harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F\left( z\right) =\dfrac{c\left( n,a\right)}{2}\int\limits_{\mathbb{R}^{2n}}\frac{\left\vert z\left( x\right) -z\left( y\right) \right\vert}{\left\vert x-y\right\vert^{n+\alpha}}^{2}dxdy-\int\limits_{\Omega}G\left(x,z\left( x\right) ,u\left( x\right) \right) dx
\left.\frac{d}{dt}\right\vert_{t=0}L^\circ_a(1 + t b) = \left.\frac{d}{dt}\right\vert_{t=0}\sum_{m \geq 0}\alpha_m (1+tb)^{m+1}&= \sum_{m \geq 0}(m+1) \alpha_m b = a* b = \left.\frac{d}{dt}\right\vert_{t=0}L^*_a(1 + t b).
\zeta(s)=\prod_{pprime}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}}\cdot\cdot\cdot
{\displaystyle V^i_{\alpha}=\pi^i_{\alpha}+ip_{\alpha\dot\alpha}\bar\theta^{\dot\alpha i}+m\theta^i_{\alpha}-ieA^i_{\alpha}\approx0,\bar V_{\dot\alpha i}=\bar\pi_{\dot\alpha i}+i\theta^{\alpha}_ip_{\alpha\dot\alpha}+m\bar\theta_{\dot\alpha i}-ie\bar A_{\dot\alpha i}\approx0,}
\|x_\alpha^A - x_\gamma^A\| = \|x_\beta^A-x_\gamma^A+h_\beta v_\beta^A \|\le \|x_\beta^A-x_\gamma^A\| +h_\beta\| v_\beta^A \| &\le M(t_\beta-t_\gamma)+M(t_\alpha-t_\beta)=M(t_\alpha-t_\gamma)
Z_h(s, \chi, D)= v(\bar{h}, D, \chi)+\sigma(\bar{h}, D, \chi)\dfrac{(1-q^{-1})q^{-s}}{1-q^{-1-s}}+Z_h(s, \chi, D_{S(h, D)}) := \dfrac{H_1(q^{-s})}{1-q^{-1-s}}+Z_h(s, \chi, D_{S(h, D)})
(\tilde u - u_h, \partial^{m}\eta)_{\Omega_h}= a_h(\tilde u - v_h, \tilde g - g_h) - \operatorname{Res}_g(\tilde u - v_h) - \operatorname{Res}_u(\tilde g - g_h) + \operatorname{Res}_u(\tilde g) &=: I_1 + I_2 + I_3 + I_4.
S\left(\tilde{\Phi}(\hat{\rho}_A)\right) - S(\hat{\rho}_A) &= S\left(\tilde{\Phi}(\hat{\gamma}_A)\right) - S(\hat{\gamma}_A) + S(\hat{\rho}_A\|\hat{\gamma}_A) - S\left(\left.\tilde{\Phi}(\hat{\rho}_A)\right\|\tilde{\Phi}(\hat{\gamma}_A)\right) &\ge S\left(\tilde{\Phi}(\hat{\gamma}_A)\right) - S(\hat{\gamma}_A)\;,
\overline{Z}_\Psi = \int\prod D\Phi^a \prod D\vartheta^b\prod \delta(d^\gamma)exp\left({i\over\hbar}W(\Phi^a,{\partial\overline\Psi\over \partial \Phi^a}, \vartheta^b,{\partial\overline\Psi\over \partial \vartheta^b}\right)
\begin{aligned}\Big(\tilde{\Omega}^{(2)}_{\mathcal{P}}(t_*) \tilde{v}, \tilde{v}\Big)_{\mathbb{C}^n}= 2 \Big( (X_2 (t_*)^t Y_2' (t_*) - Y_2^t (t_*) X_2' (t_*)) M_2 (t_*)^2 \Big{\{}X_2 (t_*)^t v_1 + Y_2 (t_*)^t v_2 \Big{\}}, & M_2 (t_*)^2 \Big{\{}X_2 (t_*)^t v_1 + Y_2 (t_*)^t v_2 \Big{\}}\Big)_{\mathbb{C}^n}.\end{aligned}
\tilde{B}(u,v)=\begin{pmatrix}\sum_{j,k=0}^{3}B_{jk}^0 u_j v_k \sum_{j,k=0}^{3}B_{jk}^1 u_j v_k \sum_{j,k=0}^{3}B_{jk}^2 u_j v_k \sum_{j,k=0}^{3}B_{jk}^3 u_j v_k \end{pmatrix}.
\zeta(s) \ll \begin{cases}|t|^{1/2 - (1-2\mu(1/2))\sigma}, \sigma\in[0, 1/2], |t|^{2\mu(1/2)(1-\sigma)}, \sigma\in[1/2, 1-\eta], |t|^{2\mu(1/2)(1-\sigma)}\log^{2/3}|t| , \sigma\in[1-\eta, 1], \log^{2/3}|t|, \sigma\ge1.\end{cases}
D^{+}( x_{1}- x_{2}) = \int \frac{d^{3}k}{( 2 \pi )^{3}2 \omega_{k}}\mathrm{e}^{- \mathrm{i}k^{+}( x_{1}- x_{2})}, \qquad \omega_{k}= \sqrt{\vec{k}^{2}+ m^{2}},
\begin{array}{c}{L ={\tilde{\sigma}}\left[ 2 c \epsilon + c{\bf e^{\prime}}_{1}^{2}+{\bf p}({\bf x^{\prime}}-{\bf e}_{+}) + \right.}\\{\left. +{\bf p}_{+}({\bf e^{\prime}}_{+}-{\bf e}_{1}) - \sum_{i , j}d_{i j}({\bf e}_{i}{\bf e}_{j}- \eta_{i j}) \right] ,}\\ \end{array}
f(\theta^{*},\omega^{*},t^{*}) =f(\theta(t^{*}),\omega(t^{*}),t^{*}) \leq e^{Kd}f(\theta(t^{*}-d),\omega(t^{*}-d),t^{*}) \leq e^{KD(\varepsilon,\kappa)}\varepsilon^{\prime},
n^r \left( f(a_i^{(n)}, b_i^{(n)}) - f(\bar a_i, \bar b_i)\right) - \sum_{i = 1}^k \left(\frac{\partial f}{\partial a_i}(\bar a_i, \bar b_i) \tilde a_i^{(n)}+ \frac{\partial f}{\partial b_i}(\bar a_i, \bar b_i) \tilde b_i^{(n)}\right) \to 0 &.
\begin{array}{l}{{\delta \theta_{\alpha}= \epsilon^{\alpha}, \qquad \delta \bar{\theta}_{\alpha}= \bar{\epsilon}^{\alpha},}}\\{{\delta x^{\bar{\mu}}= - i \bar{\epsilon}_{\alpha}\tilde{\Gamma}^{\bar{\mu}\alpha \beta}\bar{\theta}_{\beta}- i \epsilon^{\alpha}\Gamma_{\alpha \beta}^{\bar{\mu}}\theta^{\beta}, \qquad \delta x^{1 1}= 0 ,}}\\ \end{array}
A = 4 |u_j(x^*)|^2e^{\phi(x^*)}, begin{align*}1em] B = 4 \pi |u_j(x^*)|^2e^{\phi(x^*)}(H(x^*)-\partial_{\nu}\phi(x^*)), begin{align*}1em] C = |u_j(x^*)|^2 e^{\phi(x^*)}\left( \frac{8\log 2 - 6}{\pi}(H(x^*)-\partial_{\nu}\phi(x^*)) - 16R_{\partial M}^{\lambda_j}(x^*,x^*) \right).
{\cal{Z}}^{P F}( X_{1}, X_{2}; p ) = \frac{1}{( 1 - X_{1})}\frac{1}{( 1 - X_{2})}\left[ \frac{( 1 - ( X_{1}X_{2})^{p + 2})}{( 1 - X_{1}X_{2})}- \frac{({X_{1}}^{p + 2}-{X_{2}}^{p + 2})}{( X_{1}- X_{2})}\right] ~ .
I:= \int_{0<|z|\leq\delta}\psi_k(z)\, d\mu_{1,k}(z)=\int_{|z|<r}\psi_k(z)\, d\mu_{1,k}(z) +\int_{r<|z|\leq\delta}\psi_k(z)\,d\mu_{1,k}(z)=I_1+I_2, J:= \int_{0<|z|<\delta}\psi(z)\, d\mu_{1}(z)=\int_{|z|<r}\psi(z)\, d\mu_{1}(z) +\int_{r<|z|\leq\delta}\psi(z)\,d\mu_{1}(z)=J_1+J_2.
\Psi_{n}\left( x\right) :=\frac{\Phi_{n}\left( x\right)}{\left\Vert \Phi_{n}\left( .\right) \right\Vert_{H}}=\left( \begin{array}{c}\Psi_{n}\left( x\right) R^{\prime}\left( \Psi_{n}\right)\end{array}\right) .
\mathbb{E}\left[ \sum_{n=0}^{\infty}z^n e^{i t S_n}e^{i{s}\cdot{C}_n}\right]= \frac{1}{1-z \phi_{\xi_1,{\eta_1}}(t,{s})}= f^{-1}_{+}(z, t, s) f_{-}(z, t, \underline{s})
u( \cdot , t) = W_0 ( t) \phi + W_{h}h_1 +(W_{h}h_2 )\Big |_{x\rightarrow 1-x}+ i \lambda \int^t_0 W_0 ( t-\tau ) \Big ( u (\cdot ,\tau ) | u ( \cdot , \tau ) |^{p-2}\Big ) d \tau \, ,
A_{k,T}^\dagger \, A_{k,T}= P_0, A_{k,T}\, A_{k,T}^\dagger = P_{k,T}, A_{k,T}\, P_0^j = P_{k,T}^j \, A_{k,T}, A_{k,T}^\dagger\, P_{k,T}^j = P_0^j \, A_{k,T}^\dagger, A_{k,T}Q_0=Q_0 A^\dagger_{k,T}=0,
[ \mathbf{H}_{n}, \mathbf{H}_{m}] = \mathbf{k}n \delta_{n + m , 0}\quad , \quad [ \mathbf{H}_{n}, \mathbf{E}_{m}^{\pm}] = \pm \sqrt 2 \mathbf{E}_{m + n}^{\pm}\quad , \quad [ \mathbf{E}_{n}^{+}, \mathbf{E}_{m}^{-}] = \mathbf{k}n \delta_{n + m , 0}+ \sqrt 2 \mathbf{H}_{n + m}\quad
\int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y ( y - \zeta )) \; dy = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(1- y^{-1}\zeta )\; dy & = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(1- y\zeta )\; dy & = \chi(-\zeta) \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y - \zeta^{-1})\; dy.
\sum_{j=0}^{n}Y_{j}\left( \lambda \right) Y_{n-j}\left( \lambda \right) =\frac{4\left( -1\right)^{n}n!}{\left( \lambda -1\right)^{2}}\left( \frac{\lambda^{2}}{\lambda -1}\right)^{n}\sum_{j=0}^{n}\frac{1}{\left( \begin{array}{c}n j\end{array}\right)}.
X^p \frac{\partial\tilde{\Gamma}_{\,\,ij}^k}{\partial u^p}=X^p \frac{\partial\Gamma_{\,\,ij}^k}{\partial x^p}(V)+ (\nabla^V_XV)^l\frac{\partial\Gamma_{\,\,ij}^k}{\partial y^l}(V) -X^mN^l_{\,\,m}(V)\frac{\partial\Gamma_{\,\,ij}^k}{\partial y^l}(V).
f(x) = \pi^{-1}\left( h(c+\pi x)-c\right) &= \pi^{-1}\left( \sum_{r\ge 0}h^{(r)}(c) \pi^{r}x^r/r! - c\right) &= (h(c)-c)\pi^{-1}+ h'(c)x + \sum_{r\ge 2}h^{(r)}(c) \frac{\pi^{r-1}}{r!}x^r.
\sup\limits_{\textbf{t}\in \mathcal{C}}\inf\limits_{\textbf{y}\in \widehat{\mathcal{C}}_\ell}\|\textbf{t}-\textbf{y}\|\leq \max\limits_{i=1}^{k}\sup\limits_{\textbf{t}\in B(\textbf{t}_i,\frac{\epsilon}{2})}\inf\limits_{\textbf{y}\in \widehat{\mathcal{C}}_\ell}\|\textbf{t}-\textbf{y}\|.
L_2 = \int \int \tilde{K}(x+y,y)(u(x+y)-u(x))u(x) dy dx &= \frac{1}{2}\int \int (\tilde{K}(x,y) - \tilde{K}(x+y,y)) (u(x)-u(x+y))u(x) dy dx &= \frac{1}{2}\int \int (2 \tilde{K}(x,y) - \tilde{K}(x+y,y) - \tilde{K}(x-y,y)) u(x)^2 dy dx &\gtrsim -\int \int |y|^2 \| D^2_x K \|_{L^\infty}|y|^{-d-\alpha}u(x)^2 dy dx .
\frac{\kappa}{m!}\partial_z^m[z^{2k+m-1}(1-z^2)^{j}] = \frac{j!2^{j+1/2}\kappa^{2k}}{\sqrt{\pi}}\int_0^{\arccos \kappa}[\cos(j+1/2)s] & \sum_{l=0}^m\frac{1}{\Gamma(j-l+1/2)l!}\binom{2k+m-1}{m-l}\kappa^{l}(\cos s-\kappa)^{j-l}ds
S_{\epsilon,\delta}([\eta,\,T]) =\left\{i:\sup_{\substack{\eta\le s,\,t\le T,\,|s-t|\le\delta}}\left|x_{i}^{N}(t)-x_{i}^{N}(s)\right|\ge\epsilon\right\}S_{\epsilon,\delta}([0,\,\eta]) =\left\{i:\sup_{\substack{0\le s,\,t\le\eta,\,|s-t|\le\delta}}\left|x_{i}^{N}(t)-x_{i}^{N}(s)\right|\ge\epsilon\right\}
\Omega_1^{{\bf x}}{R}_2({\bf m;{y}};{\gamma}; N)=\kappa\left(m_1, \frac{{\gamma}_2-\gamma_0}{2}\right){R}_2({\bf m; y};{\gamma}; N), \Lambda_2^{{\bf x}}{R}_2({\bf m; y};{\gamma}; N)=\kappa\left(m_1+m_2, \frac{{\gamma}_3-\gamma_0}{2}\right){R}_2({\bf m; y};{\gamma}; N).
\begin{aligned}(X_1 - i Y_1)^{-1}(X_1^t + i Y_1^t)^{-1}= \Big((X_1^t + i Y_1^t) (X_1 - i Y_1) \Big)^{-1}&= \Big(X_1^t X_1 + Y_1^t Y_1 + i (Y_1^t X_1 - X_1^t Y_1) \Big)^{-1}= M_1^2.\end{aligned}
j_{m}(\rho) =\frac{q}{(2\pi)^3}\int_{|p|<(6\pi^2\rho/q)^{\frac{1}{3}}}\sqrt{|p|^2+m^2}{\rm d}p & = \frac{q}{16\pi^2}\left[\eta(2\eta^2+m^2)\sqrt{\eta^2+m^2}-m^4\ln{\left(\frac{\eta+\sqrt{\eta^2+m^2}}{m}\right)}\right], \eta=\left(\frac{6\pi^2\rho}{q}\right)^{\frac{1}{3}}.
d*_gH = 2\,\frac{h_4}{\mu}\left(e^{125}- e^{134}\right) -2\,\frac{h_3\left(a^2 b^2 +c^2\right) -c h_1\left( a^2 + b^2\right)}{\mu\left(a^2b^2 - c^2\right)}\left(e^{124}+ e^{135}\right).
t_{fric}=\frac{2.34}{ln\Lambda}\frac{\sigma_{M}^{2}}{\sigma_{s}^{3}}r_{i}=\frac{2.7Gyr}{ln\Lambda}\frac{r_{i}}{30kpc}(\frac{\sigma_{M}}{200kms^{-1}})^{2}(\frac{100kms^{-1}}{\sigma_{s}})^{3}
\|r_h^+ u - r_h u\|_{s,2}\le C h^{\sigma-s}\Big[ h^s\|r_h^+ u - u\|_{s,\eta}+ h^s\|r_h u - u\|_{s,\eta}+ \|r_h^+ u - u\|_{0,\eta}+ \|r_h u - u\|_{0,\eta}&+ \left(h^\mu \|r_h u - u\|_{\mu,\eta}\right)^{1/2}\left(\|r_h^+ u - u\|_{0,\eta}+ \|r_h u - u\|_{0,\eta}\right)^{1/2}\Big]
\Omega(N,T,V)={V^N \over N!}\Bigl\{1+{N^2 \over 2V}\int_0^{2\pi}d\theta\int_0^\infty dr\, r\Bigl({\rm e}^{-{\phi({\bf r}_{ij}) \over kT}}-1\Bigr)+\cdots \Bigr\}\ .
\rho_r(Q_{j}, Q'_{j}, t) = \int \Pi_{\alpha=1}^{N}dq_{\alpha}dq'_{\alpha}\exp \frac{i}{\hbar}[S(Q)+S_C(Q,q_{\alpha})+S_B(q_{\alpha}) -S(Q')-S_C(Q',q'_{\alpha})- S_B(q'_{\alpha})].
\mathcal{I}_1(\bar X, t) \leq \ C\delta^N(t)\int_{\bar S^N(t)}d\bar{y}\int d\bar{w}\, \frac{f^N(\bar{y},\bar{w}, t)}{|\bar X - \bar{y}|^{3}}\,=C\delta^N(t)\int_{\bar S^N(t)}d\bar{y}\, \frac{\rho^N(\bar{y}, t)}{|\bar X - \bar{y}|^{3}}.
\widehat{I}_{j}s_{j+1}\widehat{I}_{j}=(1-q)s_{j+1}\widehat{I}_{j}s_{j+1}+s_{j+1}\widehat{I}_{j}\widehat{I}_{j+1}+\widehat{I}_{j+1}\widehat{I}_{j}s_{j+1}, \widehat{I}_{j+1}s_{j}\widehat{I}_{j+1}=(1-q)s_{j}\widehat{I}_{j+1}s_{j}+s_{j}\widehat{I}_{j+1}\widehat{I}_{j}+\widehat{I}_{j}\widehat{I}_{j+1}s_{j}.
u(t,x) \geq \int_{\mathbb{R}^d}t^{-\frac{d}{2}}e^{-c_1(t+1)}e^{-\frac{|x-y|^2}{c_2t}}u_0(y)\,\mathrm dy &=(c_2\pi)^{\frac{d}{2}}e^{-c_1(t+1)}(e^{\frac{c_2}{4}t\Delta}u_0)(x).
s*^{\alpha_p}s_{a_p}*^{\alpha_{p-1}}s_{a_{p-1}}*^{\alpha_{p-2}}\cdots*^{\alpha_1}s_{a_1}*^{-\beta_1}s_{b_1}*^{-\beta_2}s_{b_2}*^{-\beta_3}\cdots*^{-\beta_q}s_{b_q}=t.
(\frac{1}{4\pi^{2}})(q^{\mu}q^{\nu}-g^{\mu\nu}q^{2})\left\{\int^{1}_{0}d\alpha2\alpha(1-\alpha)ln\left[\frac{\Lambda^{2}}{-\alpha(1-\alpha)q^{2}+m^{2}}\right]-\frac{1}{3}\right\}.
Z_{\Psi}= \int \, [ d A_{\mu}] [ d \bar{c}] [ d c ] [ d b ] [ d A_{\mu}^{\ast}] [ d \bar{c}^{\ast}] [ d c^{\ast}] \, \delta \Bigl [ \phi^{\ast}- \frac{\delta \Psi}{\delta \phi}\Bigr ] \, \exp \, \{i S \}\, .
-\partial_x \bigg( \frac{\partial_x \zeta_c}{1 - v_c^2}\bigg) + \Big( 1 - c^2 - (5 + c^2) v_c^2 + 2 v_c^4 \Big) \frac{\zeta_c}{(1 - v_c^2)^2}+ c^2 \frac{(1 + v_c^2)^2}{(1 - v_c^2)^3}\zeta_c & - c \frac{1 + v_c^2}{1 - v_c^2}\xi_c = -\tilde{\lambda}_c \zeta_c,
T_r(f,g)(x)=\left( \sum_{k=1}^N \big| \int\limits_{a_k < \xi_1<\xi_2<b_k}\hat{f}(\xi_1) \hat{g}(\xi_2) e^{2\pi i x(\xi_1 +\xi_2)}d\xi_1 d\xi_2 \big|^r \right)^{\frac{1}{r}}.
I = \frac{\pi^{2}}{6}+ 2 \alpha^{\prime}\left[ - \zeta ( 3 ) \rho - 2 \zeta ( 3 ) \alpha_{3 4}- \zeta ( 3 ) \alpha_{1 2}- 2 \zeta ( 3 ) \alpha_{2 3}- 2 \zeta ( 3 ) \alpha_{2 4}\right] +{\cal O}({\alpha^{\prime}}^{2}) \, .
F_{k1}(c_{1},\phi)=c_{1}h_{max}(1-\frac{\phi-\pi}{\beta_{1}}+\frac{\delta}{h_{max}})-\frac{h_{max}}{2\pi}(\frac{(2\pi)^{2}m_{1}\omega^{2}}{\beta_{1}^{2}}-c1)sin(\frac{2\pi(\phi-\pi)}{\beta_{1}})
w_0'' + \frac{w_0'}{r}= w_0 ( -a^2+c^2 \sum_{i=0}^2 w_i^2 - \frac{b^2}{\sqrt{6}}w_0) + \frac{b^2}{\sqrt{6}}(w_1^2 + w_2^2), &w_1'' + \frac{w_1'}{r}-\frac{k^2}{r^2}w_1=w_1\bigg(-a^2+ c^2\sum_{i=0}^2 w_i^2 + \frac{2 b^2}{\sqrt{6}}w_0\bigg), &w_2'' + \frac{w_2'}{r}-\frac{k^2}{r^2}w_2= w_2\bigg( -a^2 + c^2\sum_{i=0}^2 w_i^2 + \frac{2 b^2}{\sqrt{6}}w_0\bigg)
p(x_2,x_1) \leq hH_p(Tx_1,Tx_0)\leq h\lambda\left[p(Tx_1,x_0)+p(Tx_0,x_1)\right] &\leq h\lambda\left[p(x_2,x_0)+p(x_1,x_1)\right] &\leq h\lambda\left[p(x_2,x_1)+p(x_1,x_0)-p(x_1,x_1)+p(x_1,x_1)\right] &= h\lambda\left[p(x_2,x_1)+p(x_1,x_0)\right],
(T^m)'x \ge C_1 ( m - \ell_k + 1)^{1 + \frac{1}{\alpha\beta}}(\ell_1 + 1)^{1 + \frac{1}{\alpha}}\times \prod_{j=2}^k \max \{\lambda , C_2 ( \ell_{j}- \ell_{j-1})^{1 + \frac{1}{\alpha}}\}.
T^\ast \tilde \Phi(-) T = \sum_{i,j=1}^n \sum_{k=1}^{m_i}\sum_{l=1}^{m_j}c_i b_k^{(i) \ast}\phi(a_k^{(i)\ast}(-) a_l^{(j)}) b_l^{(j)}c_j^\ast \colon \mathfrak A \to \mathfrak B.
\zeta(s,\mathbf a,\mathbf x):=\zeta(s,\mathbf a,\mathbf x \,{}^t \mathbf a)=\sum_{0\leq m_1,m_2,\dots,m_r \in \mathbb Z}\left((x_1+m_1)a_1+\dots+(x_r+m_r)a_r\right)^{-s}.
\begin{aligned}V^{(4)}(r_0)= -\frac{84 V'(r_0)}{r_0^3}+\frac{32 V''(r_0)}{r_0^2}+\frac{16V'''(r_0)}{r_0}-\frac{8(V''(r_0))^2}{r_0V'(r_0)}&+\frac{240(V'(r_0))^2}{r_0^3(3V'(r_0)+r_0V''(r_0))}-\frac{40V'(r_0)V'''(r_0)}{r_0(3V'(r_0)+r_0V''(r_0))}&+\frac{5r_0(V'''(r_0))^2}{3(3V'(r_0)+r_0V''(r_0))}\end{aligned}
\nabla_{\vec{x}}V(\vec{x})=-\vec{S}_{\mathcal{E}}^{\top}(\vec{G}_{\mathcal{E}}\vec{H}^{-1}\vec{G}_{\mathcal{E}}^{\top})^{-1}(\vec{W}_{\mathcal{E}}+\vec{S}_{\mathcal{E}}\vec{x}),\forall \vec{x}\in \mathcal{R}(\mathcal{E}) \cap \mathrm{int}(\mathcal{X}).
\begin{array}{c c c c}{w ( t ) = 1 / t}&{\quad}&{\mathrm{w i t h}}&{k = 0 ,}\\{w ( t ) = 1 / \sin t}&{\quad}&{\mathrm{w i t h}}&{k = + 1 ,}\\{w ( t ) = 1 / \sinh t}&{\quad}&{\mathrm{w i t h}}&{k = - 1 ,}\\ \end{array}
f({\vec{\theta}}^{1},{\vec{\theta}}^{2}) =\theta_{a}^{1}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{2}\theta_{e}^{2}\theta_{f}^{2}+C_{a}\theta_{a}^{1}\theta_{b}^{2}\theta_{c}^{2}\theta_{d}^{2}+D_{a}\theta_{a}^{2}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{1}\ .
\Pi(|\sigma^2-\sigma_0^2|\leq \widetilde{W}_3\epsilon_n)\geq2\widetilde{W}_3\epsilon_n\inf_{|u-\sigma_0^2|\leq\widetilde{W_3}\epsilon_n}\pi_{\sigma}(u)=2\widetilde{W}_3\epsilon_n\pi_{\sigma}(\sigma_0^2)[1+o(1)],
I_{11221}= - \int_{0}^{t}\int_{\Gamma_1}\sqrt{g}g^{ij}\hat n^{\sigma}g^{kl}\partial_{k}v^{\tau}\hat n_{\tau}\partial_{l}\eta_{\mu}\partial_{j}\partial_{t}^{2}v^{\mu}\Pi_{\sigma}^{\alpha}\partial_{i}\partial_{t}^{2}v_{\alpha}.
V_{nm}^{<}(B) = \frac{-1}{2\pi^2 l^3}\int_{0}^{1}d\eta\int_{0}^{1}du\, \frac{u^2 \left[ f(u; r) \right]^{n-2m}\left[ g(u; r) \right]^m}{\left[ (ru +\frac{\pi}{2})^2 +\bar{c}^2 \right]^n}.
\begin{pmatrix}a^*_0 b^*_0 \xi^* c^*_1 a^*_1 b^*_1 & c^*_2 a^*_2 \cdot & \cdot \cdot \cdot & \cdot \cdot b^*_{d-1}\mathbf{0}c^*_d a^*_d \end{pmatrix},
{\rm AoI}(S_{\mathrm{z}},Q^*,F^*)= \min_{\{Q\in\mathcal{Q}\}}\min_{\{L\in\mathcal{L}\}}{\frac{E[L^2]}{2E[L]}+E[L]}\mbox{s.t.}\sum_i 2^{-l_i}\le 1 &\sum_i \int_{a_{i-1}}^{a_i}(x-c_i)^2 f(x)dx \le D &\l_i\in R^+.
{\cal L}_{f e r m i o n}~ = ~ \bar{\psi}~ \left[ i \gamma^{\mu}~ \left( \partial_{\mu}~ - ~ \sigma^{\rho \beta}v_{\rho}^{\nu}g_{\lambda \nu}\partial_{\mu}v_{\beta}^{\lambda}~ - ~ g_{\alpha \delta}\sigma^{a b}v_{a}^{\beta}v_{b}^{\delta}\tilde{\Gamma}^{\alpha}\,_{\mu \beta}\right) \right] ~ \psi
\exp \left\{- \Gamma_{\mathrm{W}}[ \tilde{A}\, ] \right\}\propto \prod_{n = - \infty}^{\infty}\left( \; \int \mathcal{D}\chi_{n}^{\dagger}\, \mathcal{D}\chi_{n}\; \exp \left\{- \mathrm{I}_{\, 3}\left[ \, \chi_{n}^{\dagger}, \chi_{n}, \tilde{A}\, \right] \right\}\; \right) \; ,
a_{0}^{2}+\lambda a_{1}=\lambda^{2}\left(\left[\int_{\mathbb{R}}\frac{1}{\alpha-t}\frac{d\mu_{1}(t)}{\lambda}\right]^{2}-\int_{\mathbb{R}}\frac{1}{(\alpha-t)^{2}}\frac{d\mu_{1}(t)}{\lambda}\right)<0.
{}^{M\!L\!}\Gamma_{p}(x){}^{M\!L\!}\Gamma_{p}(y)=4\int_{0}^{\frac{\pi}{2}}\int_{0}^{\infty}r^{2(x+y)-1}\cos^{2x-1}\theta \sin^{2y-1}\theta \times E_{\alpha, \beta}^{\gamma}\left(-r^2\cos^2\theta -\frac{p}{r^2\cos^2\theta}\right) E_{\alpha, \beta}^{\gamma}\left(-r^2\sin^2\theta -\frac{p}{r^2\sin^2\theta}\right)drd\theta
\sum_{k=1}^N\det\left((\lambda_i(t) I_N-H_{\alpha}(t)(\lambda_j(t) I_N-H_{\alpha}(t))_{k|k}\right) =\sum_{k=1}^N\prod_{p(\neq k)}(\lambda_i(t)-\lambda_p(t))(\lambda_j(t)-\lambda_p(t)) &=\begin{cases}f_{\lambda}(\lambda_i(t))^2 i=j 0 i \neq j\end{cases}.
\bigl\|({\rm I}\,+zD_{{}^\Vert})^{-1}u\bigr\|_p\le M_{p,\mu}\|u\|_p,\forall\,z\in{\mathbb{C}}\setminus S_{\mu},\ \forall\, u\in{\rm{\sf R}}^p(d,\Omega)\cap L^q(\Omega,\Lambda) ={\rm{\sf R}}^q(d,\Omega) \mbox{and}\quad \forall\, u\in{\rm{\sf R}}^p(\underline{\delta},\Omega)\cap L^q(\Omega,\Lambda) ={\rm{\sf R}}^q(\underline{\delta},\Omega).
E(\tau) :=h_\rho(\tau,q(\tau,t;y(0,t;x,\xi),\xi),p(\tau,t;y(0,t;x,\xi),\xi)) &=h_\rho(\tau,q(\tau,0;x,\eta(t,0;x,\xi)),p(\tau,0;x,\eta(t,0;x,\xi)))
\mathbf{v}_j=\mathbf{v}_{j\mathrm{t}}+\mathbf{v}_{j\mathrm{v}}+\mathbf{v}_{j\mathrm{h}}\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, (j=1,\,2).
\gamma(a;b_1,\ldots,b_l) :=((\underbrace{a^1,\ldots,a^1}_{m_1{\ \rm times}})+b_1,(\underbrace{a^2,\ldots,a^2}_{m_2{\ \rm times}})+b_2,\ldots,(\underbrace{a_l,\ldots,a_l}_{m_l{\ \rm times}})+b_l).
\limsup_{n\to\infty}\biggl|{\lambda}_{n}^{-(d-1)}\sum_{\mathbf{i}\in[\partial R_n]^{{\eta}_{n}{\lambda}_{n}}\cap\mathbb{Z}^d}\bigl|\theta_n(\mathbf{i})\bigr|^2- \sigma_{\mathrm{EE}}^2 \biggr| =0.
\lambda_{P/A}(\{e\}) = r_{P/A}(\{e\})+r_{P/A}((E-A)-\{e\})-r_{P/A}(E-A) = r_P(A\cup\{e\})-r_P(A)+r_P(E-\{e\})-r_P(A)-(r_P(E)-r_P(A)) = r_P(A\cup\{e\})-r_P(A) = r_{P/A}(\{e\}),
\sum_{n=0}^{\infty}p(N,M,n)q^{n}={(\begin{matrix}N+M\\M\end{matrix})}_{q}=\frac{(1-q^{N+M})(1-q^{N+M-1})\cdot\cdot\cdot(1-q^{N+1})}{(1-q)(1-q^{2})\cdot\cdot\cdot(1-q^{M})}
G^2(t) =g_i^2+o(1), G_tG =\frac12g_i\bar{g}_{\bar{t}}(R_i)-\frac12g_i\bar{g}_{\bar{t}}(R_i)\frac{t}{\varepsilon_i}+o(1), G_t^2+G_{tt}G =-\frac12g_i\bar{g}_{\bar{t}}(R_i)\cdot\frac{1}{\varepsilon_i}+O(1),
w(\sigma) = \frac{\|I^\sigma(I_\sigma(S_\eta^{j+1}(A)))\|}{(j+2) \cdot \|I_\sigma(S_\eta^{j+1}(A))\|_\sigma}\leq \frac{\eta^{-2^{k-j-1}}}{j+2}\cdot \|I^\sigma(I_\sigma(S_\eta^{j+1}(A)))\|
x = x^{M}+ \frac{\mu+1}{2}\tilde{g}= \Psi_{m}\left( \begin{pmatrix}1 0 0 -\mu\end{pmatrix}+ \frac{\mu+1}{1-\mu|\psi|^{2}}\begin{pmatrix}|\psi|^{2}\psi \bar{\psi}1\end{pmatrix}\right) \Psi^{*}_{m}&= \frac{1}{1-\mu|\psi|^{2}}\Psi_{m}\begin{pmatrix}1+|\psi|^{2}(\mu+1)\psi (\mu+1)\bar{\psi}1+\mu^{2}|\psi|^{2}\end{pmatrix}\Psi_{m}^{*}.
\pi_n(\theta_j^{(n)})H\left(\frac{\theta_{j+1}^{(n)}}{n\pi_n(\theta_j^{(n)})}\right) \geq\pi_n(\theta_j^{(n)})H(x_K)\geq\mathrm{e}^{-3}(1-r^{-1})^{r-1}\frac{(K^{r/\lceil K\rceil})^j}{nr}H(x_K)a_c^{(n)},
X_{t}=x_{0}+\int_{0}^{t}\alpha+\beta X_{s}+\sigma^{2}\nabla_{x}\log h(X_{s},s)\mathrm{d}s+\sigma B_{t}, X_{t}^{\prime}=x_{0}^{\prime}+\int_{0}^{t}\alpha+\beta X_{s}^{\prime}+\sigma^{2}\nabla_{x}\log h(X_{s}^{\prime},s)\mathrm{d}s+\sigma B_{t}.
\partial_{t}^{2}X_{4}- \left({\frac{l \, \rho}{2 \pi \alpha^{\prime}}}\right)^{2}\left({\frac{1}{\rho^{2}}}\partial_{\theta^{\prime}}^{2}+ \partial_{z}^{2}\right) X_{4}+ O ( l^{3}) = 0 \, .
\mathbf{E}\|U\| =\mathbf{E}\max_{k}\|U_k\| \le\max_k\mathbf{E}\|U_k\| +\mathbf{E}\max_k|\,\|U_k\|-\mathbf{E}\|U_k\|\,| &\le\max_k\mathbf{E}\|U_k\| +\sum_k\mathbf{E}|\,\|U_k\|-\mathbf{E}\|U_k\|\,| &\le\max_k\mathbf{E}\|U_k\| +\sum_k \mathrm{Var}(\|U_k\|)^{1/2}\lesssim a+b.
\int |\tau|^{-\rho}\|G_\tau\|^2_2 d\tau = \int |g^2(y)| \Big( \int |\tau|^{-\rho}\cdot |g|^2(y+\tau)d\tau\Big)dy &= \int |g|^2(y) I_\rho (|g|^2)(y)dy .
(2a_{1}^2+2a_{2}^2-1 )x_1+4a_1a_2x_2+2(a_1+a_2)\bar{a}^T\bar{x}=\frac{s_1}{\lambda}, &( 2a_{1}^2+2a_{2}^2-1 )x_2+4a_1a_2x_1+2(a_1+a_2)\bar{a}^T\bar{x}=\frac{s_2}{\lambda}, & 2(a_1+a_2)x_1\bar{a}+2(a_1+a_2)x_2\bar{a}+\bar{x}+4\bar{a}^T\bar{x}\bar{a}=\frac{\bar{s}}{\lambda}.
\phi_{U V}= 1 -{\frac{1}{4}}y^{2}\log{\frac{Y}{y}}+ y^{4}[{\frac{1}{4 8}}\log^{3}{\frac{Y}{y}}+{\frac{1}{6 4}}\log^{2}{\frac{Y}{y}}+{\frac{1}{1 2 8}}\log{\frac{Y}{y}}+ C ] + . . .
\hat{S}= \int d^{D}x \sqrt{- \hat{G}}e^{- \hat{\phi}}\left( R_{\hat{G}}+ \left( \hat{\partial}\hat{\phi}\right)^{2}- \frac{1}{1 2}\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}\hat{H}^{\hat{\mu}\hat{\nu}\hat{\rho}}\right) ,
\alpha = - q_1, \wp(\rho) = q_1^2 - 2 q_2, \wp'(\rho) = 2 (q_1^3 - 3 q_1 q_2 + 3 q_3), g_2 = 20 (q_2^2 - 2 q_1 q_3 + 2 q_4).
c(g_j^{-1}, X) :=\{h_{t_j}^{-1}|~ t_j\}, X_{t_j}:=\{x\in X: c(g_j^{-1}, x)=h_{t_j}^{-1}\}, c((g_ig_{i+1})^{-1}, X) :=\{h_{s_i}^{-1}|~ s_i\}, X_{s_i}:=\{x\in X: c((g_ig_{i+1})^{-1}, x)=h_{s_i}^{-1}\}.
\int_0^{\tau}dt\cdot \omega_\mu (t,z,\tau)+\int_0^{\tau}dt\cdot \omega_{2-\mu}(t,z,\tau)=\tilde q^{-1}\left( \int_0^{-z}dt\cdot \omega_\mu (t,z,\tau)-\int_0^{-z}dt\cdot \omega_{\mu -2}(t,z,\tau)\right)
V(t_{i})=\widehat{e_{0}}\overbrace{e_{1}e_{0}e_{1}\cdots}^{t_{i}-1}+\widehat{e_{1}}\overbrace{e_{0}e_{1}e_{0}\cdots}^{t_{i}-1}-\overbrace{\cdots e_{1}e_{0}e_{1}}^{t_{i}-1}\widehat{e_{0}}-\overbrace{\cdots e_{0}e_{1}e_{0}}^{t_{i}-1}\widehat{e_{1}}.
K=C^{2}+\sum_{i=1}^{L+1}\frac{1}{2}m_{i}\{A^{i},B\}+\frac{n}{2}\{A,B^{2}\}+\sum_{i=1}^{M+1}k_{i}A^{i}+\ell_{1}B + \ell_2 B^2.
\hat{f}(\xi)=\hat{u_*}+i\int_1^t e^{-i\frac 12\xi^2s}e^{-s}\widehat{u(s)v_*(x)}(\xi)ds+i\int_1^t\int_1^se^{-i\frac 12\xi^2s}e^{-(s-s')}\widehat{|u(s')|^2u(s)}(\xi)ds'ds,
\frac d{dt}\iint_{\Sigma_-}(\partial_t^j\partial_y^n v_k)^2\,dxdy +\int_0^L (\partial_t^j\partial_y^n v_{kx})^2\big|_{x=0}\,dy = 2\iint_{\Sigma_-}\partial_t^j\partial_y^nf \partial_t^j\partial_y^nv_k\,dxdy,
\phi_{S S}/ 2 = \tan^{- 1}\left( u \frac{\sinh ( 8 m \gamma x )}{\cosh ( 8 m \gamma u t )}\right) , \phi_{S \overline{{S}}}/ 2 = \tan^{- 1}\left( \frac{1}{u}\frac{\sinh ( 8 m \gamma u t )}{\cosh ( 8 m \gamma x )}\right) ,
\alpha_0(A)(x)= A(\hat\Phi_\cdot^\infty(\hat\Phi_\cdot^\infty(\Phi_{\cdot 0}^\infty(x),\Phi_{\cdot 0}^\infty(\Phi_{\cdot 1}^\infty(x))), \Phi_{\cdot 1}^\infty(\Phi_{\cdot 1}^\infty(x))))\;.
B'({\bf m})=\sum_{\substack{1\leq i_1 < j_1\leq n 1\leq i_2 < j_2\leq n 1\leq i_1 \leq i_2\leq n 1\leq j_1 \leq j_2\leq n j_{1}>i_2}}m_{i_{1},j_{1}}m_{i_{2},j_{2}},
\frac{\nabla{G}_r(z)+\nabla{G}^T_r(z)}{2}= \begin{bmatrix}k\nabla^2 f(x) - A^TA \frac{1}{2}(A\nabla^2f(x))^T \frac{1}{2}A\nabla^2f(x) AA^T \end{bmatrix}
\tilde{S}_0^{(n)}:=\{0\}\times[-1,1], \tilde{S}_m^{(n)}(\delta) :=\overline{S_m^{(n)}(\delta)}\quad\; m\in\{1,2,\ldots,n-2\}, \tilde{S}_{n-1}^{(n)}:=\{L\}\times[-1,1] ,
dY_{t}^{y,v}=\int_{E}v_{t}\left( \theta\right) \widetilde{q}\left(dt,d\theta\right) -A^{\ast}\left( \Gamma_{t}^{\gamma_{0}}\right)Y_{t}^{y,v}dt-\int_{E}C^{\ast}\left( \Gamma_{t}^{\gamma_{0}},\theta\right)v_{t}\left( \theta\right) \widehat{q}\left( dt,d\theta\right) ,\text{}Y_{0}^{y,v}=y\in\mathbb{R}^{N}.
\Biggl.\Biggl.+\frac{(i+1)^2\left(\zeta_{i+1}^{(i_2)}\zeta_{i}^{(i_1)}-\zeta_{i}^{(i_2)}\zeta_{i+1}^{(i_1)}\right)}{\sqrt{(2i+1)(2i+3)}(2i-1)(2i+5)}\Biggr)\Biggr] - \frac{1}{24}{\bf 1}_{\{i_1=i_2\}}{\Delta^3},
M_{\gamma\to2\gamma}= -\frac{4}{15\sqrt{\pi}}\biggl( \frac{e^4}{3 m^4}\sin \theta\left[\frac{7}{4}(\phi_1^2 + \phi_2^2) - \phi_{12}^2\right]-\frac{Qg (Q^2 - g^2)}{8 M^4}\cos \theta\,\phi_{12}^2 \biggr) \omega \omega_1 \omega_2 H \cos 2 \beta
\mathrm{G}_n(\mathbb{C})=\langle\sigma_n=\Big(\prod_{\stackrel{i=0}{i\not=0}}^{n}z_i:\prod_{\stackrel{i=0}{i\not=1}}^{n}z_i:\ldots:\prod_{\stackrel{i=0}{i\not=n}}^{n}z_i\Big),\,\mathrm{Aut}(\mathbb{P}^n_\mathbb{C})\rangle
\frac{1}{4}|1-\langle C_\varphi^*f,f\rangle|^2-\frac{1}{4}|1+\langle C_\varphi^*f,f\rangle|^2= (||f_2||^2-||f_1||^2)||f||^2 = ||f_2||^2-||f_1||^2.
\omega_{1}=\dfrac{-\mu c+\sqrt{D}}{2}<0, \omega_{2}=\dfrac{-\mu c-\sqrt{D}}{2}<0, \omega=-\dfrac{\mu c}{2}<0, D=\mu^{2}c^{2}-4k. &\bar{Z}(\xi) = B_{5}\cdot \sin [\frac{\sqrt{|D|}}{2}\xi]+ B_{6}\cdot \cos[ \frac{\sqrt{|D|}}{2}\xi]
\left(\widetilde{\boldsymbol\tau},B^{\frac{1}{p}}D^\star{\boldsymbol\varphi}\right)+{\cal R}e\, a_\star\left({\boldsymbol u},{\boldsymbol u},B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right) \mbox{for all}\{\boldsymbol\varphi}\in \boldsymbol V_B^{p}.
{\tilde{E}}_{i}= \frac{\mathrm{j}}{2}\left( \begin{array}{c c}{\tau_{i}}&{0}\\{0}&{\tau_{i}}\\ \end{array}\right) ,{\tilde{F}}_{i}= \frac{1}{2}\left( \begin{array}{c c}{0}&{\tau_{i}}\\{- \tau_{i}}&{0}\\ \end{array}\right) , i = 1 , 2 , 3 .
[X \xleftarrow{f}M \xrightarrow{g}Y] \circ \Bigl ([Y \xleftarrow{h_1}N_1 \xrightarrow{k_1}Z] + [Y \xleftarrow{h_2}N_2 \xrightarrow{k_2}Z] \Bigr ) & = [X \xleftarrow{f}M \xrightarrow{g}Y] \circ [Y \xleftarrow{h_1}N_1 \xrightarrow{k_1}Z] + [X \xleftarrow{f}M \xrightarrow{g}Y] \circ [Y \xleftarrow{h_2}N_2 \xrightarrow{k_2}Z].