harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathbb{R}^{n-1}}e^{-C|\xi|}\frac{\xi_{\alpha}^2}{|\xi|^2}\,d\xi = \frac{\Gamma(n-1)\operatorname{vol}(\mathbb{S}^{n-2})}{(n-1)C^{n-1}},C>0,\n\geqslant 2,
\bigcup_{j=1}^{\max(c_{k_i},1) q_{k_i}}B \left( j\theta ,q_{k_{i+1}}^{-\tau}\right) = \left( \bigcup_{j=1}^{\max(c_{k_i},1) q_{k_i}}B \left( j\theta ,q_{k_{i}+1}^{-\tau}\right) \right)\cap \left( \bigcup_{j=1}^{q_{k_{i}+1}}B \left( j\theta ,q_{k_{i+1}}^{-\tau}\right) \right).
\limsup_{\rho\to 0^{+}}\limsup_{k\to +\infty}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\big(|u_{k}(x)|^{p(x,y)}+ |v_{k}(x)|^{p(x,y)}\big)\frac{|\Psi_{\rho}(x)-\Psi_{\rho}(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy\,dx=0.
|\tilde{I}_2| \le c \frac{\Delta_n}{T_n}\sum_{j = j_{\delta}+1}^{j_D}(\frac{1}{t_j}+ 1) & \le c \frac{\Delta_n}{T_n}(\sum_{t_j \le 1, \, j = j_{\delta}+1}^{j_D}\frac{1}{t_j}+ \sum_{t_j > 1, \, j = j_{\delta}+1}^{j_D}1) & \le \frac{c}{T_n}(|\log D| + |\log \delta| + D).
\sum_{k\in\mathbb{T}}\partial_k f(x^{(\mathbb{T})})x_k =\sum_{k\in\mathbb{U}}\partial_k f(x^{(\mathbb{T})})x_k +\sum_{k\in\mathbb{T},l\in\mathbb{U}}\int_0^1 (1-t) \partial_{kl}f(x^{(\mathbb{U})}+t(x^{(\mathbb{T})}-x^{(\mathbb{U})}))x_kx_l\, d t.
\Gamma_{[\ell]}\Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n})=\mathbf{CK}_{x_{n}}^{\mu_{n}}\Big[\cdots \Gamma_{[\ell]}\mathbf{CK}_{x_{\ell}}^{\mu_{\ell}}\big[ \cdots \mathbf{CK}_{x_2}^{\mu_2}[x_1^{j_1}]\big]\Big]v_{s}.
\langle \, T_{--}( \sigma ) \, \rangle = - \, \frac{1}{1 2 \pi}\, \frac{( 1 + v )^{2}}{( 1 - v^{2})^{3 / 2}}\: \frac{d}{d t}\frac{v^{\prime}}{( 1 - v^{2})^{3 / 2}}\, ,
z' = \frac{2 \theta_n|g_n (s_3)| - 2 \theta_n |g_n (s_2)|}{s_3 - s_2}= \frac{2 \theta_n g_n (s_3) - 2 \theta_n g_n (s_2)}{s_3 - s_2}> 0 > -2 \theta_n g_n' (t).
\begin{array}{c}{{t_{i}e_{j}t_{i}^{- 1}= q^{a_{i j}}e_{j}}}\\{{t_{i}f_{j}t_{i}^{- 1}= q^{- a_{i j}}f_{j}}}\\{{{}[ e_{i}, f_{j}] = \delta_{i j}\frac{t_{i}- t_{i}^{- 1}}{q - q^{- 1}}}}\\ \end{array}
\begin{array}{rcl}F_{\sigma}^{(n)}(\zeta'', \zeta_{2n-1}, sx^{-1}\zeta_{2n-1})^{k\cdots k' k'\pm 1 k_{2n}}= \delta_{k',k_{2n}}\frac{d_s}{\{k_{2n}\}}F_{\sigma}^{(n-1)}(\zeta'')^{k\cdots k'}\;\;\;\; (s=\pm ), \end{array}
\phi^{4}= \left( \begin{array}{c c c c}{{k^{1}t \otimes 1_{N_{1}\times N_{1}}}}&{{0}}&{{\dots}}&{{0}}\\{{0}}&{{k^{2}t \otimes 1_{N_{2}\times N_{2}}}}&{{\dots}}&{{0}}\\{{\dots}}&{{\dots}}&{{\dots}}&{{\dots}}\\{{0}}&{{\dots}}&{{\dots}}&{{k^{K}t \otimes 1_{N_{K}\times N_{K}}}}\\ \end{array}\right)
\partial_t M^kf+v\cdot\partial_x M^kf + \mathcal LM^k f =\Gamma(f,\M^kf)+M^k\Gamma(f, f)-\Gamma(f,\M^kf)- \big[M^k,\ \mathcal L\big] f -k\partial_{v_1}^2M^{k-1}f.
+ c_{1}\lambda^{2}\, ( \overline{{{\psi}}}\Gamma^{\mu}\partial_{\mu}\psi )^{2}+ c_{2}\lambda^{2}\, \overline{{{\psi}}}\Gamma^{\mu}\partial^{\nu}\psi \overline{{{\psi}}}\Gamma_{\mu}\partial_{\nu}\psi + c_{3}\lambda^{2}\overline{{{\psi}}}\Gamma^{\mu}\partial^{\nu}\psi \overline{{{\psi}}}\Gamma_{\nu}\partial_{\mu}\psi
n \sum_{z \in \mathbb{Z}^3}\Delta \omega_{\ell,\lambda}(P_z(x)-y) + \frac{1}{2}V_{\ell}(x-y) \widetilde K(x,y) &=\frac{n}{2}\sum_{z \in \mathbb{Z}^3}\Big( \epsilon_{\ell, \lambda}- \big( V_\ell (1-\omega_{\ell}) \big) (P_z(x)-y) -V_\ell(x-y) \omega_{\ell,\lambda}(P_z(x)-y) \Big),
{\cal Y}( T , U ) \stackrel{{\Gamma^{o}( 3 )}_{T}}{\rightarrow}( i \gamma T + \delta ) \;{\cal Y}( T , U ) - i \gamma \{( \partial_{\frac{U}{3}}\eta^{- 2}( \frac{U}{3}) \; \eta^{- 2}( T ) + ( \partial_{U}\eta^{- 2}( U ) ) \; \eta^{- 2}( \frac{T}{3}) ) \}.
z^*(E(f)(x)-E(f)(y)) = z^*(E(f)(x)) - z^*(E(f)(y)) & = \upsilon_x( \langle z^*,f\rangle) - \upsilon_y(\langle z^*,f\rangle) & \leq \|z^*(f)\|_{Lip}W_1(\upsilon_x, \upsilon_y) & \leq \|f\|_{Lip}K d(x, y),
\frac{\partial F_k^\alpha}{\partial q_\beta^l}(q) A_\alpha^k A_\beta^l = |q|^{n-4}(|q|^2 \delta^{\alpha \beta}\delta_{kl}A_\alpha^k A_\beta^l + (n-2) q_\alpha^k q_\beta^l A_\alpha^k A_\beta^l) & \geq |q|^{n-2}|A|^2 & \geq 2 \nu |A|^2,
q\frac{d E_2}{dq}= \frac{E_2^2 - E_4}{12}\ \q\frac{d E_4}{dq}= \frac{E_2E_4 - E_6}{3}\ \q\frac{d E_6}{dq}= \frac{E_2E_6 - E_4^2}{2}
P_{\rm{pro}}^{\cal O}(\gamma_{th}) \buildrel \Delta \over = P_{{T_1}}^{\cal O}\left({{\gamma_{th}}}\right) + P_{{T_2}}^{\cal O}\left({{\gamma_{th}}}\right) - P_{{T_1}}^{\cal O}\left({{\gamma_{th}}}\right)P_{{T_2}}^{\cal O}\left({{\gamma_{th}}}\right)
AP=A[\begin{matrix}|&&|\\v_{1}&\cdot\cdot\cdot&v_{n}\\ |&&|\end{matrix}]=[\begin{matrix}|&&|\\Av_{1}&\cdot\cdot\cdot&Av_{n}\\ |&&|\end{matrix}]=[\begin{matrix}|&&|\\ \lambda_{1}Bv_{1}&\cdot\cdot\cdot&\lambda_{n}Bv_{n}\\ |&&|\end{matrix}]=[\begin{matrix}|&&|\\Bv_{1}&\cdot\cdot\cdot&Bv_{n}\\ |&&|\end{matrix}]D=BPD
q(T_f(r)+T_g(r))\ge 2\sum_{v=i}^q\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r) \right ) &+ (q-2n-2)\sum_{v=1}^{q}N^{[1]}_{(f,a_v)}(r)+o(T_f(r)) \ge (2+\frac{q-2n-2}{2n})\sum_{v=i}^q\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r) \right )+o(T_f(r)) \ge (2+\frac{q-2n+2}{2n})\dfrac{2q}{3(n+1)}(T_f(r)+T_g(r))+o(T_f(r)).
\begin{aligned}\left .\dfrac{\partial h(s)}{\partial s}\right |_{s=x+\delta \sin (2\pi t)}= \, \left.\dfrac{\partial h(s)}{\partial s}\right|_x &+\left.\dfrac{\partial^2 h(s)}{\partial s^2}\right|_{x}\delta \sin (2\pi t)+ O(\delta^2) \end{aligned}
L_3=\dot{x}p_x+\dot{y}p_y+\dot{z}p_z-\frac{1}{2}p^2_x-\frac{1}{2}p^2_y-\frac{1}{2}p^2_z-\frac{1}{2}\omega^2\left(x^2+y^2\right)-\xi\left[p_z+g(xp_y-yp_x)\right]\ \ \ .
+{\frac{1}{2 4}}\mathrm{t r}R^{2}\Big ( \sum_{i}( \mathrm{t r}F_{9 i})^{2}+ \sum_{i}( \mathrm{t r}F_{5 i})^{2}+{\frac{1}{2}}\sum_{i j}w_{i j}\mathrm{t r}F_{5 i}\mathrm{t r}F_{9 j}\Big )
D=diag(-\frac{\lambda_2+\lambda_3}{2 \lambda_1 \lambda_2 \lambda_3}-\gamma ,-\frac{\lambda_1+\lambda_3}{2 \lambda_1 \lambda_2 \lambda_3}-\gamma,-\frac{\lambda_1+\lambda_2}{2 \lambda_1 \lambda_2 \lambda_3}-\gamma,\frac{1}{2 \lambda_2 \lambda_3}-\gamma ,\frac{1}{2 \lambda_1 \lambda_3}-\gamma ,\frac{1}{2 \lambda_1 \lambda_2}-\gamma).
\sum_{\alpha,\beta=1}^{3}a_{\alpha\beta}x_{\alpha}x_{\beta}=0\rightarrow\begin{matrix}x^{2}+y^{2}+4k^{2}t^{2}=0&(elliptic)\\x^{2}+y^{2}-4k^{2}t^{2}=0&(hyperbolic)\end{matrix}
G_{\parallel}= \frac{V_{3}N u^{4}}{2 \pi^{2}\lambda^{2}}\left( \sqrt{1 - \frac{2 m}{u^{4}}- \frac{\lambda \Omega^{2}( u^{2}+ l^{2})}{u^{4}}+ \frac{2 \sqrt{2 m \lambda}l \Omega}{u^{4}}}- 1 - \frac{\sqrt{2 m \lambda}l \Omega}{u^{4}}\right) .
\nu \left(t,\Omega \right) = \inf_{\substack{u \in L^{1/2,2}(\Omega) u \neq 0}}\frac{\int_\Omega \left| (-\Delta+m^2)^{1/4}u\right|^2 + \int_\Omega V |u|^2}{\left( \int_\Omega |u|^t \right)^{2/t}}{}= +\infty \quad.
\Im F ( a , b ; c ; z ) = - \frac{\pi \Gamma ( c ) ( z - 1 )^{c - a - b}\theta ( z - 1 )}{\Gamma ( a ) \Gamma ( b ) \Gamma ( 1 - a - b + c )}F ( c - a , c - b ; c - a - b + 1 ; 1 - z ) ,
R^{\sf CF}_{\sf sym}(\mathbf{H}) = \min_{\ell = 1, \dots, L}\log\left( \frac{{\sf SNR}}{\sigma^2_{{\sf eff}, \ell}}\right) - \log\left( 2 \pi e G(\mathcal{L}') \right) - \log\left( \frac{\mu(\mathcal{L}, P_e)}{2 \pi e}\right)
\mathbb{E}_{P,Q}[\tilde{\phi}] = \mathbb{P}\left(|\tilde{N}_0/n - P_0(A)|>|\tilde{N}_0/n - P_1(A)|\right) = \mathbb{P}\left( 2 \{\widetilde{N}_0/n - P(A) \}< P_0(A) + P_1(A) - 2P(A) \right) \leq \mathbb{P}\left( 2 \{\widetilde{N}_0/n - P(A) \}< -\mathrm{TV}(P_0,P_1) + 2 \varepsilon \right),
o(1) = \left(\| v_{n}\|^{2}+\int_{\mathbb{R}^{N}}V(y) \gamma(v_n)^2 dy \right) &\geq\left( 1 - \frac{V_0}{m}\right) \int_{\mathbb{R}_{+}^{N+1}}|\nabla v_n|^2 +{m(m-V_0)}\int_{\mathbb{R}_{+}^{N+1}}|v_n|^2,
\theta_{j+kd^h+1}\triangleq \begin{cases}\exp\left(\frac{\pi ik \left(2j+kd^h\right)}{d^{h+1}}\right) d \exp\left(\frac{\pi ik \left(2j+\left(k+1\right)d^h\right)}{d^{h+1}}\right) d\end{cases},
W(G) \ge W(G_{n_0})+\sum_{i=n_0+1}^{n}\left( i+(r-1)^2 \right) &> \binom{n_0}{2}+ \sum_{i'=n_0}^{n-1}i' +(n-n_0)+(n-n_0)(r-1)^2 &\ge \binom{n}{2}+an_0+ (n-n_0)a &= \binom{n}{2}+an
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{\theta}{4}\epsilon^{\mu \nu \alpha}F_{\mu \nu}A_{\alpha}+ \frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi - \frac{i}{2}( \partial_{\mu}\bar{\psi}) \gamma^{\mu}\psi - m \bar{\psi}\psi + e \bar{\psi}\gamma^{\mu}A_{\mu}\psi ,
F_Y(\theta) =\frac{1}{9^k}\Bigl|\sum_{n_0,\dots,n_{k-1}\in\{0,\dots,9\}\backslash \{a_0\}}e\Bigl(\sum_{i=0}^{k-1}n_i10^i\theta\Bigr)\Bigr| &=\prod_{i=0}^{k-1}\frac{1}{9}\Bigl|\sum_{n_i\in\{0,\dots,9\}\backslash \{a_0\}}e(n_i 10^i \theta)\Bigr| &=\prod_{i=1}^k\frac{1}{9}\Bigl|\frac{e(10^{i}\theta)-1}{e(10^{i-1}\theta)-1}-e(a_010^{i-1}\theta)\Bigr|.
\upsilon_R(r) = r^{k/2}\upsilon_R(1) + r^{k/2}\frac{R^k (\sqrt\pi-\upsilon_R(1))}{R^k-1}\bigg(1-\frac1{r^k}\bigg) =r^{k/2}\frac{R^k\sqrt\pi-\upsilon_R(1)}{R^k-1}-r^{-k/2}\frac{R^k(\sqrt\pi-\upsilon_R(1))}{R^k-1}, r\in(1,R].
\mathbf{v}= \left( \frac{1}{16 \, d_2}, 0, \frac{1}{2}, \frac{(2\lambda_1\lambda_2-\lambda_1-\lambda_2+2)}{8 \, d_2}, \frac{(\lambda_1-\lambda_2)^2}{16 \, d_2}, 1, 1, 1\right).
U ( \phi ) = \frac{1}{2}\sum_{a}\frac{\partial W^{1}}{\partial \phi^{a}}\frac{\partial W^{1}}{\partial \phi^{a}}= \frac{1}{2}\left( 1 - 2 ( \phi_{1}^{2}+ \phi_{2}^{2})^{\frac{n - 1}{2}}\cos \left[ ( n - 1 ) \operatorname{a r c t a n}\frac{\phi^{2}}{\phi^{1}}\right] + ( \phi_{1}^{2}+ \phi_{2}^{2})^{n - 1}\right)
U_{n}^{-1}\Sigma = \left[\begin{array}{c|c}U_{n-1}^{-1}\mathbf{0}\hline \mathbf{0'}1 \end{array}\right], \Sigma^{-1}U_{n}= \left[\begin{array}{c|c}U_{n-1}\mathbf{0}\hline \mathbf{0'}1 \end{array}\right].
\begin{array}{ccc}\lambda:E^0\times C^0 \to{X'}^0 ((w',r),(w,u)) \mapsto (z,p,q) begin{align*}2mm] =( \dfrac{(1-x_1)^{\frac{2}{3}}(1-x_2)^{\frac{2}{3}}(1-r)(1-tu)ww'}{u(1-u)(1-p)(1-q)(1-x_1p-x_2q)},p,q).\end{array}
\eta(x,{\rm d}\theta) :=\left(\frac{c(x)}{\gamma(x)}1_{A}(x)+1_{E\setminus A}(x)\right)\delta_{0}({\rm d}\theta) +\frac{1}{\gamma (x)}1_{A}(x)1_{(0,+\infty)}(\theta)\theta\Pi^{NL}(x,{\rm d}\theta).
\limsup_{\varepsilon_1 \rightarrow 0^{+}}\limsup_{n \rightarrow \infty}\mathbb{E}_{\mu_n}\Big[ \int_0^T n^{\gamma-1}\sum_{x,y:|x-y| \geq \varepsilon n}[ \overleftarrow{\eta}_t^{\varepsilon_1 n}(y) \overrightarrow{\eta}_t^{\varepsilon_1 n}(y+1) - \eta_t^n(y) \eta_t^n(y+1) ] F(t, \tfrac{x}{n}, \tfrac{y}{n}) (c_{\gamma})^{-1}p (x-y) dt \Big] =0.
\min\Big(\alpha, \,\frac{d}{\alpha}T^{-\frac{\beta-1}{\beta}}\Big)=\min\Big(\max(\alpha, T^{-1/2+1/\beta}), \frac{d}{\sqrt{T}}, \,\frac{d}{\alpha}T^{-\frac{\beta-1}{\beta}}\Big)\enspace.
\widetilde \Delta ( x^{T}) \ = \ \int \frac{d^{3}k^{T}}{( 2 \pi )^{3}}\ \frac{e^{{\displaystyle i k^{T}. x^{T}}}}{( k^{T 2}+ i \epsilon )^{2}}\ = \ - \frac{1}{8 \pi}r \ .
\{s_{Q}, s_{Q}\}= - 2 Z , \qquad \{s_{Q}, \bar{Q}_{\mu}^{( 0 )}\}= - i P_{\mu}, \qquad \{\bar{Q}_{\mu}^{( 0 )}, \bar{Q}_{\nu}^{( 0 )}\}= - 2 \delta_{\mu \nu}\bar{Z},
F_p=-\frac{1}{4}\pi^2 \tilde{V}_3 N_3T^4 \left[ 1 - 4 \left( \frac{M}{\pi T}\right) + 10 \left(\frac{M}{\pi T}\right)^2 - 20 \left( \frac{M}{\pi T}\right)^3 + \cdots \right].
h_{1,m}\partial_{t}v^{(1,m)}_0 =h_{1,m}\partial^2_{x_2 x_2}v^{(1,m)}_0 -h_{1,m}k_1\big(v^{(1,m)}_0\big) - 2 \delta_{\alpha_1,1}\kappa_1\big(v^{(1,m)}_0\big) + 2 \delta_{\beta_1,1}g^{(1)}_0
\gamma_n(\varepsilon)=\frac{\Gamma(1+\varepsilon{}n)}{\Gamma(1+n)},\ \n=0,\,1,\,2,\,\,\ldots\,, \gamma_0(\varepsilon)=1,\ \ \gamma_n(\varepsilon)=e^{-\varepsilon{}n\log n}, \ \n=1,\,2,\,\,\ldots\,, \gamma_n(\varepsilon)=\frac{1}{\Gamma(1+\varepsilon n)}, \ \n=0,\,1,\,2,\,\,\ldots\,.
dL(G)^2 =\int_{G}|x|^2{\rm d}x= \int_{G\cap r_d B^2}|x|^2{\rm d}x+\int_{G\setminus r_d B^2}|x|^2{\rm d}x\ge\int_{G\cap r_d B^2}|x|^2{\rm d}x+ r_d^2|G\setminus r_d B^2| &\ge \int_{G\cap r_d B^2}|x|^2{\rm d}x+\int_{r_d B^2\setminus G}|x|^2{\rm d}x=dL(r_d B^2)^2.
\sqrt{- g}g^{\mu \nu}={\frac{1}{1 - \lambda_{+ +}\lambda_{--}}}\left( \begin{array}{c c}{{{2 \lambda{--}}}}&{{{1 + \lambda_{+ +}\lambda_{--}}}}\\{{{1 + \lambda_{+ +}\lambda_{--}}}}&{{{2 \lambda_{+ +}}}}\\ \end{array}\right)
\Delta E_{\mathrm{F}}^{( 2 )}= \frac{1}{( 4 \pi )^{2}}\int_{0}^{\infty}\frac{d q \, q^{2}}{( 2 \pi )^{2}}\tilde{V}( q ) \tilde{V}( - q ) \left[ - 2 + \sqrt{1 + \frac{4 m^{2}}{q^{2}}}\, \ln \frac{\sqrt{1 + \frac{4 m^{2}}{q^{2}}}+ 1}{\sqrt{1 + \frac{4 m^{2}}{q^{2}}}- 1}\right]
\pi_{( \underline{{\alpha}})}^{2}= \left( \pi_{q}^{2 -}\atop \pi_{\dot{q}}^{2 +}\right) = \left( \nabla \theta_{q}^{2 -}+ \frac 1 2 \gamma_{q \dot{r}}^{i}\theta_{\dot{r}}^{2 +}\Omega^{- 2 i}\atop \nabla \theta_{\dot{q}}^{2 +}+ \frac 1 2 \tilde{\gamma}_{\dot{q}r}^{i}\theta_{r}^{2 -}\Omega^{+ 2 i}\right) .
\bar{P}_{}^e = 0.168 \Phi\bigg(1.752{e^{{\mu_\gamma}+ \frac{{\sigma_\gamma^2}}{2}}},\frac{{{\sigma_\gamma}}}{2}\bigg)+ 0.144\Phi\bigg(1.05{e^{{\mu_\gamma}+ \frac{{\sigma_\gamma^2}}{2}}},\frac{{{\sigma_\gamma}}}{2}\bigg) + 0.002\Phi\bigg(1.206{e^{{\mu_\gamma}+ \frac{{\sigma_\gamma^2}}{2}}},\frac{{{\sigma_\gamma}}}{2}\bigg),
\sin\rho_n \pi = (-1)^m(\varepsilon_n \pi + O(\varepsilon_n^3)), m := n - M_1 - 1, \varkappa_\pi(\rho_n) = \int_{-\pi}^{\pi}f(x) e^{i(n+\varepsilon_n)x}dx = \int_{-\pi}^{\pi}f(x)e^{inx}dx + i\varepsilon_n \int_{-\pi}^{\pi}xf(x)e^{inx}dx + O(\varepsilon_n^2) = \varkappa_n + o(\varepsilon_n).
d^{2}(\phi, \psi)=d^{1}(\phi, \psi) +\max_{1\leq i\leq l_{1}}\sup_{u\in U_{x_{i}}}\Vert D_{u}^{2}(\tilde{\phi}_{x_{i}}) - D_{u}^{2}(\tilde{\psi}_{x_{i}})\Vert+\max_{1\leq i\leq l_{2}}\sup_{v\in V_{y_{i}}}\Vert D_{v}^{2}(\hat{\phi}_{y_{i}}) - D_{v}^{2}(\hat{\psi}_{y_{i}})\Vert,
\partial_{x_j}(d_X) \partial_{x_i}(d_X) + d_X \partial_{x_j}\partial_{x_i}(d_X) + \partial_{x_j}\partial_{x_i}(d_X) d_X + \partial_{x_i}(d_X) \partial_{x_j}(d_X) = \partial_{x_i}\partial_{x_j}W
D_{-\frac{i}{2}\Lambda}(-(1+i) \tau),\; \;D_{-\frac{i}{2}\Lambda}((1+i) \tau),\; \;D_{\frac{i}{2}\Lambda -1}((1-i) \tau),\; \;D_{\frac{i}{2}\Lambda -1}(-(1-i) \tau) .
M_{B, C}= \left( \begin{array}{cc}\frac{(n_1+2)(n_2+1)}{2}0 0 -\frac{(n_1+2)(n_2+1)}{2}\end{array}\right), M_{D, E}= \left( \begin{array}{cccc}\frac{n_1+n_2+3}{3}\frac{n_2-n_1-1}{3}-\frac{2n_2+2}{3}-\frac{2n_1+4}{3}\frac{2n_1+4}{3}0 \frac{n_1-n_2 + 1}{3}-\frac{n_1 + n_2 + 3}{3}\frac{2n_2 + 2}{3}\end{array}\right),
E^p[H_1,H_2](\tau_2)+\int_{\tau_1}^{\tau_2}E^{p-1,deg}[H_1,H_2](\tau)d\tau \leq C\left( E^p[H_1,H_2](\tau_1)+\int_{\tau_1}^{\tau_2}E^{-1}[H_0,P,Q](\tau)d\tau. \right).
\displaystyle{\raisebox{0.9ex}{\scriptsize{$\diamond$}}}\mbox{\hspace{-0.33ex}}\rho_\tau (Q) = \frac{\epsilon (q_0)}{\pi}\, Im \, \frac{1}{\displaystyle{\raisebox{1.1ex}{\scriptsize{$\diamond$}}}\mbox{\hspace{-0.33ex}}D_\tau (Q)}\, .
\sum_{\mathsf{j=1}}^{+\infty}\sum_{\mathsf{k=1},\mathsf{k\neq j}}^{+\infty}\left\vert \left( \frac{\mathsf{\hat{r}}_{\mathsf{j}}}{\left\Vert \mathsf{\hat{r}}_{\mathsf{j}}\right\Vert_{2}},\frac{\mathsf{\hat{r}}_{\mathsf{k}}}{\left\Vert \mathsf{\hat{r}}_{\mathsf{k}}\right\Vert_{2}}\right)_{2}\right\vert^{2}<+\infty
l_\sigma(v^{2(k+1)};0,0) = l_\sigma(v^{2k};0,0) l_\sigma(v^{2};0,0)= ((uv)^{-2k};0,2k) ( (uv)^{-2}; 0,2) & = ((uv)^{-2k}( \theta(0,2k)(u) \theta(0,2k)(v) )^{-2};0,2k + 2) & = ( (uv)^{-2k}(uv)^{-2};0,2(k+1))= ( (uv)^{-2(k+1)};0,2(k+1)).
\int d \mu_{B}\ = \ \int{\frac{d^{3}X}{( 2 \pi )^{3 / 2}}}\S_{\mathrm{c l}}^{3 / 2}\int_{0}^{2 \pi}{\frac{d \theta}{( 2 \pi )^{1 / 2}}}\{\frac{S_{\mathrm{c l}}^{1 / 2}}{M_{W}}}\ .
\psi_{0}( \vec{r}) = \sum_{n = - \infty}^{\infty}T_{n}^{0}( r , \phi ) = \sum_{n = - \infty}^{\infty}\int_{- \infty}^{\infty}d \lambda ( - i )^{| \lambda |}J_{| \lambda |}( k r ) e^{i \lambda ( \phi + \pi - \theta + 2 n \pi )}.
\bar{\mu}^{\dot{A}}= \left( X^{B \dot{A}}- i \Theta^{B}\bar{\Theta}^{\dot{A}}\right) \lambda_{B}+ 2 \bar{z}^{\dot{A}\dot{B}}\bar{\lambda}_{\dot{B}}- i \bar{\Theta}^{\dot{A}}\bar{\Theta}^{\dot{B}}\bar{\lambda}_{\dot{B}}
L^{\prime}_{ns}[\dot{x}^{\prime}(t^{\prime}),x^{\prime}(t^{\prime}),t^{\prime}]= \frac{1}{C^{\prime}_1 f^{\prime\ 2}(t^{\prime}) [f^{\prime}(t^{\prime}) \dot{x}^{\prime}(t) - a^{\prime}_o x^{\prime}(t^{\prime})+C^{\prime}_2 + v^{\prime}_o V_0]}\ .
F(x+y\rho) = f(x) + y \Delta_{yr}f(x) \cdot \rho + (g(x) + y \Delta_{yr}g(x) \cdot \rho)\rho & = f(x) + (y \Delta_{yr}f(x) + g(x) + yr\Delta_{yr}g(x)) \rho & = f(x) + (y \Delta_{yr}f(x) +g(x+yr))\rho.
\begin{aligned}\limsup_{N\rightarrow \infty}\left\Vert \left( f_N^{(s)}(t) -\int_{\mathcal{P}\left( \mathbb{R}^{2d}\right)}h(t)^{\otimes s}d\pi(h_0) \right) \mathbf{1}_{Z_s \in \mathcal{K}_s\cap \mathcal{U}_s^{\eta(\varepsilon)}}\mathbf{1}_{E_s (Z_s)\leq R^2}\right\Vert_{L^\infty_{Z_s}}= 0\end{aligned}
\sum_{t=1}^{k}\frac{z_{t}\eta_{t}}{\alpha_{t}}\left(f\left(y_{t+1}\right)-f\left(x^{*}\right)\right)-\frac{z_{t}\eta_{t}\left(1-\alpha_{t}\right)}{\alpha_{t}}\left(f\left(y_{t}\right)-f\left(x^{*}\right)\right)
(\theta_{1})^{2}(\theta_{2})^{2}=(\theta_{2})^{2}(\theta_{1})^{2}=\theta_{1}(\theta_{2})^{2}\theta_{1}=\theta_{2}(\theta_{1})^{2}\theta_{2}=-\frac{1}{2}\theta_{1}\theta_{2}\theta_{1}\theta_{2}=-\frac{1}{2}\theta_{2}\theta_{1}\theta_{2}\theta_{1},
{\Bigl({d^2\over{d\tau}^2}-{B_{ab}\over \bar T}\Bigl)}^{-1}={\Bigl({d\over{d\tau}}\Bigr)}^{-2}+{\Bigl({d\over{d\tau}}\Bigr)}^{-2}{B_{ab}\over \bar T}{\Bigl({d\over{d\tau}}\Bigr)}^{-2}+{\Bigl({d\over{d\tau}}\Bigr)}^{-2}{B_{ab}\over \bar T}{\Bigl({d\over{d\tau}}\Bigr)}^{-2}{B_{ab}\over \bar T}{\Bigl({d\over{d\tau}}\Bigr)}^{-2}+\cdots,
\|P_-v_s\|^2= (P_-^*P_-v_s, v_s) = ((P_-^*-P_-)P_-v_s, v_s) &+((P_--e^{-isH}E_-(-s))P_-v_s, v_s) &+(P_-v_s,E_-(-s)^*v) .
(i) \ \ \bigcup\nolimits^{n'}_{j=1}\overline{T^{l-1}_j}= \bigcup\nolimits^{n'}_{j=1}\overline{Y'_j}\cup \bigcup\nolimits^{l-1}_{j=1}\overline{X'_j}, \ \ \ (ii) \ \T^{l-1}_{j_1}\cap \overline{X'_{j_2}}= T^{0}_{j_1}\cap \overline{X'_{j_2}}
\rho_{\mu}( x ) = \rho_{\nu}( x ) \frac{q^{C ( \mu ) / 2}+ \epsilon ( \mu ) \epsilon ( \nu ) x q^{C ( \nu ) / 2}}{x q^{C ( \mu ) / 2}+ \epsilon ( \mu ) \epsilon ( \nu ) q^{C ( \nu ) / 2}}, \forall \mu \neq \nu .
d_{j_1}^{(n-j_1)}\cdots d_{j_k}^{(n-j_k)}d_i^{-(n-i)}= d_{i_1}^{(n-i_1)}\cdots d_{i_{l-1}}^{(n-i_{l-1})}d_{i_l}^{-(n-i_l)}d_{i_{l+1}}^{-(n-i_{l+1})}\cdots d_{i_{k+1}}^{-(n-i_{k+1})}.
-2\eta_{t}\sum_{i=1}^{n}\mathbb{E}[\langle x^{i}(t)-\bar{x}(t), g^{i}(t)-\bar{g}(t) \rangle|\mathcal{F}_{t-1}] \leq \lambda h(t-1)+\frac{\eta_{t}^{2}}{\lambda}\kappa n[C^{*}G^{2}d+\frac{3d^{2}\sigma^{2}}{2h_{t}^{2}}].
{\widehat T}={\widehat M}_{1}= \left( \begin{array}{c c c c c c}{{1}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{- 2}}&{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) \; .
Z(s, f, \phi)(g) = \frac{L^S((2n+1)s+1/2,\pi \boxtimes \chi, \varrho_{2n+1})}{L^S((2n+1)(s+1/2), \chi)\prod_{j=1}^nL^S((2n+1)(2s+1)-2j, \chi^2)}\times i^{nk}\,\pi^{n(n+1)/2}c((2n+1)s- 1/2)\phi(g),
\begin{array}{rcl}K_{\nu}(t) = \displaystyle \frac{1}{2}\left(\frac{t}{2}\right)^{\nu}\int_0^{\infty}d\alpha~\alpha^{-1-\nu}\exp\left[-\alpha-\frac{t^2}{4\alpha}\right],~|\arg t|<\frac{\pi}{2},~{\rm Re~}t^2>0.\end{array}
\frac{d}{ds}\frac{1}{\partial_x\Phi_{t,\gamma(s)}}=-\frac{1}{(\partial_x\Phi_{t,\gamma(s)})^{2}}\frac{d}{ds}\partial_x\Phi_{t,\gamma(s)}, \frac{d}{ds}\left(\frac{d\gamma(t)}{dt}\right),\quad\mbox{and}\;\partial_{s}\Phi_{t,\gamma(s)}^{-1}
\mathrm{P}_{m,i}\approx \mathrm{P}_{t,1}+\frac{4(\lambda_c\pi)^{t}}{(t-m-1)!(m-1)!}\sum^{t-m-1}_{p=0}(-1)^p{t-m-1 \choose p}\times \sum^{N}_{l=1}\frac{\pi\left(\tau_2-\tau_1\right)}{2 N}f_m\left(\frac{\tau_2-\tau_1}{2}w_l+\frac{\tau_2+\tau_1}{2}\right)\sqrt{1-w_l^2},
D_G(u,v,w)= -27a^4(u^6+v^6+w^6) + 162a^2b^2(u^4vw+uv^4w+uvw^4) & + (54a^4 - 108ab^3)(u^3v^3+v^3w^3+w^3u^3) + (-324a^3b + 81b^4)u^2v^2w^2 .
\mu(\{\eta\}) = \sum_{(u_{i})_{i \in I}\in K^{I}}\delta_{c}(\{\eta^{\mbox{\tiny \sc Time}}\}) \prod_{i \in I}\delta_{[s \mapsto (u_{i},i)]}(\{\eta^{\mbox{\tiny $\mathcal{D}$}}\}) \mu^{*}(\{(u_{i})_{i \in I}\})\;,
\mathbb D^2 \big[2 \phi \mathbb S \, \mathbb D P_n \big] = 4 m_1 \,\mathbb D^2 \big[ \phi \,\mathbb D^2 P_n \big] + 2\mathbb S \mathbb D \big[\phi \,\mathbb D^2 P_n \big] + 2\mathbb S \mathbb D (\phi) \,\mathbb D^2 P_n + 2\mathbb D^2(\phi)\,\mathbb S \mathbb D P_n.
\frac{F}{V}= \left( \frac{1}{4 \pi^{2}\alpha^{\prime}}\right)^{5}\int_{- 1 / 2}^{1 / 2}d \tau_{1}\int_{0}^{\infty}\frac{d \tau_{2}}{\tau_{2}^{6}}\left| \frac{1}{\eta ( \tau )}\right|^{2 4}2 \sum_{\stackrel{n = 1}{n \o d d}}^{+ \infty}e^{- \frac{\beta^{2}n^{2}}{4 \pi \alpha^{\prime}\tau_{2}}}\left| \theta_{2}( 0 , \tau ) \right|^{8}
W_\epsilon(x) := F_\epsilon - |F_\epsilon|^2 \geq d\alpha \omega_\epsilon^{-1}-\Big(2\alpha+4\alpha^2\Big)\omega \omega_\epsilon^{-2}+ \Big(2\alpha+4\alpha^2\Big)\omega_\epsilon^{-2}\sum_j \sin^4(x_j/2) -2\alpha\omega\omega_\epsilon^{-1}+ T_2^\epsilon(x) + T_1^\epsilon(x) - \frac{1}{2}T_3^\epsilon(x)- \frac{1}{2}T_4^\epsilon(x).
-\frac{2P_3(1)}{L}\sum_{\substack{p \leq y_2 (p,q)=1}}\frac{(\log p)P_2[p]}{p}\frac{1}{2}\sum_{w \mid q}\varphi(w)\mu(q/w) \sum_{n \equiv \pm 1 (w)}\frac{1}{n^{\frac{1}{2}}}F \left( \frac{n}{p}\right).
\mathbb{E}[\eta_1 | d(x, \hat{x}) = k] = \frac{k}{n}(n-s) \mathbb{E}[\eta_2 | d(x, \hat{x}) = k] = \frac{k}{n}(s) \mathbb{E}[\eta_1 \eta_2 | d(x, \hat{x}) = k] = (n-s)(s) \frac{k(k-1)}{n(n-1)}= s(n-s) \left( \frac{k^2}{n^2}- \frac{k(n-k)}{n^2(n-1)}\right).
\mathtt{f}_{\nu}(z) = \left(2^{\nu}\Gamma(\nu+1) \mathtt{J}_{\nu}(z)\right)^{\tfrac{1}{\nu}}, \mathtt{g}_{\nu}(z) = 2^{\nu}\Gamma(\nu+1) z^{1-\nu}\mathtt{J}_{\nu}(z), \mathtt{h}_{\nu}(z) = 2^{\nu}\Gamma(\nu+1) z^{1-\frac{\nu}{2}}\mathtt{J}_{\nu}(\sqrt{z}).
\textup{LHS}= (1+\lambda_h)\Vert u_{\mathrm{nc}}\Vert_{\delta}^2+ \lambda \Vert u-u_{\mathrm{nc}}\Vert^2_{L^2(\Omega)}+ 2\lambda b(u,u_{\mathrm{nc}}-Ju_{\mathrm{nc}}) +2a_{\mathrm{pw}}(u, Ju_{\mathrm{nc}}-u_{\mathrm{nc}}).
B_{h-k}^{n-k-l}(x) = \sum_{i=k+l}^{n}b_{h-k,i-k-l}^{(n-k-l,\alpha+2l,\beta+2k)}R_{i-k-l}^{(\alpha+2l,\beta+2k)}(x) (h=k,k+1,\ldots,n-l).
\bar{z}=z -\textbf{z}(v,w),~\bar{x}=x -\textbf{x}(v,w), \bar{\eta}=\eta -\theta (v,w)-C\bar{x}\bar{u}=u -\Psi \eta(t_{k}),~ t\in[t_{k},t_{k+1}),k\in\mathbb{S}
r(z)=\int_{0}^{z}\frac{dz}{H(z)}=\frac{1}{\sqrt{\Omega_{m}}H_{0}}\int_{0}^{z}\frac{dz}{(1+z)^{3/2}}=\frac{2}{\sqrt{\Omega_{m}}H_{0}}(1-\frac{1}{\sqrt{1+z}}).
\varphi \bigl( ([z, \theta_{s_r}(F)],[s_r, F]), ([w, \theta_{t_r}(F')], [t_r, F']) \bigr) = \varphi \bigl( [zs_rws_r^*, \theta_{s_r}(F)], [s_r t_r, F'] \bigr) &= [zs_rws_r^*s_rt_r, F'] &= [zs_rs_r^*s_rwt_r, F'] &= [z s_r wt_r, F']
\mathrm{err}_1=\frac{\|f_i-\mathcal{F}^i_2\|_{{L}^1(\mathbb{R})}}{\|f_i\|_{\mathbb{L}^1(\mathbb{R})}},\mathrm{err}_2=\frac{|\langle \cos(\cdot) f_i\rangle- \langle\cos(\cdot)\mathcal{F}^i_2\rangle|}{|\langle \cos(\cdot)f_i\rangle|},
\lim_{\Lambda_{1}\uparrow\uparrow V}\psi_{\Lambda_{1}}(b_{\Lambda})=\sum_{i,j\in I}\prod_{x\in \Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, j}^* b_{x}\right)\prod_{x\in \Lambda_{1}\setminus\Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, j}^* \right)
\max_{j_{0}}\left\Vert \overset{3}{\mathbb{P}}\left(\tilde{E}_{j_{0}}^{T}\right) \right\Vert^{2}\lesssim \left\vert \eta \right\vert^{-2}\max_{j_{0}}\left\Vert \tilde{E}^{Inc}_{j_{0}}\right\Vert^{2}+ \left\vert \eta \right\vert^{-2}\, \max_{j_{0}}\sum_{n}\, \left\vert Error(n,j_{0}) \right\vert^{2}.
n^{\frac{1}{2\sigma^2}\kappa(\underline{\gamma})-1/2}\sigma\prod_{l=1}^d2^{J_{n,l}(\sigma_0^2/\sigma^2-\mu_l)}\lesssim n^{-\mu_{\mathrm{min}}-1/2+\frac{\sigma_0^2}{\sigma^2}+\frac{1}{2\sigma^2}\kappa(\underline{\gamma})}\sigma.
\Omega_{1}= \left( \begin{array}{c c}{I_{d}}&{\gamma}\\{0}&{I_{d}}\\ \end{array}\right) \quad \Omega_{2}= \left( \begin{array}{c c}{I_{d}}&{0}\\{\alpha}&{I_{d}}\\ \end{array}\right) \quad \Omega_{3}= \left( \begin{array}{c c}{{A^{- 1}}^{T}}&{0}\\{0}&{A}\\ \end{array}\right) ,
M = \frac{V_p \Omega_{d-2}2\pi R_T}{16 \pi G}r_0^{d-3}\frac{d-1}{2}+ \sqrt{\frac{1}{4}\left( \frac{V_p \Omega_{d-2}2\pi R_T}{16 \pi G}(d-3) r_0^{d-3}\right)^2 + Q^2}
\left|\frac{m_j(x,\theta)}{\sigma_{P,j}}-\frac{m_j(x,\theta')}{\sigma_{P,j}}\right| =\frac{\Vert(\theta'-\theta)\Vert}{\sigma_{P,j}},~~\ell=0,1, D_{P,\ell}(\theta) =\frac{(-1)^{(1-\ell)}}{\sigma_{P,\ell}},~~\ell=0,1.
\frac{\partial u^{\epsilon}}{\partial t}(t,x) = \frac{\partial^2 u^{\epsilon}}{\partial x^2}(t,x) +\sqrt{\epsilon}\sigma(t, x, u^\epsilon(t,x))\frac{\partial^2W}{\partial t \partial x}(t,x) & +\frac{\partial}{\partial x}g(t,x,u^{\epsilon}(t,x))+ f(t,x,u^\epsilon(t,x)),
\langle\delta^2 k_R\rangle =\langle k_R^2\rangle -\langle k_R\rangle^2 =\lambda^2\left(\frac{n+2}{Nn}\right)^2\left\{\left\langle \left(\sum_{\bf i}\Vert\varphi_{\bf i}\Vert^2\right)^2\right\rangle-\left(\sum_{\bf i}\langle\Vert\varphi_{\bf i}\Vert^2\rangle\right)^2\right\}