harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rclrcl}\left[L_1,L_{-2}\right] = L_{-1}, \quad\left[L_1,L_2\right] = 0, \left[L_2,L_{-2}\right] = N + \frac{D+1}{2}, \quad\left[L_0,L_n\right] = 0, \left[L_1,L_{-1}\right] = L_0, \quad\left[N,L_n\right] ={}-nL_n,\quad n=0,\pm 1,\pm 2, \end{array}
\langle \overline{p}(T),x(T)-\overline{x}(T) \rangle = \int_{T-\tau}^T \langle \dot{\overline{p}}(s), x(s)-\overline{x}(s) \rangle\ds +\int_{T-\tau}^T \langle \overline{p}(s),\dot{x}(s)-\dot{\overline{x}}(s) \rangle \d s &= (I) + (II)
h_R(x_1^{i_1}x_2^{i_2}\cdots x_{n-1}^{i_{n-1}})=\sum_{k=1}^{n-1}(-1)^{|x_1^{i_1}\cdots x_{k-1}^{i_{k-1}}||x_k|}\bigtriangleup(i_k)y_kx_1^{i_1}\cdots x_{k-1}^{i_{k-1}}x_k^{i_k-1}x_{k+1}^{i_{k+1}}\cdots x_{n-1}^{i_{n-1}}.
\lim_{\lambda \rightarrow0}\int^{\infty}_{\lambda}\frac{\lambda}{\sigma}\bar{u}(\sigma)d\sigma =\lim_{\lambda \rightarrow0}\int^{\infty}_{\lambda}\frac{\lambda}{\sigma^2}\sigma\bar{u}(\sigma)d\sigma= \lim_{\sigma \rightarrow 0}\sigma\bar{u}(\sigma),
q_{1}= \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{{- 1}}}\\ \end{matrix}\right) ; \quad q_{2}= \left( \begin{matrix}{{0}}&{{{- 1}}}\\{{0}}&{{0}}\\ \end{matrix}\right) ; \quad q_{3}= \left( \begin{matrix}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{matrix}\right)
|G:\mho_1(G)| \leq |G:\mho_1(C_G(A))|= |G:C|\cdot |C:C_G(A)|\cdot |C_G(A):\mho_1(C_G(A))| \leq 2^{\binom{k}2+2k+k}=2^{\frac{k(k+5)}2}.
\frac{1}{(1-t^2)^{r+1}}\begin{bmatrix}\frac{t^3}{1+t}[t((1+t)^r-1)+((1-t^2)^r-1)] -\frac{t^3}{1+t^3}[t^3((1+t)^r-1)+((1-t^4)^r-1)] \end{bmatrix}.
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{p}\leq (\delta^{-3/2}R'^2)^{2/p-1}(\delta^{1/2}R'^{-1/2})^{2-2/p}(\# T_1\#T_2)^{1/2}= \delta^{5/2-4/p}(R')^{5/p-3}(\# T_1\# T_2)^{1/2}.
M^{\mu\nu}= \left[ \vert \tau_1 -\tau_2 \vert- \Delta_\theta\,(t_2+t_3)(\tau_1-\tau_2)^2 -(\epsilon-\eta)^2{\theta^2\over4}\Delta_\theta\, (t_2+t_3)\, \right]^{\mu\nu}\ .
P(g,\psi) = \lim \limits_{\epsilon\to 0}\limsup\limits_{n\to+\infty}\frac{1}{n}\log \sum\limits_{x\in F_n(\epsilon)}\exp(S_n \psi(x)) = \lim \limits_{\epsilon\to 0}\liminf\limits_{n\to+\infty}\frac{1}{n}\log \sum\limits_{x\in F_n(\epsilon)}\exp(S_n \psi(x)),
t = t_{0}+{\frac{1}{5}}\Big ({\frac{2^{7}}{m^{5}n^{4}}}\Big )^{1 / 3}y^{5 / 2}F \left( \Big [{\frac{5}{6}},{\frac{2}{3}}\Big ] , \Big [{\frac{1 1}{6}}\Big ] , -{\frac{4 y^{3}}{m^{2}n^{2}}}\right) .
\begin{cases}\frac{d}{dt}Z_s (t) = \left( V_s (t), 0\right) \;\;\textnormal{if}\;\; Z_s (t) \notin \partial \mathcal{D}_s Z_s (t^+) = \left(Z_s (t^-) \right)^* \;\; \textnormal{if}Z_s (t) \in \partial \mathcal{D}_s \end{cases}
G_{0}(0;r\vec{x},\vec{s}) = r^{-\frac{\sigma_{3}}{4}}\begin{pmatrix}1 0 - i\frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}R(0) \begin{pmatrix}1 0 id_{1}1\end{pmatrix}x_{1}^{-\frac{\sigma_{3}}{4}}N \widehat{c}^{-\sigma_{3}},
\rho_{i}= \frac{\rho}{2}+ \frac{\rho}{2^{i+1}}, \rho_0 = \rho, \ \rho_{\infty}= \frac{\rho}{2}, k_{i}=\frac{k}{2}+ \frac{k}{2^{i+1}}, k_{0}=k, \k_{\infty}=\frac{k}{2}, K_{i}= K_{\rho_i}, Q_{i}=K_{i}\times [-\theta k^{-\alpha}\rho_{i}^{2}, 0].
\widetilde{X}_n(t)=\int_{\{\tau_{{E'}}(\xi)\notin(d_{n-1},d_n]\}}\bigl(\mathrm{e}^{\mathrm{i}\langle t,\xi\rangle}-1 \bigr) \frac{\widetilde{\mathcal{M}}(\mathrm{d}\xi)}{\psi(\xi)^{1 + Q/2}}
x^-_{\theta}(1)=x_{n\,1}^-(1)=[\,x_1^-(0),\ldots, x_{n-1}^-(0),\,x_{n}^-(1)\,]_{(r^{-2},\,\ldots,r^{-2})}, x_{\theta}^+(-1)=\tau(x^-_{\theta}(1))=x_{n\,1}^+(-1) =[\,x_{n}^+(-1),\,x_{n-1}^+(0),\,\ldots, x_1^+(0)\,]_{\langle s^{-2},\,\ldots, s^{-2}\rangle}.
\max_{q(x_1,x_2)}\min_{p(y_2|x_1,x_2)\in \mathcal{P}}I(Y_2;X_1|X_2) \geq \max_{x_2}\max_{q(x_1|x_2)}\min_{p(y_2|x_1,x_2)\in \mathcal{P}}I(Y_2;X_1|X_2=x_2) &= \max_{x_2}\bar C(\mathcal G_{x_2}),
\psi^\dagger_{1\xi}(\epsilon)\psi_{2\eta}(-\epsilon)=ie^{\frac{i}{2}[-\theta(\eta)-\theta(\xi)+\frac{1}{2}R_2(\epsilon,\xi,\eta)]}e^{-i\pi\int d^2y \frac{1}{e}\partial_iE_i(y)}e^{iF_2(A,E)}
\begin{aligned}|G_{\omega,E,\Lambda_L(x)}(a,b)| \leq |G_{\omega,E_0,\Lambda_L(x)}(a,b)|+|E-E_0| \cdot |G_{\omega,E,\Lambda_L(x)}(a,b)| \cdot |G_{\omega,E_0,\Lambda_L(x)}(a,b)| \leq e^{-m_0|a-b|}+e^{-m'L}e^{L^{1-\eta}+L^{1-\eta}}\leq e^{-m|a-b|}\end{aligned}
\mathcal{G}^*(x,t):=t^{p^*(x)}+\mu(x)^{\frac{q^*(x)}{q(x)}}t^{q^*(x)}\quad\mathcal{T}^*(x,t):=t^{p_*(x)}+\mu(x)^{\frac{q_*(x)}{q(x)}}t^{q_*(x)}.
c(h,q,s)=A(n)\chi(\det(q_{\mathbf{h}}))^{-1}\det(q_{\infty})^s|\det(q)|_{\mathbf{h}}^{2m-1-s}\alpha_{\mathfrak{n}}(q_{\mathbf{h}}^{\ast}hq_{\mathbf{h}},s,\chi)\xi(q_{\infty}q_{\infty}^{\ast},h,s+l,s-l).
\displaystyle s_1=\inf\left\{\frac{\int_{\Sigma}\left(|\nabla\varphi|^{2}-q\varphi^2\right)e^{-f}d\sigma}{\int_{\Sigma}\varphi^2e^{-f}d\sigma}; \varphi\in C_0^{\infty}(\Sigma), \int_{\Sigma}\varphi^2e^{-f}d\sigma\neq 0\right\}.
E\max_{1\leq k\leq n-1}\left\vert \sum\nolimits_{i=k}^{n-1}P^{i}(S_{i})\right\vert^{p}\ll E\left\vert \sum\nolimits_{i=1}^{n-1}P^{i}(S_{i})\right\vert^{p}.
\sum_{n = 1}^{\infty}e^{- M \sqrt{n^{2}\beta^{2}+ \tilde{p}_{2}^{2}}}= - \frac{1}{2}e^{- M \tilde{p}_{2}}+ \sqrt{2 \pi}[ \frac{1}{2}F ( 0 ) + \sum_{n = 1}^{\infty}F ( 2 \pi n ) ] ,
|\lambda_{r-k}| b_{r-k,r-k}\leq r+\sum_{j=r-k+1}^{r}|\lambda_j| b_{j,r-k}& \leq r + \left(\sum_{j=r-k+1}^{r}|\lambda_j|\right) (b_{r-k,r-k}-1) & \leq r+(1+1+2+\ldots+2^{k-2})r(b_{r-k,r-k}-1) & = r+2^{k-1}r(b_{r-k,r-k}-1) & \leq 2^{k-1}r b_{r-k,r-k},
\underset{\left\{\pmb{F}_k\right\}_{k=1}^K,\left\{\pmb{W}_k\right\}_{k=1}^{K}}{}\,\,\,\,\,\sum\limits_{k=1}^K||\pmb{\Xi}_k^{(\kappa)}\pmb{J}_k||_{*}+||\pmb{\Phi}_e^{(\kappa)}\pmb{S}_e||_{*}\,\,\,\,\, \qquad\pmb{S}_k \succeq \pmb{0},\,\,\, \sigma_{min}(\pmb{S}_k) \geq \epsilon.
J( u_n) =J(\Vert u_n\Vert v_n ) \leq g^+\max\left\lbrace \Vert u_n\Vert^{g^-}\Vert v_n\Vert^{g^-},\Vert u_n\Vert^{g^+}\Vert v_n\Vert^{g^+}\right\rbrace - \int_{\mathbb{R}^{d}}K(x)F(x,\Vert u_n\Vert v_n) dx \leq g^+\max\left\lbrace \Vert u_n\Vert^{g^-},\Vert u_n\Vert^{g^+}\right\rbrace - \int_{\mathbb{R}^{d}}K(x)F(x,\Vert u_n\Vert v_n) dx,
\int_M d f_k\wedge d^c_B f_k\wedge \wedge \omega_\phi^{n-1}\wedge \eta=\int_M f_k (dd^c_B u-dd^c_B u_k)\wedge \omega_\phi^{n-1}\wedge \eta\leq \int_M f_k \omega_u\wedge \omega_\phi^{n-1}\wedge \eta.
- \frac{1}{2 L}\frac{1}{\sqrt{J}}\partial_{p}J \partial_{p}\frac{1}{\sqrt{J}}= - \frac{1}{2 L}\partial_{p}\partial_{p}+ \frac{1}{2 L}\frac{1}{\sqrt{J}}\left( \partial_{p}\partial_{p}\sqrt{J}\right)
f=\sum_{r,s,t\in{\mathbb F}_p}a_{r,s,t}T_{1k}^rT_{1k+1}^sT_{1k+2}^t=\sum_{r,s,t\in{\mathbb F}_p}a_{r,0,0}T_{1k}^rT_{1k+1}^sT_{1k+2}^t=\sum_{r\in{\mathbb F}_p}a_{r,0,0}T_{1k}^r\times \sum_{s\in{\mathbb F}_p}T_{1k+1}^s\sum_{t\in{\mathbb F}_p}T_{1k+2}^t=f_k(T_{1k})C(T_{1k+1})C(T_{1k+2})=p^{-2}f_k(T_{1k})c(T_{1k+1})c(T_{1k+2}),
\sum_{j=0}^\infty \int_{T^{-m}[b_{\ell+1}, 1] \cap [b_{j+1}, b_j]}\frac{|u'(x)|}{(T^m)'x}\, dx = \sum_{j=p}^\infty \int_{T^{-m}[b_{\ell+1}, 1] \cap [b_{j+1}, b_j]}\frac{|u'(x)|}{(T^m)'x}\, dx + \int_{T^{-m}[b_{\ell+1}, 1] \cap [b_{p}, \frac12]}\frac{|u'(x)|}{(T^m)'x}\, dx
\partial_{(1)}^{A}\otimes\partial_{(2)}^{A}=\partial^{A}\otimes1+(\mathcal{L}_{\partial}){^{A}}_{B}\otimes\partial^{B}, \hat{\partial}_{(1)}^{A}\otimes\hat{\partial}_{(2)}^{A}=\hat{\partial}^{A}\otimes1+(\mathcal{\bar{L}}_{\partial}){^{A}}_{B}\otimes\hat{\partial}^{B}.
\lim_{M\rightarrow\infty}|S_{m,N}|^{\frac{1}{q}-\frac{1}{p}}\gamma\left|\bigcap_{i=M}^{\infty}\{k\in S_{m,N}:|x^{(i)}_{k}-x^{(j)}_{k}|>\gamma\}\right|^{\frac{1}{p}}<\epsilon.
\sum_{n=1}^{i_{k}}\binom{i_{k}}{n}(-1)^{n-1}\zeta^{\ast}_{n}(\{1\}_{b_1-1},w_{2,r})=\frac{1}{i_k^{b_1-1}}\sum_{j=1}^{i_k}(-1)^{j-1}\binom{i_{k}}{j}\zeta^{\ast}_{j}(w_{2,r}).
{\cal D}X^\mu=DX^\mu-\frac{l^2(k_\rho D X^\rho)-(k\cdot l)(l_\rho D X^\rho)}{k^2l^2-(k\cdot l)^2}\,k^\mu\ - \frac{k^2(l_\rho D X^\rho)-(k \cdot l)(k_\rho D X^\rho)}{k^2l^2-(k \cdot l)^2}\,l^\mu,
\|F_n(\mathbb U(t))-F_n(\tilde U(t))\|^2 \le C\sum_{k=1}^{n-1}\big(1+\mathbb U_k(t)^2+\tilde U_k(t)^2\big)^2|\mathbb U_k(t)-\tilde U_k(t)|^2 &\le Cn\big(1+\|\mathbb U(t)\|_{l^{4q^\prime}_n}^4+\|\tilde U(t)\|_{l_n^{4q^\prime}}^{4}\big)\|\mathbb U(t)-\tilde U(t)\|^2_{l_n^{2q}},
e^F\bigl(\Delta + M_V\bigr)e^{-F}=e^{F}\bigl(\frac{4\pi^2}{T^2}(A+\frac{\beta}4)^2+ Q_0+M_V\bigr)e^{-F}= \frac{4\pi^2}{T^2}(A+\frac{\beta}{4})^2 + B+S,
e_1=\left|\begin{array}{c}\overrightarrow{e_1}0\end{array}\right| ,\quad e_2=\left|\begin{array}{c}\overrightarrow{e_2}0\end{array}\right| ,\quad e_3=\left|\begin{array}{c}\overrightarrow{e_3}0\end{array}\right| ,\quad e_4=\left|\begin{array}{c}0 1\end{array}\right| .
\frac{h_{H}(f^{n}(P))}{\delta_{f}^{n}}=h_{H}(P)+\sum_{k=0}^{n-1}\frac{h_{E'_{1}}(f^{k}(P))}{\delta_{f}^{k+1}}-\sum_{k=0}^{n-1}\frac{h_{Z_{1}}(p^{-1}f^{k}(P))}{\delta_{f}^{k+1}}.
{\cal L}= - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}- \frac{\kappa}{2}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}+ \sum_{a}\bar{\psi}_{a}[ \gamma_{a}^{\mu}( i \partial_{\mu}+ q_{a}A_{\mu}) - m_{a}] \psi_{a}\, \, ,
\rho_{\psi_\kappa}^T:=\frac{1}{T^{k/2}}\sum_{j_1 \neq \ldots \neq j_k}\sigma_{j_1}\ldots \sigma_{j_k}\langle \Lambda^{(k)}(x^{j_1}+\xi^{j_1},\ldots,x^{j_k}+\xi^{j_k};T),\psi_\kappa \rangle,
E(t,X(t)) = -\nabla (\phi * \rho(t,\cdot))(X(t)) = -\nabla (\phi * (G *_{s,x}\mathcal S + \mathcal S))(X(t)) = \mathcal{F}^R(t) + \mathcal{E}^R_1(t) + \mathcal{E}_2(t)+\mathcal{E}_3(t),
L_B=-{a\over 4}C_{\mu\nu}^aC^{\mu\nu a}-{1\over{2g}}B_\mu^aB^{\mu a}+\alpha\epsilon^{\mu\nu\lambda}C_\mu^a(\partial_\nu C_\lambda^a+{1\over 3}f^{abc}C_\nu^bC_\lambda^c).
-\partial_{tt}v_{3,1,2}+\partial_{rr}v_{3,1,2}+\frac{1}{r}\partial_{r}v_{3,1,2}- \frac{v_{3,1,2}}{r^{2}}= F_{0,1}^{\lambda_{1}}(t,r) - F_{0,1}^{\lambda_{2}}(t,r)
\bar{g}(s)=\frac{g s^{1/4}}{1-\frac{\pi A(y)b(y)g}{y}(s^{1/4}-1)}=\frac{g_{IR}^* |\tilde{g}|s^{1/4}}{1+|\tilde{g}|s^{1/4}},\ \ \ \ \ \ \tilde{g}=\frac{g}{g-g_{IR}^*}\ \in \ \ ( -\infty,0).
\int_2^{e^{|h-h'|^{-1}}}\frac{\cos(|h-h'|\log t)}{t\log t}\,dt =\int_{\log 2}^{|h-h'|^{-1}}\frac{\cos(|h-h'|v)}{v}\,dv &=\int_{\log 2}^{|h-h'|^{-1}}\frac{1}{v}\,dv+O\left(|h-h'|^2\int_{\log 2}^{|h-h'|^{-1}}v\,dv\right) &=\log |h-h'|^{-1}-\log\log 2+O(1).
{}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}=1+\frac{ab}{c}\frac{z}{1!}+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^{2}}{2!}+\cdot\cdot\cdot.
\begin{aligned}R^+ = \begin{pmatrix}r_n^+ r_{n-1}^+ \dots r_1^+\end{pmatrix}S^+ (\lambda) = \begin{pmatrix}\mu_1^+ r_n^+ \mu_2^+ r_{n-1}^+ \dots \mu_n^+ r_1^+\end{pmatrix}. \end{aligned}
\frac{d^{l-m}}{dx^{l-m}}(x^{2}-1)^{l}=(x^{2}-1)^{m}\sum_{k=m}^{l}(\begin{matrix}l-m\\k-m\end{matrix})l(l-1)...(1-l+k)\frac{(x+1)^{l}}{(x+1)^{l+m-k}}
\begin{array}{lll}V_1\cap W_1=\{t_1,t_{2}\ldots,t_i \}, V_1\cap W_2=\{t_{i+1},t_{i+2},\ldots,t_{2i}\}, &V_2\cap W_1=\{t_{2i+1},t_{2i+2},\ldots,t_{2i}\}, V_2\cap W_2=\{t_{3i+1},t_{3i+2},\ldots,t_{4i}\}. \end{array}
\begin{cases}\min\{\frac{1}{2}+ \frac{\alpha}{2\beta}(r+\lambda-\kappa), 1 \}, \mbox{if}\alpha + \gamma = 1, \min\{\frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\}+ (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}, \mbox{if}\alpha + \gamma \neq 1\end{cases}
{\cal R}={\frac{p}{\tilde{L}^{2}}}\left[ ( p + 1 ) ( \tilde{\ell}^{2}- \tilde{E}^{2}) \left({\frac{\tilde{L}}{a}}\right)^{2 ( p + 1 )}+ ( p - 1 ) \tilde{\ell}^{2}\left({\frac{\tilde{L}}{a}}\right)^{2 p}\right] .
\begin{matrix}{{I^{2}= J^{2}= K^{2}= - 1 , \, \, \,}}&{{I J = - J I = K \, \, \, \, \, \, \& \, \, \mathrm{c y c l i c}\, \, \mathrm{p e r m u t a t i o n s}.}}\\ \end{matrix}
\frac{1}{2}\sum_{d\leq x}\frac{\mu(d)\log d}{d}\sum_{l\leq x/d}1 =\frac{x}{2}\sum_{d\leq x}\frac{\mu(d)}{d^2}\log d + O\left(\sum_{d\leq x}\frac{\log d}{d}\right) &=\frac{\zeta'(2)}{2\zeta^{2}(2)}x + O\left(\log^{2}x\right),
((u-1)^2+v^2)^2 =\dfrac{16({\rho}_0)^4 (1-\beta)^4}{(1+({\rho}_0)^2-2{\rho}_0 \cos t)^2},\intertext{and}\left(\dfrac{u-1}{1-\alpha}\right)^2+ \left(\dfrac{v}{1+\alpha}\right)^2 =\dfrac{4({\rho}_0)^2(1-\beta)^2 \left[ ({\rho}_0-\cos t)^2(1+\alpha)^2 + (\sin t)^2 (1+\alpha)^2 \right]}{(1+({\rho}_0)^2-2{\rho}_0 \cos t)^2 (1-\alpha)^2 (1+\alpha)^2}.
x_{t}^i(k)=\begin{cases}x_{t}^{i-1}(k), \t\in I_n^{i}, \k\le \sum\limits_{j< i}n_j, 0, \t\in I_n^i, \ \sum\limits_{j< i}n_j< k\le 1+n+ \sum\limits_{j< i}n_j, t, \t\in I_n^i, \k>1+n+ \sum\limits_{j< i}n_j.\end{cases}
\|\vartheta(t) - \mu(t) \|_E = \biggl\| \int_{0}^{\sigma(t)}k(t,s)v_1(s)\,ds - \int_{0}^{\sigma(t)}k(t, s)v_2(s)\,ds\biggr\|_E &\leq \int_{0}^{\sigma(t)}k(t, s)\, ds \| v_1(s) - v_2(s) \|_E &\leq \sup_{t, s \in J}k(t, s) \varphi(\Delta(v_1, v_2)) v_1, v_2\in S^1_F(x).
d_{\lambda}^{(1)}= d_{(\lambda_1+1,\lambda_2,\cdots \lambda_j)}+ d_{(\lambda_1,\lambda_2+1,\cdots \lambda_j)}+ \cdots + d_{(\lambda_1,\lambda_2,\cdots \lambda_j+1)}+ ( M-j )d_{(\lambda_1,\lambda_2,\cdots \lambda_j,1)}.
\begin{aligned}\frac{1}{m}>\bar{f}_R^{P_m}(x,y) &\geq 1-\frac{|((W_R(x,y))^c\cup h^{-1}(W_R(x,y))^c)\cap[0,R]|}{R}- &\frac{1}{R}\sum_{j\geq0}|C_{j,R,m}(x,y)\cap W_R(x,y)\cap h^{-1}(W_R(x,y))| &\geq\frac{9}{10}-\frac{1}{R}\sum_{j\geq0}|C_{j,R,m}(x,y)\cap W_R(x,y)\cap h^{-1}(W_R(x,y)) |\end{aligned}
\eta_k := \max\{\beta_k, \|g_1^k\|, \|g_2^k\|\}, \ \lambda_k := \dfrac{\beta_k}{\eta_k}, y^k := \arg\min\{\lambda_k f_1(x^k, y) + \dfrac{1}{2}\|y - x^k\|^2 \mid y \in C\}, z^k := \arg\min\{\lambda_k f_2(x^k, y) +\dfrac{1}{2}\|y - y^k\|^2 \mid y \in C\}, x^{k+1}:= \gamma z^k + (1-\gamma)T(x^k).
\frac 1{\sqrt{- G}}\partial_{\mu}\left( \sqrt{- G}G^{\mu \nu}\partial_{\nu}\phi \left( x \right) \right) = \frac 1{F^{d}}\partial_{\mu}\left( F^{d - 2}\eta^{\mu \nu}\partial_{\nu}\phi \left( x \right) \right) = 0 .
p(2+p)(4+p)\cdots (j-1+p) &=2^{\frac{j+1}{2}}(0+\frac{p}{2})(1+\frac{p}{2})(2+\frac{p}{2})\cdots (\frac{j-1}{2}+\frac{p}{2}) &=2^{\frac{j+1}{2}}\prod_{k=1}^{\frac{j+1}{2}}\Big(k-1+\frac{p}{2}\Big)=2^{\frac{j+1}{2}}\frac{\Gamma((j+1)/2+p/2)}{\Gamma(p/2)}.
(n+1) \Delta_{P, n}^{(\alpha, \beta; \,\ell+2)}(x) + (\ell+1) \Delta_{P, n+1}^{( \alpha, \beta; \,\ell+1)}(x) + (\ell +1)(1-x) \Delta_{P, n+1}^{( \alpha+1, \beta; \ell)}(x) + (n + \frac{\alpha + \beta}{2}+ 1) (1-x) \Delta_{P, n}^{( \alpha + 1, \beta; \ell + 1)}(x)= \alpha \Delta_{P, n}^{(\alpha, \beta; \,\ell+1)}(x).
z(t, \cdot) = \sum_{k = 0}^{s}\frac{z^{k}t^{k}}{k!}+ P_{z}(t, \cdot) t^{s + 1}, \theta(t, \cdot) = \sum_{k = 0}^{s-1}\frac{\theta^{k}t^{k}}{k!}+ P_{\theta}(t, \cdot) t^{s},
\|\log(\gamma_M^{-1}\cdot\gamma_L)\|_{}\leq \|\sigma_L-\sigma_M\|_{}+K(\alpha^{\frac1r}A)^2\leq \mathcal{V}_r(\sigma;[M,N])+KA^2 &=A(1+KA)\leq A(1+2K\delta)=O(A),
f'(Q_n \cdot (\exp(H), 1), s)=\Big(e^{d+(2n+1)({\rm Re}(s)+1/2)}\Big)^{a_1+\cdots+a_n}f'(\left[\begin{smallmatrix}I_n &I_n &A I_n A I_n\end{smallmatrix}\right]\left[\begin{smallmatrix}I_n & I_n & I_n &-I_n\end{smallmatrix}\right],s)
S=\frac{1}{2}\int d^{4}x \epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg[ ({^+}\tau -{^-}\tau) F^{~~ab}_{\mu\nu}F^{~~cd}_{\alpha\beta}+ ({^+}\tau +{^-}\tau) F^{~~ab}_{\mu\nu}\tilde F^{~~cd}_{\alpha\beta}\bigg].
\varepsilon^{\mu \nu \lambda}\partial_{\mu}X^{9}\bar{S}\gamma_{9}\gamma_{\nu}\psi_{\lambda}+ \sqrt{- G}\partial_{\mu}X^{9}\bar{S}\gamma_{9}\psi^{\mu}= - 2 \partial_{i}X^{9}S^{T}\gamma_{9}\psi_{j}\varepsilon^{i j},
\aligned b_N(\xi) = -4\pi^2 \sum_{k,l\in \Lambda_N}\textbf{1}_{\Lambda_N}(k+l) \xi_k \xi_l \frac{k\cdot l^\perp}{|l|^2}\tilde e_{k+l}= -4\pi^2 \sum_{j\in \Lambda_N}\bigg[\sum_{l\in \Lambda_N}\textbf{1}_{\Lambda_N}(j-l) \xi_l \xi_{j-l}\frac{j\cdot l^\perp}{|l|^2}\bigg] \tilde e_j , \endaligned
\{S^{p},T^{r}\}=\left(\sum_{k_{p}=0}^{r}...\sum_{k_{2}=0}^{k_{3}}\sum_{k_{1}=0}^{k_{2}}\alpha^{k_{1}}\alpha^{k_{2}}...\alpha^{k_{p}}\right) S^{p}T^{r}.
| a_n| r^n + 2 \Re( a_0 ) \leq \frac{1}{\pi}\int_0^{2 \pi}| \Re( f(r e^{i \theta}) ) | + | \Re( f(r e^{i \theta}) | d\theta \leq 4 \,\, \underset{|z| = r}{max}(\Re( f(z)), 0 ).
J_{1-r^2(1-l_1^2)/4}\left(\frac{d}{2},\frac{1}{2}\right)=\frac{\int_0^{1-r^2(1-l_1^2)/4}t^{\frac{d}{2}-1}(1-t)^{\frac{1}{2}-1}dt}{\int_0^{1}t^{\frac{d}{2}-1}(1-t)^{\frac{1}{2}-1}dt}.
M_{5}^{\mu \nu}\equiv S p \left\langle 0 \right| \gamma^{\mu}\frac{1}{{\cal P}\! \! \! \! / - m + i \varepsilon}\gamma^{\nu}\gamma^{5}\frac{1}{{\cal P}\! \! \! \! / - k \! \! \! / - m + i \varepsilon}\left| 0 \right\rangle ,
2\mathbb{G}^\alpha(T_o) \frac{\partial r}{\partial z^\alpha}= (\hat{\mathbb{G}}^{\alpha}(T) + \sqrt{-1}\hat{\mathbb{G}}^{\alpha+n}(T))(\frac{\partial r}{\partial x^\alpha}-\sqrt{-1}\frac{\partial r}{\partial x^{\alpha+n}}) &=\hat{\mathbb{G}}^{k}(T)\frac{\partial r}{\partial x^k}-\sqrt{-1}\hat{\mathbb{G}}^{i}(T)J^k_i\frac{\partial r}{\partial x^k}.
\begin{aligned}\dfrac12\lim_{m\to \infty}\mathcal L\sum_{i,j,k,p}( h^{p^{\ast}}_{ijk})^2(p_m) &=\bar H^2\biggl[-\dfrac{\bar H^4}2+\dfrac{3\bar H^2}2 -S+\dfrac{(\bar H^2-1)\sqrt{(4S-3\bar H^2)\bar H^2}}2 \biggl ]. \end{aligned}
\P(X_t< N/2, \, \forall t\in[0,T_N]|X_0=0)\ge \P\left(\cap_{k=1}^{T_N}\{\sigma_k-\sigma_{k-1}< N/2\}|X_0=0\right) &\ge \left(1-c_1^{-1}e^{-c_1 N}\right)^{T_N}\ge 1-c_1^{-1}T_Ne^{-c_1N}=1-c_1^{-1}e^{-\alpha N}.
\theta ( \vec{x}) \mid U_{\pm}= \int_{N}\frac{- 2 i S_{0}}{1 + z_{\pm}( \vec{x}) \overline{{z}}_{\pm}( \vec{x})}\overline{{z}}_{\pm}( \vec{x}) \delta z_{\pm}( \vec{x}) d \mu ( \vec{x}) .
{\cal H}_{I}( \phi_{r}, \Phi , \mu ) = -{\frac{1}{2}}\mu^{2}\phi_{r}^{2}-{\frac{m^{2}}{\beta^{2}}}\cos ( \beta ( \phi_{r}+ \Phi ) ) \exp \{{\frac{\beta^{2}}{2}}I_{( 1 )}({\cal M}^{2}) \}.
\begin{array}{c c c c c c}{{0}}&{{x_{1}}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{0}}&{{x_{2}}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{x_{3}}}&{{\cdots}}&{{0}}\\{{\vdots}}&{{\vdots}}&{{\vdots}}&{{\vdots}}&{{\vdots}}&{{\vdots}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{\cdots}}&{{x_{p - 1}}}\\{{x_{0}}}&{{0}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\ \end{array}
g_\alpha(p) = (n-1)p^{2n-k-5}+ (n-k-1)(n-2) p^{n-3}(p-1)- (n-k-1)p^{n-3}&+\left((n-k-2) + \binom{n-k-2}{2}\right)p^{n-3}(p-2)(p-3) .
a_{n,j}=\frac{1}{(n+2-j)!}\sum_{m=0}^{n+2-j}\binom{n+2-j}{m}(-1)^m(n+2-m)(n+1-m) &=\frac{1}{(n+2-j)!}\frac{d^2}{dx^2}\left.\left\{\sum_{m=0}^{n+2-j}\binom{n+2-j}{m}x^{n+2-m}(-1)^m\right\}\right|_{x=1}&=\frac{1}{(n+2-j)!}\frac{d^2}{dx^2}\left.\left\{x^j(x-1)^{n+2-j}\right\}\right|_{x=1}.
\phi_{2}^{\ast}\!=\!\min\!\bigg(\!\underbrace{\frac{1}{2^{Q_1}}\!+\!\frac{\sigma_1^2}{P|h_{1}|^{2}2^{Q_{1}}}\!-\! \frac{\sigma_1^2}{P|h_{1}|^{2}}}_{\phi_2^{\dag}},~~\phi_2^{\ddag}\!\bigg),\! &\phi_{1}^{\ast}=1-\phi_{2}^{*}, R_{E}^{\ast}=R_E^{\dag}(\phi_2^{\ast}),
[\sum_{i=2m}^{\infty}\nu_i^{(m)}u^iY_i,t_k]=\sum_{\substack{l\geq 0 l\geq m-k-1}}(-1)^{l+1}c_{k-m+l+1}^{(k,m)}u^{2k+2l+2}\frac{\partial}{\partial t_l}.
v_2''(t) + Av_2'(t) = \kappa v'(t) - \sum\limits_{j=1}^{m}\omega_{\alpha_j}(t)B_ju_0 - \sum\limits_{j=1}^{m}\int_0^t \omega_{\alpha_j}(s)B_j \frac{\partial v(t-s)}{\partial t}ds,
\left|A_0-B_{0,n-1}A_{n-1}\right|+\left|B_{0,N}A_N-B_{0,n}A_n\right|\le C_{|A|}\sum_{j=1}^{N}\frac{c^{A}_J}{\lambda}\hat{b}_{j-1,j}\exp \left(\frac{c^{A}_J}{\lambda}\sum_{k=1}^N\hat{b}_{k-1,k}\right) \quad.
a =(I+\mathcal{K}^*)^{-1}\operatorname{Re}\left\{ie^{i\theta}\left( [z_t,\mathfrak{H}]\frac{\bar{z}_{t\alpha}}{z_\alpha}+e^{i\theta}(I-\mathfrak{H}){\bf 1}+(I-\mathfrak{H})\bigl(\frac{\partial_\alpha P}{|z_\alpha|}e^{-i\theta}\bigr)\right)\right\}.
\int_{-\infty}^{\infty}\mathcal{L}_sf(z,w)e^{i\lambda w}\,dw=(2\pi)^{-n}|\lambda|^n\sum_{k=0}^{\infty}(2|\lambda|)^{s}\frac{\Gamma\big(\frac{2k+n}{2}+\frac{1+s}{2}\big)}{\Gamma\big(\frac{2k+n}{2}+\frac{1-s}{2}\big)}f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z),
= \;\|\, u(\cdot,t_0) \,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}^{\:\!q}\!\;\!+\:q \,(q - 1)\!\int_{\mbox{}_{\scriptstyle t_{\mbox{}_{0}}}}^{\:\!t}\!\;\!\int_{\mbox{}_{\scriptstyle \mathbb{R}^{n}}}\!\!|\,u\,|^{\:\!q - 2}\,\langle \,\mbox{\boldmath $f$}(x,\tau,u), \;\! \nabla u\,\rangle\;dx\,d\tau,
e_{\epsilon}(\tilde{u}_{\epsilon})=e_{\epsilon}(u_{\epsilon})-\frac{1}{2}|j\phi_{\epsilon}|^2-\langle d^*\xi_{\epsilon}+h_{\epsilon}',j\phi_{\epsilon}\rangle-\frac{1}{2}(1-|u_{\epsilon}|^2)|j\phi_{\epsilon}|^2,
\hat{\cal M}= \left( \hat{\cal M}_{i j}\right) = \frac{1}{\Im \mathrm{m}\lambda}\left( \begin{array}{r r}{{| \hat{\lambda}|^{2}}}&{{- \Re \mathrm{e}\hat{\lambda}}}\\{{- \Re \mathrm{e}\hat{\lambda}}}&{{1}}\\ \end{array}\right) \, ,
d s_{\mathrm{B T Z}}^{2}= -{\frac{( \rho^{2}- \rho_{+}^{2}) ( \rho^{2}- \rho_{-}^{2})}{\rho^{2}\ell^{2}}}d t^{2}+ \rho^{2}\left( d \phi -{\frac{J}{2 \rho^{2}}}d t \right)^{2}+{\frac{\rho^{2}\ell^{2}}{( \rho^{2}- \rho_{+}^{2}) ( \rho^{2}- \rho_{-}^{2})}}d \rho^{2}.
STDDEV(X)=s(x)=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}=\sqrt{\frac{1}{N}(\sum_{i=1}^{N}x_{i}^{2})-(\overline{x})^{2}}=\sqrt{SUM(x^{2})/COUNT(x)-AVG(x)^{2}}
\|\mathcal{A}(\mathcal{M}_\ell)^\nu\|_{\mathcal{L}\times \mathcal{L}}\le \sqrt{2}\, \eta(\nu-1) \, \|\mathcal{D}_d \|_{\mathcal{L}\times \mathcal{L}}\mathcal{D}_d = \begin{bmatrix}\hat{A}0 0 \hat{S}\end{bmatrix}.
k_{n}=(-1)^{n}t [c_{1}\alpha_{1}(\alpha_{1}-1)(\alpha_{1}-2)\cdots(\alpha_{1}-n+1){\lambda_{1}}^{\alpha_{1}-n}+c_{2}\alpha_{2}(\alpha_{2}-1)(\alpha_{2}-2) &\cdots (\alpha_{2}-n+1){\lambda_{2}}^{\alpha_{1}-n}].
\sum_{n_1,n_2 \geq 0}\frac{q^{\frac{3}{2}n_1^2 + 4 n_1 n_2 + 4 n_2^2 - \frac{1}{2}n_1}}{(q;q)_{n_1}(q^4;q^4)_{n_2}}= \frac{1}{(q,q^4,q^7;q^8)_\infty}, \sum_{n_1,n_2 \geq 0}\frac{q^{\frac{3}{2}n_1^2 + 4 n_1 n_2 + 4 n_2^2 + \frac{3}{2}n_1 + 4n_2}}{(q;q)_{n_1}(q^4;q^4)_{n_2}}= \frac{1}{(q^3,q^4,q^5;q^8)_\infty}.
(\theta, \boldsymbol u) \in \,\mathcal C([0, T], H^{s_1})\times \mathcal C([0, T], H^{s_2}), (\theta_t, \boldsymbol u_t) \in L^2(0, T; H^{s_1-\beta})\times L^2(0, T; H^{s_2-\alpha}).
\beta_{s,a^2}^2=\frac{\left[r^2(s,a_s^1,a^2)-r^2(s,a_s^1,a_s^2)\right]}{\left[r^2(s,a_{s}^1,a^2)-\sum_{s'\in S}p(s'|s,a_s^1,a^2)r^2(s',a_{s'}^1,a_{s'}^2)\right]}.
d((3,1)) =kd_1, d((2,2)) =\begin{cases}kd_1/2 (kd_1: \textrm{even}), kd_1 (kd_1: \textrm{odd}),\end{cases}d((1,3)) =\begin{cases}kd_1/3 (k\equiv 0 \mod 3), kd_1 (k\not\equiv 0 \mod 3),\end{cases}
\dim \delta H_3^g = \; 6 + 9 (k+2) +(k+1)(k+2) + 3\,k(k+1)/2 &\;+{(k-1)k(k+1)}/{6}-{(k+4)(k+5)(k+6)}/{6}= \; k+6, \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g)=\; k+3,
A_{4}^{L}={\frac{\Gamma ( - \hat{s}/ 8 + M^{2}/ 2 ) \Gamma ( - \hat{t}/ 8 )}{\Gamma ( 1 - \hat{t}/ 8 - \hat{s}/ 8 + M^{2}/ 2 )}}K^{L}( 1 , 2 , 3 , 4 )
\partial_{t}( v - \langle v \rangle ) + v \partial_{x}v - \frac{{\Delta u}^{2}+{\Delta v}^{2}}{{\Delta q}^{2}}( x - \langle q \rangle ) + \frac{1}{2}\partial_{x}\bigg ( u^{2}+ \frac{\hbar}{m}\partial_{x}u \bigg ) \, = \, 0 \: .
\sum_{i, j=1}^NV^i_j \, (V^j-\ell_j)\, n^i =-\frac{1}{|U|}\sum_{i, j, k=1}^NV^i_j\, (V^j-\ell_j)\, (V^k-\ell_k)\, u_{k i}&=- \frac{1}{|U|}\,\sum_{i,j,k,h=1}^N F_{i h}(Du)\, u_{h j}\, (V^j-\ell_j)\, u_{i k}\, (V^k-\ell_k) &= -\frac{1}{|U|}\, \big(D^2F(Du)\, U, U\big)< 0,
\hat{X}_{\mu}\hat{W}\equiv - \frac{\partial}{\partial p_{\nu}}\left( \int_{0}^{1}( 1 - \frac{1}{2}s ) e^{- \frac{i ( 1 - s )}{2}\bigtriangleup}F_{\mu \nu}\hat{W}d s + \hat{W}\int_{0}^{1}\frac{1 - s}{2}e^{\frac{1}{2}i s \bigtriangleup}F_{\mu \nu}d s \right)
\sup_{\theta_{0}\in l_{0}[q_{n}]}E_{\theta_{0}}\sum_{i=1}^{n}Var(\theta_{0i}|X_i) \lesssim q_n\log\big(\frac{1}{\tau^{2a}}\big)+(n-q_n)\tau^{2a}\sqrt{\log\big(\frac{1}{\tau^{2a}}\big)}.
2 i{\frac{( 1 2 ) ( 1 3 ) ( 2 3 )}{x_{1 2}^{4}x_{3 4}^{4}}}\left[{\frac{c}{s}}+ \beta_{0}{\frac{1 +{\frac{\beta_{1}}{\beta_{0}}}U +{\frac{\beta_{2}}{\beta_{0}}}U^{2}}{1 +{\frac{1 + s - t}{s}}U +{\frac{1}{s}}U^{2}}}\right] X_{2}^{\mu}
\mathrm{J}_3=\int_{\vert x-x(t)\vert<R_0}\{p_x*u_x^3\}u\Psi'+\int_{\vert x-x(t)\vert>R_0}\{p_x*u_x^3\}u\Psi'=:\mathrm{J}_3^1+\mathrm{J}_3^2.
\frac{\partial E}{\partial z^i_{\epsilon}}v^i= 2b^2f(x^1) \cos\left( x^2\right) \left[ -\delta_{\epsilon 1}f(x^1)f'(x^1) \cos\left( x^2\right) +\delta_{\epsilon 2}\left\lbrace 1 +f'^2 \left( x^1\right) \right\rbrace \sin\left( x^2\right) \right].
\hat{H}_{\mathrm{r e l}}^{\prime}= - \frac{1}{2 \mu}\left[ \frac{\partial^{2}}{\partial r^{2}}+ \frac{1}{r}\frac{\partial}{\partial r}+ \frac{1}{r^{2}}\left( \frac{\partial}{\partial \theta}+ i \omega_{j}\right)^{2}\right] .
\Delta(\lambda) \phi_1(0) = \int^0_{-r}e^{\lambda(-r-\tau)}\alpha A_1\psi_1(\tau)d\tau + \int^0_{-r}\beta(\theta) \int^0_{\theta}e^{\lambda(\theta-\tau)}A_2\psi_1(\tau)d\tau d\theta+\psi_0.
K^2 +b = \frac{1}{4 x^2}\:\big( x^4 -2bx^2 + b^2 + 4 b x^2 \big) = \frac{1}{4 x^2}\:\big( x^2 +b \big)^2 = \frac{1}{4 x^2}\:\big( -a +b \big)^2 A = \frac{\sqrt{K^2+b}}{x}= \frac{-a+b}{2x^2}= \frac{a-b}{2a}B = \frac{K}{x}= \frac{1}{2x^2}\:\big( x^2-b \big) = -\frac{1}{2a}\:\big( -a-b \big)= \frac{a+b}{2a}\:.