harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Xi_{[s]}(R_1) ={\bf 1}\otimes R_{\binom{n}{0}}\otimes{\bf1}, \Xi_{[s]}(R_2) ={\bf 1}\otimes R_{\binom{0}{n}}\otimes{\bf1}, \Xi_{[s]}(R_3) ={\bf 1}\otimes R_{\binom{-}{n,0}}\otimes{\bf1}, \Xi_{[s]}(R_4) ={\bf 1}\otimes R_{\binom{n,0}{-}}\otimes{\bf1}.
\mathcal{H}_{\varphi}(v)=\frac{1}{\mathcal{F}}\left\lbrace \frac{\partial^2 \mathcal{F}}{\partial z^i_{\epsilon}\partial z^j_{\eta}}\frac{\partial^2 \mathcal{\varphi}^j}{\partial x^{\epsilon}\partial x^{\eta}}\right\rbrace v^i.
(v^m)''+\frac{n-1}{r}(v^m)'+\alpha v+\beta rv'=0, \,\, v>0\quad\forall r>0 \Leftrightarrow\quad (r^{n-1}(v^m)')'+\alpha r^{n-1}v+\beta r^nv'=0,\,\, v>0\qquad\forall r>0.
x \in H^{0}( P ,{\mathcal O}_{P}( 1 ) \otimes p^{*}{\mathcal L}^{2}) , \; y \in H^{0}( P ,{\mathcal O}_{P}( 1 ) \otimes p^{*}{\mathcal L}^{3}) , \; z \in H^{0}( P ,{\mathcal O}_{P}( 1 ) ) .
I_{1}= \frac{u_{a_{j}}( a_{i})}{(\frac{\lambda_{i}}{\lambda_{j}}+ \lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2-n}}( a_{i}) )^{\frac{n-2}{2}}}\int_{\mathcal{A}}\frac{1}{(1+r^{2})^{\frac{n+2}{2}}}
2^{2}\times 3 . 1 4 = 1 2 . 5 6 ( m^{2}) 1 2 . 5 6 \times 1 . 8 = 2 2 . 6 0 8 ( m^{3}) 2 2 . 6 0 8 \div 3 = 7 . 5 3 6 ( m^{3})
\hat{\omega}_{1 1}= \ln | k_{2}| \, , \quad \hat{\omega}_{2 2}= \ln | k_{1}| \, , \quad \hat{\omega}_{1 2}= \hat{\omega}_{2 1}= - \ln \left| \frac{( u_{1}- u_{2}) ( v_{1}- v_{2})}{( u_{1}- v_{2}) ( v_{1}- u_{2})}\right| \, .
\int_{-1}^{1}e^{xt}P_{k}^{(\alpha,\beta)}(x)\,d\mu_{\alpha,\beta}(x) =\frac{(-1)^{k}}{2^{k}k!}\int_{-1}^{1}e^{xt}\frac{d^{k}}{dx^{k}}\left((1-x)^{\alpha+k}(1+x)^{\beta+k}\right)\,dx &=\frac{t^{k}}{2^{k}k!}\int_{-1}^{1}e^{xt}(1-x)^{\alpha+k}(1+x)^{\beta+k}\,dx,
\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz \leq C_8 \| y_1\|_{m_1}^{\kappa_1 + 1}\| y_2\|_{m_2}^{\kappa_2 + 1}\leq C_{9}\|y_1\|_{1, p}^{\kappa_1 + 1}\|y_2\|_{1, p}^{\kappa_2 + 1},
\lim_{x\rightarrow x^{\ast}-}\frac{1}{\sqrt{x^{\ast}-x}}\frac{d\mu}{dx}(x)=\lim_{x\rightarrow x^{\ast}-}\frac{1}{2\pi i x}\frac{w_{-}(x)-w_{+}(x)}{\sqrt{x^{\ast}-x}}=\frac{a_1}{2\pi x^{\ast}}>0,
\int_0^T |u_{n-1}|^2 \,dt \le C\Big(\sum_{i=1}^{m-1}e_{1-i}(U_{n-i})(0) + \sum_{p=1}^{m-1}|A^{-p/2}w_{n-m}^{m-p}(0)|^2 &+ e_1(W^{m}_{n-m})(0) + \int_0^T |w_{n-m-1}^{m}|^2\,dt\Big) \,.
\log\Upsilon(x)=\int\limits_0^\infty\frac{dt}{t}\;\biggl[\biggl(\frac{Q}{2}-x\biggr)^2e^{-t}-\frac{\sinh^2\bigl(\frac{Q}{2}-x\bigr)\frac{t}{2}}{\sinh\frac{bt}{2}\sinh\frac{t}{2b}}\biggr].
t^{n_{e_j}}\varphi^{2d-2}e_j =\varrho b^{-1}e_j, t^{n_{f_j}}\varphi^{2d-2}f_j =\varrho b^{-1}f_j, \gamma(x)e_j =x^{-s_{e_j}}e_j, \gamma(x)f_j =x^{-s_{f_j}}f_j
\frac{\delta C(\kappa)}{\delta \sigma(x)}|_{\sigma=0}= +2 \kappa^2 \sqrt{g(x)}\Psi_{\kappa}(x) -2 \int d \mu C(\mu) \Psi_{\kappa}(x) \Psi_{\mu}(x) \frac{\kappa^2}{\kappa^2 -\mu^2}\sqrt{g(x)}
\| x_{i}(\widetilde{w}_{2n}-\widetilde{w}_{2n-2}) \| = \| x_i w_{2n}v_{2n}\widetilde{w}_{2n-2}v_{2n}^*- x_i \widetilde{w}_{2n-2}v_{2n}v_{2n}^*\| & \leq \| x_i w_{2n}[v_{2n},\widetilde{w}_{2n-2}]\| + \| x_{i}(w_{2n}-1)\widetilde{w}_{2n-2}v_{2n}\| & < \frac{1}{2^{2n-1}}
R(V) \geq \omega \sum_{j\in V}\beta_j -\omega \sum_{j=1}^m \beta_j q_j - 2\omega^2 \sum_{j=1}^m \beta_j^2\geq \omega A -\omega \sum_{j=1}^m \beta_j q_j -2\omega^2 \sum_{j=1}^m \beta_j^2.
\varphi(t,s)= \frac{r}{2}\Big(\cos\Big(\frac{t}{r}\Big)\Big(1+\cos\Big(\frac{2s}{r}\Big)\Big),\sin\Big(\frac{2s}{r}\Big)\cos\Big(\frac{t}{r}\Big), &\sin\Big(\frac{t}{r}\Big)\Big(1+\cos\Big(\frac{2s}{r}\Big)\Big),\sin\Big(\frac{2s}{r}\Big)\sin\Big(\frac{t}{r}\Big),0\Big).
d s^{2}= -{\frac{4 \pi}{A \tau^{2}}}d t^{2}+ \left({\frac{A}{4 \pi}}\right) \left[{\frac{d \tau^{2}}{\tau^{2}}}+ ( d \theta^{2}+ \sin^{2}\theta d \varphi^{2}) \right] \ .
\begin{array}{l l}\min z begin{align*}1ex] \textnormal{s.t.}\bar c_j^\top x +\tilde c_j\leq z,j=1,\ldots,m begin{align*}1.2ex] x=\sum_{v\in V^k}\alpha_v v \alpha\ge 0 \sum_{v\in V^k}\alpha_v = 1\;. \end{array}
\|\mathcal{H}(f)\|_{A^p}= \left(\int_{\mathbb{B}}\left|\mathcal{H}(f)(z)\right|^pd\mu(z)\right)^\frac1p = \left(\int_{\mathbb{B}}\left|\int^1_0\mathcal{T}_t(f)(z)dt\right|^pd\mu(z)\right)^\frac1p \leq \int^1_0\left(\int_{\mathbb{B}}\left|\mathcal{T}_t(f)(z)\right|^pd\mu(z)\right)^\frac1pdt = \int^1_0\|\mathcal{T}_t(f)\|_{A^p}dt,
u^{\pm}= \frac{1}{\sqrt{2}}\left( u^{0}\pm u^{D - 1}\right) \ , \u^{i}\ , \i = 1 , 2 , \cdots , D - 2 \ , \u \cdot v = - u^{+}v^{--}u^{-}v^{+}+ u^{i}v^{i}\ ,
v=b,u=v-\frac F{\sqrt{F-V}},q=1-p=\frac VF, E_{Y_{u,v,p}^2}=q(b-u)^2=q(v-u)^2=\frac VF\,\Big(\frac F{\sqrt{F-V}}\Big)^2=\frac{VF}{F-V}=E_{V,F},
-2a_1 (\bar{\Lambda}_1^0)^2(D_\zeta\sigma_1)+ 2 (D_{\zeta}\sigma_1) \, \mathcal Poly_4(0,\zeta^0,\bar\Lambda_0) = 2 (D_{\zeta}\sigma_1) \Bigl[-a_1 (\bar{\Lambda}_1^0)^2 + \mathcal Poly_4(0,\zeta^0,\bar\Lambda_0) \Bigr].
H = \frac{{\bf P}^{2}}{2 m}- \frac{{\bf P}^{2}}{2 m^{2}c^{2}}\left( \frac{{\mathrm{\boldmath ~ \p i ~}}^{2}}{2 \mu}+ U ( q ) \right) - \frac{{\bf P}^{4}}{8 m^{3}c^{2}}+ \frac{{\mathrm{\boldmath ~ \p i ~}}^{2}}{2 \mu ( q )}+ v ( q ) \, .
Z = T r \exp [ - \beta \epsilon_{k}( \bar{\Phi}_{1 , k}\Phi_{1 , k}+ \bar{\Phi}_{2 , k}\Phi_{2 , k}) ] \exp [ \beta \mu ( \bar{\phi}_{1 , k}\phi_{1 , k}+ \bar{\phi}_{2 , k}\phi_{2 , k}) ] .
\langle D\phi, Y\rangle = \langle D\phi^{\ell}, Y\rangle = \lim_{n\to\infty}\langle D\phi_n^{\ell},Y\rangle = \lim_{n\to\infty}\langle D\phi_n, Y\rangle &= \lim_{n\to\infty}\langle V_n, QY\rangle = \lim_{n\to\infty}\langle V_n^{\ell}, QY\rangle = \langle V^{\ell}, QY\rangle = \langle V, QY\rangle,
\sum_{r \in \mathbb{Z}}c^F_r\big(\widetilde{U},\widetilde{W'}\big)q_F^{-\frac{r}{2}}X^{-r}= \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},\overline{\psi}_F^l\big)\sum_{r \in \mathbb{Z}}c^F_r(U,W')q_F^{\frac{r}{2}}X^{r},
F ( x^{+}, x^{-}, 0 ) = \frac{1}{2 \pi}\sum_{K , N_{\perp}, \alpha}\frac{1}{2 L}\frac{1}{l}\left( \frac{\pi K}{L}\right)^{3}e^{- i P_{\alpha}^{-}x^{+}- i P^{+}x^{-}}\frac{| \langle u | \alpha \rangle |^{2}}{l K^{3}| N_{u}|^{2}}\, .
\mathbb{P}_{\Lambda;q,\mathbf{p}}^{\pi}\left( \omega\right) :=\frac{1}{\mathcal{Z}_{\Lambda}^{\pi}\left( q;\mathbf{p}\right)}\prod_{e\in\mathbb{E}^{\Lambda}}p_{e}^{\omega_{e}}\left( 1-p_{e}\right)^{1-\omega_{e}}q^{\kappa_{\Lambda}\left( \omega\right)}\mathbf{1}_{\Omega_{\Lambda}^{\pi}}\left( \omega\right) \ ,
p_j :=(\operatorname{id}\otimes\omega_{e_j,c^*\zeta'})(W)=(\operatorname{id}\otimes\omega_{e_j,\zeta'}(c\,\cdot\,))(W)\in{\mathfrak N}_{\psi}, q_j :=(\operatorname{id}\otimes\omega_{e_j,d^*\xi'})(W)=(\operatorname{id}\otimes\omega_{e_j,\xi'}(d\,\cdot\,))(W)\in{\mathfrak N}_{\psi}.
\left(\begin{array}{c c c}j_{1}j_{2}j_{3}-m_{1}-m_{2}-m_{3}\end{array}\right)_{q}=\left(\begin{array}{c c c}j_{2}j_{1}j_{3}m_{2}m_{1}m_{3}\end{array}\right)_{q}=(-)^{j_{1}+j_{2}+j_{3}}\left(\begin{array}{c c c}j_{1}j_{2}j_{3}m_{1}m_{2}m_{3}\end{array}\right)_{q^{-1}}
\begin{array}{c}L={\tilde\sigma}\left[2c\epsilon+ c{\bf e'}^2_1+{\bf p}({\bf x'}-{\bf e}_+)+\right. \left.+{\bf p}_+({\bf e'}_+ -{\bf e}_1)-\sum_{i,j}d_{ij}({\bf e}_i{\bf e}_j -\eta_{ij})\right],\end{array}
m(\varphi)=\frac{a+b}{2}(B_{0}\varphi-B_{2}sin2\varphi+B_{4}sin4\varphi-B_{6}sin6\varphi+B_{8}sin8\varphi-\cdot\cdot\cdot-\frac{2nsin2\varphi}{\sqrt{1+2ncos2\varphi+n^{2}}}).
B_{1,\phi_{2}}(k):=\int_{0}^{t}\phi_{2}(k)iske^{-isk^{2}}\iiint u^{\#}(\ell)\overline{u^{\#}}(n)u^{\#}(m)\,\mu_{R,1}^{\#,(2)}\left(k,\ell,n,m\right)\,d\ell dmdn\,ds.
-\frac{1}{N}\sum_{i=1}^N \nabla^T f(y_{i,t}) (y_{i,t}-x^{\ast})= -\frac{1}{N}\sum_{i=1}^N (\nabla^T f(y_{i,t})-\nabla^T f(x^{\ast})) (y_{i,t}-x^{\ast}) \leq -2\kappa \overline{F}_t= -2\kappa(U_t+\overline{V}_t).
b''_n = - \sum_{\substack{k > 1 P_1 \cup \cdots \cup P_k = \{1,\ldots, n\}P_i}}b''_{|P_1|}\cdots b''_{|P_k|}\frac{(k-1)!}{2}\sum_{j=0}^{k-1}a_ja_{k-1-j}\left(\sum_{i=1}^{k}\binom{|P_i|}{2}+ \binom{n}{2}\right)\frac{(j+1)(k-j)}{\binom{n}{2}}
\langle V_{v / r^{8}}\rangle ={\frac{1 0 5 \sqrt{2}\tilde{R}^{7}}{r^{9}N^{2}}}( r_{i}v_{i + 3}- r_{i + 3}v_{i}) \left( 1 -{\frac{1 0}{3 N^{2}}}+{\frac{7}{3 N^{4}}}\right) \ .
K^{\alpha}_{R,m}(z,w) = \frac{(\alpha+1)}{\pi R^{2}}\left\{\sum\limits_{n=0}^{m-1}\frac{\Gamma(\alpha+2+n)}{\Gamma(n+1)\Gamma(\alpha+2)}\left(\frac{z\overline{w}}{R^{2}}\right)^{n}+ \frac{(z\overline{w})^{m}}{(m!)^2}{_3F_2}\left( \begin{array}{c}1,1 ,\alpha+2 m+1,m+1\end{array}\bigg| \frac{z\overline{w}}{R^{2}}\right) \right\}.
P_{+}= \int_{-{\infty}}^{\infty}d x^{-}{\theta}_{+}^{\; \; +}( x ) - \int_{-{\infty}}^{\infty}d x^{+}B ( x ) \frac{1}{m^{2}- n_{-}{\partial}_{+}^{\; \; 2}}{\partial}_{+}C ( x ) ,
g\left(J_i (r), J_j (r)\right)=s_k(r)^2 g\left(V_i(r), V_j(r)\right)= s_k(r)^2 g\left(V_i(0), V_j(0)\right) = s_k(r)^2 g\left(J_i'(0), J_j'(0)\right) = s_k(r)^2 \overline g\left(J_i'(0), J_j'(0)\right) = s_k(r)^2 \overline g\left(\frac{\partial}{\partial \theta^i}, \frac{\partial}{\partial \theta^j}\right).
\oint\limits_{|\theta|\le n^{-\delta}}\frac{p_{r,s}(z)}{z^{n+1}}\,dz= \,i\exp\left(bn^{1/2}+\frac{1}{2}\log\frac{c}{2\pi\sqrt{n}}- h-w\right) &\times\frac{(\pi c)^{1/2}}{n^{3/4}}\bigl(1+O(n^{-1}(h+w+1)^2)\bigr).
\Phi = \left( \begin{array}{cccc}0 \phi_1 \otimes M_e + \phi_2 \otimes M_{\nu}0 0 \phi^*_1 \otimes M^*_e + \phi^*_2 \otimes M^*_{\nu}0 0 0 0 0 0 0 0 0 0 0 \end{array}\right)
\ell_q(Q_j) = \frac{-Q_j \theta_q'(Q_j)}{\theta_q(Q_j)}\theta_q(Q_j) = \prod_{r \geq 0}(1-q^{r+1})(1+q^r Q_j)\left( 1 + \frac{q^{r+1}}{Q_j}\right)
f(s,m)=-\binom{m+1}{m-s+1}N_{\mathrm{d}\,m-s+1}- \binom{m+1}{m-s}N_{\mathrm{d}\,m-s}\mathcal{C}_{s}^{s+1}-\cdots- \binom{m+1}{m+1-(s+r)}N_{\mathrm{d}\,m+1-(s+r)}\mathcal{C}_{s}^{s+r}-\cdots -\binom{m+1}{2}N_{\mathrm{d}2}\mathcal{C}_{s}^{m-1}-\binom{m+1}{1}N_{\mathrm{d}1}\mathcal{C}_{s}^{m},
\alpha^{( 1 )}= - \frac{2 \sqrt{2}\pi}{v^{2 / 3}}\left( \frac{\kappa}{4 \pi}\right)^{2 / 3}\int_{{\cal{C}}_{\omega}}{( ( 1 0 S + 5 ( r + 3 ){\cal{E}}) \sigma + 9 6 F )}.
- \frac{1}{a(\tau)}+ \frac{1}{a(\tau_1)}\leq \sum_{i=1}^m \frac{(n_i+1)^2 q_i^2}{4n_ip_i^2}\left (- \frac{1}{\tau +c_i}+ \frac{1}{\tau_1 +c_i}\right ) & \leq \left ( \sum_{i=1}^m \frac{(n_i+1)^2 q_i^2}{4n_ip_i^2}\right) \left(\frac{1}{\tau_1}\right).
\langle a_m,\dots,a_1;\langle b_n,\dots,b_1;c\rangle\rangle = \sum \langle V_{2n},\langle V_{2n-1};b_n\rangle,\dots,V_4,\langle V_3;b_2\rangle, V_2,\langle V_1;b_1\rangle,V_0;c\rangle,
\sum^{[k/2]}_{j=1}\frac{(1-q^{jc})(1-q^{(n-j)c})}{(1-q^j)(1-q^{n-j})}T_{m,n,r,s}(j;k)(1-\frac{\delta_{j\frac k{2}}}{2}) \geq \frac{(1-q^{i_0 c})(1-q^{(n-j)[k/2]})}{(1-q^{[k/2]})(1-q^{n-[k/2]})}\sum^{[k/2]}_{j=1}T_{m,n,r,s}(j;k)(1-\frac{\delta_{j\frac k{2}}}{2}).
A = \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right) .
\begin{array}{c}{{{\dot{P}}}_{ICE}}=-\frac{1}{{{\tau}_{ICE}}}{{P}_{ICE}}+\frac{1}{{{\tau}_{ICE}}}P_{ICE}^{\max}\left({{\omega}_{ICE}}\right) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot eng\left({{\omega}_{ICE}}\right)\cdot{{u}_{ICE}}(t) \end{array}
s_{4}-s_{5}+1= -((s_{1}-s_{4})+(s_{6}-s_{1})+(s_{3}-s_{6})+ (s_{7}-s_{3})+ (s_{2}-s_{7})+(s_{5}-s_{2}))+1 = -((-r_{6}+r_{4}+r_{3}-r_{2}+1)+(r_{6}-r_{3}+r_{2})+(r_{3}-r_{2})+r_{2}+ &(-r_{5}-r_{4}-r_{3})+ (-r_{6}+r_{5}+r_{4}+r_{3}-r_{2}-r_{1}))+1 = r_{6}-r_{4}-r_{3}+r_{2}+r_{1}=p_{3,5}.
\begin{array}{l}|P^{\flat}_m|\leq (k+m-1)/(k+m-1+k+m+3/2+W_m/3)\leq 6/m~~~~~~~~~~\,{\rm and}|(P^{\flat}_m)'|\leq (W_m/3+k+m+3/2)|P_m^{\flat}/q|+(k+m-1)q^{k+m-2+k+m+3/2+W_m/3}\leq 2+k+m-1=k+m+1~.\end{array}
\frac{2e^{x^{\prime}\Sigma^{-1}\mu}}{(2\pi)^{\frac{k}{2}}|\Sigma|^{0.5}}(\frac{x^{\prime}\Sigma^{-1}x}{2+\mu^{\prime}\Sigma^{-1}\mu})^{\frac{v}{2}}K_{v}(\sqrt{(2+\mu^{\prime}\Sigma^{-1}\mu)(x^{\prime}\Sigma^{-1}x)})
\frac{d}{dk}\log(\Gamma(k)) = d_1 ~=~ \log(k) - \frac{1}{2k}+ O(k^{-2}) \frac{d^2}{dk^2}\log(\Gamma(k)) = d_2 - d_1^2 ~=~ \frac{1}{k}+ \frac{1}{2k^2}+ O(k^{-3}) \frac{d^4}{dk^4}\log(\Gamma(k)) = d_4 - 3d_2^2 - 6d_1^4 - 4 d_3 d_1 + 12 d_1^2d_2 ~=~ O(k^{-3})
\lim_{d\rightarrow\infty}E_d(Q) = \frac{1}{2}\bigg(1 + \frac{1}{q}\bigg)\bigg(\frac{1}{1 - \frac{1}{q}}\bigg)^2 - \frac{1}{2}\bigg(1 - \frac{1}{q}\bigg)\bigg(\frac{1}{1 - \frac{1}{q^2}}\bigg) = \frac{2}{q}+ \frac{2}{q^2}+ \frac{4}{q^3}+ \frac{4}{q^4}+ \frac{6}{q^5}+ \frac{6}{q^6}+ \frac{8}{q^7}+ \frac{8}{q^8}+ \frac{10}{q^9}+ \ldots
(\triangle_{\alpha,q}f)(x,y) =\partial_{x,q}^2f(x,y)+q^{2\alpha+1}\partial_{y,q}^2f(x,y)-\frac{q[-2\alpha-1]_{q}}{y}\partial_{y,q}f(x,y) &=\partial_{x,q}^2f(x,y)+q^{2\alpha+1}\partial_{y,q}^2f(x,y)-{q[-2\alpha-1]_{q}}\int_{0}^{1}\partial_{y,q}^{2}f(x,yt)d_qt.
=\frac{im^2}{\pi^2}\Bigl( g^{\mu \nu}- \frac{k^\mu k^\nu}{k^2}\Bigr)\Bigl( \frac{1}{k^2 \sqrt{1-4m^2/k^2}}\log \frac{\sqrt{1-4m^2/k^2}+1}{\sqrt{1-4m^2/k^2}-1}+ \frac{1}{2m^2}\Bigr) \quad ,\quad k^2>4m^2,k^0>0 \, .
d s^{2}= - d t^{2}+ \tilde{R}^{2}( \tilde{x}^{5}) d \Omega_{k}^{2}+ e^{\tilde{\mu}( \tilde{x}^{5})}[ d \tilde{x}^{5}+ \kappa \tilde{A_{0}}( t ) d t ]^{2}
\begin{array}{ll}(p^d-1)\sum\limits_{(\alpha,\beta)\in N_1}S(\alpha,\beta)+(p^{2d}-1)\sum\limits_{(\alpha,\beta)\in N_2}S(\alpha,\beta)=(p^d-1)p^{\frac{m}{2}}((\sqrt{q^*}+1)(n_{1,1,1}-n_{-1,-1,1})+ &(\sqrt{q^*}-1)(n_{1,-1,1}-n_{-1,1,1})+(p^d+1)^2(n_{1,1,2}-n_{-1,-1,2})+(p^{2d}-1)(n_{1,-1,2}-n_{-1,1,2})). \end{array}
( A^T E_r - \sum_{s=1}^p \alpha_{rs}E_s, e_k)= (E_r, Ae_k)- \sum_{s =1}^p \alpha_{rs}(E_s, e_k) = \sum_{s=1}^p \alpha_{sk}(E_r,e_s)-\alpha_{rk}= \alpha_{rk}-\alpha_{rk}=0,
T_{4,4}= B\int e^{i\Theta(\eta,\nu)}\frac{\eta-2\nu}{\eta+\nu}\Bigg(e^{2ia\ln|(\eta+\nu)/2|}\left(1+\frac{2ia}{(\eta+\nu)^3}\right)\chi\left( \frac{\eta+\nu}{2}\right) &\qquad+ e^{-2ia\ln|(\eta+\nu)/2|}\left(1-\frac{2ia}{(\eta+\nu)^3}\right)\chi\left( -\frac{\eta+\nu}{2}\right)\Bigg) S\left(\frac{\eta-\nu}{2}\right)d\nu
\begin{aligned}\displaystyle \partial_t f + \partial_{\theta}(v[f] f) = 0, (\theta, \omega) \in \mathbb T \times \mathbb R,~~t > 0, & v[f](\theta, \omega, t) = \omega - K \iint_{\mathbb T \times \mathbb R}\sin(\theta-\theta_*) f(\theta_*, \omega_*, t) ~ d\theta_*d\omega_* .\end{aligned}
\tau^+(0)\overline{\tau}^+{=}\sum_{(\alpha, \lambda)=-1, -2}c_{\alpha}d_{\lambda}e^{\alpha}(0)e^{\lambda}&{=}-\frac{1}{2}\left\{\sum_{\alpha\in X}c_{\alpha}d_{h-\alpha}\varepsilon(h, \alpha)\right\}h(-1)e^h{=}h(-1)e^h=Te^{-h},
\bar{\Delta}V_{n}=\frac{a_{nn}P_{n}\lambda_{n}}{np_{n}}s_{n}+\sum_{v=1}^{n-1}\frac{P_{v}\lambda_{v}}{vp_{v}}\Delta_{v}(\hat{a}_{nv})s_{v}+\sum_{v=1}^{n-1}\hat{a}_{n,v+1}\lambda_{v}\Delta\left( \frac{P_{v}}{vp_{v}}\right)s_{v}+\sum_{v=1}^{n-1}\hat{a}_{n,v+1}\frac{P_{v+1}}{(v+1)p_{v+1}}\Delta\lambda_{v}s_{v}\bar{\Delta}V_{n}=V_{n,1}+V_{n,2}+V_{n,3}+V_{n,4}.
Y_{m + k + 1 - n , n - 1}= \Delta^{2}Z_{m + k + 1 - n , n}- \Delta^{1}Z_{m + k + 2 - n , n - 1}+ \xi_{1}u^{i}C_{i ( 2 n - m - k - 1 ) ( 1 )}({\frac{\partial}{\partial u_{1}^{*}}})^{2 n - m - k - 1}Q_{1}({\frac{\partial}{\partial u_{2}^{*}}})^{1}Q_{2},
\frac{\partial L}{\partial f}-\frac{d}{dx}(\frac{\partial L}{\partial f^{\prime}})+\frac{d^{2}}{dx^{2}}(\frac{\partial L}{\partial f^{\prime\prime}})-...+(-1)^{k}\frac{d^{k}}{dx^{k}}(\frac{\partial L}{\partial f^{(k)}})=0
\delta^{i_1i_2}G_{i_1i_2i_3}^{\eta j_2j_3}\delta_{j_2j_3}\tilde\gamma_{\eta k}+2\delta^{i_1i_2}G_{i_1i_2i_3}^{j_1j_2j_3}\gamma_{j_1j_3}\delta_{j_2k}= \delta_{i_3k}
S^{ij}(\theta -\sigma_{ij})=\left( \begin{array}{ccc}\eta_{ij}(-1)_{\theta}\eta_{ij}^{2}(-2)_{\theta}\eta_{ij}^{3}(-3)_{\theta}\eta_{ij}^{2}(-2)_{\theta}-(-3)_{\theta}(-1)_{\theta}\eta_{ij}^{2}(-2)_{\theta}\eta_{ij}^{3}(-3)_{\theta}\eta_{ij}^{2}(-2)_{\theta}\eta_{ij}(-1)_{\theta}\end{array}\right),
\frac{1}{2}\Delta|\nabla^4 W|^{2}= |\nabla^{5}W|^{2}+ \langle \nabla^4 W, \nabla \Delta \nabla^3 W \rangle +\frac{R}{4}|\nabla^4 W|^{2}+ 8 W_{ijkl,trsu}W_{pjkl,trsv}R_{piuv}+2W_{ijkl,trsu}W_{ijkl,prsv}R_{ptuv}+2W_{ijkl,trsu}W_{ijkl,tpsv}R_{pruv}+2W_{ijkl,trsu}W_{ijkl,trpv}R_{psuv}\,.
S \! = \! \! \int \! \! \! \sqrt{- g}d^{D}\! x \! \left[ - \sum_{k = 1}^{N}\bar{\psi}_{k}\gamma^{\alpha}\nabla_{\alpha}\psi_{k}+ \frac{\lambda_{0}}{2 N}\left( \sum_{k = 1}^{N}\bar{\psi}_{k}\psi_{k}\right)^{2}\right] ,
\lambda^s \sigma(\{x \in Q \setminus H\colon\, V_{\sigma, Q}b(x) > \lambda\}) &=\lambda^s \sigma(\{x \in Q \setminus H\colon\, V_{Q}\nu(x) > \lambda\}) &\lesssim \lambda^s\mu(\{x \in Q \setminus U_Q\colon\, V_{Q}\nu(x) > \lambda\}) \le B_2\|\nu\| = B_2\sigma(Q).
A_t^n(f) = \int_0^t n^{\alpha -1/2}\sum_{x,y}a(y-x) \bar{\eta}_s^n(y) \bar{\eta}_s^n(x) \big( f\big(\tfrac{y-(1-2\rho)m_n^\alpha s}{n}\big) - f \big( \tfrac{x-(1-2\rho)m_n^\alpha s}{n}\big)\big)ds.
\frac{1}{\prod_{n=1}^N ( \lambda - \lambda_n^{(0)})}= \sum_{n=1}^N \frac{1}{\lambda - \lambda_n^{(0)}}\frac{1}{\frac{d}{d \lambda}\prod_{n'=1}^N ( \lambda - \lambda_{n'}^{(0)}) \vert_{\lambda = \lambda_n^{(0)}}}& = \sum_{n=1}^{N}\frac{1}{\lambda - \lambda_n^{(0)}}\frac{1}{\prod_{n'\neq n}(\lambda_{n}^{(0)}- \lambda_{n'}^{(0)})}
H = \sum_{k}\omega_{k}A_{1 , 1}\left( k \right) + \sum_{n , \left\{m \right\}, \left\{m^{\prime}\right\}}\sum_{k_{1}, \ldots k_{n}}^{\prime}V_{\left\{m \right\}, \left\{m^{\prime}\right\}}\left( k_{1}, \ldots k_{n}\right) \prod_{i = 1}^{n}A_{m_{i}, m_{i}^{\prime}}\left( k_{i}\right) .
\bigl( g_1,(g_2 ,g_3) \bigr) \cdot \bigl( h_1,(h_2,h_3) \bigr) =\bigl(Z_1^{d(g_2,g_3) \, d(h_1)}g_1 h_1, (g_2,g_3)\cdot (h_2,h_3)\bigl) & = \bigl(Z_1^{(d(g_2) + d(g_3)) \, d(h_1)}Z_1^{d(g_3)d(h_2)}g_1 h_1, g_2 h_2,g_3 h_3 \bigr) & = \bigl( Z_1^{d(g_2) d(h_1) + d(g_3) d(h_1) + d(g_3) d(h_2)}g_1 h_1, g_2 h_2,g_3 h_2 \bigr)
M_{p l}^{2}={\frac{M_{X}^{3}}{k}}e^{2 \pi k \rho - 2 c}\left( 1 - e^{- 2 k y_{2}}+ e^{- 2 k ( \pi \rho - y_{4}+ y_{2})}- e^{- 2 k ( 2 \pi \rho - y_{4})}\right) . \,
\sup_{\gamma \in [0,\theta]}\| (-A)^{(\gamma-\theta)}\|_{L(H)}= \sup_{\gamma \in [0,\theta]}\| (-A)^{-\gamma}\|_{L(H)}&= \sup_{\gamma \in [0,\theta]}\left[ (\nu \pi^2)^{-\gamma}\right] = \max\!\left\{\frac{1}{( \nu \pi^2 )^{\theta}}, 1 \right\}< \infty
\rho_F(a)\rho_F(acd)+\rho_F(c)\rho_F(d) =\rho_F(a)\rho_F(acd)+\rho_F(c)\rho_F(d)+\rho_F(ac)\rho_F(ad) =\rho_F(a)\rho_F(ac)+\rho_F(a)\rho_F(d)+\rho_F(c)\rho_F(d)+\rho_F(ac)\rho_F(ad) =[\rho_F(a)\rho_F(ac)+\rho_F(ac)\rho_F(ad)]+[\rho_F(a)\rho_F(d)+\rho_F(c)\rho_F(d)] =\rho_F(d)\rho_F(ac)+\rho_F(d)\rho_F(ac)=0,
B_1(\lambda) := P_\ell \lambda^\ell +P_{\ell-1}\lambda^{\ell-1}+\cdots + P_1\lambda + P_0, &B_j(\lambda) := P_{\ell j}\lambda^\ell+P_{\ell j-1}\lambda^{\ell-1}+\cdots + P_{\ell(j-1)+1}\lambda,\mbox{for}j=2,\hdots,k.
\int_{\Omega}{(2(p+q)|\psi_{\uparrow}|^2+2(p-q)|\psi_{\downarrow}|^2-2p)}dx =-2\pi p N_1 \int_{\Omega}{(2(p-q)|\psi_{\uparrow}|^2+2(p+q)|\psi_{\downarrow}|^2-2p)}dx =-2\pi pN_2.
f_v(\bar c) = r(v){\textstyle\sqrt{\frac{k-3}{3}}}+ (k-r(v)){\textstyle\sqrt{\frac{3}{k-3}}}\le &\le 3{\textstyle\sqrt{\frac{k-3}{3}}}+ (k-3){\textstyle\sqrt{\frac{3}{k-3}}}=2\sqrt{3(k-3)},
[ a ( q , \lambda ) , a^{\dagger}( q^{\prime}, \lambda^{\prime}) ] = \delta ( q^{+}- q^{\prime +}) \delta^{2}( q_{\perp}- q_{\perp}^{\prime}) \delta_{\lambda , \lambda^{\prime}}= \delta^{3}( q - q^{\prime}) \delta_{\lambda , \lambda^{\prime}}.
\lim_{n\to \infty}\mu_p (\mathcal{T}_n(\ell)) = Z(p,\ell)=1-C_p^{-1}p^{-(\ell+1)^2}\left[ 1- \frac{2}{p^2-p}\cdot p^{-\ell}+O\left( p^{-2\ell}\right) \right] \ \{\mathrm{as}}\ \ \ell\to \infty\,;
M_{f}^{\mathfrak{m}}(t) = f\bigl(\hat{X}(t),\hat{W}(t)\bigr) - f\bigl(\hat{X}(0),0\bigr) - \int_{\Gamma\times[0,t]}\! \mathcal{A}^{\mathfrak{m}}_{\gamma,s}(f)\bigl(\hat{X}(s),\hat{W}(s)\bigr)\hat{\rho}(d\gamma,ds),
\sum_{p_1,p_2,p_3,p_4 \in{\mathcal P}: P \leq p_1,p_2,p_3,p_4 \leq 2P}\int \dotsi \int g(p_1) g(p_2) \overline{g(p_3)}\overline{g(p_4)}\theta(x_1) \theta(x_2) \overline{\theta(x_3)}\overline{\theta(x_4)}&\cdot \sum_{m \leq X/(dp_i), x_i/(dp_i) \leq m \leq (x_i+H)/(dp_i) \forall i=1,2,3,4}e(m(p_1+p_2-p_3-p_4) \alpha) dx_1 dx_2 dx_3 dx_4.
\kappa_{1, n}={\frac{1}{6}(s_n^2-s_{k_n}^2 )^{-3/2}}{(s_n^{(3)}- s_{k_n}^{(3)})}, \kappa_{2, n}= \frac{1}{72}(s_n^2-s_{k_n}^2 )^{-3}(s_n^{(3)}- s_{k_n}^{(3)})^2, \kappa_{3, n}= \frac{1}{24}(s_n^2-s_{k_n}^2 )^{-2}\sum_{j=k_n}^{n-1}\Big(\sigma_j^{(4)}-3\big(\sigma_j^{(2)}\big)^2 \Big).
\left[ \frac \delta{\delta \Phi \left( y \right)}, w \left( x \right) \right] = - d_{\mathrm{W}}\left( \Phi \right) \delta \left( x - y \right) \frac \delta{\delta \Phi \left( x \right)}\, \, , \Phi = \left( b^{a}, \, h^{a}\right) \, \, ,
\widetilde{S}={\frac{1}{2}}\left( \begin{matrix}{{1}}&{{1}}&{{1}}&{{1}}\\{{1}}&{{1}}&{{- 1}}&{{- 1}}\\{{1}}&{{- 1}}&{{- 1}}&{{1}}\\{{1}}&{{- 1}}&{{1}}&{{- 1}}\\ \end{matrix}\right)
\widehat{P}_{}(\hbar):=\frac{i}{2}\left(\widehat{P}_{\hbar}^*-\widehat{P}_{\hbar}\right)=\frac{i}{2}\left(\widehat{Q}_1^*-\widehat{Q}_1\right)+\frac{i\hbar}{2}\left(\widehat{Q}_2^*-\widehat{Q}_2\right),
\begin{array}{rcl}x^{+}= F_{x}(y_{[1,R]})\circ\Psi(x,z,v_{[0,R-A]}) z^{+}= y_{c,[1]}\circ\Psi(x,z,v_{[0,R-A]}) v_{[0,R-A-1]}^{+}= y_{[A+1,R]}\circ\Psi(x,z,v_{[0,R-A]})\,.\end{array}
\lim_{t\rightarrow \infty}\, \lim_{l\rightarrow \infty}\ \ \sup_{\xi\geq 0 , u\in K}\Biggl[ \frac{1}{l}\int_{\xi}^{\xi+l}\bigl \| q(t+s,u)\bigr\|^{p}\, ds\Biggr]^{1/p}=0.
\frac{d^2y}{dx^2}=\left (\frac{1}{2y}+\frac{1}{y-1}\right )\left (\frac{dy}{dx}\right )^2-\frac{1}{x}\frac{dy}{dx}+(1-2\Theta)\frac{y}{x}-\frac{y(y+1)}{2(y-1)},~~\Theta\in\mathbb{R}\setminus \mathbb{Z},
\begin{array}{l}\lim\limits_{t\rightarrow \infty}\mathbb{E}[g(e_{t-t_k},e_{t-t_{k-1}},...,e_{t-t_1})]= \mathbb{E}[g(e^*_0,e^*_{t_k-t_{k-1}},...,e^*_{t_k-t_1})].\end{array}
G_{D}^{( n )}({\bf r^{\prime \prime}},{\bf r^{\prime}}; E ) = \sigma^{n}\left[ G_{D}^{( 0 )}({\bf 0},{\bf 0}; E ) \right]^{n - 1}G_{D}^{( 0 )}({\bf r}^{\prime \prime},{\bf 0}; E ) G_{D}^{( 0 )}({\bf 0},{\bf r}^{\prime}; E ) ,
\kappa_{\ell,0}= \frac{\ell^{\frac32}}{(\arctan\frac{1}{\ell})^{\frac12}}\left( \frac{1}{\ell^2+\tau^2},0\right),\nu_{\ell,0}=\frac{\ell^{\frac32}}{(\arctan\frac{1}{\ell})^{\frac12}}\left(0,\frac{1}{\ell^2+\tau^2}\right),
v_B^{-1}\triangleright \chi^{\epsilon}_{s+1}= -\epsilon q^{-s-\frac{1}{2}}\overset{\mathcal{X}^+(2)}{W}\!\!\!\!\!_B \triangleright \big(v_B^{-1}\triangleright \chi^{\epsilon}_{s}\big) - q^{-2s}v_B^{-1}\triangleright \chi^{\epsilon}_{s-1}.
A^+_{ji}:=\sum_{s\geq 1}\sum_{j=j_1<\ldots<j_{s+1}=i}(-1)^{s-1}\tilde{e}^{(0)}_{j_1 j_2}\cdots\tilde{e}^{(0)}_{j_s j_{s+1}}, A^-_{ij}:=\sum_{s\geq 1}\sum_{j=j_1<\ldots<j_{s+1}=i}(-1)^{s-1}\tilde{f}^{(0)}_{j_{s+1}j_s}\cdots\tilde{f}^{(0)}_{j_2 j_{1}}
\mathcal{B}^{t}\varphi(\rho_{-})=-H'(\rho_{-})b(\rho_{-},t)\varphi'(\rho_{-})-\int_{\rho_{-}}^{\infty}[H]_{\rho_{-},\rho_{+}}(\varphi(\rho_{+})-\varphi(\rho_{-}))f(\rho_{-},\rho_{+},t)d\rho_{+}
\pi_t (m^1,m^2) =\frac{Q(y_t|e^1_t(m^1),e^2_t(m^2))\pi_{t-1}(m^1,m^2)}{\sum\limits_{m^1,m^2}Q(y_t|e^1_t(m^1),e^2_t(m^2))\pi_{t-1}(m^1,m^2)}
\mathbf{P}_{R_s}=\sum\limits_{k=R_l+1}^R{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k, \mathbf{P}_{R_l}=\sum\limits_{k=1}^{R_l}{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k.
p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}A_{k}(n)\sqrt{k}\cdot\frac{d}{dn}(\frac{1}{\sqrt{n-\frac{1}{24}}}sinh[\frac{\pi}{k}\sqrt{\frac{2}{3}(n-\frac{1}{24})}]).
P_{n}(t, S)=~ p(t, S) + \int_{0}^{t}rK e^{-r(t-\xi)}\aleph (-d_{2}(S, t -\xi, \mathcal{B}_{n}(\xi)))\mathrm{d}\xi - \int_{0}^{t}\delta S e^{-\delta (t- \xi)}\aleph (-d_{1}(S, t- \xi, \mathcal{B}_{n}(\xi))) \mathrm{d}\xi,
T_0 \ge\begin{cases}\displaystyle C_{\ast}\varepsilon^{-\frac{1}{(\frac{1}{p-1}-\frac{n}{2})(1+\beta)}}(\beta \in (-1,1)), \displaystyle \exp \left( C_{\ast}\varepsilon^{-\frac{1}{\frac{1}{p-1}-\frac{n}{2}}}\right) (\beta = -1).\end{cases}
\frac{\partial g}{\partial N_\ell}\,=\, h_L^{-\alpha/\alpha'}h_\ell^{-d}- \frac{\mu}{\lambda}N_\ell^{-1/\lambda-1}h_\ell^{\beta}\,=\, 0 \qquad\mbox{for}\ell=0,\ldots,L\,. \frac{\partial g}{\partial \mu}\,=\, \sum_{\ell=0}^L N_\ell^{-1/\lambda}h_{\ell}^{\beta}- c_2 h_L^{2\alpha}= 0\,.
N_d(x,y) =\frac{\displaystyle\int_0^{1}t^{m-1}\mathrm{Tr}(e^{-t\Delta_d^{(N)}})dt}{(m-1)!\omega_d}\cdot(\max\{\rho(x),\rho(y)\})^{2m-d}+\frac{\Gamma(m-\frac{d}{2})}{(4\pi)^{\frac{d}{2}}(m-1)!}+ &\ \ \ \ \frac{\mathrm{Tr}(e^{-\Delta_d^{(N)}})}{\omega_d}\cdot(\min\{\rho(x),\rho(y)\})^{-d}.
[\begin{matrix}I\\P\\T\end{matrix}]=[\begin{matrix}0.4000&0.4000&0.2000\\ 4.4550&-4.8510&0.3960\\ 0.8056&0.3572&-1.1628\end{matrix}][\begin{matrix}L^{\prime}\\M^{\prime}\\S^{\prime}\end{matrix}]
\frac{d}{du}\left(\begin{array}{c}\partial\beta/\partial v \partial t/\partial v \end{array}\right)=\left(\begin{array}{cc}a_{11}a_{12}a_{21}a_{22}\end{array}\right)\left(\begin{array}{c}\partial\beta/\partial v \partial t/\partial v\end{array}\right)\textrm{for}v=0,