harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{B}_{p}(F,f,L)=\sup\left\{\dfrac{1}{\left\vert Q\right\vert}\int_{Q}(M_{d}\phi)^{p}:\operatorname{Av}_{Q}(\phi^{p})=F,\operatorname{Av}_{Q}(\phi)=f,\sup\limits_{R:Q\subseteq R}\operatorname{Av}_{R}(\phi)=L\right\}
\mathrm{T r}\left[ \gamma^{0}\langle \psi ( \vec{x}_{0}, t ) \bar{\psi}( \vec{x}_{0}, t ) \rangle_{\beta}\right] = \frac{\Delta_{n}}{2 \pi}\int_{- \infty}^{\infty}d \omega \; F_{1 / 2 , n}( \omega ) .
a_1 = -a_{22}, a_2 = -(a_{11}a_{33}+ a_{12}a_{32}+ a_{13}a_{31}+ a_{21}a_{23}), a_3 = a_{12}a_{23}a_{31}+ a_{13}a_{21}a_{32}- a_{11}a_{23}a_{32}- a_{12}a_{21}a_{33}.
\begin{aligned}\rho(x)\partial_t^2 w_t^n - \operatorname{div}(K(x)\nabla w_t^n) \rightarrow 0~ \hbox{in}~H^{-1}_{loc}(\Omega \times (0,T)), \rho(x)\partial_t^2 z_t^n - \operatorname{div}(K(x)\nabla z_t^n) \rightarrow 0~ \hbox{in}~H^{-1}_{loc}(\Omega \times (0,T)),\end{aligned}
\lim\limits_{\lambda_s^{}\to 1}{\pi_r(B_r)}=\lim\limits_{\pi_s(0) \to 0}\frac{\mathrm{C}_{B_r}(1-\pi_s(0))^{B_r}}{\sum\limits_{k=0}^{B_r}{\mathrm{C}_k}(1-\pi_s(0))^k}& =\frac{\mathrm{C}_{B_r}}{\sum\limits_{k=0}^{B_r}{\mathrm{C}_k}}=\frac{n-2}{n-2+B_r}.
\left| \Phi \right\rangle = \left\{\phi_{0}, \phi_{1}^{( \mu_{1})}\left( p_{1}\right) , \phi_{2}^{( \mu_{1}) ( \mu_{2})}\left( p_{1 ,}p_{2}\right) , . . . , \phi_{n}^{( \mu_{1}) . . . ( \mu_{n})}\left( p_{1}, . . . , p_{n}\right) , . . . \right\},
\Pi_{S^{1}}( q^{2}; R ) \simeq -{\frac{1}{2 4 \pi^{2}}}\left[ \ln \left( 2 \pi R \sqrt{q^{2}}\right) -{\frac{4}{3}}\right] + \mathrm{a n a l y t i c ~ i n ~ q^{2}~ R^{2}~}.
\hat{\varrho}_{\mathrm{f l d}}= \hat{\varrho}_{m}\left[ 1 + \zeta \left({\frac{\hat{c}}{c}}\right)^{2}\right] + \hat{\varrho}_{r}\left({\frac{\hat{c}}{c}}\right)^{2}, \ \hat{\wp}_{\mathrm{f l d}}={\frac{1}{3}}\hat{\varrho}_{r}\left({\frac{\hat{c}}{c}}\right)^{2},
\sum_{}\sum_{1 \leq i \leq \ell}{\ell + 1\choose i}{n - \ell - 1\choose k - i}\alpha_{A_i}+{\ell + 1\choose \ell + 1 - i}{n - \ell - 1\choose i - 1}\beta_{B_i}&= (\ell + 1)! (|\mathcal F| + |\mathcal G|)
\nu_{D_1}(z)=\nu_{D_{i_0}}(z)\ge\sum_{i\ne i_0}\left (\max\{0,\nu_{(f\tilde a_i)}^{0}(z)-k\}-(k+1)\bigl (2\nu_{c_{si}}^{\infty}(z)+\nu^0_{a_{i0}}(z)\bigl )\right ).
\frac{d\eta_{b}}{d\rho}=\frac{(1-\rho^{2}+2\rho cos\alpha_{1})(4cos\alpha_{1}-4\rho)-2\rho(2cos\alpha_{1}-\rho)(-2\rho+2cos\alpha_{1})}{(1-\rho^{2}+2\rho cos\alpha_{1})^{2}}=0
\Vert w\Vert_{\infty,K}= \sup_{1\le k \le K}\Vert w(y^k) \Vert_V, \; &\Vert w \Vert_{p,K}= \big(\sum_{k=1}^K \omega^k \Vert w(y^k) \Vert_V^p\big)^{1/p}p<\infty,
f_r(f_s(m_i)) -f_s(f_r(m_i)) =\Bigl((i+s+4r+4)(i+4s+4)-(i+r+4s+4)(i+4r+4)\Bigr)m_{i+s+r}=(s-r)(i+4(r+s)+4)m_{i+s+r}=[f_r,f_s](m_i).
\dim H^1(G_T, W_{[n]}) \leq \sum_{i=1}^{n}\left( \dim Z_{[i]}+ \dim H^2(G_T, Z_{[i]}) - \dim Z_{[i]}^{+}\right) = \dim W_{[n]}+ \sum_{i=1}^{n}\dim H^2(G_T, Z_{[i]}) - \sum_{i=1}^{n}\dim Z_{[i]}^{+}.
T =2\pi\xi \,n+\int \!{\rm d}^2 x\left\{\frac{1}{2g^2}\left[F_{12}+g^2\left(|\varphi|^2-\xi\right)\right]^2+ \left|\left({\cal D}_1+i{\cal D}_2\right)\varphi\right|^2 \right\}.
X_{1}=~\{x, T^{*3}x ,\ldots, T^{*3(3j-n-r)}x, y, T^{*3}y ,\ldots, T^{*3(3j-n-2r)}y\}X_{2}=~\{T^{*3(3j-n-r+1)}x ,\ldots,T^{*3j}x, T^{*3(3j-n-2r+1)}y ,\ldots, T^{*3(j-1-r)}y, z,\ldots, T^{*3(n-2j-2+r)}z\}.
(u^2+2tu+2t+8u+8)([3+t+u]b_2-[1+2t]c_2-ub_3) & =(u^2+4tu+4t+4u+2)([2+2t+u]c_2-[2+t]b_2-uc_3) & +2u(u+1)b_2+u^2b_3+u^2c_3
\overline{{c}}_{- 1}^{( 1 )}\equiv \zeta ( 2 ) \frac{4 8 \zeta ( 2 ) \zeta ( k ) - 1 6 \zeta ( 4 ) \zeta ( k ) - 4 8 \zeta ( 4 ) \zeta ( k - 2 )}{\zeta ( 2 ) \zeta ( k ) - \zeta ( 4 ) \zeta ( k - 2 )}
b=e^{\frac{i}{2}\ln(\frac{2\hbar}{\theta})D}\left(\frac{1}{\sqrt{2}}\left(\frac{\tilde{x}_1+i\tilde{x}_2}{\sqrt{2\hbar}}\right)+\frac{i}{\sqrt{2}}\left(\frac{\tilde{p}_1+i\tilde{p}_2}{\sqrt{2\hbar}}\right)\right)e^{-\frac{i}{2}\ln(\frac{2\hbar}{\theta})D}
\phi_j(s)=\frac{1}{\sqrt{T-t}}\begin{cases}1,\ \{\rm if}\j=0 ~ \sqrt{2}{\rm sin}(2\pi r(s-t)/(T-t)),\ \{\rm if}\j=2r-1 ~ \sqrt{2}{\rm cos}(2\pi r(s-t)/(T-t)),\ \{\rm if}\j=2r\end{cases},\ \ \r=1,\ 2,\ldots
\begin{array}{c c}{\bar{\theta}_{i}= \theta^{i \dagger}\tilde{\gamma}^{0}= ( \theta^{j})^{t}\bar{\cal E}_{j i}}&{\bar{\tilde{\theta}}_{i}= \tilde{\theta}^{i \dagger}\gamma^{0}= ( \tilde{\theta}^{j})^{t}\bar{\cal E}_{j i}}\\ \end{array}
l_\sigma(u^r;0,0) = l_\sigma(u^{-r};0,0)^{-1}= ((B u^{-1})^{-r}B^r ; -r,0)^{-1}= ( ( \theta(r,0)( (Bu^{-1})^{-r}B^r ) )^{-1}; r,0 ) & = ( ( ( \theta(r,0)(B) \theta(r,0)(u)^{-1})^{-r}\theta(r,0)(B)^r )^{-1};r,0 ) & = ( ( ( B (B^r u B^{-r})^{-1})^{-r}B^r )^{-1}; r,0 ) = ( (B u^{-1})^{r}B^{-r};r,0 ).
w_\alpha(x) = r_\alpha^{n-2}\mu_\alpha^{-\frac{n-2}{2}}u_\alpha\left(\exp_{x_\alpha}(r_\alpha x) \right), &g_\alpha(x) = \left(\exp_{x_\alpha}^\star g\right)(r_\alpha x),~ &\tilde{h}_\alpha(x) = h_\alpha\left(\exp_{x_\alpha}(r_\alpha x) \right),
\begin{array}{r}\dfrac{s^{n+1}-s^{n}}{\Delta t}-\dfrac{r_{xx}^{n+1}+r_{xx}^{n}}{2}+q\dfrac{((r^{2}+s^{2})r)^{n+1}+((r^{2}+s^{2})r)^{n}}{2}=0 \dfrac{r^{n+1}-r^{n}}{\Delta t}+\dfrac{s_{xx}^{n+1}+s_{xx}^{n}}{2}+q\dfrac{((r^{2}+s^{2})s)^{n+1}+((r^{2}+s^{2})s)^{n}}{2}=0\end{array}
|v|^{2\rho-1}\int_{|t|\geqslant|v|^{-\sigma}}J_{1,2}dt=|v|^{2\rho-1}\left(\int_{|v|^{-\sigma}\leqslant|t|<|v|^{1-2\rho}}+\int_{|t|\geqslant|v|^{1-2\rho}}\right)J_{1,2}dt.
\pi_{i_1}\pi_J = \pi_{J_1}\cdots \pi_{J_{k-1}}(\pi_{i_1}\pi_{J_k})\pi_{J_{k+1}}\cdots \pi_{J_m}= \pi_{J_1}\cdots \pi_{J_{k-1}}\left( \binom{|J_k|+1}{1}\pi_{J'(i_1)}\right) \pi_{J_{k+1}}\cdots \pi_{J_m}
S=\int d^{2}ze[\frac{1}{2}\nabla_{a}X^{\mu}\cdot\nabla_{a}X^{\mu}-\frac{1}{2}i\overline{\psi}^{\mu}\gamma^{a}\nabla_{a}\psi^{\mu}+\frac{1}{2}i(\overline{\chi}_{a}\gamma^{b}\gamma^{a}\psi^{\mu})(\partial_{b}X^{\mu}-\frac{1}{4}i\overline{\chi}_{b}\psi^{\mu})],
\int_{K_t}\omega_\phi^n\wedge \eta\leq \frac{1}{t}\int_M -\psi \omega_\phi^n\wedge \eta \leq \frac{1}{t}\left(\int_M -\psi (\omega^T)^n\wedge \eta+n \int_M (\omega^T)^n\wedge \eta\right)
\left<u, \tilde{L}^\alpha_t(\varphi)x^\star \right>_{L^2([0,t]:H)}=\int_0^t \left<u(s), (t-s)^{-\alpha}[S(t-s)G(s,\varphi(s))]^\star x^\star \right>_Hds = \int_0^t\left<(t-s)^{-\alpha}S(t-s) G(s,\varphi(s))u(s), x^\star\right>_{E_1,E_1^\star}ds =\left<L^\alpha_t(\varphi)u,x^\star \right>_{E_1,E_1^\star}.
D_t^2\dot{u}=\dot{k}\nu-k^2\dot{u}, D_t^3\dot{u}=\ddot{k}\nu+\dot{k}(-k\dot{u})-2\dot{k}k\dot{u}-k^2(k\nu)=\ddot{k}\nu-3\dot{k}k\dot{u}-k^3\nu.
(-1)^{l-3}n^{l+m-2}2^{l+m-2}P_{l-2}(m) \frac{(l+m-2)!}{(l+m-3)!}\alpha_p(1)^{l+m-3}\alpha_p(3) = (-1)^{l-3}n^{l+m-2}2^{l+m-2}P_{l-2}(m)(l+m-2)\alpha_p(3).
\tilde D_{k+1}^{\ell}(\Gamma_{k+1}^{\ell}) = \sum_{i \in I_k}\mathbb E \big[ d({\cal E}_k^{\ell}(x_k^{i},Z_{k+1}), \Gamma_{k+1}^{\ell})^2 \big] \mathbb P\big(\hat X_k =x_k^{i}\big)
\Psi(f,g) = \int_0^{2\pi}\left(d\widehat F(t) f(t), g(t)\right)_{\mathcal{H}}:= \lim_{\delta\to +0}\sum_{k=1}^N \left(\widehat F(\Delta_k) f(t_k), g(t_k)\right)_{\mathcal{H}},
\begin{array}{l}\vert 0, z\rangle_{+k}=\displaystyle\exp_{q^{-2}}(-\frac{1}{2}|q^2z|)\sum_{m=0}^\infty \frac{(-q^2z)^m}{[m]_{q^{-2}}!}({\cal D}_{+k})^m\vert 0 \rangle_{+k}~,~~~~~{\cal X}^{+k}\vert 0\rangle_{+k}=0~,{\cal X}^{+k}\vert 0,z\rangle_{+k}=z\vert 0,z\rangle_{+k}~.\end{array}
|Q_{-j,m}|^{-\frac{1}{p}}\Big(\sum_{k:Q_{0,k}\subset Q_{-j,m}}\!\!\! |\lambda_k|^p\Big)^\frac{1}{p}\leq \|Q_{-j,m}|^{-\frac{1}{p}}\|\lambda|\ell_\infty\|\ |Q_{-j,m}|^{\frac1p}= \|\lambda|\ell_\infty\|,
V_0=\left(\int_\Omega|v_0|^{2\,p/(p-1)}\,dx\right)^{-1/p}|v_0|^{2/(p-1)}\mbox{and}U_0=\left(\int_\Omega|u_0|^{2p/(p+1)}\,dx\right)^{1/p}|u_{0}|^{-2/(p+1)}.
\int_{|z|>r_{\epsilon}}|\mathsf{P}(f)(z)|^{4}(1-|z|^{2})^{3}|\sigma_{a}'(z)|^{p}dm(z) &\leq{\epsilon}^{2}\int_{\mathbb{D}}|\mathsf{P}{(f)}(z)|^{2}(1-|z|^{2})|\sigma_{a}'(z)|^{p}dm(z) &\leq{\epsilon}^{2}\kappa\int_{\mathbb{D}}|(\mathsf{P})(f))'(z)|^{2}(1-|z|^{2})^{3}|\sigma_{a}'(z)|^{p}dm(z) &={\epsilon}^{2}\kappa I_{a}.
D=a_1 \frac{\partial}{\partial x_1}+a_2 \frac{\partial}{\partial x_2}=\begin{bmatrix}0 -\frac{\partial}{\partial x_1}+i \frac{\partial}{\partial x_2}\frac{\partial}{\partial x_1}+i \frac{\partial}{\partial x_2}0\end{bmatrix}
E\big[|\textit{\textbf{S}}(t)|^2\big] = (1-\lambda)\beta_{1,k}^2 M^2 + \mathcal{O}\Big(\frac{M^2}{\tau}\Big) \approx (1-\lambda)\beta_{1,k}^2 M^2, E\big[|\textit{\textbf{N}}(t)|^2\big] = \sigma_n^2\beta_{1,k}M + \mathcal{O}\Big(\frac{M}{\tau}\Big) \approx \sigma_n^2\beta_{1,k}M.
\tilde L_1^{(1)}\tilde u_1=0\ \ \mbox{in}\ \ \hat\sigma_0^+, \tilde u_1\big|_{x_0=t_1+\delta}=\frac{\partial\tilde u_1}{\partial x_0}\Big|_{x_0=t_1+\delta}=0, \tilde u_1\big|_{(t_1+\delta,t_2-\delta)\times\gamma_0}=f,\ \ \ \tilde u_1\big|_{\partial \hat\sigma_0^+}=0,
2\Phi_{xx}= -\frac{1}{\Phi_{xq}}[\Phi_{xy}(\Phi_{yq}+\Phi_{yp})+\Phi_{xq}(\Phi_{pq}+\Phi_{qp})], 2\Phi_{qq}= \frac{1}{\Phi_{xq}}[\Phi_{xy}(-\Phi_{yq}+\Phi_{yp})+\Phi_{xq}(-\Phi_{pq}+\Phi_{qp})].
\begin{array}{rl}H_1= \langle(1,2,4)(3,11,13,6,7,14)(5,29,22,16,10,26)(8,18,20,9,17,21)(12,24,35) &(15,33,41,37,28,46)(19,32,44)(23,25,36,40,27,38)(30,42,43)(31,34,45,47,39,48), &(1,3)(2,5)(4,8)(6,12)(7,15)(9,19)(10,23)(11,18)(13,25)(14,26)(16,30) &(17,31)(20,33)(22,34)(24,36)(27,39)(32,45)(35,46)(37,47)(38,44)(41,42)(43,48)\rangle\end{array}
( K^{- 1})_{S}^{S}= \frac{1 - A | z |^{2}}{P^{\prime \prime}( y )}, \qquad ( K^{- 1})_{T}^{T}= \frac{( T + T^{*})^{2}( 1 - A | z |^{2})}{3 ( 1 - \frac{\bar{n}}{3}B ( 1 - A | z |^{2}) \| \Pi \| )}
\mathcal{F}_{D,m}(z,s) := \Gamma\biggl(\frac{s+1}{2}\biggr)\sum_{n|m}\biggl(\frac{D}{n}\biggr)P_{\frac{1}{2}, \frac{m^2D}{n^2}}\biggl(z,\frac{s}{2}+\frac{1}{4}\biggr).
P_{3}=(\begin{matrix}1&0\\ 0&0\end{matrix});\overline{P}_{3}=(\begin{matrix}0&0\\ 0&1\end{matrix});P_{3}e_{1}=(\begin{matrix}0&1\\ 0&0\end{matrix});e_{1}P_{3}=(\begin{matrix}0&0\\ 1&0\end{matrix}).
[Z_j]_{\alpha, \alpha}= \sum_{i = 1}^{n(T, j+1)}[D_{j+1}]_{i, \alpha}\ [D_{j+1}]_{i, \alpha}\ [R_{j+1}^{-1}]_{i, i}= \sum_{i = 1}^{n(T, j+1)}[D_{j+1}]_{i, \alpha}\ [R_{j+1}^{-1}]_{i, i}\, .
\Theta=\bigcup\limits_{l_i>1}\{\alpha_{\tilde{l}_{i-1}+1},...,\alpha_{\tilde{l}_i-1}\}\bigcup\limits_{l_i>1}\{\alpha_{\tilde{l}_{i-1}+1},...,\alpha_{\tilde{l}_i-1}\}\cup\{\alpha_l\}
\begin{aligned}(k, v) \cdot_0 (x, 0 ) \cdot_0 (k, v)^{-1}= (k, v) \cdot_0 (x, 0 ) \cdot_0 (k^{-1}, -k\cdot v) = (k\cdot x, x^{-1}\cdot v ) \cdot_0 (k^{-1}, -k \cdot v) = (k\cdot x \cdot k^{-1}, kx^{-1}\cdot v - k \cdot v).\end{aligned}
\frac{[z^n] \frac{\partial^2}{\partial x_h^2}(N_1+N_2) |_{x_1=x_2= 1}+ [z^n] \frac{\partial}{\partial x_h}(N_1+N_2) |_{x_1=x_2= 1}}{[z^n] (N_1+N_2) |_{x_1=x_2 = 1}},
\omega =|V(W)\backslash (A\cup B)|+|V(W)\cap (A\cup B)|\leq |V(C)\backslash (A\cup B)|+|V(W)\cap (A\cup B)| &\leq |V(C)|-|A\cup B|+|V(W)\cap A|+|V(W)\cap B|\leq c-(d-1)(t-1).
[y_{p},y_{(1\,l)}]_c=\begin{cases}\lambda_{112}(1-\xi^2)y_{(3\,l)}, p=1; \lambda_{122}(1-\xi)y_{(3\,l)}, p=2; 0, 2<p<l; (1-\xi)y_ly_{(1\,l)}, p=l. \end{cases}, l\geq 3.
{\frac{\alpha}{\pi}}< F_{\mu \nu}^{a}F_{\mu \nu}^{a}> = N M^{4}{\frac{1}{2}}\left( -{\frac{1}{4 0 \pi^{2}}}+{\frac{1}{2 4 \pi^{2}}}\right) ={\frac{N}{1 2 0 \pi^{2}}}M^{4}= 0 . 0 0 8 G e v^{4}
P_n^*(f, \mathcal{F}, \epsilon) = \sup \left\{\sum_{x \in E}e^{\Phi_n(x)}: E \mbox{is $(f, n, \epsilon)$-separated}\right\}&\geq \sup \left\{\sum_{x \in E}e^{S_m \Phi_k(x)}e^{\Phi_r(f^{mk}x)}: E \mbox{is $(f^k, m,\delta)$-separated}\right\}&\geq P_m(f^k, \Phi_k, \delta) e^C.
\lim_{\theta\to 0^+}\frac{f(\theta)}{g(\theta)^2}= \lim_{\theta\to 0^+}\frac{-2 \|P(1, e^{i\theta}, 1,\dots, 1, 0,\dots)\| \big(\frac{d}{d\theta}\|P(1, e^{i\theta}, 1,\dots, 1, 0,\dots)\| \big)}{4\sin(\theta/2)\cos(\theta/2)}.
\ \ \left. \left. + \frac{3}{2}\gamma \varphi_{0}^{2}\right) \dot{\phi}\left( \frac{\dot{a}}{a}\right) \right] + e^{- 2 k \phi}\left[ \frac{\gamma}{2}\varphi_{0}^{2}k^{2}\dot{\phi}^{2}- V_{0}\varphi_{0}^{\alpha}- e^{\psi ( 2 - \alpha )}\right] \ .
g(\Delta u, e^1) =\frac{1}{s^2}\{\partial^2_\theta u_r-c\partial_\theta u_\theta+s\partial_r^2(s u_r)\}&=\frac{1}{s^2}\partial^2_\theta u_r-\frac{c}{s^2}\partial_\theta u_\theta\mp a^2u_r+2\frac cs\partial_r u_r+\partial^2_r u_r,
g(r) = A(r) = C_1^{(1)}\sum_{n=0}^{\infty}\frac{(\alpha)_n(\beta)_n}{n! \left( \frac52 \right)_n}\, r^{2n}+ C_2^{(1)}r^{-3}\sum_{n=0}^{\infty}\frac{\left( \alpha-\frac32 \right)_n\left( \beta-\frac32 \right)_n}{n! \left( -\frac12 \right)_{n}}\,r^{2n}, \qquad k=+1.
\langle z_\pm,j | n_{12},j \rangle = \iint d\lambda_{x_\pm}d\lambda_{y_\pm}\langle z_\pm,j | x_\pm y_\pm \rangle \displaystyle\sum\limits_{n=0}^{n_{12}+j}\langle x_\pm |n\rangle \langle y_\pm | n_{12}+j-n \rangle\langle n, n_{12}+j-n |n_{12},j\rangle,
\|{\widehat{G_{3}}}\|_{L^{q}(d \mu)}^q= \int |\widehat{G_{3}}((M^{\gamma,d}_{0})^{-t}x)|^{q}d\mu_\circ(x) \sim \int |\widehat{G_{2}}(x)|^{q}d\mu_\circ(x) \gtrsim T^{-\frac{q}{2}}\lambda^{\frac{q}{4}}\lambda^{\frac{d-\alpha}{2}}|S| \gtrsim \lambda^{\frac{q(d-\alpha)}{4}}.
\Biggl [ -{\frac{g^{2}}{( 2 \pi )^{2}}}{\frac{d^{2}}{d z^{2}}}+{\frac{g^{2}}{2}}\bigl [ C ( z ) + D_{1 / 2}( z ) \bigr ] \biggr ] \bar{\zeta}( z ) = \epsilon_{1}\bar{\zeta}( z ) \; .
I^{\tilde G}(\bar x) :=\{l\in\mathcal Q\,|\,\tilde G_l(\bar x)=0\,\land\,\tilde H_l(\bar x)\neq 0\}, I^{\tilde H}(\bar x) :=\{l\in\mathcal Q\,|\,\tilde G_l(\bar x)\neq 0\,\land\,\tilde H_l(\bar x)= 0\}, I^{\tilde G\tilde H}(\bar x) :=\{l\in\mathcal Q\,|\,\tilde G_l(\bar x)=0\,\land\,\tilde H_l(\bar x)= 0\}.
\frac{m_{i}}{m_{j}}= \frac{\sin \left[ ( \eta_{k}- \eta_{j}) \pi \vartheta_{h}+ ( \hat{\eta}_{k}- \hat{\eta}_{j}) \pi \vartheta_{H}\right]}{\sin \left[ ( \eta_{i}- \eta_{k}) \pi \vartheta_{h}+ ( \hat{\eta}_{i}- \hat{\eta}_{k}) \pi \vartheta_{H}\right]}.
P(dx_3dx_2x_1) = P(dx_3x_1dx_2) + P(dx_3[dx_2,x_1]) = P(x_1dx_3dx_2) + P([dx_3,x_1]dx_2) + P(dx_3[dx_2,x_1]) P(x_2dx_3dx_1) = P(dx_3x_2dx_1) + P([x_2,dx_3]dx_1) = P(dx_3dx_1x_2) + P(dx_3[x_2,dx_1]) + P([x_2,dx_3]dx_1) P(dx_2dx_1x_3) = P(dx_2x_3dx_1) + P(dx_2[dx_1,x_3]) = P(x_3dx_2dx_1) + P([dx_2,x_3]dx_1) + P(dx_2[dx_1,x_3]).
\langle x,y|p_x,p_y\rangle=\,\frac{1}{(2\pi\hbar)^{4N/2}}\,\exp\left(\,\frac{\imath}{\hbar}\left(\sum^N_{k=1}\,p_{x k \mu}\,x^\mu_{k}+\,p_{y k \mu}\,y^\mu_k\right)\right).
\dot{\tilde{\mathcal{F}}}(\eta,\phi_{\partial},\Sigma)=\{\tilde{\mathcal{F}},\tilde{H}\}_{D}(\eta,\phi_{\partial},\Sigma)-\int_{\Gamma}E(\frac{\delta\tilde{\mathcal{F}}}{\delta\Sigma})\wedge\ast\boldsymbol{n}(d\phi),
c_1(\xi):= \frac{1}{96}\left(64e^{i(\xi_1+2\xi_2)}-208e^{i(\xi_1+\xi_2)}+203e^{i\xi_1}-83e^{i(\xi_1-\xi_2)}-62e^{2i\xi_2}+199e^{i\xi_2}-58+109e^{-2i\xi_2}\right. &\left.-17e^{i(\xi_2-\xi_1)}+49e^{-i\xi_1}-4e^{_2i\xi_1}\right), c_2(\xi):= (1-c_1(\xi)/c_1(0))^2,c_3(\xi):=\frac{c_2(\xi)^2-1}{c_1(\xi)}.
V_{n}(T,x)= 2\pi\lambda\sum_{k=1}^{4}b_k \int_{\psi_\mathrm{L}(x)}^{\infty}F\left(N_\mathrm{N},\frac{nC_\mathrm{N}\eta_\mathrm{L}\bar{a}_kTx^{\alpha_\mathrm{L}}}{C_\mathrm{L}N_\mathrm{N}t^{\alpha_\mathrm{N}}}\right) &(1-p(t))t\mathrm{d}t,
{\bar{U}}_{K}\doteq \sup_{n\in \mathbb{N}}{{E}}\left\{\sum_{k=K}^{\infty}\int_{[0,T]\times \lbrack 0,1]}k\varphi_{k}^{n}(s,y){{1}}_{[0,r_{k}(\bar{\boldsymbol{X}}^{n}(s)))}(y)\,ds\,dy\right\}.
\sum_{k=1}^{p+1}S_{2,\lambda}(p+1,k)x^{k}=\phi_{p+1,\lambda}(x)=e^{-x}\sum_{n=1}^{\infty}(n)_{p+_1,\lambda}\frac{x^{n}}{n!}&=e^{-x}\sum_{n=0}^{\infty}(n+1)_{p+1,\lambda}\frac{x^{n+1}}{(n+1)!}=e^{-x}\sum_{n=0}^{\infty}(n+1-\lambda)_{p,\lambda}\frac{x^{n+1}}{n!}.
L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}{(\hat{k}_{AF})}_{\kappa}F_{\mu\nu}-\frac{1}{4}F_{\kappa\lambda}{(\hat{k}_{F})}^{\kappa\lambda\mu\nu}F_{\mu\nu},
a+b = \begin{cases}H_p\mbox{if}a=b,\,a,b\ne0 \{a,b\}\mbox{if}a\ne b,\,a,b\ne0 \{a\}\mbox{if}b=0 \{b\}\mbox{if}a=0 \end{cases}a\cdot b =k\mbox{where}0\le k<p\mbox{and}k\equiv ab\mbox{mod p}.
M=\eta \left( \begin{array}{cc}e^{2b\pi}0 0 e^{-2b\pi}\end{array}\right) \quad \eta =\pm 1\quad ;\qquad H_{M}=\left\{\left( \begin{array}{cc}a 0 0 a^{-1}\end{array}\right) \quad \vert \, \, a\neq 0\right\}\, \, .
\Lambda_{T}^Vg=(w_g(T),w_g'(T))\,, &R(T)=\left \{(u_g(T),u_g'(T))\,,\g\in L^2(0,T;L^2(\Gamma))\right \}\,, & R_V(T)=\left \{(w_g(T),w_g'(T))\,,\g\in L^2(0,T;L^2(\Gamma))\right \}\,.
\hat{q}^{(2)}_{k+1}(a_{k+1}|a^k)={\hat{q}_{k+1}^{(2)}(a^{k+1})\over \hat{q}_{k}^{(2)}(a^{k})}={\hat{p}^{(k+1)}(a^{k+1}|[X^n]_b)\over \hat{p}^{(k)}(a^k|[X^n]_b)}.
\lim_{\mu_1 \searrow 0}\lim_{L\to \infty}\mathbb E \langle m^1 \rangle_\beta =\lim_{\mu_1 \searrow 0}\lim_{L\to\infty}\Big[ \frac{1}{\beta}\frac{\partial}{\partial \mu_1}p_L(\beta, \Delta_1, \Delta_2, \mu_1, \mu_2) \Big]_{\Delta_1 = \sqrt{\frac{\mu_1}{\beta_{\rm N}}}}=0,
\underset{n\rightarrow +\infty}{\liminf}\ \mathcal{W}_2(\delta_{\mu^n},\P) \geq \int \underset{n\rightarrow +\infty}{\liminf}\W_2^2(\mu_n,\nu)d\P(\nu) \mbox{by Fatou's Lemma}& \geq \int W_2^2(\mu,\nu)d\P(\nu)=\mathcal{W}_2^2(\delta_{\mu},\P)\qquad
S^Y(\omega) =\lim_{n\to\infty}n^{-1}\int_Y|\sum_{j=0}^{n-1}e^{i\omega r_j}V_\omega\circ F^j|^2\,d\mu_Y =\lim_{n\to\infty}n^{-1}\int_{Y\times S^1}|\sum_{j=0}^{n-1}U_\omega\circ F_\omega^j|^2\,d\nu_Y=\hat\Sigma_\omega^{11}+\hat\Sigma_\omega^{22},
= \sum_{j=1}^{k}\binom{2k+1}{j}\frac{\Bigl[ (c_1t-\beta)^{j}(c_2t+\beta)^{2k+1-j}- \Bigl(\frac{c_2}{c_1}\Bigr)^{2k+1-2j}(c_1t-\beta)^{2k+1-j}(c_2t+\beta)^{j}\Bigr]}{\bigl[(c_1+c_2)t \bigr]^{2k+1}}
\left\langle \int d^{4}x \sqrt g s \Delta_{4}f_{\epsilon}( x , x_{0}) \right\rangle_{_V}= \left\langle \int d^{4}x \sqrt{\overline{g}}s \overline{\Delta}_{4}e^{\beta_{8}\sigma}\overline{f}_{\epsilon}( x , x_{0}) \right\rangle_{_V}\sim s V^{\frac{\beta_{8}}{\beta_{0}}}
\left.\begin{array}{l}\partial_{\phi}F^{a}=0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial_{\pi_{\phi}}G^{a}=0, F^{b}\partial_{b}G^{a}- G^{b}\partial_{b}F^{a}+\pi_{\phi}\partial_{\phi}G^{a}- 2 (e^{\phi}- e^{-2\phi}) \partial_{\pi_{\phi}}F^{a}=0,\end{array}\right.
\frac{\log_{\lambda}(1+t)}{t}(1+t)^{x}=\sum_{k=0}^{\infty}\beta_{k,\lambda}(x)\frac{1}{k!}\big(\log_{\lambda}(1+t)\big)^{k}&=\sum_{k=0}^{\infty}\beta_{k,\lambda}(x)\sum_{n=k}^{\infty}S_{1,\lambda}(n,k)\frac{t^{n}}{n!}&=\sum_{n=0}^{\infty}\bigg(\sum_{k=0}^{n}\beta_{k,\lambda}(x)S_{1,\lambda}(n,k)\bigg)\frac{t^{n}}{n!}.
\sum_{x}\frac{1}{2}\sum_{C}\oint_{C}d x^{t}\frac{\delta^{2}\chi [ C ]}{\delta \sigma^{\nu t}( x ) \delta \sigma^{\mu t}( x )}\epsilon^{1 , \cdots , 7 \nu \mu t}\times \frac{1}{7 !}W_{1 , \cdots , 7}( x ) ,
d^2\!f(\overline{x}|\overline{v})(w) \ge d^2\vartheta(F(\overline{x})\,|\,\overline{\xi})(dF(\overline{x})(w)) +\langle \overline{\xi},d^2 F(\overline{x})(w)\rangle = d^2 \vartheta(F(\overline{x}))(dF(\overline{x})(w)) +\langle \overline{\xi},d^2 F(\overline{x})(w)\rangle.
\frac{c \cdot (s-c+1)}{s}= \frac{\alpha(I_{s,c})}{\widehat{\alpha}(I_{s,c})}= \frac{\alpha(I_{V_{\lambda, c}})}{\widehat{\alpha}(I_{V_{\lambda, c}})}\leq \rho(I_{V_{\lambda, c}}) \le \rho(I_{s, c}) = \frac{c \cdot (s-c+1)}{s},
{\bf{V}}_1^{[1]}(n) = \left[{\begin{array}{*{20}{c}}{\frac{{h_1^{[21]}(1)}}{{h_1^{[21]}(n - 2)}}}0 0{\frac{{h_2^{[21]}(1)}}{{h_2^{[21]}(n - 2)}}}\end{array}}\right],{\bf{V}}_2^{[1]}(n) = \left[{\begin{array}{*{20}{c}}{\frac{{h_1^{[11]}(6)}}{{h_1^{[11]}(n - 2)}}}0 0{\frac{{h_2^{[11]}(6)}}{{h_2^{[11]}(n - 2)}}}\end{array}}\right].
A_{t}={\frac{2 \sqrt{2}m a \cos \delta \cos \theta}{r^{2}- a^{2}\cos^{2}\theta}}, A_{\varphi}= -{\frac{2 \sqrt{2}m a r \sin \delta \sin^{2}\theta}{r^{2}- 2 m r - a^{2}\cos^{2}\theta}}.
[ \hat{C}^{i b}, \hat{Q}^{a}]_{+}= i \varepsilon^{a b}\hat{\pi}_{( B )}^{0 i}, [ \hat{\pi}_{( C ) i b}, \hat{Q}^{a}]_{+}= i \varepsilon_{0 i j k}\delta_{b}^{a}( \partial^{j}\hat{A}^{k}+ \frac{1}{2}\hat{A}^{j}\wedge \hat{A}^{k}) ,
\vec{\mathbb{I}}^2 = \langle \langle \mathbb{I}^2_{\gamma}\colon \gamma \leq \delta \rangle, \langle \dot{\mathbb{J}}^2_{\gamma}\colon \gamma < \delta \rangle \rangle \dot{\vec{\mathbb{I}}}^1 = \langle \langle \dot{\mathbb{I}}^1_{\gamma}\colon \gamma \leq \delta \rangle, \langle \dot{\mathbb{J}}^1_{\gamma}\colon \gamma < \delta \rangle \rangle
\theta^N_X:C^{N}\Omega^N X \rightarrow \Omega^NX \left[d_{\underline n},\gamma^{\underline n}\right] \mapsto \left(t\mapsto\begin{cases}\gamma^i(d_i^{-1}(t)), t\in d_i(I^N) \ast_c, t\not\in d_{\underline n}\left(\coprod_{\underline n}I^N\right) \end{cases}\right)
[\begin{matrix}1&a&c\\ 0&I_{n}&b\\ 0&0&1\end{matrix}]\cdot[\begin{matrix}1&a^{\prime}&c^{\prime}\\ 0&I_{n}&b^{\prime}\\ 0&0&1\end{matrix}]=[\begin{matrix}1&a+a^{\prime}&c+c^{\prime}+a\cdot b^{\prime}\\ 0&I_{n}&b+b^{\prime}\\ 0&0&1\end{matrix}]
\begin{cases}\dot{\bar z}_i=\bar f_i(\bar z_i, e_i, v, w,\mu) \dot{\hat \eta}_i=F_i\hat\eta_i+F_iG_ie_i-G_i\bar g_i(\bar z_i,e_i, v, w,\mu) \dot{e}_i=\bar g_i(\bar z_i,e_i, v, w,\mu)-\Psi_i\hat \eta_i-\Psi_iG_ie_i+u_i^r.\end{cases}
\left( x^- \over x^+ \right)^2 F(x^-,x^+) = \left. \left|{L \over \pi}\langle 0 | T^{++}(k) | n \rangle \right|^2{M_n^4 \over 8 \pi^2 k^3}K_4(M_n r) \right|_{M_n =0}=\left|{L \over \pi}\langle 0 | T^{++}(k) | n \rangle \right|^2_{M_n=0}{6 \over k^3 \pi^2 r^4}.
\frac{\Gamma(s)\,x^{2s}}{\left(x^{2}+t^{2}\right)^{s}}-\frac{\Gamma(s)\,x^{2s}}{t^{2s}}=\sqrt{\pi}2^{\frac{1}{2}-s}x^{s+\frac{1}{2}}\,\intop_{0}^{\infty}y^{s-\frac{1}{2}}\left\{J_{s-\frac{1}{2}}(xy)-\frac{2^{\frac{1}{2}-s}x^{s-\frac{1}{2}}}{\Gamma(s+\frac{1}{2})}y^{s-\frac{1}{2}}\right\}\,e^{-ty}dy.
\Lambda(f \otimes \mathrm{Ad}(g), k) = 2^{k+1-\nu(M)}C(N,M,\chi) \frac{\langle f,f \rangle}{\langle h,h\rangle}\frac{|\langle (\mathrm{id}\otimes U_M) F_{\chi |\mathcal H \times \mathcal H}, g \times g\rangle|^2}{\langle g,g\rangle^2},
x^{\alpha}y^{\beta}z^{\gamma}+x^{\beta}y^{\gamma}z^{\alpha}+x^{\gamma}y^{\alpha}z^{\beta}-x^{\beta}y^{\alpha}z^{\gamma}-x^{\gamma}y^{\beta}z^{\alpha}-x^{\alpha}y^{\gamma}z^{\beta}= (x-y)(x-z)(y-z)\sum_{i=0}^{\beta-\gamma-1}\sum_{j=0}^{\alpha-\beta-1}\sum_{k=0}^{\alpha-\beta-1-j+i}x^{\gamma+i+j}y^{\beta-1-i+k}z^{\alpha-2-j-k}.
\begin{aligned}|T_{N}'(x)|= |\sin\left(N \arccos (x) \right)|\cdot N \cdot \frac{1}{\sqrt{1-x^{2}}}=\frac{N\cdot |\sin\left(N\pi u_{0}\right)|}{\sqrt{1-\cos^{2}\left(\pi u_{0}\right)}}= \frac{N\cdot |\sin\left(N\pi u_{0}\right)|}{|\sin\left(\pi u_{0}\right)|}\end{aligned}
\overset{4}{H}(X,Y)Z = \overset{4}{R}(X,Y)Z - \overset{4}{'R}(AY,X)AZ + \overset{4}{'R}(AZ,Y)AX - \frac{1}{n-1}( \overset{4}{R}ic (Y,Z)X - \overset{4}{R}ic (X,Z)Y - \overset{4}{'R}(AY,AZ)X + \overset{4}{'R}(AX,AZ)Y )
P_{s}\varphi(v) = \int_{\Gamma_{\mu}}\|g v\|^{s}\varphi( g \!\cdot\! v ) \mu(dg), P_{s}^{*}\varphi(v) = \int_{\Gamma_{\mu}}\|g^{\mathrm{T}}v\|^{s}\varphi(g^{\mathrm{T}}\!\cdot\! v) \mu(dg),
_cD_{0t}^{\alpha}y(x) =\,_cD_{0t}^{\alpha}\left( b_0E_{\alpha, 1}(\lambda x^{\alpha})\right)+_cD_{0t}^{\alpha}\left(b_1xE_{\alpha, 2}(\lambda x^{\alpha})\right)+_cD_{0t}^{\alpha}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right).
s_k =-\frac{f(x,y)}{k}\binom{p-2}{k-1}f(x,v)^{k-1}f(y,v)^{p-k-1}v =-\frac{1}{k}\binom{p-2}{k-1}f(x,v)^{k-1}f(y,v)^{p-k-1}f(x,y)v.
\begin{aligned}\overline H(f) = \sup_{\phi}\, \sup_{\tilde\pi\in\widetilde\Pi}\,\inf_{g}\bigg[\int \d\P_0(\omega)\,\phi(\omega) \bigg\{\sum_{e\in \mathcal U_d}\tilde\pi(\omega,e) \big(F(\omega,e) -\log\widetilde\pi(\omega,e)\big)\bigg\}\bigg],\end{aligned}
f(g_0,\cdots,g_n) = g_0(\psi^n)^{-1}(\phi)(g_0^{-1}g_1,\cdots,g_{n-1}^{-1}g_n) &=-g_0\cdot g_0^{-1}g_1\cdot (\psi^n)^{-1}(\phi)((g_0^{-1}g_1)^{-1},g_0^{-1}g_1g_1^{-1}g_2,\cdots,g_{n-1}^{-1}g_n) &=-g_1\cdot (\psi^n)^{-1}(\phi)(g_1)^{-1}g_0,g_0^{-1}g_2,\cdots,g_{n-1}^{-1}g_n) &= -f(g_1,g_0,\cdots,g_n).
Z_{1}( 0 ) = - 2 \, \zeta ( - 1 ) \int_{0}^{\infty}d y \ln \frac{y}{\sqrt{1 + y^{2}}}= - 2 \left( - \frac{1}{1 2}\right) \left( - \frac{\pi}{2}\right) = - \frac{\pi}{1 2}.
\int_{B_{r}(x)}\sum_{i=1}^{m}\sum_{\ell=1}^{Q}\left( \langle D_{i}u_{\ell}, D_{i}(Y(y, u_{\ell})) \rangle + \langle A_{u_{\ell}}(D_{i}u_{\ell}, D_{i}u_{\ell}), Y(y, u_{\ell}) \rangle \right) \, \mathrm{d}y = 0.
T=-(\tau_{ij})\mathrm{d}Z^i\mathrm{d}\overline{Z}^j+\frac{\chi}{2\pi}\sum_{q_0\in \mathbb{Z}-\{0\}}\sum_{n>0}e^{-2\pi \mathrm{i}nq_0\zeta^0}q_0^2K_0(2\pi n\tau_2|q_0|)|\mathrm{d}Z^0|^2\,.
\mathbf{v}_{l,k}^{\left[n\right]}=\sqrt{P_{T}/2N}\left\langle \mathbf{H}_{k}^{-1}\mathbf{e}_{m}^{\left[n\right]}\right\rangle ,\mathbf{v}_{k,l}^{[n]}=\sqrt{P_{T}/2N}\left\langle \mathbf{H}_{l}^{-1}\mathbf{e}_{m}^{\left[n\right]}\right\rangle ,
\|\mathcal{T}(V^{m})-\mathcal{T}(V^{n})\|_{y}\leq\frac{\beta_{0}}{2^{[\tfrac{n}{2}]}}\left(\frac{1}{2^{[\tfrac{m-1}{2}]-[\tfrac{n}{2}]}}+\frac{1}{2^{[\tfrac{m-2}{2}]-[\tfrac{n}{2}]}}+\cdots+\frac{1}{2^{[\tfrac{n+1}{2}]-[\tfrac{n}{2}]}}+1\right) &\leq\frac{\beta_{0}}{2^{[\tfrac{n}{2}]}}\left(1+2\sum_{k=0}^{\infty}\frac{1}{2^{k}}\right).