harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pi_{\mu}= \frac{1}{2 \pi \alpha^{\prime}}[ ( g_{\mu \nu}+ k^{2}A_{\mu}A_{\nu}){\dot{x}}^{\nu}+ k R \lambda A_{\mu}] , \hspace{2 0 m m}\Pi_{5}= \frac{a_{\tau}}{2 \pi{\alpha^{\prime}}}.
\frac{R}{r}\ge\frac{abc+a^{3}+b^{3}+c^{3}}{2abc}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-1\ge\frac{2}{3}(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})\ge2.
=\sum_{l=0}^{|k|-1}\frac{2|k|-(2l+1)}{2k^2h^2}\left[ \phi(x+(l-1)h)-2\phi(x+lh) + \phi(x+(l+1)h) \right] &=\frac{1}{2k^2h^2}\left( \phi(x+kh)+\phi(x+(k-1)h)+(2k-1)\phi(x-h)-(2k+1)\phi(x) \right),
\frac{w_i'w_i}{d}\,-\, \sum_{j=0}^k \frac{w_i'N_{(i-1)}}{d}\left( I_{i-1}- \frac{N_{(i-1)}'N_{(i-1)}}{d}\right)^j\frac{N_{(i-1)}'w_i}{d}\,+\, r_i
r_0 = 3y^{2}+z^{2}-\frac{17}{4}y^{3}+\frac{9}{4}y^{4}+\frac{3}{4}z^{4}+\frac{7}{4}w^{4}-\frac{3}{8}y^{5}-\frac{3}{8}yz^{4}-\frac{3}{8}yw^{4}.
\begin{aligned}\varphi (X_t)-\varphi(X_0)=\int_0^t 1_{(a_k,b_k]}(X_{s-})dX_s +\sum_{0<s\leq t}\left[\varphi(X_s)-\varphi(X_{s-})-1_{(a_k,b_k]}(X_{s-})\Delta X_s \right] +\frac{1}{2}L(t,a_k)-\frac{1}{2}L(t,b_k). \end{aligned}
\varepsilon'(\Theta^{\star}(d_1),d_1) =\frac{P_{\texttt{total}}'(\Theta^{\star}(d_1),d_1)-\frac{P_\texttt{total}(\Theta^{\star}(d_1),d_1)}{\Theta^{\star}(d_1)}}{B\Theta^{\star}(d_1)}&=0.
H_{n,k,m}: =H_{0,m}\left(\sum P_{n,k}(\ell)a_{\ell}z^{\ell}\right) &=\begin{pmatrix}P_{n,k}(0)a_0 P_{n,k}(1)a_{1}\ldots P_{n,k}(m)a_{m}P_{n,k}(1)a_{1}P_{n,k}(2)a_{2}\ldots P_{n,k}(m+1)a_{m+1}\ldots P_{n,k}(m)a_{m}P_{n,k}(m+1)a_{m+1}\ldots P_{n,k}(2m)a_{2m}\end{pmatrix},
\begin{cases}\dot{x}_{i}^{N}(t) = \frac{1}{N}\stackrel[j=1]{N}{\sum}m_{j}^{N}(t)\partial_{x}V(x_{j}(t)-x_{i}(t)), x_{i}^{N}(0)=x_{i}^{0,N}\dot{m}_{i}^{N}(t) = \psi_{i}^{N}(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)), m_{i}^{N}(0)=m_{i}^{0,N}.\end{cases}
-\int^{Q_{\{a,p\}}(y)}_{Q_{\{a,p\}}(x)}\frac{dt}{P(t)}=\frac{2}{\sqrt[3]{4}}B\left(y^2;\frac{1}{6},\frac{2}{3}\right)-\frac{2}{\sqrt[3]{4}}B\left(x^2;\frac{1}{6},\frac{2}{3}\right),
(A-A)\cap (E_2-E_1)\ =\ (A\cap E_2) - (A\cap E_1) =\T_{E_2}(S_2) - T_{E_1}(S_1) =\ (M(S_2) + \mathbf{b}_2) - (M(S_1) + \mathbf{b}_1) =\M(S_2-S_1) + (\mathbf{b}_2-\mathbf{b}_1) =\T_{E_2-E_1}(S_2-S_1),
|\mathcal{E}_d^{}(\rho)| \leq \begin{dcases}C \rho^2 \widehat{g}(0)^3 \rho R^2 \log(\widehat{g}(0)), d=2, C\rho^2 \widehat{g}(0)^3 \rho R^2 \sqrt{\rho \widehat{g}(0)^3}, d=3.\end{dcases}
R_{0,0}= 1, R_{1,0}= x+ y + \hat{q}y^{-1}x^{-1}, R_{1,1}= x^{-1}+ y^{-1}+ \hat{q}yx, R_{1,2}= y^{-1}x + \hat{q}^3y^2x + \hat{q}^3y^{-1}x^{-2}, R_{1,3}= y^{-1}x^2 + \hat{q}^5y^3x + \hat{q}^8y^{-2}x^{-3}, R_{2,0}= x^2+y^2+\hat{q}^4y^{-2}x^{-2}, R_{3,0}= x^3 + y^3 + \hat{q}^9y^{-3}x^{-3}.
r_d(G):=\sum_{F \in \mathcal{F}_G^{+}}\frac{R_d(F)}{d_F^{2}}=\sum_{F \in \mathcal{F}^{+}_{G}}\frac{\mathrm{Res}_{s=1}\zeta_F(s)}{2^d d_F^2 \zeta_F(2)}> 0.
C_{k,n}= \max_{k \le n_0 \le n}\Biggl\{(n_0 -1)\Biggl(\frac{F(\beta_k|\beta_k)}{k-1}+ \sum_{l=k}^{n_0-1}\frac{F(\beta_{l+1}|\beta_k) -F(\beta_{l}|\beta_k)}{l}\Biggr) \Biggr\}
G^{(n),1}(s):=r^{(n)}\left(s \right), G^{(n),5}(s;x,y) :=\sum_{i}\sum_{j\leq i}d^{(n)}_{ij}\left(s \right)f_i(x)f_j(y), &G^{(n),3}(s):=p^{(n)}\left(s \right), G^{(n),6}(s;x) :=\sum_i\mu^{(n)}_i\left(s \right)f_i(x)=\mu^{(n)}\left(s ;x\right),
\mathrm{T r}e^{- s Q - s{\cal M}^{2}}= \frac{e^{- s{\cal M}^{2}}}{1 6 \pi^{2}s^{2}}\sum_{n = 0}^{\infty}s^{n}\biggl \{\int d x^{4}\sqrt{g}\:{\cal L}^{( n )}\biggr \}.
\mathrm{rank}\ \widetilde H_i (\Delta ( P(a,b) ); \mathbb Z )= 2 \sum_{t=0}^i \binom{a-3-i}{t-1}\bigg[ \binom{i}{t}\binom{b-2-i}{i-t}+ \binom{i}{t-1}\binom{b-3-i}{i-t}\bigg],
\frac{\partial}{\partial \theta^\alpha}= \frac{\partial\theta_\eta}{\partial \theta^\alpha}\frac{\partial}{\partial\theta_\eta}+ \frac{\theta'_{\eta}}{\partial\theta^\alpha}\frac{\partial}{\partial\theta_{\eta'}}= -\eta_\alpha\frac{\partial}{\partial\theta_\eta}- \eta'_\alpha\frac{\partial}{\partial \theta'_{\eta}},
\widetilde{I}_{3k}^{(m)}(F)= \int_{B\eta}^1 \frac{B}{y(B-y)}\idotsint_{\mathcal{R}_{k-1}}\left( \int_0^{T_m(y)}F(x_1,\dots,x_k) \, dx_m\right)^2 dx_1 \dots dx_{m-1}\dots dx_k dy
\sum_{\substack{n \in[x, x + x^\theta] n\equiv c\pmod{W}}}g(n) \rho^+(n) e\left(\gamma n \right) = \frac{\alpha^+ + o(1)}{\log x}\cdot \frac{[W,q]}{\varphi([W,q])}\sum_{\substack{n \in[x, x + x^\theta] n\equiv c\pmod{W}(n, q) = 1}}g(n) e\left(\gamma n \right) + O\left(\frac{x^{\theta}}{\log^Ax}\right),
T_{t_{-\alpha_2}}\big(E_{2,k+2}\big) = -T_{t_{-\alpha_2}}T_{k+1}\big(E_{2,k+1}\big) = -T_{k+1}T_{t_{-\alpha_2}}\big(E_{2,k+1}\big) \hphantom{T_{t_{-\alpha_2}}\big(E_{2,k+2}\big)}{}= T_{k+1}\big(E_{2,k+1}\big)(1) = -E_{2,k+2}(1).
N_q(X) - N_q(Y) = \theta_q(\frac{n-1}{2}) \bigl( (d-1)q^{\frac{n-1}{2}}+1 \bigr) -\Bigl(\theta_q(\frac{n-3}{2})q^{\frac{n-1}{2}}(d-1) + \theta_q(\frac{n-1}{2}) \Bigr) = (d-1)q^{n-1}= q^{n-\frac{1}{2}}>0.
\int_{Q_{e,V}}\bar{\sigma}_{\underline{\vec{a}}_1}\cup \bar{\sigma}_{\underline{\vec{a}}_2}= \sum_{e_1 + e_2 = e}\sum_{\vec{b}}\left( \int_{Q_{e_1,V_1}}\bar{\sigma}_{\underline{\vec{a}}_1}\cup \bar{\sigma}_{\vec{b}}\right) \left( \int_{Q_{e_2,V_2}}\bar{\sigma}_{\underline{\vec{a}}_2}\cup \bar{\sigma}_{\vec{b}^c}\right).
\tilde{\Gamma}^2_{22}=\Gamma^2_{22}+\frac{\gamma_y}{\gamma},\tilde{\Gamma}^2_{12}=\Gamma^2_{12}+\frac{\gamma_x}{2\gamma},\tilde{\Gamma}^2_{11}=\Gamma^2_{11}, \tilde{\Gamma}^1_{22}=\Gamma^1_{22},\tilde{\Gamma}^1_{12}=\Gamma^1_{12}+\frac{\gamma_y}{2\gamma},\tilde{\Gamma}^1_{11}=\Gamma^1_{11}+\frac{\gamma_x}{\gamma}.
\langle \hat{\phi}^{2}\rangle ={\frac{b ( m )}{1 6 \pi^{2}}}~ ~ , ~ ~ \langle \hat{\bar{\psi}}\hat{\psi}\rangle = 4 m \langle \hat{\phi}^{2}\rangle ~ ~ , ~ ~ \langle \hat{V}_{\mu}\hat{V}^{\mu}\rangle = 3 \langle \hat{\phi}^{2}\rangle .
| \psi \rangle = \Psi_{e \bar{e}}( x , \vec{k}_{\perp}; \lambda_{1}, \lambda_{2}) b_{\lambda_{1}}^{\dagger}( x , \vec{k}_{\perp}) d_{\lambda_{2}}^{\dagger}( 1 - x , - \vec{k}_{\perp}) | 0 \rangle
M^{1}(\Omega) := \{u\in L^2(\Omega)| \tilde{Y}_ju \in L^2(\Omega) \mbox{for}j\in \{1,\dots,2n\}\}, M^{2}(\Omega) := \{u\in L^2(\Omega)|\tilde{Y}_ju, \tilde{Y}_{k}\tilde{Y}_\ell u\in L^2(\Omega) j,k,\ell\in \{1,\ldots, 2n\}\}.
\beta_i^{\mathbb{Z}_p}((I+(w))^q) - \beta_i^\mathbb{Q}((I+(w))^q) = \sum_{\ell=1}^q \left[\beta_i^{\mathbb{Z}_p}(I^{\ell})+\beta_{i-1}^{\mathbb{Z}_p}(I^{\ell})-\beta_i^{\mathbb{Q}}(I^{\ell})-\beta_{i-1}^{\mathbb{Q}}(I^{\ell})\right] & \geq \sum_{\ell=1}^h \left[\beta_{i-1}^{\mathbb{Z}_p}(I^{\ell})-\beta_{i-1}^{\mathbb{Q}}(I^{\ell})\right]+|B| \geq |B|,
\frac{R_n}{{n \choose 2}}=\frac{1}{{n \choose 2}}\sum_{1\leq i< j \leq n}{\bf 1}_{A(i,j)}\frac{Q_n}{{n \choose 2}}=\frac{1}{{n \choose 2}}\sum_{1\leq i< j \leq n}{\bf 1}_{B(i,j)}
\Phi^{-1}(p) ~ \sim ~ \frac{1}{A}\left[ 1 ~-~ \frac{A^\prime}{(p^2)^{\omega}}\right] ~~~,~~~ \sigma^{-1}(p) ~ \sim ~ \frac{1}{B}\left[ 1 ~-~ \frac{B^\prime}{(p^2)^{\omega}}\right]
\min \;F(d,X)=\eta_1 F_1(RX)+\eta_2F_2(d) s.a. \;\; T(X)+M^T\alpha-\beta=0 &\;\;\Gamma d-MX=0 &\;\;\beta^T X=0 &\;\;\beta\geq 0, X\geq 0, d\geq 0
H_t \left[ \frac{1}{2}(\vec{v}_t + \vec{w}_t) \right] = \frac{1}{2}(\lambda_t \vec{v}_t + \mu_t \vec{w}_t) H_t \left[ \vec{v}_t - \vec{w}_t \right] = \lambda_t \vec{v}_t - \mu_t \vec{w}_t
c_{\lambda}^{+}({\bf p}) | 0 > = 0 \; , ~ ~ | f > = \int{\tilde{f}}^{( \lambda )}({\bf p )}c_{\lambda}^{+}({\bf p )}| 0 > \; , ~ ~{\tilde{f}}^{( \lambda )}({\bf p )}= \left. \int \frac{d{\bf p}}{\left( 2 \pi \right)^{3 / 2}}e^{- i{\bf p x}}e_{k}^{( \lambda )}({\bf p}) f^{k \perp}( x ) \right|_{x^{0}= 0}\; ,
\frac{\mathrm d^k}{\mathrm d t^k}\, \Phi_{a,b}(t)= \sum_{j=0}^k C_{a,j,k}t^{a -(k-j)}A^{b+j}S(t)Q^\frac{1}{2}= \sum_{j=0}^k C_{a,j,k}\Phi_{a-(k-j),b+j}(t),
t^{(5,2,2,1)}+ t^{(4,3,2,1)}(1-t_1) + t^{(3,3,2,2)}(1-t_1) &+ t^{(4,2,2,2)}(1-t_1)(1-t_2) + t^{(3,2,2,2,1)}(1-t_1)(1-t_2) + t^{(3,3,2,1,1)}(1-t_1)(1-t_4) + t^{(4,2,2,1,1)}(1-t_1)(1-t_2)(1-t_4)
\int_{ext}\left<T_{00}(\vec{x})\right>_B = -\frac1{16\pi^2}\left(a\int_{ext}\zeta_B(2|0,x_2) + b\int_{ext}\zeta_B(2|x_1,0)\right) +\frac1{32\pi^2}\int_{ext}\zeta_B(2|x_1,x_2)
+\frac{1}{\sqrt{2\pi}}\sum_{n=1}^{\infty}\frac{\int\limits_0^tu_n(s)e^{\lambda^-_n(t-s)}ds}{l^{(1)}_n(\lambda^-_n)}\psi_n(x)+\frac{1}{\sqrt{2\pi}}\sum_{n=1}^{\infty}\sum\limits_{k=0}^{N-1}\left(\frac{\int\limits_0^tu_n(s)e^{-q_{k,n}(t-s)}ds}{l^{(1)}_n(-q_{k,n})}\right)\psi_n(x).
{\cal H}_{B_{n}}^{u n t w i s t e d}={\frac{1}{2}}p^{2}+{\frac{g_{l}^{2}}{2}}\sum_{\alpha \in \Delta_{l}}\wp ( \alpha \cdot q ) +{g_{s}^{2}}\sum_{\lambda \in \Delta_{s}}\wp ( \lambda \cdot q ) .
F_{1}=\sum_{s=0}^{1}\sum_{k=1}^{m}\sum_{t=0}^{1}\left(\Delta_{a_{1},\iota(t)}^{\alpha,\beta}-\Delta_{\iota^{c_{k}}(a_{s+1}),\iota^{c_{k}+k+s}(t)}^{\alpha,\beta}\right)e_{a_{1}}W(t;c_{k}+k+s-2)g_{k,s}=\sum_{s=0}^{1}\sum_{k=1}^{m}\sum_{t=0}^{1}\left(\Delta_{a_{1},\iota(t)}^{\alpha,\beta}-\Delta_{a_{s+1},\iota^{k+s}(t)}^{\alpha,\beta}\right)Y_{k,t}(s).
\frac{\mathbb{E}(X_p Y_p )}{\mathbb{E}(X_p) \mathbb{E}(Y_p)}= \frac{\Big( 1-\frac{1}{p}\Big)^{-(k-1)}+\Big( 1-\frac{1}{p}\Big)^{-(\ell-1)}-1}{\Big( 1-\frac{1}{p}\Big)^{-(k-1)}\Big( 1-\frac{1}{p}\Big)^{-(\ell-1)}}= \Big( 1-\frac{1}{p}\Big)^{k-1}+ \Big( 1-\frac{1}{p}\Big)^{\ell-1}- \Big( 1-\frac{1}{p}\Big)^{k+\ell-2}.
\sum_{k=\tau_{s_1}}^{\tau_{s_1+1}-1}{A_{il_1}(k)}\geq \frac{1}{n^2},\,\, \sum_{k=\tau_{s_2}}^{\tau_{s_2+1}-1}{A_{l_1 l_2}(k)}\geq \frac{1}{n^2},\,\, \ldots,\,\, \sum_{k=\tau_{s_{r}}}^{\tau_{s_{r}+1}-1}{A_{l_{r-1}j}(k)}\geq \frac{1}{n^2}.
\nu_\varepsilon\cdot\nabla\lambda = \frac{(-1)^{i+1}k_d^{-1}}{\sqrt{1+\varepsilon^2|\bar{\tau}_\varepsilon^i|^2}}\Bigl(-\overline{V_\Gamma}+\varepsilon^2\bar{g}_i\bar{\tau}_\varepsilon^i\cdot\nabla\overline{V_\Gamma}\Bigr) \quad\Gamma_\varepsilon^i(t)
\lim_{k\rightarrow\infty}\frac{\frac{1}{k}\sum_{i=1}^k\lambda_i}{k^{\frac{2}{n}}}=\lim_{k\rightarrow\infty}\frac{\Big(\frac{1}{k}\sum_{i=1}^k\lambda_i\Big)+\frac{n^2H_0^2+\eta_0^2+2\bar{\eta_0}}{4}}{k^{\frac{2}{n}}}&=\lim_{k\rightarrow\infty}\frac{\frac{1}{k}\sum_{i=1}^k(\lambda_i+\frac{n^2H_0^2+\eta_0^2+2\bar{\eta_0}}{4})}{k^{\frac{2}{n}}}.
\vert \hat g(t)-g(t)\vert =\vert g(\langle a,t\cdot\hat a\rangle)-g(t)\vert \leq c_0 \left\vert t \left(1-\langle a,\hat a\rangle\right)\right\vert=c_0 \vert t\vert\cdot\left\vert\langle a,a-\hat a\rangle\right\vert &\leq c_0\cdot\|a-\hat a\|_{l_2^d}.
\lim_{N\rightarrow\infty}{\prod_{i=1}^m{\big(t^{N-1+1}; q\big)}_{\theta(i-1)}}= 1, \lim_{N\rightarrow\infty}{\frac{1}{\prod\limits_{i=1}^m{\big(x_i t^{m-1}q; q\big)_{\theta(N-m+1)-1}}}}= \frac{1}{\prod\limits_{i=1}^m{\big(x_it^{m-1}q; q\big)_{\infty}}}.
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}}=\frac{\sqrt{\pi}\,2^{\frac{1}{2}-s}}{\Gamma(s)x^{s-\frac{1}{2}}}\intop_{0}^{\infty}y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(xy)\,\sigma_{p}(y)\,dy,\,\,\,\,(s)>\frac{1}{2}.
\lambda_{k,n}(\alpha_p) = \max_{w \in W_p}\frac{\int_0^L \frac{|w'|^2 V_{\alpha}}{|\alpha_p'|}+ \frac{k^2 |w|^2 | \alpha_p'|}{V_{\alpha}}\,dt}{\int_0^L |w|^2 V_\alpha |\alpha_p'| \,dt}
\begin{cases}|\nabla\zeta_{k,h}|+h^{\varrho_\perp}|D^2\zeta_{k,h}|\leq Ch^{-\varrho_\perp}B(b_k,h^{\varrho}/2) |\nabla\zeta_{k,h}|+h^{\varrho}|D^2\zeta_{k,h}|\leq Ch^{-\varrho}B(b_k,h^{\varrho}) \end{cases}\,.
\mathrm{div}^{\overline{\nabla}}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})=\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})+(-1)^{r+1}\widehat{W}(S_{r+1}) +(-1)^{r+1}(-S_{1}S_{r+1}+(r+2)S_{r+2})-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W}).
I_1 = (-\infty, \omega_{\xi}^{\ast}- \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}], I_2 = [\omega_{\xi}^{\ast}- \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}, \omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}], I_3 = [\omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}, 0], \mbox{and}I_4 = [0, \infty).
X^{\mu}= X_{0}^{\mu}+{\frac{1}{\pi T}}P^{\mu}\tau +{\frac{i}{2 \sqrt{\pi T}}}\sum_{n \neq 0}{\frac{1}{n}}\bigg \{\alpha_{n}^{\mu}e x p \bigg ( - i 2 n ( \tau - \sigma ) \bigg ) + \tilde{\alpha}_{n}^{\mu}e x p \bigg ( - i 2 n ( \tau + \sigma ) \bigg ) \bigg \}
R(u_1 -u_2 ) t^{c}_{b}(u_1 )\otimes t^{b}_{a}(u_2 )=\sum_{d}W\left( \left. \begin{array}{cc}c d b a \end{array}\right| u_1 -u_2 \right) t^{d}_{a}(u_1 )\otimes t^{c}_{d}(u_2 ).
\left.\begin{array}{l}\delta_a\bigl([S_a]^{-1}[S_2]\bigl)=\delta_a\bigl((D_2)(\gamma_2)(D_1)^{-1}\bigl)=0, \delta_b\bigl([S_b][S_1]\bigl)=\delta_b\bigl((D_1)(\gamma_1)\bigl)=0,\end{array}\right.
dC^{(n)}= F^{(n+1)}= (-1)^{\frac{n(n+1)}{2}}\hat{\star}F^{(9-n)}= (-1)^{\frac{n(n+1)}{2}}\hat{\star}dC^{(8-n)}= (-1)^{\frac{n(n-1)}{2}}d\hat{\star}C^{(10-n)},
M_{b c a}^{( s , t , r ) \d}\equiv \hat{G}_{a b}\delta_{c}^{d}\delta_{r + s , 0 m o d \lambda}+ \hat{G}_{a c}\delta_{b}^{d}\delta_{r + t , 0 m o d \lambda}- f_{a b}^{e}f_{e c}^{d}.
A\mathcal{X}^x_t =e^{At}Ax+\int^t_0e^{A(t-s)}A F(\mathcal{X}^x_s) \, ds+\int^t_0e^{A(t-s)}A B(\mathcal{X}^x_s) \, dW_s &=e^{At}Ax+\int^t_0e^{A(t-s)}A F(A^{-1}(A\mathcal{X}^x_s)) \, ds+\int^t_0e^{A(t-s)}A B(A^{-1}(A\mathcal{X}^x_s)) \, dW_s.
\begin{gathered}D_{q,\rho}\,\rho^{1-\nu}\int_0^\infty u^{\alpha}\left[\int_0^b x \Phi_1(x)J_{\nu-\alpha}(ux; q^2)\,d_qx + \int_b^\infty x \Psi_1(x)J_{\nu-\alpha}(ux; q^2)\,d_qx \right] \times J_{\nu-1}(u\rho q^{-1}; q^2) \,d_q u = - \frac{\rho^{1-\nu}q^{1-\nu}G(\rho)}{(1-q)}, \rho\in B_{q,b}. \end{gathered}
-i\frac{3a^{2}}{2\beta \hslash^{2}}\lambda a_{2}+ a_{1}= \lambda^{2}a_{1}\, \Rightarrow \, a_{2}= i \frac{2\beta \hslash^{2}}{3a^{2}}. \frac{\lambda^{2}-1}{\lambda}a_{1}
P^\mu_{\ \nu}= -{1\over (d-2)^2}\left[ R^\mu_{\ \lambda}R^\lambda_{\ \nu}-{d\over 2(d-1)}R R^\mu_{\ \nu}-{1\over 2}\delta^\mu_\nu R^\alpha_{\ \beta}R^\beta_{\ \alpha}+{d+2\over 8(d-1)}\delta^\mu_{\ \nu}R^2 \right] \ .
\begin{array}{c c}{\hat{M}_{0}( z ) Z_{0}= Z_{0}H ( z )}&{H ( z ) = \left( \begin{array}{c c}{\tilde{\hat{\omega}}( z ) - \frac{1}{2}\hat{\lambda}( z )}&{2 \hat{\rho}^{j}( z )}\\{0}&{\hat{T}_{i}^{j}( z )}\\ \end{array}\right)}\\ \end{array}
I_1=\mathbb{E}[\int_0^T(\int_D\{h(t,x)-\widehat{h}(t,x)\}\mathbb{E}[\delta_Z(z)|\mathcal{F}_t]dx)dt], I_2=\mathbb{E}[\int_D\{k(x)-\hat{k}(x)\}\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]dx].
\prod_{i=0}^{j-h}(H_a-i+1)^{-1}\,\cdot\,\sigma_a(x)\,\cdot\,\prod_{i=0}^{j-h}(H_a+i+1)\,= &\prod_{i=1}^{j+h}(H_a+i+1)\,\cdot\,\sigma_a(x)\,\cdot\, \prod_{i=1}^{j+h}(H_a-i+1)^{-1}\,.
e_{H}(B_r) =\, \left(\frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- 2(q-1)^r+3^r \right) T + \left(- \frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- (q-1)^r+3^r \right) N ,
H_{m, n}^+(x) = \frac{4}{\pi}\int_{t = 0}^\infty \int_{\zeta = -\infty}^\infty \cos(x \cosh \zeta) e\Bigl( \frac{t \zeta}{\pi}\Bigr) e^{- \frac{(t - T)^2}{M^2}}V(m^2 n, t) t \, d \zeta \, d t + O(T^{-A})
\tilde{{\mathcal H}}^{\,^{\bullet}}\ : = \, d i s p l a y s t y l e \bigoplus_{k = 0}^{\infty}\ \tilde{{\mathcal H}}^{k}\, \quad \tilde{{\mathcal H}}^{k}\ : = \ \overline{{{\tilde{\Omega}^{k}({\mathcal A})}}}^{( \cdot , \cdot )}\ .
\begin{array}{c c}{{\gamma^{a}= \left( \begin{array}{c c}{{0}}&{{\rho^{a}}}\\{{\bar{\rho}^{a}}}&{{0}}\\ \end{array}\right) \, ,}}&{{~ ~ \rho^{a}\bar{\rho}^{b}+ \rho^{b}\bar{\rho}^{a}= 2 \delta^{a b}\, .}}\\ \end{array}
{\|f\|}_{\dot{H}^{N,b}}:= \Big\{\sum_{k\in\mathbb{Z}}{\|P_kf\|}^2_{L^2}(2^{2Nk}+2^{2kb}) \Big\}^{1/2}\approx{\big\| (|\partial_x|^N + |\partial_x|^b)f \big\|}_{L^2}, &{\|f\|}_{\dot{W}^{N,b}}:= \sum_{k\in\mathbb{Z}}{\|P_kf\|}_{L^\infty}(2^{Nk}+2^{bk}),
\int \prod_{i > j}[ d c^{i j}] [ d \bar{c}^{i j}] [ d A_{\mu}^{i j}] e^{- S_{\mathrm{o f f - d i a g}}}= \exp \left[{\frac{1}{8 \pi}}\int_{\Sigma_{g}}R \sum_{i > j}\log ( \lambda_{i}- \lambda_{j}) \right] ~ .
p_{W_\mathcal{I},X^n, Y_\mathcal{I}^n, \hat W_\mathcal{I}}=p_{W_\mathcal{I},X^n}\left(\prod_{k=1}^n p_{Y_{\mathcal{I},k}|X_k}\right) p_{\hat W_1 |Y_1^n}p_{\hat W_2 |Y_2^n}.
f(w; k \lambda) = f\left(w; (k \lambda)^{\vee\vee}\right) = \check{C}\cdot \delta^{k w}\cdot \dfrac{\prod_{i=0}^{k r-1}\varGamma\left(w + \frac{i}{k r}\right)}{\prod_{i=1}^{k r}\varGamma(w+ \xi_i)},
X_{1}Y_{2}Z_{3}=[\begin{matrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{matrix}]
\;|\langle A^2 u^N,u^N\cdot \nabla u^N\rangle| \leq \;C\| A u^N\|_{L^{\frac{2n}{n-2s}}}\|\nabla u^N\|_{L^{\frac{n}{s}}}(\|Au^N\|_{L^2}+\|\nabla^2 u^N\|_{L^2}) \leq \;C\|u^N\|_{D(A^{1+s/2})}\|u^N\|_{D(A^{(1+\frac{n}{2}-s)/2})}\|u^N\|_{D(A)}\leq \;C\|u^N\|_{D(A^{1+s/2})}\|u^N\|_{D(A)}^2.
T = \ln(C)+\frac{1}{\sigma_n^2}\left(\sum_{m=1}^{\tau}\textbf{y}^H[m]\textbf{G}[m]\right) \times \left(\sum_{m=1}^{\tau}\textbf{G}^H[m]\textbf{G}[m]+ \textbf{D}\right)^{-1}\!\left(\sum_{m=1}^{\tau}\textbf{G}^H[m]\textbf{y}[m]\right).
S_{k}= \Phi_{B_{m}}^{\dot{A}_{n}}\left( i \partial_{\beta_{m + 1}\dot{\alpha}_{n}}\Lambda_{\dot{A}_{n - 1}}^{B_{m + 1}}+ i \partial_{\dot{\alpha}_{1}}^{\beta_{1}}\cdots i \partial_{\dot{\alpha}_{n}}^{\beta_{n}}\Gamma^{\beta_{n + 1}\cdots \beta_{m}}\right) \ ,
\begin{aligned}Ef(a+dX) =\int\prod_{j=1}^n\cos (a_j+dy_j)\mathrm{d}\mu_X(y) &= \int\prod_{j=1}^n\left(\cos(a_j)\cos (dy_j)-\sin(a_j)\sin(dy_j)\right)\mathrm{d}\mu_X(y). \end{aligned}
(\lambda_{j_p}X^{-j_p}+ p H(X))^p = \sum^p_{j=0}\binom{p}{j}(\lambda_{j_p}X^{-j_p})^j(p H(X))^{p-j}\equiv \sum^p_{j=p-1}\binom{p}{j}(\lambda_{j_p}X^{-j_p})^j (p H(X))^{p-j}\pmod{p^2}= p(\lambda_{j_p}X^{-j_p})^{p-1}(p H(X)) + (\lambda_{j_p}X^{-j_p})^p \equiv \lambda_{j_p}^p X^{-p j_p}\pmod{p^2}
D[\Theta^{}_1(\Lambda)](0) = A_c D k_c(0) + Dg_c(K{}{}(0))DK{}{}(0) -Dk_c((A_c + r)(0))D(A_c + r)(0) &=A_c Dk_c(0) + Dg_c(0)DK{}{}(0) - Dk_c(0)(A_c + Dr)(0).
{\cal V}_{N ; g}\, = \,{\cal C}_{g}< \Omega | \int [{\mathrm{d}}m ]_{N}^{g}\exp \left\{\frac{1}{2}\sum_{i = 1}^{N}\sqrt{\frac{\alpha^{\prime}}{2}}p^{( i )}\cdot \left. \! \left[ \sqrt{\frac{\alpha^{\prime}}{2}}p^{( i )}+{\alpha}_{1}^{( i )}\partial_{z}+ \bar{\alpha}_{1}^{( i )}\partial_{\bar{z}}\right] \log | V_{i}^{\prime}( z ) |^{2}\right|_{z = 0}\right\}
\kappa_\pi(T_1,\ldots, T_n) = \sum_{\sigma \leq \pi}\mathcal{E}_\sigma(T_1, \ldots, T_n) \mu_{BNC}(\sigma, \pi)\mbox{and}\mathcal{E}_\pi(T_1, \ldots, T_n) = \sum_{\sigma \leq \pi}\kappa_\sigma(T_1, \ldots, T_n).
\sum_{a=0}^n \epsilon_a \zeta^0_a[g] \gamma_0 \cdot \gamma_a \cdot \partial_t \widetilde{\phi}= -\sum_{i}\sum_{a=0}^n \epsilon_a \zeta^{i}_a[g] \gamma_0 \cdot \gamma_a \cdot \partial_{i}\widetilde{\phi}- \sum_{a=0}^n \epsilon_a \gamma_0 \cdot \gamma_a \cdot \omega(s_a) \cdot \widetilde{\phi}.
\Pi_{a b}( k ) = i g D_{a a^{\prime}}^{- 1}( k ) \tau_{b x}^{3}\left( \int \frac{d^{6}p}{( 2 \pi )^{6}}\Gamma_{x a^{\prime}b}( p - k / 2 , k , - ( p + k / 2 ) ) \right) .
\begin{aligned}a_1 = 8n_1^3+6t^2n_1^2n_2+6t^2n_1n_2^2+8n_2^3-2n_1-2n_2+a_2, b_1 = 8n_1^3+2t(3t-4)n_1^2n_2-2t(3t-4)n_1n_2^2 & -(8t-8)n_2^3-2n_1+(2t-2)n_2+a_2, b_2 = 8tn_1^3+4(3t-2)tn_1^2n_2+8tn_1n_2^2-2tn_1+a_2, d_1 = -4n_1^2-(3t^2-4)n_1n_2-4n_2^2+1, d_2 = 4n_1^2-(3t^2-12t+4)n_1n_2+4n_2^2-1,\end{aligned}
l_R(dy)l_R(dz) =l_R(d)l_R(d)+l_R(d)l_R(z)+l_R(d)l_R(y)+l_R(y)l_R(z) =l_R(d)l_R(d)+l_R(d)[l_R(z)+l_R(y)]+l_R(y)l_R(z) =l_R(d)l_R(d)+l_R(d)l_R(yz)+l_R(y)l_R(z) =l_R(d)l_R(d)+l_R(d)l_R(ab)+l_R(a)l_R(b) =l_R(d)l_R(d)+l_R(d)l_R(a)+l_R(d)l_R(b)+l_R(a)l_R(b) =l_R(da)l_R(db),
a^{1234}=a^{1324},a_{1}^{1234}=a_{1}^{1324},a_{2}^{1234}=a_{2}^{1324},a_{3}^{1234}=a_{3}^{1324},a_{4}^{1234}=a_{4}^{1324}, a_{12}^{1234}=a_{12}^{1324},a_{13}^{1234}=a_{13}^{1324},a_{23}^{1234}=a_{23}^{1324},a_{24}^{1234}=a_{24}^{1324},a_{34}^{1234}=a_{34}^{1324}, a_{123}^{1234}=a_{123}^{1324},a_{234}^{1234}=a_{234}^{1324}
\langle\|x\|x-y\|y\|,y-z\rangle =\langle\|x\|(x-y)+y(\|x\|-\|y\|),y-z\rangle &\leq 2\big\|x-y\|\big|\big\|y-z\big\| &\leq \|x-y\|^2+\|y-z\|^2.
\times\left|H_{n-k}\left(\frac{(T+1)z+\left[\sigma_{pp}-\sigma_{qq}-2i\sigma_{pq}\right]z^{\star}}{\left\{(2T+4d+1)\left[\sigma_{pp}-\sigma_{qq}-2i\sigma_{pq}\right]\right\}^{\frac 12}}\right)\right|^2.
\Psi(u)= \frac{AA'(B-B')}{A+A'}u(A+A',B+B')-\frac{A(AB'+A'B)}{A+A'}u(A+A',B) &+\frac{A'(AB'+A'B)}{A+A'}u(A+A',B')+AB'u(A,B)-A'Bu(A',B'),
DH(v)u = \beta P|v|^2 u + 2\beta P (u\cdot v)v + \lambda_0 P (u\cdot\nabla)v + \lambda_0 P (v\cdot\nabla)u - 2 P \sum_{i,k=1}^n a_{jk}(u^j v^k+u^k v^j).
[\widehat{P}]_{i,j}^{}= \hat{p}_{i,j}^{}= \begin{dcases}\frac{n-i}{n}\mbox{if}\, j = i \neq n; \frac{n}{n}\mbox{if}\, j = i = n; \frac{i}{n}\mbox{if}\, j = i + 1; 0 \mbox{otherwise};\end{dcases}
\begin{array}{rcl}f(x)\!\! \!\!>\!\! \!\!Q_{13}(x)\!=\!1\!-\!8x^2\!-\! \overline T_{12}^{\cos ,0}(4x)\!-\!8 x^2 \overline T_8^{\cos ,0}(2x)\!+\!P_6(x) \underline T_7^{\sin ,0}(2x) ,\end{array}
0 = \liminf_{t \to + \infty}\int_{|x|> t}\left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x &= \liminf_{t \to - \infty}\int_{|x|>-\varepsilon -t}\left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x.
\frac{1-\delta_1\delta_2 R_{12}}{1+\delta_1\delta_2(R_{12}-\beta T_1)}\frac{(1-\beta\tilde T_1)^{-m+1}}{2-\beta\tilde T_1}\frac{(1-\beta\tilde T_2)^{-m+1}}{2-\beta\tilde T_2}\cdot d_\lambda(1)\, d_\lambda(2) = 0,
\int_0^T s||w(s)||^2_{H^2}ds = \int_0^T\!\!\! ( \int_t^T ||w(s)||^2_{H^2}ds) dt \le C \int_0^T ||w (t) ||^2_{H^1_0(0,L)}dt \le C ||w_0||^2_{L^2_{xdx}}.
\theta\bigl((\operatorname{id}\otimes\omega_{\Lambda_{\tilde{\varphi}}(x),\Lambda_{\varphi}(p)})(W)\bigr)=\bigl\langle\Lambda_{\tilde{\varphi}}(x),\Lambda_{\tilde{\varphi}}(\pi[(\bar{\theta}\otimes\operatorname{id})(\Delta p)])\bigr\rangle.
I_{q,\tilde x}^{(x)}(f(x)) = \int_0^{\tilde x}{\int_0^{{\sigma_{q - 1}}}{\ldots \int_0^{{\sigma_2}}{\int_0^{{\sigma_1}}{f({\sigma_0}){\mkern 1mu}d{\sigma_0}d{\sigma_1}\ldots d{\sigma_{q - 2}}d{\sigma_{q - 1}}}}}},
h_j \cdot g a_1^j C(c)=ge_1^{k+1-j}a_1^{-j}g^{-1}ga_1^jC(c)=ge_1^{k+1-j}C(c) \hbox{and}h_j \cdot gC(b)=ge_1^{k+1-j}a_1^{-j}C(b)=ge_1^{k+1-j}C(b).
c^t_{\Gamma'}=-[4t(1-t)]^{n-2}\sum_{e\neq (1,2)}(-1)^{e-1}\int_{\mathrm{Conf}_{n-2}(\mathbb C\smallsetminus\{0,1\})}\beta_e \omega^{\frac 1 2}_{\Gamma' \smallsetminus \{e,(1,2)\}}.
R^{\pm}( \zeta ) = \left( \begin{array}{c c c c}{{a^{\pm}( \zeta )}}&{{}}&{{}}&{{d^{\pm}( \zeta )}}\\{{}}&{{b^{\pm}( \zeta )}}&{{c^{\pm}( \zeta )}}&{{}}\\{{}}&{{c^{\pm}( \zeta )}}&{{b^{\pm}( \zeta )}}&{{}}\\{{d^{\pm}( \zeta )}}&{{}}&{{}}&{{a^{\pm}( \zeta )}}\\ \end{array}\right) .
\frac{d^2 f}{d x^2}+ \frac{a^2-6a+2 - (a^2 - 8a +2) x^{a}}{2x (1-x^{a})}\frac{d f}{d x}+\frac{\lambda^2}{x^{4-a}(1-x^{a})^2}f- \frac{l(l+a)}{x^2 (1-x^{a})}f =0\,.
dz=\phi^{*}\omega=(n-2)\frac{x_0(z,\tau)\frac{\partial x_k}{\partial z}(z,\tau)-x_k(z,\tau)\frac{\partial x_0}{\partial z}(z,\tau)}{(\Omega_{\tau})_{0,k}(x_0(z,\tau),\ldots,x_{n-1}(z,\tau) )}\cdot dz .
J_{\alpha\beta}^{i(1)}=\left(P(J_{\alpha\beta}^{i})\right)^{(1)}=\big{(}x^\gamma X_\gamma(J_{\alpha\beta}^{i})+z^j Z_j(J_{\alpha\beta}^{i})\big{)}^{(1)}=x^\gamma X_\gamma(J_{\alpha\beta}^{i})(q)
\int_{[D_R(0)\setminus\cup^s_{j=1}D_{2R'\lambda^j_\alpha/\mu^k_\alpha}(\tilde{y}^j_\alpha)]\cap D_{1/2\tilde{R}\mu^l_\alpha/\mu^k_\alpha}(\tilde{x}^l_\alpha)}|u^k|^{2^*}\,d\sigma_{\tilde{h}_\alpha}=o(1)+\epsilon_\alpha(R').
\frac{\partial P(A,t)}{\partial t}=\sum_{i,j}\frac{\partial}{\partial A_{i}}(-k_{B}T[A_{i},A_{j}]\frac{\partial H}{\partial A_{j}}+\lambda_{i,j}\frac{\partial H}{\partial A_{j}}+\lambda_{i,j}\frac{\partial}{\partial A_{j}})P(A,t).
B =\pi \frac{2\pi^{m/2}}{\Gamma(m/2)}\sum_{I=0}^L \frac{(2L)!}{(2I)!(2L-2I)!}\frac{(2I)!}{2^I I!}\frac{2^I(2L-2I)!L!}{(2L)!(L-I)!}\frac{(2L)!}{2^{2L}(L!)^2}(-1)^{L-I}&=\frac{1}{2^{2L}}\pi \frac{2\pi^{m/2}}{\Gamma(m/2)}\binom{2L}{L}\sum_{I=0}^L \binom{L}{I}(-1)^{L-I}.
p_k(x_1,\ldots,x_{d-1},x_d) = p_k[x_d](x_1,\ldots,x_{d-1}) = \sum_{i_1,\ldots,i_{d-1}=0}^{n}c_{i_1,\ldots,i_{d-1}}(x_d)\prod_{s=1}^{d-1}\phi_{i_s}(x_s),
\phi(y)=\sum_{s ; z_{i+1}\in I\left(z_{i}^{\prime}\right)}G_{I(y)}\left(y, z_{1}\right) G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime}, z_{2}\right) \cdots G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime}, z_{t+1}\right) \phi\left(z_{t+1}^{\prime}\right),
0=\int_0^T(PU_{\lambda},u)dx_0=\int_0^T(U_{\lambda},Pu)dx_0-i(D_0^2U_{\lambda}(T),u(T)) -i(D_0U_{\lambda}(T), D_0u(T))-i(U_{\lambda}(T),D_0^2(T))+i(U_{\lambda}(0),D_0^2u(0)) +i((D_1^2+x_1^2D_n^2)U_{\lambda}(T),u(T)).
\begin{array}{ccccccc}f_1=0\;, \quad \frac{\partial f_1}{\partial\lambda}=0\;, \quad \frac{\partial f_1\;\;\;\;}{\partial x_{2,3}}\neq0\;, \quad \frac{\partial^2 f_1}{\partial\lambda^2}\neq0\;,\end{array}\;,