{curl}( \nu \, \textbf{curl}\, \boldsymbol{y})- \boldsymbol{u}\big) \boldsymbol{p}\, d\boldsymbol{x}dt,
\Gamma_{\phi}^{i j}( \nu ) = \frac{\omega^{4}}{\sigma \nu ( \nu^{2}- \omega^{2})}\left( \delta_{i j}\cot \left( \pi \frac{\nu}{\omega}\right) + ( 1 - \delta_{i j}) \csc \left( \pi \frac{\nu}{\omega}\right) \right) .
E (t, \tau) = E_1 (t, \tau) + E_2 (t, \tau), E_1 (t, \tau) := e^{-i\tau A(t)^{1/2}}F(t)^{\perp}P - F(t)^{\perp}e^{-i\tau (t^2 S)^{1/2}P}P, E_2 (t, \tau) := e^{-i\tau A(t)^{1/2}}F(t) P - F(t) e^{-i\tau (t^2 S)^{1/2}P}P.
v_{2}(t,r) = c_{b}\int_{0}^{\infty}d\xi \sin(t \xi) J_{1}(r\xi) \chi_{\leq \frac{1}{4}}(\xi)\cdot \begin{cases}\frac{1}{\log^{b-1}(\frac{1}{\xi})}, b \neq 1 \log(\log(\frac{1}{\xi})), b=1 \end{cases}
\lim_{t\to\infty}\tfrac 1t \log \mathbb P(P_t \geq xt) = -(1 - x + x\log x) x>1, \lim_{t\to\infty}\tfrac 1t \log \mathbb P(P_t \leq xt) = -(1 - x + x\log x) x\in(0,1).
\mathcal{L}_{p,F}= \dfrac{l_\lambda eTr_1\left(\left(Tr_2(\mathcal{E}_c^{\chi\chi_{-1}}*\Theta_\chi\mid[\mathfrak{l}^2])\right)\mid\Psi_2\right)}{\Delta(X,Y)\mathcal{E}_2(X,Y)H},
B_{z_1,z_2,z_3,z_4}(n) = \prod_{p^m \| n}\left(\sum_{j\geq 0}\frac{\sigma_{z_1,z_2}(p^{j + m}) \sigma_{z_3,z_4}(p^j)}{p^j}\right)\left(\sum_{j\geq 0}\frac{\sigma_{z_1,z_2}(p^{j}) \sigma_{z_3,z_4}(p^j)}{p^j}\right)^{-1}
\dot{\bar{z}}= \bar{f}(\bar{z},e,v,w) \dot{\bar{\eta}}= M \bar{\eta}+M c_{1}e-c_{1}\bar{g}_{1}(\bar{z},e,v,w)-c_{2}\bar{g}_{2}(\bar{z},e,v,w) \dot{\xi}= A_{c}\xi+B\Psi\bar{\eta}+\bar{G}(\bar{z},e,v,w)+B\bar{u}+B\Psi\tilde{\eta}
\big|x+\eta\big(b(x)-b(0)\big)\big|^{2}= |x|^{2}+2\eta \left\langle b(x)-b(0),x\right\rangle + \eta^{2}|b(x)-b(0)|^{2}\leq \left(1-2\theta_{1}\eta+\theta_{2}^{2}\eta^{2}\right) |x|^{2}+ 2 K\eta,
P^i_{\mathbb{I}}=\sum_{k=1}^n\left(\frac{\partial P_\mathbb{I}^{ik}}{\partial x^k}+\frac{\partial P_\mathbb{I}^{in+k}}{\partial x^{n+k}}\right)=-\frac{i}{4}\sum_{k=1}^n\left(\frac{\partial P^{ik}}{\partial z^k}-\frac{\partial \overline{P^{ik}}}{\partial \overline{z}^k}\right)
i \frac{\partial}{\partial t}\phi_{v}( \vec{r^{\prime}}, t ) = \left[ - \frac{{\nabla}_{r^{\prime}}^{2}}{2 m}- \frac{m}{2}\left( \frac{\ddot{a}}{a}\right){\vec{r^{\prime}}}^{2}\right] \ \phi_{v}( \vec{r^{\prime}}, t ) .
\tau_0 = \inf\{j \geq 0: R_j = s\}\mbox{and}\tau_{k+1}= \inf\{j > \tau_k: R_j = s\}~,~k \geq 0; & \tau'_0 = \inf\{j \geq 0: S_j = s\}\mbox{and}\tau'_{k+1}= \inf\{j > \tau'_k: S_j = s\}~,~k \geq 0.
1=|w_{n}^{+}|_{\varepsilon,p}=\frac{1}{t_{\varepsilon}(w_{n})^{p-2}}\|w_{n}\|_{\varepsilon}^{2}+\omega G_{\varepsilon}(t_{\varepsilon}(w_{n}))\ge\frac{1}{t_{\varepsilon}(w_{n})^{p-2}}\|w_{n}\|_{\varepsilon}^{2}.
\begin{array}{lll}\mathcal{Q}_{t}(x_{1:t-1})=\rho_t\Big( \mathfrak{Q}_{t}(x_{1:t-1}, \xi_t) \Big) \geq \rho_t \Big( \mathfrak{Q}_{t}^{k-1}(x_{1:t-1}, \xi_t) \Big) & \geq \displaystyle \sup_{p \in \mathcal{P}_t}\;\displaystyle \sum_{j=1}^{M}p_j \Phi_{t, j}\mathfrak{Q}_{t}^{k-1}(x_{1:t-1}, \xi_{t, j}).\end{array}
\bigsqcup_{t\in H_P}\sigma_t(U_P) = \bigcup_{t\in \mathcal{O}_S}\sigma_t(U_P) = \bigcup_{\widetilde{t}\in \mathcal{O}_P}\sigma_{\widetilde{t}}(U_P) = \bigsqcup_{\widetilde{t}\in G_P}\sigma_{\widetilde{t}}(U_P),
e^{{\frac{i\hbar}{2}}(\stackrel{\leftarrow}{\partial}_{x}\stackrel{\rightarrow}{\partial}_{p}-\stackrel{\leftarrow}{\partial}_{p}\stackrel{\rightarrow}{\partial}_{x}) +\hbar \stackrel{\leftarrow}{\partial}_{\theta}\stackrel{\rightarrow}{\partial}_{\theta}}~.
T^{(i)}(q) =\begin{cases}q^{i-\ell(\lambda^{(i)})}, \hbox{if the $i$th box is in the first column,}q^{i-{\lambda_{j}^{(i)}}'}- q^{i-1-{\lambda_{j-1}^{(i)}}'}, \hbox{if the $i$th box is in the $j$th column, $j\geq2$.}\end{cases}
J(u(\cdot))=\mathbb{E}\left[ \int_{0}^{T}\left( A_{4}(t)X(t)^{2}+B_{4}(t)Y(t)^{2}+C_{4}(t)Z(t)^{2}+D_{4}(t)u(t)^{2}\right) dt+GX(T)^{2}+Y(0)^{2}\right] ,
|m^G(\xi)| \le(\pi|\xi|)^{-1}\varphi_{\zeta}^G(0)+ (2\pi|\xi|)^{-1}\int_{-u_{\zeta}}^{u_{\zeta}}|(\varphi_{\zeta}^G)'(u)|{\rm d}u &=(\pi|\xi|)^{-1}\varphi_{\zeta}^G(0)-(\pi|\xi|)^{-1}\int_{0}^{u_{\zeta}}(\varphi_{\zeta}^G)'(u){\rm d}u &\le 6\pi^{-1}(L|\xi|)^{-1}.
2^{m}(J_{m}-a^2a_{m}^2I) =2^{m}\int_0^{\frac{\pi_4}{2}}\frac{a_{m}^4\cos_4^4{\theta}+b_{m}^4\sin_4^4{\theta}-a_{m}^4}{(a_{m}^4\cos_4^4{\theta}+b_{m}^4\sin_4^4{\theta})^{\frac34}}\, d\theta &=2^{m}c_{m}^4\int_0^{\frac{\pi_4}{2}}\frac{-\sin_4^4{\theta}}{(a_{m}^4\cos_4^4{\theta}+b_{m}^4\sin_4^4{\theta})^{\frac34}}\, d\theta.
\delta_\perp \kappa_1 = \kappa_3 \kappa_2 \Psi_3+ 2 \kappa_2 \Psi_2{}' + \kappa_2' \Psi_2 + \Psi_1{}'' - \left( \kappa_1^2 + \kappa_2{}^2 \right) \Psi_1\,.
\Phi_\mu(G_{i,i+1}M_{1,1}) = \widetilde{\Phi}_\mu(U_\mu^{-1}G_{i,i+1}M_{1,1}U_\mu) =\widetilde{\Phi}_\mu(G_{i,i+1}M_{1,1}) = E_{\chi_1^\mu}g_{i,i+1}E_{\chi_1^\mu}.
{\operatorname*{l i m}_{\left| k_{2 \mu}\right| \rightarrow \infty}}T_{\mu \nu}^{( 3 ) a b c d}T_{\mu^{\prime}\nu^{\prime}}^{( 3 ) a^{\prime}b^{\prime}c d *}Q^{\mu \mu^{\prime}}( k_{1}) Q^{\nu \nu^{\prime}}( k_{2}) = \widetilde{S}
r_{T_0+1}\leq r_1 + \sum_{t=1}^{T_0}\left(\frac{8}{\alpha^2T_0}\left({\sf b}Lh_t^{\beta-1}\right)^2 + \frac{16}{\alpha^2 T_0^2}\left({\sf v_2}\bar{L}^2h_t^{2}+{\sf v_3}\sigma^2h_t^{-2}\right)\right)\enspace.
m \, [ \: x^\mu, \dot x^\nu \: ]= - \mbox{i}\hbar \, \{g^{\mu \nu}(x)+ h^{\mu \nu}_{\:\;\;\;\rho}(x) \, \dot x^\rho+ k^{\mu \nu}_{\:\;\;\;\rho \sigma}(x) \, \dot x^\rho \, \dot x^\sigma+ \cdots \}.
\frac{2\sqrt{\pi}\,c^{\frac{1}{2}-s}\Gamma\left(s-\frac{1}{2}\right)}{\Gamma(s)}\zeta_{p^{\prime}}(2s-1)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\left(2C_{p^{\prime}}^{(1)}-\log\left(4c\right)\right)+O\left(s-1\right).
(X_n\otimes Y_n)^{c_1,\dots,c_n}=\bigoplus_{\substack{{a_1,\dots,a_n}{b_1,\dots,b_n}{a_i+b_i=c_i}}}X_n^{a_1,\dots,a_n}\otimes Y_n^{b_1,\dots,b_n}
\frac{N_{\mathrm{c}m}}{(2m)!!}=\frac{1}{2^{n+1}}\sum_{i=0}^{m}(-1)^{i}\sum_{a_{1},\cdots,a_{i+1}=1}^{\infty}\delta_{a_{1}+\cdots+a_{i+1},m+1}\prod_{j=1}^{i+1}\frac{(2a_{j})!}{a_{j}!},
d(Jx,Jy) = H(\{Jx\},\{Jy\}) \leq H(\{Jx\},Fx) + H(Fx,Fy) + H(Fy,\{Jy\}) \leq \frac{2}{n}+ \alpha [d(Jx,Fx ) + d(Jy,Fy)] \leq \frac{1}{n}(2+2\alpha).
\nabla^i \nabla_i f_t + |\alpha_t|^2 \nabla^i \nabla_i f_t - (\alpha_t)_i (\alpha_t)_j \nabla^i \nabla_j f_t & = -\nabla^i (\alpha^*)_i - |\alpha_t|^2 \nabla^i (\alpha^*)_i + (\alpha_t)_i (\alpha_t)_j \nabla^i (\alpha^*)_j.
u_{- 1}= - 5 u_{1}+ v_{- 1}+ 2 w_{1}a^{- \gamma}+ ( \mathrm{\frac{9}{2}}u_{1}^{2}- \mathrm{\frac{1}{2}}v_{- 1}^{2}- 3 u_{1}v_{- 1}+ w_{2}) a^{- 2 \gamma}
a_i^{(k)}= \begin{cases}\beta r_i^{S_{k-1}}/ (1-\beta) i \in S_{k-1}\beta r_i^{S_{k-1}}i \in S_{k-1}^c\end{cases}b_i^{(k)}= \begin{cases}\beta w_i^{S_{k-1}}/ (1-\beta) i \in S_{k-1}\beta w_i^{S_{k-1}}i \in S_{k-1}^c.\end{cases}
\begin{array}{c c c}{F_{i j}}&{=}&{\partial_{i}a_{j}- \partial_{j}a_{i}+ i ( c_{j}^{*}c_{i}- c_{i}^{*}c_{j}) ,}\\{}&{=}&{\partial_{i}b_{j}- \partial_{j}b_{i}- i ( c_{j}^{*}c_{i}- c_{i}^{*}c_{j}) .}\\ \end{array}
\left|p((\ell+i+1)\delta_K)-p((\ell+i)\delta_K)\right|= p((\ell+i)\delta_K)\frac{\left|p((\ell+i+1)\delta_K)-p((\ell+i)\delta_K)\right|}{p((\ell+i)\delta_K)}\leq \frac{\|p\|_{}\delta_K}{\underline{p}}p((\ell+i)\delta_K)
\left|a_{u}^{'}-a_{v}^{'}\right|=\left|b_{i}\gamma_{2}-b_{j}\gamma_{1}\right|\leq \left|b_{i}-b_{j}\right|~for~all~b_{i}^{'}\in B_{i}^{'}~~and~a_{j}^{'}\in B_{j}^{'}~for~i,j\in\{1,\ldots,s\}
\frac{1}{\tilde{u}}\frac{\partial \tilde{u}}{\partial \tilde{t}_1}=-\frac{2}{\tilde{t}_1}\big(\tilde{p}_1\tilde{q}_1+\tilde{p}_2\tilde{q}_2+\tilde{\theta}^\infty_1\big),\frac{1}{\tilde{u}}\frac{\partial \tilde{u}}{\partial \tilde{t}_2}=-2\tilde{p}_2\tilde{q}_1.
\sum_{i=(k-1)\delta n+1}^{\max(t,k\delta n)}q_{X,i}=\sum_{h=1}^{t-1}Y_h =\sum_{i=(k-1)\delta n+1}^{\max(t,k\delta n)}p_{X,i}\pm\tfrac{2\cdot 10^5\delta m \delta_{k\delta n}}{\gamma^7}\pm \delta^2 n =\sum_{i=(k-1)\delta n+1}^{\max(t,k\delta n)}p_{X,i}\pm\tfrac{10^6\delta m \delta_{k\delta n}}{\gamma^7}\,.
\mathbb{E}_{\alpha}[H(X_i|X_j=x_j )] \leq \mathbb{E}_{\alpha}[ \log(N_i)] = \mathbb{E}_{\alpha_i}[ \mathbb{E}_{\alpha_j:j \ne i}[\log(N_i)]] \leq \mathbb{E}_{\alpha_i}[ \log(\mathbb{E}_{\alpha_j:j \ne i}[N_i])].
q^{2 ( \lambda , \lambda + 2 \rho_{0})}= \sum_{( \mu , 2 \kappa , - s ) \in D^{+}}\sum_{s = 0}^{\infty}( m_{\mu , s}^{+}- m_{\mu , s}^{-}) q^{( \mu , \mu + 2 \rho_{0}) / 2 - s ( 2 \kappa + g )}\frac{D_{q}[ ( \mu , 2 \kappa , - s ) ]}{D_{q}[ ( \lambda , \kappa , 0 ) ]}
[ B,A ]= \begin{pmatrix}i f^{'}_{2}f_{2}+ if_{2}f^{'}_{2}+ [f_{2},z]_{-}- i \hbar \delta \lambda - i f^{'}_{2}f_{2}- if_{2}f^{'}_{2}[z, f_{2}]_{-}+ i \hbar \end{pmatrix}
c^{>}= ( - C_{G}+ C_{M}) / 1 6 \pi^{2}; \hspace{0 . 3 i n}c^{<}= - C_{G}/ 1 6 \pi^{2}; \hspace{0 . 3 i n}c^{>}- c^{<}= b_{0}^{>}- b_{0}^{<}= - 3 / 2 ( b^{>}- b^{<}) = C_{M}/ 1 6 \pi^{2}.
f(x)-f(y) =f(x)-f(u_0)+f(v_0)-f(y)+f(u_0)-f(v_0) > -d(x,u_0)-d(v_0,y)+(1-\alpha/2+\delta)d(u_0,v_0) >d(x,y)-\alpha/2 d(u_0,v_0) \ge (1-\alpha)d(x,y).
\begin{aligned}ric_g-\frac{scal_g}{n}g=\frac{n-2}{h}\Bigg[(\nabla^2h)_{g_0}-\frac{(\Delta h)_{g_0}}{n}g_0+ +dh\otimes d(\log\rho_\kappa)+d(\log\rho_\kappa)\otimes dh-\frac{2}{n}g_0\left((\nabla h)_{g_0},(\nabla\log\rho_\kappa)_{g_0}\right)g_0\Bigg], \end{aligned}
\dot g(j_1)_{|i(g)}= \prod_{k=1}^{\kappa(j_1)}\left(s_{\iota(k)}\right)_{|\, i(g)-\sigma(k-1)}\ne \prod_{k=1}^{\kappa(j_2)}\left(s_{\iota(k)}\right)_{|\, i(g)-\sigma(k-1)}= \dot g(j_2)_{|i(g)},
{\bf h}_{u,m,k}=\sum_{l_m=1}^{L_m}\sum_{k=1}^{K}{\alpha_{l_m}}e^{-\mathrm{j}\frac{2\pi kd}{K}}{\bf a}_t(\phi_{l_m},f_k)r(kT_s-\tau_{l_m}),\;\in\mathbb{C}^{{N_m}\times1},
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2\cdot \frac{(2^{(n_{2,\nu_2}+1)/2}+1)(2^{(n_{2,\nu_2}-1)/2}+1)}{2^{n+1}-1}.
\max \biggl\{\frac{\partial f_{i}}{\partial x_{i}}(t,y(t),\alpha_{i2},\ldots ,\alpha_{ii},u(t)),\ldots ,\frac{\partial f_{n}}{\partial x_{n}}(t,y(t),\alpha_{n2},\alpha_{n3},\ldots ,\alpha_{nn},u(t))\biggr\}\le 0, &\forall \alpha_{j\nu}\in{\mathbb R},\, j=i,\ldots ,n;\, \nu =2,3,\ldots ,j
m(\xi,\eta) = O\big( f(|\xi|,|\xi-\eta|,|\eta|) \big)\, \Longleftrightarrow \,{\| m^{k,k_1,k_2}(\xi,\eta) \|}_{S^\infty}\lesssim f(2^k,2^{k_1},2^{k_2}) \mathbf{1}_{\mathcal{X}}(k,k_1,k_2).
\mu(\gamma_p^{c}B_s \cap B_t) = \mu( \gamma_p^{c}\theta(B_{s,1}) \cap \theta(B_{t,1})) + \mu( \gamma_p^{c}\theta(B_{s,2}) \cap \theta(B_{t,2})) = \mu( \theta( \gamma_p^{a}B_{s,1}\cap B_{t,1}) ) + \mu( \theta( \gamma_p^{b}B_{s,2}\cap B_{t,2}) ) = t \mu(\gamma^{a}_p \cap B_{s,1}\cap B_{t,1}) + (1-t) \mu(\gamma^{b}_p \cap B_{s,2}\cap B_{t,2})
f_{\tau^{(_c)}_1\vert M^{(_c)}\geq 1}\left(s\right) = \sum_{k=1}^{\infty}\frac{k s^{k-1}}{T_{M}^k}\frac{\left( \lambda T_{M}\right)^{k}}{k!}\frac{\exp \left( -\lambda T_{M}\right)}{1-\exp\left( -\lambda T_{M}\right)}&=\frac{\lambda \exp\left( \lambda s\right)}{\exp\left( \lambda T_{M}\right) -1}, \;\;s\in \left(0,T_{M}\right],
\begin{cases}\partial_t(\alpha \partial_t\Lambda) - \partial_x(\alpha \partial_x\Lambda) = 2\alpha \sinh{2\Lambda}((\partial_t\phi)^2-(\partial_x\phi)^2) \partial_t(\alpha \sinh^2 \Lambda \partial_t\phi )- \partial_x(\alpha \sinh^2 \Lambda \partial_x\phi )= 0,\end{cases}
\nabla_i R =\frac{1}{2}(\nabla_i H_{jkl}-\nabla_k H_{jil}+\nabla_l H_{jik})H_{jkl}&=\frac{1}{4}\nabla_i |H|^2-\nabla_k H_{jil}H_{jkl}&=\frac{1}{4}\nabla_i |H|^2-2\nabla_i R
1 > \sum_{i=1}^r \frac{(1-\lambda) + (m_i-1) + [((1-\lambda) + (m_i-1))^2 + 4\lambda m_i]^{1/2}}{2m}= \sum_{i=1}^r\frac{(m_i-\lambda)+(m_i+\lambda)}{2m}= 1,
\delta_{H}={\frac{N^{3 / 4}2^{- 1 / 4}}{5 \sqrt 3 \pi^{3}}}\,{\frac{g_{s}( M_{s}L )^{1 / 2}}{\beta^{1 / 4}( 2 \pi R \, M_{s})^{9 / 2}}}= 1 . 9 1 \times 1 0^{- 5}\, ,
\! \! \! \!{\cal L}_{D S M}= \frac 1{1 + 4{\Phi}^{2}}\left( \phantom{\frac I I}\!{\textstyle \frac 1 2}\left( \delta^{i j}+ 4 \Phi^{i}\Phi^{j}\right) \partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}- \varepsilon^{\mu \nu}\varepsilon^{i j k}\Phi^{i}\partial_{\mu}\Phi^{j}\partial_{\nu}\Phi^{k}\right) ~ .
\psi^{\, \prime}(t) \, = \, \left[ \, F(\bar{x}+ t(x - \bar{x})) - F(\bar{x}) \, \right]^T( x - \bar{x}) \, = \, t \, F^{\, \prime}(\bar{x};x - \bar{x})^T( \, x - \bar{x}\, ).
\frac{b_{p,q}(x)-b_{p,q}(x_n)}{x-x_n}=\frac{\pi_q(b_1b_2\cdots)-\pi_q(b_1\cdots b_{n-1}0^{\infty})}{\pi_p(b_1b_2\cdots)-\pi_p(b_1\cdots b_{n-1}0^{\infty})}&=\frac{\pi_q(0^{n-1}1b_{n+1}\cdots)}{\pi_p(0^{n-1}1b_{n+1}\cdots)}&\ge \frac{\pi_q(0^{n-1}10^{\infty})}{\pi_p(0^{n-1}1^{\infty})}=\frac{p-1}{p}\left(\frac{p}{q}\right)^n
\hat{\cal H}\prod_{j = 1}^{r}\, ( B_{f_{j}}^{+})^{n_{j}}\, e^{W}= \left({\cal E}_{0}+ \omega \sum_{k = 1}^{r}n_{k}f_{k}\right) \prod_{j = 1}^{r}\, ( B_{f_{j}}^{+})^{n_{j}}\, e^{W}.
\frac{d}{dt}(f_{t}^{H_{0}}(f_{t}^{H_{1}^{t}}(z))) =X_{H_{0}}(f_{t}^{H_{0}}f_{t}^{H_{1}^{t}}(z))+X_{H_{1}}(f_{t}^{H_{0}}f_{t}^{H_{1}^{t}}(z)) & =X_{H_{0}+H_{1}}(f_{t}^{H_{0}}(f_{t}^{H_{1}^{t}}(z)))
\|P(t)-y(t)\| \le\|P(t)-p(t)\|+\|p(t)-y(t)\| &\le \sqrt{m}B^2+B[1+F^2]^{\frac{1}{2}}[e^{Lt}-1] &= B\{\sqrt{m}B+[1+F^2]^{\frac{1}{2}}[e^{Lt}-1]\}
{\mathrm d}\widetilde{Z}_{t}^{\varepsilon}\, =\, \frac{Z_{t}^{-1}b(t, X_{t}, F_{t})}{\, (1 + \varepsilon Z_{t}^{-1})^{2}\,}{\mathrm d}X_{t}- \frac{\varepsilon Z_{t}^{-2}\lvert b(t, X_{t}, F_{t}) \rvert^{2}\,}{( 1 + \varepsilon Z_{t}^{-1})^{3}\,}{\mathrm d}t \, ; t \ge 0 \, .
F(x,u,u_1,u_2,u_{11}, u_{12}, u_{22}) :=\sum_{i,j=1}^2 \Big[h^{ij}\left( u_{ij}+ \Gamma_{ij}^3 + u_i\Gamma_{3j}^3 + u_j\Gamma_{3i}^3 + u_i u_j\Gamma_{33}^3 \right) &\ \ \ \ \ \ \ \ \ -\sum_{m=1}^2u_m h^{ij}\left( \Gamma_{ij}^m + u_i\Gamma_{3j}^m + u_j\Gamma_{i3}^m + u_i u_j\Gamma_{33}^m \right)\Big].
x_R [n] = \arg \mathop{\max}\limits_x \Pr (x_A [n] \oplus x_B [n] = x|r) = \arg \mathop{\max}\limits_x \sum\limits_{x_A^n x_B^n :x_A [n] \oplus x_B [n] = x}{\Pr (x_A^n x_B^n |r)}.
(\dot{\mathrm{H}}^{s_0,p}(\mathbb{R}^n),\dot{\mathrm{H}}^{s_1,p}(\mathbb{R}^n))_{\theta,q}=(\dot{\mathrm{B}}^{s_0}_{p,q_0}(\mathbb{R}^n),\dot{\mathrm{B}}^{s_1}_{p,q_1}(\mathbb{R}^n))_{\theta,q}=\dot{\mathrm{B}}^{s}_{p,q}(\mathbb{R}^n)
\left\{(g_j, g, \bar{g}, h, \bar{h})\begin{tabular}{|l}$h\bar{h}=h_1, g\bar{g}=g_j$ $g_j\in c'(h_1, Y), h\in c(g, X), \bar{h}\in c(\bar{g}, X)$ $gX_{\bar{g}, \bar{h}}\cap X_{g, h}\neq \emptyset$\end{tabular}\right\}.
\iota_{+}^\star\circ\Phi(E) =\iota_{+}^\star\circ((f_+)_{\star}f_{-}^\star(E)) &=(F_{+})_{\star}\circ \tilde{\iota}^{\star}\circ f_{-}^\star(E) &=(F_{+})_{\star}\circ F_{-}^\star\circ \iota_{-}^\star(E) &=\Psi\circ \iota_{-}^\star(E).
G:\:\overline{ran}(C\otimes I_d) \rightarrow \overline{ran}(A\otimes I_d), \mbox{such that}G =(A\otimes I_d)^{1/2}(B\otimes I_d)(C\otimes I_d)^{1/2}= (A^{1/2}\otimes I_d)(B\otimes I_d)(C^{1/2}\otimes I_d) = ((A^{1/2}B)\otimes I_d)(C^{1/2}\otimes I_d) = (A^{1/2}BC^{1/2})\otimes I_d = G_M\otimes I_d
\left< \tilde{u}^{n}_{t},\varphi\right> - \left< \tilde{u}^{n}_{0},\varphi\right> = \int_0^t \left< \tilde{u}^{n}_{z}, \tilde{L}_z^n \varphi\right> dz + \tilde{W}^n_t(R\varphi),