harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
1 \leftrightarrow \bar{1}, \quad 2 \leftrightarrow \bar{2}, \quad \alpha \rightarrow \alpha^{\prime}, \quad \phi ( z^{1}, z^{2}, z^{3}) \rightarrow \phi^{\prime}( \bar{z}^{1}, \bar{z}^{2}, z^{3}) , \quad \psi \rightarrow \psi^{\prime}
V_{a}^{t}=\sum_{i}|A_{ai}|^{2}\nu_{i}^{t}, Z_{a}^{t}=\sum_{i}A_{ai}\hat{x}_{i}^{t}-\frac{V_{a}^{t}}{\Delta_{0}^{t}+V_{a}^{t-1}}\bigl(y_{a}-Z_{a}^{t-1}\bigr).
{\bar{e}}_{\alpha}^{i}{\bar{e}}_{\beta}^{j}\left[{B}_{i j}^{\bar{g}^{\prime}( 2 )}\right] ={\bar{e}}_{\alpha}^{i}{\bar{e}}_{\beta}^{j}\frac{\partial}{\partial \bar{g}^{i j}}[ \Gamma^{( 2 )}] +{K}_{\alpha \beta}^{g^{\prime}}
\mathrm{d}s_{K}^{2}= \sum_{m , n \neq k}^{\infty}{\frac{\partial^{2}K ( z_{( k )},{\bar{z}}_{( k )})}{\partial z_{( k )}^{m}\partial{\bar{z}}_{( k )}^{n}}}\, \mathrm{d}z_{( k )}^{m}\mathrm{d}{\bar{z}}_{( k )}^{n}.
R_{1}=\sum_{i=1}^{N}E\left[\max_{1\leq k\leq d}\left|\sum_{j=1}^{N}\gamma_{k}(i,j)W^{(i)}_j\right|^3\right](E[|Y_i|^{3}]+E[|G_i|^3]), R_{2}=\max_{1\leq k,l\leq d}\left|\mathfrak{C}(k,l)-E[Q_{k}(G)Q_{l}(G)]\right|, R_{3}=\max_{1\leq k\leq d}\sqrt{E[Q_{k}(G)^4]-3E[Q_{k}(G)^2]^2}.
\mathcal{TS}^{op}=\{\pi^*(B_1),\ldots,\pi^*(B_{k-1})-E,E,\pi^*(B_{k})-E,\ldots,\pi^*(B_n)\}=\{\pi^*(A_{n-1}),\ldots,\pi^*(A_{n-k+1})-E,E,\pi^*(A_{n-k})-E,\ldots,\pi^*(A_1),\pi^*(A_n)\}
\phi_{D}\biggl(\genfrac..{0pt}{}{a;\, b_{1}, \dotsc, b_{r}}{c}; x_{1}, \dotsc, x_{r}\biggr) := \sum_{i_{1}= 0}^{\infty}\dotsm \sum_{i_{r}= 0}^{\infty}\frac{(a)_{i_{1}+ \dotsm + i_{r}}(b_{1})_{i_{1}}\dotsm (b_{r})_{i_{r}}}{(c)_{i_{1}+ \dotsm + i_{r}}(q)_{i_{1}}\dotsm (q)_{i_{r}}}x_{1}^{i_{1}}\dotsm x_{r}^{i_{r}}.
\sum_{\lambda \in \mathcal{P}}t^{h_{1,1}(\lambda)}q^{|\lambda|}= \sum_{\lambda\in \mathcal{P}}\prod_{i\geq 1}\left[\lambda_i-\lambda_{i+2}+1\atop \lambda_{i+1}-\lambda_{i+2}\right]_{t}q^{{\lambda_1\choose 2}+|\lambda|}.
|F(xn_{k+1})-F(xn_k)| = |F^\prime (\xi )| |xn_{k+1}-xn_k| &\leq \frac{\xi +2}{\xi^2}|x|(n_{k+1}-n_k) &\leq \frac{xn_{k+1}+2}{x^2n_k^2}|x|(n_{k+1}-n_k) &=\frac{2n_{k+1}}{n_k^2}(n_{k+1}-n_k).
\left(W^{(m)}_{m_1,m_2}f \right)(z)=\left(\frac{r}{z+m_1\omega_1 +m_2\omega_2}\right)^2 \overline{f\left(a_m+ \frac{r^2}{\overline{z-a_m+m_1\omega_1 +m_2\omega_2}}\right)}
\begin{array}{rcl}\hbox{and}\nabla z A \nabla z = r \nabla z_{1}A \nabla z_{1}+ (1-r) \nabla z_{2}A \nabla z_{2}-r(1-r)(\nabla z_{1}- \nabla z_{2})A(\nabla z_{1}- \nabla z_{2}) &\leq r \nabla z_{1}A \nabla z_{1}+ (1-r) \nabla z_{2}A \nabla z_{2}. \end{array}
u (t)=\vartheta_{2}(\check{\xi}_{2}(t_{k}))+\Psi \eta (t_{k}) \dot{\hat{\xi}}(t)=A_{o}\hat{\xi}(t)+\lambda e(t_{k})+B(u(t)-\Psi \eta (t_{k})) \dot{\eta}(t)=M \eta (t)+N u (t),~\forall t\in[t_{k},t_{k+1}),~k\in\mathbb{S}
u(t,\cdot;s,u_0) = e^{-A(t-s)}u_0 -\chi \int_s^t e^{-A(t-\tau)}\nabla \cdot \left(\frac{u(\tau,\cdot;s,u_0)}{v(\tau,\cdot;s,u_0)}\nabla v(\tau,\cdot;s,u_0) \right) d\tau + \int_s^t e^{-A(t-\tau)}u(\tau,\cdot;s,u_0)\big[{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u(\tau,\cdot;s,u_0)\big] d\tau
\frac{d}{ds}\mathcal{F}_\Sigma (u+s\xi)\big|_{s=0}=-\int \left\langle \vec{H}+\frac{X^\perp(u)}{2}, \xi^A\frac{\partial}{\partial y^A}\right\rangle e^{-|X(u)|^2/4}\sqrt{\det g_u}\,dx^1\wedge\dots\wedge dx^n,
\begin{array}{l}\frac{M}{A_{8-p}}=\frac{8-p}{2}r_{H}^{7-p}+\frac{7-p}{2}\alpha, \left(\frac{Q_{0}}{(7-p)A_{8-p}}\right)^{2}=\alpha \left(\alpha+r_{H}^{7-p}\right),\end{array}
q = \left( \begin{array}{c c c c c}{{q_{( 1 )}}}&{{}}&{{}}&{{0}}&{{}}\\{{}}&{{q_{( 2 )}}}&{{}}&{{}}&{{}}\\{{0}}&{{}}&{{\cdots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{q_{( \tilde{N}_{c})}}}&{{}}\\ \end{array}\right) ,
\langle +\infty|-\infty\rangle_{\lambda+\delta \lambda}=\langle +\infty|-\infty\rangle_{\lambda}-\delta\lambda \frac{1}{\tilde{N}}{\int}d\mu [B(t)X^{\omega}(t) +ig\bar{c}(t)Y^{\omega}(t)c(t)]\exp\{i{\int}({\cal L}+{\cal L}_{g})dt\}
d_n (f)(g_1, \ldots , g_{n+1}) = g_1 \cdot f(g_2, \ldots, g_{n+1}) & + \sum_{i=1}^n (-1)^i f(g_1, \ldots, g_i g_{i+1}, \ldots, g_{n+1}) & + (-1)^{n+1}f(g_1,\ldots,g_n),
C^{-1}\Lambda^{-1}\int_{B}\int_{B}(v(x)-v(y))^{2}k_{0}(x,y)dxdy \leq c_{n,\beta}\int_{B}\int_{B}\frac{(v(x)-v(y))^{2}}{|x-y|^{n+2\beta}}dxdy & \leq C\Lambda \int_{B}\int_{B}(v(x)-v(y))^{2}k_{0}(x,y)dxdy, B = B_{\rho}(x_{0}).
|f(x)-p(x)| =|f(x)-\frac{1}{2}(p_1(x)+p_2(x))| &=|\frac{1}{2}(f(x)-p_1(x))+\frac{1}{2}(f(x)-p_2(x))| &\leq\frac{1}{2}|f(x)-p_1(x)|+\frac{1}{2}|f(x)-p_2(x)|.
\Phi_{t}\left(x,s\right)\coloneqq\begin{cases}\left(x,s+t\right) ,0\leq t<\varphi\left(x\right)-s \left(\theta^{N_{s+t}\varphi\left(x\right)-1}\left(x\right),s+t-S_{N_{t+s}-1}\varphi\left(x\right)\right) ,\varphi\left(x\right)-s\leq t.\end{cases}
\int_{e^{-(2+\delta)y}}^\infty t^{a+2b+3c-\frac{3}{2}-m-\frac{\epsilon}{2}}\left[e^{y}V ( \kappa-\Lambda)\right]^{a+3b+4c-2m}e^{-te^{2y}V^2( \kappa-\Lambda)^2}dt &= O\left(e^{-(j+c -1 - \epsilon ) y}\right)+ O\left( e^{\frac{\delta}{2}y-(j+c-1 - \epsilon )(1+\frac{\delta}{2}) y}\right),
[ \nabla_{\mu}, \nabla_{\nu}] \phi_{\mu_{1}\dots \mu_{k}}= - \sum_{i = 1}^{k}R_{\mu_{i}\mu \nu}^{\lambda}\phi_{\mu_{i}\dots \mu_{i - 1}\lambda \mu_{i + 1}\dots \mu_{k}}+ W_{\mu \nu}\phi_{\mu_{1}\dots \mu_{k}},
\left\|\left( T_{\varphi}(x_{j})\right)_{j=1}^{n}\right\|_{Y^{\rm dual}(F^*)}= \sup_{(y_{j})_{j=1}^{\infty}\in B_{Y(F)}}\left|\sum_{j=1}^{n}\varphi(x_{j}\otimes y_{j})\right|\leq \left\|\varphi \cdot \right\|\left\|(x_{j})_{j=1}^{n}\right\|_{X(E)}.
\|\begin{matrix}M_{1}^{1}&M_{1}^{2}&M_{1}^{3}\\M_{2}^{1}&M_{2}^{2}&M_{2}^{3}\\M_{3}^{1}&M_{3}^{2}&M_{3}^{3}\end{matrix}\|=\|\begin{matrix}1&0&0\\ 0&cos(\theta)&sin(\theta)\\ 0&-sin(\theta)&cos(\theta)\end{matrix}\|
S(z) = \left(\mathbb{I}+\mathcal{O}\left(\frac{1}{z}\right)\right)\begin{pmatrix}1 0 0 0 z^{\frac{1}{4}}0 0 0 z^{-\frac{1}{4}}\end{pmatrix}\begin{pmatrix}1 0 0 0 \frac{1}{\sqrt{2}}\frac{i}{\sqrt{2}}0 \frac{i}{\sqrt{2}}\frac{1}{\sqrt{2}}\end{pmatrix}.
\bordermatrix{\cr \nu^{{\pi^{*,\infty}}}(0)\cr \nu^{{\pi^{*,\infty}}}(1)\cr}=\bordermatrix{\cr \nu^{\pi^{*,\infty}}(0|0) \nu^{\pi^{*,\infty}}(0|1)\cr \nu^{\pi^{*,\infty}}(1|0) \nu^{\pi^{*,\infty}}(1|1)\cr}\bordermatrix{\cr \nu^{{\pi^{*,\infty}}}(0)\cr \nu^{{\pi^{*,\infty}}}(1)\cr}
X(t)=\frac{1}{\Gamma(\beta)}\left(\int_0^{t}(t-s)^{\beta-1}[AX(s)+h(X(s))]ds-\int_0^{\tau_0}(\tau_0-s)^{\beta-1}[AX(s)+h(X(s))]ds\right),t\le\tau_0.
1\otimes c_sE(\lambda n_s^{-1}) = 1\otimes c_sE(\lambda_1)E(\lambda_2 n_s^{-1}) & = 1\otimes E(\lambda_1)(\lambda_1^{-1}\cdot c_s)E(\lambda_2 n_s^{-1}) & = (sw\widetilde{\chi})(E(\lambda_1))\otimes(\lambda_1^{-1}\cdot c_s)E(\lambda_2 n_s^{-1}).
\sum_{e=n+1}^{n+m}\sum_{f=n+1}^{n+m}\langle \gamma_b, \gamma_e, g^{ef}\gamma_f,\gamma_c\rangle_{0,1}=(n+5)\frac{\partial^2 F}{(\partial s_1)^2}(0) \delta_{b,c}+\frac{\partial F}{\partial s_2}(0)\delta_{b,c}.
h_{12}^2=-h_{13}^3, &\omega_{11}^2=-\frac{h_{11}^2 (\cos (c-2 \Lambda )+\cos (2 c))}{\sin (c-2 \Lambda )+\sin (2 c)}, &\omega_{11}^3=-\frac{h_{11}^3 (\sin (2 c)-\sin (2 \Lambda +c))}{\cos (2 c)-\cos (2 \Lambda +c)}.
[\mathcal{M},\tilde{\mathcal{M}}]=0,\, [\mathcal{M},\mathcal{F}_A]=0,\, [\tilde{\mathcal{M}},\mathcal{F}_A]=0, & [F_A,\mathcal{F}_B]=0,\,[\mathcal{M},\mathcal{F}_{AB}]=0,\, ....,
f = 2 \zeta ( 3 ) \tau_{2}^{3 / 2}+{\frac{\tau_{2}^{3 / 2}}{\Gamma ( 3 / 2 )}}\sum_{n \neq 0 , p}\int_{0}^{\infty}d y y^{1 / 2}\exp \left\{- y ( p + n \tau ) ( p + n \bar{\tau}) \right\}
v_{x+y}(t) = \int_t^1\frac{(s-t)^{y-1}}{\Gamma(y)}\frac{v_x(s)}{(1-s)^\alpha}(1-s)^\alpha ds &\le C \int_t^1\frac{(s-t)^{y-1}}{\Gamma(y)}\frac{v_x((1-\lambda)+\lambda s)}{\lambda^\alpha(1-s)^\alpha}(1-s)^\alpha ds &=C \lambda^{-(\alpha+y)}\int_{t'}^1\frac{(u-t')^{y-1}}{\Gamma(y)}{v_x(u)}du &= C\lambda^{-(\alpha+y)}v_{x+y}(t').
U_{j-\frac{1}{2}}^{+}=Q_j(x_{j-\frac{1}{2}}), U_{j+\frac{1}{2}}^{-}=Q_j(x_{j+\frac{1}{2}}), \bar{U}_{j+q}=\frac{1}{\Delta x}\int_{x_{j+q-\frac{1}{2}}}^{x_{j+q+\frac{1}{2}}}Q_j(x)dx,
\sqrt[4]{\int_{0}^\mathrm{T_0}\Big[|\nabla_{\textbf{tan}}\Phi^{\textbf{e}}(\xi,t;z,\tau)|^2 + \Phi^{\textbf{e}}(\xi,t;z,\tau)^2 + \Big(\partial_{t}^{\frac{1}{2}}\Phi^{\textbf{e}}(\xi,t;z,\tau)\Big)^2\Big] d\tau}= \mathcal{O}\Bigg(\sqrt[4]{\mathcal{K}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-z|^{\frac{3}{2}-\mathrm{r}}}\Bigg).
F(p^{(\varphi,\psi,\rho)})=F\left(\left(p^{j\,(\varphi_1,\psi_1,\rho_1)}\right)^{(\varphi,\psi,\rho)}\right)=F(p^{j\,(\varphi\varphi_1,\psi\psi_1,\rho\rho_1)})=(\psi\psi_1)q_j(\rho\rho_1)
P(x_1x_2x_4x_3) = \P(x_2x_1x_4x_3) + P (dx_2dx_1x_4x_3) + P ([x_1,x_2]x_4x_3) P(x_1x_2dx_4dx_3) = \P(x_2x_1dx_4dx_3) + P (dx_2dx_1dx_4dx_3) + P ([x_1,x_2]dx_4dx_3) P(x_1x_2[x_3,x_4]) = \P(x_2x_1[x_3,x_4]) + P (dx_2dx_1[x_3,x_4]) + P ([x_1,x_2][x_3,x_4]).
D(\alpha, \alpha)w_{\theta,\frac{\pi}{2}-\theta}D(\alpha, \alpha) = w_{\phi,\phi'}D(\beta, \gamma)w_{\phi,\phi'}, D(\alpha, \alpha)w_{-\theta,\theta-\frac{\pi}{2}}D(\alpha, \alpha) = w_{-\phi,-\phi'}D(\beta, \gamma)w_{-\phi,-\phi'},
\sum_{m=1}^M\psi_\mu(m,\chi)= \sum_{m=1}^M\sum_{n<m}\chi(n)\Lambda(n)(m-n)^{\mu-1}=\sum_{n=1}^M\chi(n)\Lambda(n)\sum_{m=n+1}^M(m-n)^{\mu-1}= \sum_{n=1}^M\chi(n)\Lambda(n)\sum_{m=1}^{M-n}m^{\mu-1}=\sum_{n=1}^M\chi(n)\Lambda(n)\int_0^{M-n}u^{\mu-1}du+O(M).
\hat{V}_{\mu\nu\rho}^{(1\,x,y)\,AB}=-2\cdot 2\,{\rm Tr}\,T^AT^B\,(Y_R^{(d)}-Y_L^{(d)})\, V_{\mu\nu\rho}^{(1)}=4\,{\rm Tr}\,T^AT^B\,V_{\mu\nu\rho}^{(1)}\, .
\times \left[ \frac 1 2 ( d - 3 )^{2}\zeta_{R}\left( 2 z + 2 k - l ,{\frac{d + 1}{2}}\right) + ( d - 1 ) \zeta_{R}\left( 2 z + 2 k - l - 1 ,{\frac{d + 1}{2}}\right) \right] .
= m^{( 1 )}c^{( 1 )}: \cos \Bigg ( 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi \! + \! g N}}U_{1 b}\Phi^{( 1 )}( x ) + 2 \sqrt{\pi}\sum_{I = 2}^{N - 1}U_{I b}\Phi^{( I )}( x )
S=\int d^2\sigma \sqrt{g}\left( \frac{1}{4\pi\alpha'}g^{\alpha\beta}\partial_\alpha X^{\mu}\partial_\beta X^{\nu}g_{\mu\nu}(X) + \frac{i}{2\pi}\epsilon^{\alpha\beta}B_{\mu\nu}(X) \partial_\alpha X^{\mu}\partial_\beta X^{\nu}- \frac{1}{4\pi}\phi(X) R \right)
G ( \omega , \vec{k}) = - \frac{\phi_{1}f_{2}\partial_{r}\phi_{2}^{( - )}- \phi_{2}^{( - )}f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}\phi_{2}^{( + )}- \phi_{2}^{( + )}f_{1}\partial_{r}\phi_{1}}\ .
|W|q^{d^2-1}\theta_d=|M| \le\delta(\alpha+4)q^{d^2+d-2}+\sum_{l\in \Omega_1}| l\cap W|q^{d^2-1}&\hphantom{\le{}}\mathrel{+}\omega_2 |W|q^{d^2-d-1}+\sum_{\mathclap{(P,H)\in \Omega_3}}|W|q^{d^2-d}+|\widehat{\Omega}_4|.
\Omega_{ij}=\Omega_{ji}=\left( \begin{array}{cc}\frac{4\varepsilon+0.9}{3.6\varepsilon-0.09}\frac{\varepsilon}{1.8\varepsilon-0.045}\frac{\varepsilon}{1.8\varepsilon-0.045}\frac{\varepsilon}{3.6\varepsilon-0.09}\end{array}\right),
R_k(s;\alpha,v)=\frac1{k!}\left(\frac{\Delta}{2s\,S_{\nu\nu}''(\nu_-)}\right)^k\left(f(\nu,\mu)e^{sS_1(\nu,\mu;\nu_-,\mu_-)}\right)\Big|_{(\nu,\mu)=(\nu_-,\mu_-)},
\big|[b,\mathcal I_{\alpha}]f_2(x)\big| \leq\big|b(x)-b_B\big|\int_{\mathbb R^d}\big|\mathcal K_{\alpha}(x,y)f_2(y)\big|\,dy+\int_{\mathbb R^d}\big|b(y)-b_B\big|\big|\mathcal K_{\alpha}(x,y)f_2(y)\big|\,dy &:=\xi(x)+\zeta(x).
|| v ||_{\min}= \sup_k || v_k ||^k_{\min}= \sup_k \left\{\sup \{| f_k (v_k) | \colon f_k \in \mathfrak{S}(V_k)\}\right\}&=\sup \left\{| f_k \circ p_{\infty, k}(v) | \colon k \in \mathbb{N}, f_k \in \mathfrak{S}(V_k)\right\}&\leq \sup \{|f (v) | \colon f \in \mathfrak{S}(V_{\infty})\}= || v ||_{m}.
\gamma^n( \{e_x\}\times [0,t]) - \beta^n_{(2,3)}(\{e_x\}\times [0,t]) = n^{-1}\int_0^{t}\psi_e(t_n -s ) \left( \bar \Lambda^n(e_x \mid t_n - s) - \bar \xi^n_{(2)}( e_x \mid t_n - s)\right) ds &= n^{-1}\int_{t_n-t}^{t_n}\psi_e(s) \left( \bar \Lambda^n ( e_x \mid s) - \bar \xi^n_{(2)}(e_x \mid s) \right) ds.
\int_{K \backslash G /H}\int_N f(x) \cdot 1_{A\cap q(N)}(x \cdot \ddot{y}) d\omega(x) d\mu (\ddot{y})=\int_N \int_{K \backslash G /H}f(x) \cdot 1_{A\cap q(N)}(x \cdot \ddot{y})d\mu(\ddot{y})d\omega(x).
J_{y,z}(\theta_1,\theta_2) = 16 \pi^2 (\cos(2 \pi (\theta_1 -3\theta_2)) + \cos(2 \pi (3\theta_1 + \theta_2)) - \cos(2 \pi (\theta_1 + 3\theta_2)) & - \cos(2 \pi (3\theta_1 - \theta_2)).
\deg(p_x(y,\cdot))\le\sum_{u\in S_x}I(x) - 2, p_x(y,u)=\begin{cases}1 u\in (S_x-\{x\}))\cap[-1,y) 1 u=y 0 (S_{x}-\{x\})\cap(y,1] \end{cases}, p'_{x}(y,u)=0,u\in(S_{x}-\{x\})\cap(-1,1).
\begin{array}{c}{\displaystyle \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}D_{m}^{2}( k_{1}) D_{m}( k_{2}) D_{m}( k_{1}+ k_{2}) \overline{{D ( k_{1}+ p )}}K_{1}.}\\ \end{array}
U_1 S_1 V_1^{\top}= \begin{pmatrix}\cos \phi_{R0}\delta d_0 \sin \phi_{R0}\end{pmatrix}\frac{1}{1 + (\delta^2 d_0^2-1) \sin^2 \phi_{R0}}\begin{pmatrix}\cos \phi_{R0}\delta^2 d_0^2 \sin \phi_{R0}\end{pmatrix}
\omega_{\mu}{}^{a b}( e ) = - e^{\nu [ a}\left( \partial_{\mu}e_{\nu}{}^{b ]}- \partial_{\nu}e_{\mu}{}^{b ]}\right) - e^{\rho [ a}e^{\sigma b ]}\left( \partial_{\sigma}e_{c \rho}\right) e_{\mu}{}^{c}\ .
N(k) =(q^k-1)N(k-1)= &=(q^k-1)q^{\frac{k-1}{2}\left(\frac{k-1}{2}+1\right)}\prod_{i=1}^\frac{k-1}{2}(q^{2i-1}-1)= &=q^{\frac{k-1}{2}\frac{k+1}{2}}\prod_{i=1}^\frac{k+1}{2}(q^{2i-1}-1).
\dot{\tilde{w}}=\dot{w}-\dot{\hat{w}}=\frac{1}{\rho}\tilde{p}-k_{1}(\bar{y}-\hat{\bar{y}})\,, \dot{\tilde{p}}=\dot{p}-\dot{\hat{p}}=T\tilde{w}_{zz}-k_{2}(\bar{y}-\hat{\bar{y}})\,,
\Psi_{c}( x , t ) = \left[ \frac{\Delta \hat{q}\Gamma ( 1 / 4 )}{2}\right]^{- \frac{1}{4}}\exp \left\{- \frac{( x - \langle \hat{q}\rangle )^{4}}{8 ( \Delta \hat{q})^{4}}+ \frac{i}{\hbar}\left( \langle \hat{p}\rangle x - \frac{1}{2}\langle \hat{q}\rangle \langle \hat{p}\rangle \right) \right\}.
G'(0) = -\frac{n}{r_n}I_0(f^*)V_n(X^n)+I_1(f^*)V_{n-1}(X^n) & = -n\sqrt{n}\frac{2^n}{n!}+2\sqrt{n}\frac{2^n}{(n-1)!}\frac{1}{2}& = 0,
L_{+}^{D}= \left( \begin{array}{l l}{{( q t^{- 1})^{S^{3}}t^{Z}}}&{{\Omega S^{-}}}\\{{0}}&{{( q t )^{- S^{3}}t^{- Z}}}\\ \end{array}\right) \ \ \ , \ \ \L_{-}^{D}= \left( \begin{array}{l l}{{( q t )^{- S^{3}}t^{Z}}}&{{0}}\\{{- \Omega S^{+}}}&{{( q t^{- 1})^{S^{3}}t^{- Z}}}\\ \end{array}\right) \ ,
\begin{aligned}\nabla_{ii}\phi = \, \delta \nabla_{ij}u \nabla_{ij}u + \delta \nabla_{j}u \nabla_{iij}u + b \nabla_{ii}\eta \geq \, \frac{\delta}{2}U_{ii}^2 - C \delta + \delta \nabla_{j}u \nabla_{iij}u + b \nabla_{ii}\eta. \end{aligned}
F_{r, m,n,s}(x; q, c) = (-1)^{m+n}\Delta\psi^{(m-1)}_q(x;c)\Delta\psi^{(n-1)}_q(x;c) - \alpha_{r,m,n,s}(-1)^{r+s}\Delta\psi^{(r-1)}_q(x;c)\Delta\psi^{(s-1)}_q(x;c),
\{\Theta_{1}, \Theta_{2}\}^{\pm}= r_{+}^{1 2}\Theta_{1}\Theta_{2}+ \Theta_{1}\Theta_{2}r_{+}^{1 2}- \Theta_{1}r_{\mp}^{1 2}\Theta_{2}+ \Theta_{2}r_{\pm}^{1 2}\Theta_{1}.
\tau^{n,\omega^\prime}_R:=\mathbf{1}_{\left\lbrace 3\left\lvert z^{\omega^\prime}\right\rvert_\infty<R\right\rbrace}\cdot\inf\left\lbrace t\ge 0: \left\lvert X^{n,\omega^\prime}_t\right\rvert > \frac{R}{3}\right\rbrace
\varphi((v_Z^u)^m) = \sum_{\stackrel{k_i \geq 0}{\scriptscriptstyle{\sum_i k_i \leq m}}}(k_1, \ldots, k_{l(u)}, m - \sum_i k_i)! \; \varphi(s^{r_1^u}\otimes s^{r_2^u}) & = \sum_{\stackrel{k_i \geq 0}{\scriptscriptstyle{\sum_i k_i \leq m}}}(k_1, \ldots, k_{l(u)}, m - \sum_i k_i)! \; \delta_{r_1^u, 0}\; \delta_{r_2^u, 0},
\mathbf{i}_\alpha \widetilde\alpha \equiv( i_{W\alpha}\otimes 1 + 1\otimes i_\alpha)\widetilde\alpha=0\quad,\quad\mathbf{L}_\alpha \widetilde\alpha \equiv(L_{W\alpha}\otimes 1 + 1\otimes L_\alpha)\widetilde\alpha =0 \quad,
\lambda_1 =(\frac{\theta_{1100}\theta_{1000}}{\theta_{0100}\theta_{0000}})^2,\;\lambda_2=(\frac{\theta_{1001}\theta_{1100}}{\theta_{0001}\theta_{0100}})^2,\;\lambda_3=(\frac{\theta_{1001}\theta_{1000}}{\theta_{0001}\theta_{0000}})^2,
I(a, b)=\int_{u}^v \frac{(qx/u, qx/v, \alpha ax, \beta b x; q)_\infty}{(ax, bx, cx, dx; q)_\infty}d_q x=\sum_{n=0}^\infty \lambda_n \Phi_n^{(\alpha, \beta)}(a, b|q).
B_k \supset \bigcap_{i=0}^{J_k}\left\{\sup_{t\in M_i^k}\frac{Q_{X}(t)}{f_p(t)}<1\right\}\supset \bigcap_{i=0}^{J_k}\left\{\sup_{t\in M_i^k}Q_{X}(t)< y_i^k\right\}= \bigcap_{i=0}^{J_k}\left\{\sup_{\substack{s\in \tilde M_i^k \tau\ge 0}}Z_{y_i^k}(s,\tau)< m(y_i^k)\right\}.
Q_1 :=\int_0^1 \int_0^1 a(y_1,y_2) R_1'(y_1+y_2)\mathrm{d}y_1\mathrm{d}y_2, Q_2 :=\int_0^1 \int_0^1 a(y_1,y_2) R_2'(y_1-y_2)\mathrm{d}y_1\mathrm{d}y_2.
\int_{B_{\rho}(\zeta_i)}E_i =6\,\int_{B_{\rho}(\zeta_i)}U_i^5\,Q_i+15\int_{B_{\rho}(\zeta_i)}U_i^4\,Q_i^2+\mathcal{R}&=6\,\sum_{j\neq i}\int_{B_{\rho}(\zeta_i)}U_i^5\,U_j+15\sum_{j\neq i}\sum_{m\neq i}\int_{B_{\rho}(\zeta_i)}U_i^4\, U_j\, U_m+\mathcal{R}.
\sum_{{\bf q}_k}\frac{1}{\sqrt{2 E_{q_k}V}}e^{- i \overline{Q}_k \cdot z_2}\sum_{p_j}\frac{1}{\sqrt{2 E_{p_j}V}}e^{i \overline{P}_j \cdot z_1}\langle b^\dagger_{-{\bf p}_j}b_{-{\bf q}_k}\rangle
i \ln W_{q \overline{{q}}}^{\mathrm{S R}}= \int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d t \left\{- \frac{4}{3}\frac{\alpha_{s}}{r}+ \frac{1}{2}\frac{4}{3}\frac{\alpha_{s}}{r}( \delta^{h k}+ \hat{r}^{h}\hat{r}^{k}) \dot{z}_{1}^{h}\dot{z}_{2}^{k}\right\}.
i_{\ast}\mathcal{E}_{d, v}=i_{\ast}p_{\ast}(\mathcal{O}_{\mathcal{Z}}\otimes \mathbb{C}(m_1, \ldots, m_d)) &\cong p_{\ast}(i_{\ast}\mathcal{O}_{\mathcal{Z}}\otimes q^{\ast}\mathcal{O}_{\mathcal{Y}(d)^{\lambda}}(m_1, \ldots, m_d)).
T^{H_0,\ldots,H_n}_{f_k^{[n]}}(V_1,\ldots,V_n)= \sum_{p=0}^n\sum_{0<j_1<\cdots<j_{p}\leq n}\!(-1)^{n-p}\, T^{H_0,H_{j_1},\ldots,H_{j_p}}_{(f_ku^p)^{[p]}}(\tilde{V}_{0,j_1},\ldots,\tilde{V}_{j_{p-1},j_p})\,\tilde{V}_{j_{p},n}\,
f_{I_{2}(m)}^{N}(q,\mu)=\sum_{j=1}^{m}g_{|j|}\,g_{|j+N|}\,x_{|j+N|}(\alpha_{j+N}\cdot q,\alpha_{j+N}^{\vee}\!\!\cdot\mu)\,x_{|j|}(\alpha_{j}\cdot q,-\alpha_{j+2N}^{\vee}\!\!\cdot\mu).
\sum_{i \in I}v_{i}^{2}\langle \pi_{Z_{i}}C^{\prime}f, \pi_{Z_{i}}C f\rangle=\Vert v_{i}(C^{*}\pi_{{Z}_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}_{2}\leq(\dfrac{(1+\lambda_{1})\sqrt{B}+\Vert\beta\Vert_{2}}{1-\lambda_{2}}\Vert f \Vert)^{2}.
f =\sum_{j=1}^{\infty}\lambda_{j}^{1}a_{j}^{1}=\sum_{j=1}^{\infty}\lambda_{j}^{1}\Pi_{l}(g_{j}^{1},h_{j,1}^{1}, h_{j,2}^{1}) +\sum_{j=1}^{\infty}\lambda_{j}^{1}\big(a_{j}^{1}- \Pi_{l}(g_{j}^{1}, h_{j,1}^{1},h_{j,2}^{1}) \big) =: M_{1}+ E_{1}.
\frac1{2\pi}\,\int_0^{2\pi}\frac{d\theta}{(\sigma_1\cos^2\theta + \sigma_2\sin^2\theta)^{m+1}}= \frac1{2^{2m}}\,\sum_{j=0}^m{2j\choose j}{2(m-j)\choose m-j}\,\sigma_1^{-\frac12-j}\sigma_2^{-\frac12-(m-j)};
|U'(q_k)| = |A\alpha q_k^{\alpha-1}- B \beta q_k^{-\beta -1}+ \phi'(q_k)| = \begin{cases}A\alpha q^{\alpha -1}(1 + o(1)) \,q_k\rightarrow \infty B \beta q^{-\beta-1}(1 + o(1)) \, q_k\rightarrow 0^+\end{cases}.
u_{2G}(v) = \frac{1-e^{-\frac{v^2}{2\theta^2}}}{v^{2h}}, u_{2e}(v) = \frac{1-e^{-\frac{v}{2\theta^2}}}{v^{2h}}, u_{2C}(v) = \frac{1-\frac{1}{1+\frac{v^2}{\theta^2}}}{v^{2h}}, u_{2t}(v) = \frac{1-\frac{1}{1+v^{\theta}}}{v^{2h}}, u_{2i}(v) = \frac{\frac{1}{\theta}-\frac{1}{\sqrt{v^2+\theta^2}}}{v^{2h}}.
{\rm Res}_{\bar{z}^{\prime}=\bar{z}}^{\Psi}\bar{\partial}^{\Psi}\!:\!\psi(z^{\prime})\! :\,\bar{\partial}^{\Psi}\!:\!\phi(z)\! :=-:\!(\phi_{;\mu\eta}-iC_{\eta\nu\mu}\phi_{;\nu})\psi^{;\eta}\bar{\partial}^{\Psi}\! X^{\mu}\!:+O(\epsilon^5) .
\varphi_1^2(x) = \frac{d}{d t}\Big|_{t=0}u_{\frac{\pi}{3}-t, \frac{\pi}{3}, \frac{\pi}{3}}, \varphi_2^2(x) = \frac{d}{d t}\Big|_{t=0}u_{\frac{\pi}{3}, \frac{\pi}{3}-t, \frac{\pi}{3}},\varphi_3^2(x) = \frac{d}{d t}\Big|_{t=0}u_{\frac{\pi}{3}, \frac{\pi}{3}, \frac{\pi}{3}-t}.
\rho(\tt e)=\begin{bmatrix}0 0 1 0 -1 0 1 0 0\end{bmatrix}\mbox{and}\rho(\tt p)=\begin{bmatrix}e^{i\alpha}\sqrt{-2\cos{3\alpha}}e^{-2i\alpha}0 e^{-2i\alpha}-\sqrt{-2\cos{3\alpha}}e^{-i\alpha}0 0 e^{i\alpha}\end{bmatrix},
b_\varepsilon \ge b_{1,\varepsilon}+b_{2,\varepsilon}\shortintertext{with}b_{1,\varepsilon}= (1-\delta) D^2 - \varepsilon (V^0(x) + \delta |x|^{-2q}),b_{1,\varepsilon}=\sigma D^2 - \varepsilon U(x)
\sigma_0=\sigma_2=\sigma_4=\sigma_6=(01)(27)(36)(45), &\sigma_1=\sigma_3=\sigma_5=\sigma_7=(07)(16)(25)(34), &\sigma_0\sigma_1=(0246)(1357), &(\sigma_0\sigma_1)^2=(04)(15)(26)(37), &\sigma_1\sigma_0=(\sigma_0\sigma_1)^3=(0642)(1753), &\sigma_0\sigma_1\sigma_0=(03)(12)(47)(56) &\sigma_1\sigma_0\sigma_1=(05)(14)(23)(67).
\mathrm{d}M = \prod_{i = 1}^{N}\mathrm{d}\lambda_{i}^{2}[ \mathrm{d}V ] [ \mathrm{d}U ] | \Delta ( \lambda^{2}) |^{2}\ , \ \mathrm{w h e r e}\ \Delta ( \lambda^{2}) = \prod_{i < j}( \lambda_{i}^{2}- \lambda_{j}^{2}) \ .
\left(\begin{array}{c}{[\phi_1]_0}{[\phi_5]_0}\end{array}\right) = \frac{1}{\lambda}B\,\,\mathbf{U}^H_+(b; p_{z*},0,\omega_*) + \left(\begin{array}{c}{[\widetilde{\phi}_1]_0}{[\widetilde{\phi}_5]_0}\end{array}\right).
S_3(X_3(2))=\{9,5,9,5\ldots \}S_3(X_3(7))=\{29,45,69,105,53 \ldots \}S^5_5(X_5(7))=\{57,145,73,185,465\}S^6_7(X_7(6))=\{73,37,133,469,1645,5761\}
({\cal D}^2-D_1\bar{D}_1+\frac{1}{2}D_1^2\bar{P}_{11}-\frac{1}{2}P_{11}\bar{P}_{11}+ d\lambda^2)|\chi\rangle+\lambda(\bar{D}_2-D_1\bar{P}_{12})|\Phi\rangle=0\,.
\boldsymbol{\nu}^{\gamma}(x)=\boldsymbol{\nu}N,\Omega^{\gamma}=(A\Omega +B)N,\nabla_{x}^{\gamma}=\nabla_{x}, &\omega^{\gamma}(x,y) =\omega(x,y)-\frac{1}{2}\sum_{1\le a\le b\le 2}\left(\nu_a(x)\nu_b(y)+\nu_b(x)\nu_a(y)\right)\frac{\partial}{\partial \Omega_{ab}}\log\det M, &s^{\gamma}(x ) =s(x )-6\,\nabla_x\log\det M.
\ell_{01}=\ell_{02}= \frac{2 \lambda}{1 + \lambda^2}\, , \ell_{03}=1 \, , p_{01}=p_{13}=-p_{02}=-p_{23}= \frac{1- \lambda^2}{1 + \lambda^2}\, , p_{03}=p_{12}=0 \, ,
r_n(i,j) = R_h\left(\frac{i+k}{n}, \frac{j+k}{n}\right) - R_h\left(\frac{i}{n}, \frac{j+k}{n}\right) - R_h\left(\frac{i+k}{n}, \frac{j}{n}\right) + R_h\left(\frac{i}{n}, \frac{j}{n}\right).
\mathbf c(k_1^*\operatorname{diag}(\exp(\chi),\exp(-\chi)),k_2)=\mathbf c\left(\left(\begin{matrix}e^{\chi}y+y_0 0 e^{-\chi}\end{matrix}\right),\left(\begin{matrix}1 x^*+x_0 0 1\end{matrix}\right)\right).
R_{m}=\left(\begin{array}{cccc}\frac{k_{3}^{2}}{m^2}0 0 \frac{\left| k_{3}\right| \left( m^{2}+k_{3}^{2}\right)^{1/2}}{m^2}0 0 0 0 0 0 0 0 \frac{\left| k_{3}\right| \left( m^{2}+k_{3}^{2}\right)^{1/2}}{m^2}0 0 \frac{\left( m^{2}+k_{3}^{2}\right)}{m^2},\end{array}\right)
\frac{J ( \nu , z )}{\phi ( \nu / 2 )}= M e_{\nu}( z , h^{2}) , \; ( - 1 )^{n}\frac{\phi ( n + \nu / 2 )}{\phi ( \nu / 2 )}= \frac{c_{2 n}^{\nu}( h^{2})}{c_{0}^{\nu}( h^{2})}
\omega \cdot(u\cdot \nabla) u +u \cdot (u\cdot \nabla)\omega = \div\big( u(u\cdot \omega) \big) u\cdot (\omega \cdot \nabla) u = \frac{1}{2}\div \big( |u|^2 \omega \big) \omega \cdot \nabla p = \div( p \, \omega )\,.
\lim_{n\rightarrow +\infty}(P_t\phi)(x^n)=\lim_{n\rightarrow +\infty}\mathbb{E}\left[\phi(u_t^{x^n})\right]=\mathbb{E}\left[\lim_{n\rightarrow +\infty}\phi(u_t^{x^n})\right]=\mathbb{E}\left[\phi(u_t^{x})\right]=(P_t\phi)(x)\, ,
CT\frac1{(xyz)^{p-1}}\prod_{cyc}(y+z)^{p-1}=CT\frac1{(xyz)^{p-1}}\prod_{cyc}\sum_{i=0}^{p-1}\binom{p-1}iy^iz^{p-1-i}=\sum_{j=0}^{p-1}\binom{p-1}j^3 &\equiv_{p^2}\sum_{j=0}^{p-1}(-1)^j\left[1-3pH_j\right] =1-\frac{3p}2H_{p'}\equiv_{p^2}1+3p\,q_p(2).
D_x(A^q\eta)(x) = D_x\left(\int_{x-r}^{x+r}(\eta(y)-\eta(x))\cos(2\pi q (y-x))\ \d y\right) &=(\eta(x+r)-\eta(x))\cos(2\pi q r) - (\eta(x-r)-\eta(x))\cos(2\pi q r) &+\int_{x-r}^{x+r}-\partial \eta(x)\ \cos(2\pi q (y-x)) + 2\pi q (\eta(y)-\eta(x))\sin(2\pi q (y-x)) \ \d y.
m_1\alpha_1+m_2\alpha_2+m_3\alpha_3+m_4\alpha_4=m, &m_2\alpha_1+m_1\alpha_2+m_4\alpha_3+m_3\alpha_4=m, &m_3\alpha_1+m_4\alpha_2+m_1\alpha_3+m_2\alpha_4=m, &m_4\alpha_1+m_3\alpha_2+m_2\alpha_3+m_1\alpha_4=m.
q^{m}\sum_{n=1}^{\frac{m-1}{2}}\frac{[3n](q;q^2)_{n}(q^{m};q^2)_{n}^2 q^{-{n+1\choose 2}-\frac{(2n+1)(m-1)}{2}}}{(q;q)_{n}^2 (q^2;q^2)_{n}}\equiv 0\pmod{[m]\Phi_m(q)^2}.
\beta_i:=\alpha_i|_{\frak{t}^\sigma}\ (1\leq i\leq m-1), \beta_m:=(\alpha_m+\alpha_{m+1}+\cdots+\alpha_{2m})|_{\frak{t}^\sigma}, &\beta_{m+i}:=\alpha_{2m+i}|_{\frak{t}^\sigma}\ (1\leq i\leq n-1), \beta_{m+n}:=\alpha_{2m+n}|_{\frak{t}^\sigma}.
D^{1 \; \alpha}\, J_{\alpha \alpha_{2}\ldots \alpha_{2 j_{1}}; \ \dot{\alpha}_{1}\ldots \dot{\alpha}_{2 j_{2}}}^{[ a_{1}, a_{2}, a_{3}]}= \bar{D}_{4}^{\dot{\alpha}}\, J_{\alpha_{1}\ldots \alpha_{2 j_{1}}; \ \dot{\alpha}\dot{\alpha}_{2}\ldots \dot{\alpha}_{2 j_{2}}}^{[ a_{1}, a_{2}, a_{3}]}= 0
\left| k = 0 , m \right>_{\mathrm{I I A}}\simeq \mathrm{T r}\left[ \mathcal{G}_{J + \frac{m N_{2}}{2}, J - \frac{m N_{2}}{2}}\left( \mathbf{A}, \mathbf{B}; 1 \right) \right] .
c_{l_j}[i] = \frac{sgn(s_{l_j})\bar{s}_{l_j}'[i]}{\sigma_{n}\sqrt{{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}=\frac{sgn(s_{l_j})\Re\left[{\boldsymbol w}^\emph{H}_j[i]\boldsymbol{\bar{r}}_l[i]\right]}{\sigma_{n}\sqrt{{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}.
\sum_{i=1}^{J}\alpha p_{0i}=\sum_{i=1}^{J}\lambda_{i}-\sum_{i=1}^{J}\sum_{j=1}^{J}\lambda_{j}p_{ji}=\sum_{i=1}^{J}\lambda_{i}-\sum_{j=1}^{J}\lambda_{j}(1-p_{j0})=\sum_{i=1}^{J}\lambda_{i}p_{i0}