harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I = \int_0^\infty u(a) \int_0^{a^{p/r}}v(b)^{1+\lambda/n}db da \geqslant \frac1{(pr)^{1+\frac \lambda n}}\bigg(\frac\lambda n \frac r{1-r}\bigg)^{-\frac \lambda n}\bigg(\int_0^\infty p u(a) a^{p-1}\int_0^{a^{p/r}}r v(b)b^{r-1}db da\bigg)^{1+ \frac \lambda n}.
E\langle M_{\infty}, N_{\infty}\rangle = E\Bigl\langle M_{\infty}, \sum_{k=1}^d N^k_{\infty}y_k\Bigr\rangle = \sum_{k=1}^d\mathbb E\langle M_{\infty}, y_k\rangle N^k_{\infty}\stackrel{(*)}= 0,
{}\int_{0}^{T}{\bf a}^{\top}(t){\bf z}^{*}(t)dt=\int_{0}^{T}{\bf a}^{\top}(t)\bar{\bf z}^{(0)}(t)dt=\lim_{r\rightarrow\infty}\int_{0}^{T}{\bf a}^{\top}(t)\bar{\bf z}^{(\epsilon_{k_{r}})}(t)dt=\lim_{r\rightarrow\infty}M(\epsilon_{k_{r}})=M(0+).
[ a_{m}, a_{m}^{\dagger}] - [ \psi_{\dot{\alpha}}, \psi_{\dot{\alpha}}^{\dagger}]_{+}= [ \tilde{a}_{m}, \tilde{a}_{m}^{\dagger}] - [ \tilde{\psi}_{\dot{\alpha}}, \tilde{\psi}_{\dot{\alpha}}^{\dagger}]_{+}= B - F - 1 + \mid 0 \rangle \langle 0 \mid .
\sum_{j=0}^{m}\frac{d u_j(t)}{d t}J_{L,j}^{\beta,\gamma}(x)=d\sum_{j=0}^{m}\sum_{k=0}^{j}\sum_{i=0}^{k}u_j(t)\times \Psi_{i,j,k}^{\alpha,\beta,\gamma}\times J_{L,j}^{\beta,\gamma}(x)-e\sum_{j=0}^{m}u_j(t)\frac{d}{dx}J_{L,j}^{\beta,\gamma}(x)+s(x,t).
\beta(1+Dr^m)(1+Er^m)^{\frac{D-2E}{E}}+(1-\beta)r^m(1+Er^m)^{\frac{D-E}{E}}&+\sum_{n=2}^{\infty}\prod_{k=0}^{n-2}\frac{|E-D+Ek|}{k+1}\phi_{n}(r) =(1-E)^{\frac{D-E}{E}}-\phi_{1}(r),
K_{I}'(\theta)=\ell'\;\;\frac{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}{\textrm{cn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]\textrm{dn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}\,\,\,.
\|e^{t^{\frac{1}{\alpha}}\Lambda_{1}-t\Lambda^{\alpha}}\Delta_{j}u_{0}\|_{L^{p}}= \|e^{t^{\frac{1}{\alpha}}\Lambda_{1}-\frac{t}{2}\Lambda^{\alpha}}e^{-\frac{t}{2}\Lambda^{\alpha}}\Delta_{j}u_{0}\|_{L^{p}}\lesssim \|e^{-\frac{t}{2}\Lambda^{\alpha}}\Delta_{j}u_{0}\|_{L^{p}}\lesssim e^{-\kappa2^{\alpha j}t}\|\Delta_{j}u_{0}\|_{L^{p}}.
\frac{1}{|B_{1}|}\int_{B_{1}}|X_{i}X_{j}h(x)-(X_{i}X_{j}h)_{B_{1}}|dx \leq\frac{c}{|B_{\Lambda}|}\sum_{s=1}^{q}\int_{B_{\Lambda}}|X_{s}X_{i}X_{j}h(x)|dx\ & \leq c\sum_{s=1}^{q}\sup_{B_{\Lambda}}\left\vert X_{s}X_{i}X_{j}h\right\vert \ & \leq\frac{c}{k}\sum_{i,j}^{q}\frac{1}{|B_{k}|}\int_{B_{k}}|X_{i}X_{j}u(x)|dx,
\left| \beta_k\right|=\frac{\left| 3m+i\right|}{\sqrt{\pi}}\frac{2^{n_s-3}\left|\Gamma \left( n_s\right)\right|}{|n_1(1-n_1)|}z_S^{m-3/2}\left( \frac{g_S}{g_1}\right)^{-l}\left( \frac{\omega}{\omega_s}\right)^{2-n_s}\left( \frac{M}{M_s}\right)^{-n_1/2}_{.}
esv^{-2}~\stackrel{(\ref{2.8})}{=}[(v^{\ast})^{-1}es^2]v^{-2}= (v^{\ast})^{-1}(es^2v^{-1})v^{-1}~\stackrel{(\ref{2.8})}{=}(v^{\ast})^{-1}[(v^{\ast})^{-1}es^2]v^{-1}=(v^{\ast})^{-2}es^2v^{-1}\stackrel{(\ref{2.8})}{=}(v^{\ast})^{-2}es,
\hat{A}_*(\theta) = e^{-\theta}\hat{A}_{-1}+ \hat{A}_0 + e^{\theta}\hat{A}_1= \begin{pmatrix}e^{-\theta}A_{-2}+ A_0 + e^{\theta}A_2 e^{-\theta/2}(e^{-\theta/2}A_{-1}+ e^{\theta/2}A_1) \cr e^{\theta/2}(e^{-\theta/2}A_{-1}+ e^{\theta/2}A_1) e^{-\theta}A_{-2}+ A_0 + e^{\theta}A_2\end{pmatrix},
W_{K_1K_2K_3K_4K_5K}(s,s') :=\frac{|\beta_h(s)|^2\overline{\beta_\sigma(s)}|\beta_h(s')|^2\beta_h(s')}{(2\pi)^{12}\big(1+4isz(s)\big)\big(1-4is'\widetilde z(s')\big)}e^{c(s,s')+\frac{b(s,s')\cdot b(s,s')}{4a(s,s')}}
\partial_{\mu}\left( \begin{matrix}{\sqrt{g}( F^{-}- F^{+})^{I \mu t}}\\{\noalign{\vskip 2 m m}\sqrt{g}( G^{-}- G^{+})_{I}^{\mu t}}\\ \end{matrix}\right) = 4 i \pi \sum_{A}\delta ( \vec{x}- \vec{x}_{A}) \left( \begin{matrix}{p_{A}^{I}}\\{\noalign{\vskip 2 m m}q_{A I}}\\ \end{matrix}\right) .
\bar{\delta}_{k}^m\colon \frac{\xi_{i,j}}{d}\mapsto \begin{cases}\frac{\xi_{i,j+1}}{d}k<j, \frac{\xi_{i,j}}{d}+\frac{\xi_{i,j+1}}{d}k=j, \frac{\xi_{i,j}}{d}k>j,\end{cases}
\frac{1}{N}\sum_k^{(i)}\frac{\partial d_j}{\partial g_{ik}}\mathbf{e}_k^* X_i G\mathbf{e}_i=O_\prec\big(\Psi^2\Pi_i^2\big)\,,\qquad \frac{1}{N}\sum_k^{(i)}\frac{\partial d_j}{\partial g_{ik}}\mathbf{e}_k^* X_i \mathring{\mathbf{g}}_i=O_\prec\big(\Psi^2\Pi_i^2\big)\,,
\ \sum_{j,k\ge 3, j+k=n}(2j-1)(2j-2) (2k-1)(2k-2)\zeta(2j)\zeta(2k) = \frac{1}{15}(n-1)(4n^2-1)(2n^2-13n+30)\zeta(2n)- 24(n-2)(2n-5) \zeta(4)\zeta(2n-4), n\ge 4.
\alpha(H(q))=(\begin{matrix}a+id&-c+ib\\c+ib&a-id\end{matrix})=-(\begin{matrix}a+id&-c+ib\\c+ib&a-id\end{matrix})=(\begin{matrix}-a-id&c-ib\\ -c-ib&-a+id\end{matrix})=\alpha(H(-q))
p_{w}(\omega) = \frac{1}{\mathrm{i}\, \pi}\int_{-\alpha}^{\alpha}v(x)\, \mathrm{e}^{-\mathrm{i}\, \omega\, x}\, \mathrm{d}x, q_{w}(\omega) = -\frac{4d\, v(\alpha)}{\pi}\frac{\pi \sin (\alpha \omega) + 2d \omega \cos(\alpha \omega)}{(\pi^{2}+ 4d^{2}\omega^{2})}.
|R_j(x,y)| \lesssim |y_j-x_j|\int_0^\infty \frac1{V(x,y,\sqrt t)}\frac t{\|x-y\|^2}e^{-cd(x,y)^2/t}\frac{dt}{t\sqrt t}&\leqslant \frac1{\|x-y\|}\bigg(\int_0^{d(x,y)^2}+ \int_{d(x,y)^2}^\infty\bigg) \frac1{V(x,y,\sqrt t)}e^{-cd(x,y)^2/t}\frac{dt}{\sqrt t}&=: I_1 +I_2.
\mathbb{E}(T_{i-1}Y_{i-1}) = \sum_{j,l=0}^k\left(\mathbb{E}\left(T_{i-1}|\Psi_j^{i-1}\Psi_l^{i-2}\right)\mathbb{E}\left(Y_{i-1}|\Psi_j^{i-1}\Psi_l^{i-2}\right)\right. \qquad\left.\vphantom{\mathbb{E}\left(T_{i-1}|\Psi_j^{i-1}\Psi_l^{i-2}\right)}\times\mathbb{P}(\Psi_j^{i-1})\mathbb{P}(\Psi_l^{i-2})\right) .
{\tilde{F}}(\zeta ) =\zeta +\int_w^{w+ka^{-p}w^{p+1}+o(w^{p+1})}t_\lambda '(\omega )\,d\omega &=\zeta +\int_w^{w+ka^{-p}w^{p+1}+o(w^{p+1})}\frac 1{\omega^{2}}+\frac \lambda \omega \,d\omega &=\zeta +ka^{-p}w^{p-1}+o(w^{p-1}) &=\zeta +(-1)^{p-1}ka^{-p}\frac{1}{\zeta^{p-1}}+o\left(\frac 1{\zeta^{p-1}}\right)
\mu_{T}\{\lambda\in\mathbb{C}: |\lambda|\leq r\}= \begin{cases}0, r<\lambda_1(\mu_{|T|}); 1+S^{\langle -1 \rangle}_{\mu_{T^*T}}(r^{-2}), \lambda_1(\mu_{|T|})<r<\lambda_2(\mu_{|T|}); 1, r\geq \lambda_2(\mu_{|T|}). \end{cases}
- \nabla \cdot \Big(\varepsilon(x) \nabla u^* \Big) + e^{u^* - \hat{v}}- e^{\hat{w}- u^*}= k_1, u^* = \hat{u}\mbox{on}\partial \Omega_D, \nabla u^* \cdot \nu = 0 \mbox{on}\partial \Omega_N,
{\cal L}_{E}= \bar{\psi}_{a}( i \partial \! \! \! / + m - \frac{e}{\sqrt{N}}A \! \! \! / ) \psi_{a}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- i \frac{\kappa}{8 \pi}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}.
V_{ef}\left( \phi_{c}\right) = \frac{2\alpha^{2}}{4!}\phi_{c}^{4}+\frac{\hbar}{24\pi a}\alpha^{3}\phi_{c}^{3}-\frac{\hbar}{16\pi^{2}a^{4}}\sum_{N=1}^{m}\left(\frac{1}{N}\right)^{4}+ \frac{\hbar}{16\pi^{2}a^{2}}\alpha^{2}\phi_{c}^{2}\sum_{N=1}^{m}\left(\frac{1}{N}\right)^{2}.
P_{i j}[ \alpha ( t ) , \alpha ( \mu ) ] \equiv \left( \begin{array}{l l}{p}&{r}\\{r}&{q}\\ \end{array}\right) , \qquad F_{k l}[ \alpha ( t ) , s ] \equiv \left( \begin{array}{l l}{k}&{h}\\{h}&{g}\\ \end{array}\right)
\left( \frac{\partial}{\partial u}\right)^{n_1}a_{\mu}^{m_1}\left( \frac{\partial}{\partial \bar{u}}\right)^{n_2}\bar{a}_{\mu}^{m_2}= \left( \frac{\partial}{\partial u}\right)^{n_1}\left( \frac{\partial}{\partial \bar{u}}\right)^{n_2}a_{\mu}^{m_1}\bar{a}_{\mu}^{m_2},
\lim_{\Omega \to 0}\Phi^{\Delta , J}( t , \bar{t}, \psi ; \tau , \bar{\tau}, \phi ) = e^{i ( \mu \hat{\psi}+ \lambda \hat{\phi})}\sum_{m , n = 0}^{\infty}c_{m n}t^{m}\tau^{n}e^{- \frac{1}{2}| t |^{2}- \frac{1}{2}| \tau |^{2}}.
G_{s}(E,P)=\frac{\{f\in E^{\mathbb{N}}|\forall p\in P,\exists m\in\mathbb{Z}:p(f_{n})=o(n^{m})\}}{\{f\in E^{\mathbb{N}}|\forall p\in P,\forall m\in\mathbb{Z}:p(f_{n})=o(n^{m})\}}.
S = \int d^{2}x \left[{\frac 1 2}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+{\frac 1 2}{\lambda}_{{\mu}{\nu}}\left({\partial}^{\mu}{\phi}-{\epsilon}^{{\mu}{\sigma}}{\partial}_{\sigma}{\phi}\right) \left({\partial}^{\nu}{\phi}-{\epsilon}^{{\nu}{\rho}}{\partial}_{\rho}{\phi}\right) \right] ,
\sum_{k,m \geq 0}\frac{w^k q^{m^2 + 2mk + 2k^2 + m-k}}{\left(q^2; q^2\right)_m \left(q^2; q^2\right)_k}= \left(-q^2, -wq^3, -q^4; q^4\right)_\infty, \sum_{k,m \geq 0}\frac{w^k q^{m^2 + 2mk + 2k^2 + m+k}}{\left(q^2; q^2\right)_m \left(q^2; q^2\right)_k}= \left(-wq, -q^2, -q^4; q^4\right)_\infty.
\begin{array}{l}{{\int d y_{1}d y_{2}d y_{3}d y_{4}\langle \phi^{2}( y_{1}) h^{n}( y_{1}) \partial h ( y_{1}) \partial h ( y_{1}) \phi^{2}( y_{2}) h^{m}( y_{2}) \partial h ( y_{2}) \partial h ( y_{2})}}\\{{\phi^{2}( y_{3}) h^{r}( y_{3}) \partial h ( y_{3}) \partial h ( y_{3}) h ( x_{1}) h ( x_{2}) \rangle}}\\ \end{array}
\aligned Y_{t}=x_0+\int_{0}^tb(s,Y_{s})ds+\int_{0}^t\sigma(s,Y_{s})d\bar{B}_s &\quad+\int_{0}^{t+}\int_Uf_1(s,Y_{s-},u)\tilde{N}_k(ds,du)+\int_{0}^{t+}\int_Uf_2(s,Y_{s-},u)N_k(ds,du).\endaligned
\frac{d\mu_Y}{d\mu_B}(B_0^T) = \int \frac{d\mu_{Y|W=w}}{d\mu_B}(B_0^T) d\mu_W(w) &= \int \exp \left(\int_0^T g(s, w_0^s, B_0^s) dB(s)-\frac{1}{2}\int_0^T g^2(s, w_0^s, B_0^s) ds \right) d\mu_W(w),
(\mathbb{A'}^{-1}_L)_{12}=(\Delta')^{-1}\left( -b-([d,b]-[d,a]a^{-1}b)(d-ca^{-1}b)^{-1}\right) &=-(\Delta')^{-1}\left( b(d-ca^{-1}b)+([d,b]-[d,a]a^{-1}b)\right)(d-ca^{-1}b)^{-1}&=-(\Delta')^{-1}\left( -bc +ad\right)a^{-1}b(d-ca^{-1}b)^{-1}&= - (db^{-1}a-c)^{-1}
G_0(x,y)=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}e^{ik\cdot (h(x)-h(y))}\eta(k)\frac{\phi_{k+i\beta_s}(x)\overline{\phi_{k-i\beta_s}(y)}}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}dk.
\mathbb{E}\left [ g\left ( X \right ) \right ]=\frac{c_{0}}{\Gamma \left ( \alpha \right )}\mathbb{E}\left [ X^{\alpha -1}\right ]+\frac{\mathbb{E}\left [ X^{\alpha}\right ]}{\Gamma \left ( \alpha +1\right )}\mathbb{E}\left [ D_{0}^{\alpha}g\left ( X_{\alpha}^{\left ( 1 \right )}\right ) \right ],
E_{Q}(\mu)-E_{Q}=\int Q d\mu -\iint \log|z-w|\mu(dz)\mu(dw)-\int Qd\mu_{Q}+\iint \log|z-w|\mu_{Q}(dz)\mu_{Q}(dw) &\ge -\iint \log|z-w|(\mu-\mu_{Q})(dz)(\mu-\mu_{Q})(dw).
y^k=y^0 + \sum_{s=1}^k \sum_{j=1}^N \frac{1}{N}\left( x^s_j - x^{s-1}_j\right)=\sum_{\ell=1}^{N}\frac{1}{N}v_\ell^0+\sum_{s=1}^k \sum_{j=1}^N \frac{1}{N}\left( x^s_j - x^{s-1}_j\right),
R_{f}=\sum_{i=0}^{N^{(n)}-1}f(t_{i})(x_{i+1}^{(n)}-x_{i}^{(n)})\le lim_{n\rightarrow\infty}L_{f,P^{(n)}}+lim_{n\rightarrow\infty}\frac{1}{n}=lim_{n\rightarrow\infty}L_{f,P^{(n)}}.
({\cal H}_{V})_{c l}= \sum_{{\vec{n}},{\vec{m}}}\sum_{s , s^{\prime}}\kappa_{\vec{m}}\left[ ( a_{\vec{n}}^{s \dagger}\bar{u}_{\vec{n}}^{s}+ b_{-{\vec{n}}}^{s}\bar{v}_{-{\vec{n}}}^{s}) ( a_{\vec{m}}^{s^{\prime}}\gamma^{0}u_{\vec{m}}^{s^{\prime}}- b_{-{\vec{m}}}^{s^{\prime}\dagger}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}}) \right]_{c l}F ({\vec{n}}-{\vec{m}}, X ) ,
s = \langle \mathrm{\boldmath ~ n ~}\cdot \mathrm{\boldmath ~ j ~}_{0}\rangle = - \frac{1}{2}\left. \frac{\partial \mathrm{~ \cal ~ L ~}_{\mathrm{e f f}}}{\partial V_{0}^{3}}\right|_{V_{\mu}^{3}= 0}= \frac{e B}{4 \pi}.
\Delta_1 = \int d^3x \overline \psi D^2\psi\, , \qquad \Delta_2=\int d^3x(\overline \psi \psi)^2 \quad \mbox{and}\quad \Delta_3=\int d^3x (\overline \psi \gamma^\mu \psi) (\overline \psi \gamma_\mu \psi)\, ,
ber(\sqrt{A^*XA})) = \sup_{\lambda \in \Omega}|\langle \sqrt{A^*XA}\hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle| & < \sup_{\lambda \in \Omega}|\langle{A^*XA}\hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle|^{1 \over 2}& =\sqrt{ber(A^*XA}).
\mbox{est-}\hat{B}(n) h = - \frac{1}{4}\sum_{j=1}^\infty{\left( \frac{3}{2}+ \frac{(n-3) \sum_{i=1}^{n-2}{PX^j(i)^2}}{(n-2){\sum_{i=1}^{n-3}{PX^j(i) PX^j(i+1)}}}\right)^{-1}\langle h,e_j \rangle e_j}
S = - \frac{1}{\lambda_{s}^{2}}\int d^{4}x ~ \sqrt{- g}~ e^{- \phi}\biggl [ R + g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi - \frac{1}{2}e^{2 \phi}g^{\alpha \beta}\partial_{\alpha}\psi \partial_{\beta}\psi \biggr ] .
u_\pm ={1 \over \sqrt{2\pi}}\int_{-\infty}^\infty dk\hspace{1mm}b_\pm(k) e^{-{\rm i}k x^-}\quad \mbox{and}\quad\phi_{\pm}={1 \over \sqrt{2\pi}}\int_{-\infty}^\infty{dk \over\sqrt{2|k|}}\hspace{1mm}a_{\pm}(k) e^{-{\rm i}k x^-}
g_{\Sigma}\big(\nabla^{\Sigma}_uv,w\big) = g_{\Sigma}\big(\nabla^{\mathrm{R}}_u\mathrm{P}_Hv,\mathrm{P}_Hw\big) + g_{\Sigma}\big(\nabla^{\mathrm{R}}_u\mathrm{P}_Vv,\mathrm{P}_Vw\big)
\mu(i,j;c_2) := [n^2-(i-1)n-(j-1)]!^{\frac{n^2-c_2n}{n^2-(i-1)n-(j-1)}}, \nu(j;c_2) := [n^2-(j-1)n]!^{\frac{n^2-c_2n}{n^2-(j-1)n}}.
\lambda \triangleq \sum_v \beta_v\tilde\lambda_v = \sum_v \beta_v \frac{\rho_v \tilde{p}_v \chi\left( \gamma \right)e^{- \gamma}}{e^{- \gamma}+ \tau\left( 1 - e^{- \gamma}\right)}& = \frac{\gamma \chi(\gamma) e^{-\gamma}}{e^{- \gamma}+ \tau\left( 1 - e^{- \gamma}\right)}= f(\gamma) \,.
f_{\varepsilon}= q(x_{2},x_{3}) + \varepsilon \ell_{0}(x_{2}a_{0,2}+ x_{3}a_{0,3}) + \varepsilon \ell_{1}(x_{2}a_{1,2}+ x_{3}a_{1,3}) + \sum_{i = 4}^{r}\varepsilon \ell_{i}g_{i}.
=\begin{cases}O\left(|\log(T/n)|(\log T)^\alpha\right), \beta=1, T^{1-\beta}(\log T)^\alpha+O(T^{1-\beta}(\log T)^{\alpha-1})+O(n^{1-\beta}(\log n)^\alpha), \beta\in[0,1), O((\log n)^\alpha/n^{\beta-1})+O((\log T)^\alpha/T^{\beta-1}), \beta\in\mathbb{R}_{>1}. \end{cases}
\mathbf{w}^k_{s,t}= \mathbf{w}^k_{s,u}+ \sum_{i=1}^{k-1}\mathbf{w}^{k-i}_{s,u}\otimes \mathbf{w}^{i}_{u,t}+ \mathbf{w}^k_{u,t},1 \le k \le [p], \,\, s \le u \le t.
\widehat Q_j^\mu(x)= \frac{1}{x_j+\tau_j}\Big(\tau_j I_-^\mu \big\{(1+x)^{-\mu}\hat h_j(x)\big\}+ I_-^\mu \big\{(1+x)^{-\mu}x \hat h_j(x)\big\}\Big), \tau_j=-\frac{I_-^\mu \big\{(1+x)^{-\mu}x \hat h_j(x)\big\}\big|_{x=1}}{I_-^\mu \big\{(1+x)^{-\mu}\hat h_j(x)\big\}\big|_{x=1}}.
t_1^6 + t_2z^6 + z^9=0, w(t_1)=\frac{3}{10},w(t_2)=\frac{3}{5},w(t_3)=1,w(z)=\frac{1}{5}, g_1=t_1\big({-}80t_2^2 + 910t_3z + 165t_2z^3 + 63z^6\big)/(910z), g_2= \big(4t_2t_3 - 12t_2^2z^2 - 36t_2z^5 - 27z^8\big)/4, g_3= \big(-560t_1^{18}+ 595t_3^2z^{17}+ 7140t_2^2z^{21}- 8160t_2z^{24}- 15113z^{27}\big)/\big(1190z^{17}\big).
e_{1}= \left| \begin{array}{c}{{\overrightarrow{e_{1}}}}\\{{0}}\\ \end{array}\right| , \quad e_{2}= \left| \begin{array}{c}{{\overrightarrow{e_{2}}}}\\{{0}}\\ \end{array}\right| , \quad e_{3}= \left| \begin{array}{c}{{\overrightarrow{e_{3}}}}\\{{0}}\\ \end{array}\right| , \quad e_{4}= \left| \begin{array}{c}{{0}}\\{{1}}\\ \end{array}\right| .
-2m\lambda^nT_{m,0}-3m\lambda^m\frac{\partial T_{n,0}}{\partial M_0}M_0-3m\lambda^{m+2n}\frac{\partial T_{-n,0}}{\partial M_0}M_0-n\lambda^mT_{n,0}+n\lambda^{m+2n}T_{-n,0}&=-(m+n)T_{m+n,0}-(m-n)\lambda^{2n}T_{m-n,0}.
\frac{d^n}{dx^n}(G\circ F)(0) = \sum_{k_1+2k_2+\cdots+nk_n=n}\frac{n!}{k_1!\cdots k_n!}G^{(r)}(F(0))\prod_{i=1}^n \bigl(\frac{F^{(i)}(0)}{i!}\bigr)^{k_i},
F_i(v)=\begin{cases}\sum_{j=0}^{v-1}\dfrac{1}{w(Z_i (j))\cdot w(Z_i (j+1))}v>0; \sum_{j=v}^{-1}\dfrac{1}{w(Z_i (j))\cdot w(Z_i (j+1))}v<0. \end{cases}
d s^{2}\, = \,{\frac{\Lambda^{2}}{z^{\prime \, 2}}}\, \Big \lbrack \, \,{\frac{\delta^{2}}{( z^{\prime}- \delta )^{2}}}d z^{\prime \, 2}\, + \, \,{\frac{( z^{\prime}- \delta )^{2}}{\delta^{2}}}\Big ( ( d \vec{x})^{2}\, - d t^{2}\, \Big ) \, \Big \rbrack
(\varphi^{-1}\varphi_u)_u+(\varphi^{-1}\varphi_v)_v-{\rm ad}(\varphi^{-1}\varphi_u)^*(\varphi^{-1}\varphi_u)-{\rm ad}(\varphi^{-1}\varphi_v)^*(\varphi^{-1}\varphi_v)=0
\begin{aligned}d_Y(G(y,x),G(v,u))\leq a\d_Y(y,G(v,u))+ \b\d_Y(v,G(y,x))+c\d_Y(y,v); & \forall x\leq_{P_1}u,\y\geq_{P2}v;\ 2a+c<1\end{aligned}
\begin{array}{crl}\delta_t\Delta_1 u^\alpha (x)^{(i,j)}= \Delta_1 \delta_vu^\alpha (x)^{(i,j)}+\delta x^{\nu (i+1,j)}\cdot\Delta_1\Delta_\nu u^\alpha (x)^{(i,j)}, \delta_t\Delta_2 u^\alpha (x)^{(i,j)}= \Delta_2 \delta_vu^\alpha (x)^{(i,j)}+\delta x^{\nu (i,j+1)}\cdot\Delta_2\Delta_\nu u^\alpha (x)^{(i,j)}. \end{array}
\bigg ( \frac{\partial^{2}}{\partial r^{2}}+ \frac{g^{2}}{4}e^{\mp 2 i \theta}( \lambda_{2}- \lambda_{1}) \bigg ) \psi_{\pm}^{( 1 )}= 0 , \psi_{\pm}^{( 2 )}= \frac{2 e^{\pm i \theta}}{g ( \sqrt{\lambda_{1}}+ \sqrt{\lambda_{2}})}\frac{\partial}{\partial r}\psi_{\pm}^{( 1 )}.
h^{0}(X,L)=\chi(X,L)={}\frac{1}{24}\int_{X}c^{4}_{1}(L)+\frac{1}{12}\int_{X}c^{3}_{1}(L) c_{1}(X)+\frac{1}{24}\int_{X}c^{2}_{1}(L) c^{2}_{1}(X){}+\frac{1}{24}\int_{X}c^{2}_{1}(L) c_{2}(X)+\frac{1}{24}\int_{X}c_{1}(L) c_{1}(X) c_{2}(X)+\chi(X,\mathcal{O}_{X}).
I_{r+1,r}(x) = -\sinh(x)\biggl(\frac{1}{r-2}\tanh^{r-2}(x)+\frac{r-1}{(r-2)(r-4)}\tanh^{r-4}(x) & +\frac{(r-1)(r-3)}{(r-2)(r-4)(r-6)}\tanh^{r-6}(x)+\dots + \frac{(r-1)!!}{3(r-2)!!}\tanh^2(x)\biggr) + \frac{(r-1)!!}{(r-2)!!}I_{3,2}(x).
d s^{2}= H_{1}^{1 / 4}H_{5}^{3 / 4}\left[{\frac{d u}{H_{1}H_{5}}}( - d v + K d u + 2 A_{i}d y^{i}+ 2 A_{a}d x^{a}) +{\frac{d y^{i}d y_{i}}{H_{5}}}+ d x^{a}d x_{a}\right]
(\delta\delta')^{-2\Delta}\langle V^{\delta}_{\beta}(z) \overline{V^{\delta'}_{\beta}(w)}\rangle_{D}= (\delta\delta')^{-2\Delta}\exp \big( -\lambda (\kappa_{z,w}^{\delta}+ \tau^{\delta}_{z,{w}}+\tau^{\delta'}_{w,{z}}+\tau^{\delta',\delta}_{w,z}) \big).
\sum_{X\in I_0}J_{0,2}^l(X,Y)D_{\le l}(Z,X)=\sum_{X\in I_0}J_{0,2}^l(X,Y)D_{\le l}(X,Z)=0, &(\forall Y,Z\in I_0,l\in \{0,-1,\cdots,N_{\beta}\}),
\displaystyle Z_{\scriptscriptstyle C P I}^{\scriptscriptstyle B F V}= \int{\cal D}\Lambda_{\scriptscriptstyle A}{\cal D}\xi^{\scriptscriptstyle A}{\cal D}\bar{\Gamma}_{\scriptscriptstyle A}{\cal D}\Gamma^{\scriptscriptstyle A}e^{i \int d^{4}x \widetilde{\cal L}^{\scriptscriptstyle B F V}}
r_1 = -(R_{1212}R_{1313}-R_{1213}{}^2)S_{12232}, & r_2 = -(R_{1212}R_{1313}-R_{1213}{}^2)(S_{12233}+S_{13232}), & r_4 = -(R_{1212}R_{1313}-R_{1213}{}^2)S_{13233}.
h(t,l,\eta) \equiv 0 l \not \in \{k_0-1,k_0,k_0+1\}, h(t,k_0,\eta) = h(T,k_0,\eta), h(t,k_0\pm 1, \eta) = \int_{T}^t k_0 \hat{W}(k_0) h(T,k_0,k_0t) ((\eta-k_0t)\mp t) \hat{\psi}(\eta-k_0t) dt.
\tau_{p}:=\frac{(|a'|^{\frac{2}{p-1}}+|b'|^{\frac{2}{p-1}})|a'b'|^{\frac{p-2}{p-1}}}{(|a'|^{\frac{p}{p-1}}+|b'|^{\frac{p}{p-1}})^{\frac{2p-2}{p}}}
\mathbb{P}_{x,k}^{\epsilon,\hat{v}}\Bigl\{\zeta^{\epsilon}(\tau_D^{\epsilon,\hat{v}}) = k_0 \,\bigl \vert \tau_D^{\epsilon,\hat{v}}< \infty \Bigr\}= 1\rightarrow 0 \epsilon \rightarrow 0,
\begin{aligned}B(t) =\int_{0}^{t}e^{-(t-s)}g_{in}(s)ds=\int_{0}^{t}e^{-(t-s)}(g_0+g_1N(s))ds. C(t) =2\int_{0}^{t}e^{-2(t-s)}a(s)ds=2\int_{0}^{t}e^{-2(t-s)}(a_0+a_1N(s))ds.\end{aligned}
{\cal I}= - \int_{0}^{1}\! \! d \sigma \int_{0}^{\infty}\! \!{\frac{d \rho}{\rho}}{\frac{\log ( 1 + \rho )}{( 1 + \rho )^{3}}}{\frac{[ \rho ( x - z ) + ( 1 - \sigma ) ( x - y ) ]_{+}[ \rho ( y - z ) + \sigma ( y - x ) ]_{+}^{2}}{[ \rho \sigma ( x - z )^{2}+ \rho ( 1 - \sigma ) ( y - z )^{2}+ \sigma ( 1 - \sigma ) ( x - y )^{2}- i \epsilon ]}}\ .
\frac{d}{dz}[z^{1-a_{1}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z)]=z^{-a_{1}}G_{p,q}^{m,n}(\begin{matrix}a_{1}-1,a_{2},...,a_{p}\\b_{q}\end{matrix}|z),n\ge1,
\langle 0 | \phi_{>}( x ) \phi_{>}( y ) | 0 \rangle = \int_{\mathrm{s h e l l}}\!{\frac{d^{d - 1}k}{( 2 \pi )^{d - 1}}}{\frac{1}{2 \omega_{k}}}\exp [ i k \cdot ( x - y ) ] \quad .
c_s^{(n)}= -\frac{1}{2(1+\cos(\frac{2s-1}{n}\pi))}s = 1, 2, \ldots, \Big\lfloor\frac{n}{2}\Big\rfloor p_s^{(n)}= -\frac{1}{2(1+\cos(\frac{2s\pi}{n+2}))}s = 1, 2, \ldots, \Big\lfloor\frac{n+1}{2}\Big\rfloor.
\gamma' = \begin{pmatrix}\frac{b' ( v_1 y_2 - v_2 y_1)}{y_2}+ \frac{v_2}{b}\frac{b' ( u_1 y_2 - u_2 y_1)}{y_2}+ \frac{u_2}{b}- \frac{b ( v_1 y_2 - v_2 y_1)}{y_2}- \frac{b ( u_1 y_2 - u_2 y_1)}{y_2}\end{pmatrix}
\limsup_{N\rightarrow+\infty}\frac{N}{\gamma_N^2}\log P(\frac{N}{\gamma_N}(\tau_c^N-\tau_c)>x) \leq -x^2J_{contra, \tau_c}(1)\left(\frac{d}{dt}\sum_{k=1}^3\mu_{t,k}(f_k)\Bigg|_{t=\tau_c}\right)^2 &=-J_{hit}(x).
\Gamma ( p , P ) = - e^{2}\int \mathrm{~ \, ~ \frac{d^{3}q}{( 2 \p i )^{3}}~ \, ~}D ( p - q ) \gamma_{\mu}S ( \mathrm{~ \, ~ \frac{1}{2}~ \, ~}P + q ) \Gamma ( q , P ) S ( - \mathrm{~ \, ~ \frac{1}{2}~ \, ~}P + q ) \gamma_{\mu},
f_1 =- \frac{i}{2}\Bigg(\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2 (k+p)^2}- \sum\limits_i c_i \int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2-M_i^2)\Big((k+p)^2-M_i^2\Big)}\Bigg)
|\widetilde B[s,x,\mu] - \widetilde B[s,x,\nu]|= |\sigma^{-1}(s,x)\int (b(s,x,y)\mu(dy) - b(s,x,y)\nu(dy)])| = |\sigma^{-1}(s,x)\int (b(s,x,y)(\mu(dy) - \nu(dy))])|\le \int |\sigma^{-1}(s,x)b(s,x,y)|\,|\mu(dy) - \nu(dy)| \le \|\sigma^{-1}(s,x)b(s,x,\cdot)\|_B \|\mu-\nu\|_{TV}.
L_{e}(h^{M}) (X, Y) = e h^{M}(X, Y) - h^{M}( [e, X], Y) - h^{M}( X, [\bar{e}, Y]) &= e h^{M}(X, Y) - h^{M}( [e, X], Y) &= e h^{end}(C_{X}, C_{Y}) - h^{end}(C_{[e,X]}, C_{Y}) &= h^{end}(D_{e}(C_{X})- C_{[e, X]}, C_{Y}) =0,
G_{0}^{( 2 )}( \vec{\rho}, u ) = \frac{g}{L}\frac{1}{( 2 \pi )^{d - 1}}\sum_{n = 1}^{\infty}\cos ( \frac{n \pi u}{L}) \int d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{2}+ ( \frac{n \pi}{L})^{2}+ m^{2})},
\{X^{i},X^{j}\}_{MB}=\lim\limits_{\stackrel{\alpha^\prime\to\alpha}{\beta^\prime\to\beta}}\sin\lambda(\partial_{\alpha}\partial_{\beta^\prime}-\partial_{\alpha^\prime}\partial_{\beta})X^{i}(\alpha,\beta,\sigma)X^{j}(\alpha^\prime,\beta^\prime,\sigma) .
\mathcal R_0 =-\frac{1}{2 \vert \Lambda \vert}\big( \widehat g (0) + \widehat{g \omega}(0) \big)\big(4n_{0}-2\big)-\widehat{g\omega}(0)\frac{n_{+}}{\vert \Lambda\vert}& - \frac{1}{\vert \Lambda \vert}\sum_{k \in \Lambda^*}\Big( (\widehat g(k) + \widehat{g \omega}(k)) a_k^\dagger a_k + \frac 1 2 \widehat g_k ( a_k a_{-k}+ h.c.) \Big),
\int_{\mathbb{R}}\mathbb{E}[|A(\{X_k\}_{k=1}^r)|\mid T=t](1+|t|)^{n}h_{a_j}(t)\mathrm{d}t=\mathbb{E}^{a_j}[|A(\{X_k\}_{k=1}^r)|(1+|T|)^{n}] \leq \mathbb{E}^{a_j}[(A(\{X_k\}_{k=1}^r))^2]^\frac{1}{2}\mathbb{E}^{a_j}[(1+|T|)^{2n}]^\frac{1}{2}.
\sigma_{j_{m+1}+m+1}\dots \sigma_{j_k+k}(j_m+k)= j_m+m, \sigma_{j_{m}+m}(j_m+m)= j_m + m + n-1, \sigma_{j_1+1}\dots \sigma_{j_{m-1}+m-1}(j_m + m + n-1) = j_m+n.
\varphi_n(t)= C(t,0)x_0+S(t,0)y_0+\int_0^tS(t,s)f_n(s)ds+\int_0^t S(t,s)BB^{*}S^{*}(b,t)R(a,\Upsilon_0^b)\Big[x_b &-S(b,0)x_0-\int_0^b S(b,\eta)f_n(\eta)d\eta\Big](s)ds.
\begin{array}[]{c|c c c c}{\rm Field}SU(n_c)_0 SU(n_c)_{N+1}U(1) U(1)_R \hline Q_0 n_c 1 1 -1 Q_i 1 1 (-1)^i -1 Q_N 1 \bar n_c (-1)^N -1 \lambda_j 1 1 0 1 \end{array}
{\cal I}_{U(1)}-\delta{\cal I}_{U(1)}=\lim_{\beta\rightarrow 0}2^{N/2-1}{\rm Tr'}\;(-1)^Pe^{(\beta/2)\nabla^2}=\lim_{\beta\rightarrow 0}2^{N/2-1}\; \int [dX]\;\langle -X\vert \,e^{(\beta/2)\nabla^2}\,\vert X\rangle.
\lim_{b \uparrow \infty}\frac{C_a^{(q,r)}(b)}{W^{(q+r)}(b)}= \frac{\sigma^2}{2}\left(Z^{(q) \prime}(-a, \Phi(q+r)) - r \frac{W^{(q)\prime}(-a)}{\Phi(q+r)}\right).
f(\mathbf{x})=r(\mathbf{y})+y_1g_1(\mathbf{z})+\cdots+y_sg_s(\mathbf{z})+q(\mathbf{z})+\mathbf{y}^TB\mathbf{w}+\mathbf{z}^TC\mathbf{w}+p(\mathbf{w}),
\begin{cases}u(c_{k_0}t-\frac{N-1}{c_{k_0}}\log t,t)\geq U_{k_0}\left(H_0\right)-\frac{\log t}{t^2}, u(c_{k_0}t-\frac{N-1}{c_{k_0}}\log t,t)\leq U_{k_0}\left(H^0\right)+\frac{\log t}{t^2}.\end{cases}
C_5 := KE\int_{t_0}^{u \wedge \pi_R}\int_Z \left(|x_{s}|^{p-2}|\gamma_s( x_s,z)|^2+|\gamma_s( x_s,z)|^p \right)\ \nu (dz)ds & \leq KE\int_{t_0}^{u \wedge \pi_R}\left\{|x_{s}|^{p-2}(M+|x_{s}|^2)+M'+|x_s|^p \right\}ds & \leq K EM^\frac{p}{2}+K EM' +EK\int_{t_0}^{u \wedge \pi_R}|x_r|^pds < \infty.
\begin{aligned}A_{a}(x_a,\xi_a)=\frac{1}{2}|\nabla_{x_a}\varphi_{a}(x_a,\xi_a)|^2+I^L_{a}(x_a,0)-\frac{1}{2}|\xi_a|^2-\frac{1}{2}i\Delta_{x_a}\varphi_{a}(x_a,\xi_a)\end{aligned}
\begin{aligned}R_{e,k}^{Oip}=\left(1-\frac{K}{2T}\right)(1-\eta)\log_2\left(1+\frac{\lambda_{k,h}^O\gamma_{k,h}^O M}{p_d\beta_h^k+1}\right), & k=1,\ldots, K/2\end{aligned}
L^{2}= - \left( \frac{d \tau}{d \lambda}\right)^{2}+ l^{2}\cosh^{2}\left( \frac{\tau}{l}\right) \sum_{j = 1}^{d - 1}\left( \prod_{i = 1}^{j - 1}\sin^{2}\theta_{i}\right) \left( \frac{d \theta_{j}}{d \lambda}\right)^{2},
\dfrac{1}{n}\sum_{k=0}^{n-1}\left(\partial_{\theta}b(\theta_0,X_{t_k})\right)^{\ast}(\sigma\sigma^{\ast})^{-1}(X_{t_k})\partial_{\theta}b(\theta_0,X_{t_k})\overset{\P^{\theta_0}}{\longrightarrow}\Gamma(\theta_0),
F ( u ) = \left( \prod_{j = 1}^{m}\frac{[ v_{j}- v_{j - 1}+ \frac{1}{2}- \pi_{j}+ u ]}{[ - \pi_{j}+ u ]}\right) \frac{[ v_{0}- v_{m}- \frac{m}{2}+ 1 - \pi_{\mu}+ u ]}{[ 1 - \pi_{\mu}+ u ]}.
2(|n - p - 1| + 1) + \sum_{j \neq i}\frac{4x^1_i}{x^1_i - x^1_j}- 2 - \sum_{j \neq i}\frac{4x^1_i}{x^1_i - x^1_j}= 2 (|n - p| + 1) + \sum_{j \neq i + 1}\frac{4 y_{i + 1}}{y_{i + 1}- y_j}- \sum_{j \neq i + 1}\frac{4 y_{i + 1}}{y_{i + 1}- y_j}.
\Pr( (X_n,Y_n) = (x_n,y_n) ) = \frac{1}{2\pi \sigma_X\sigma_Y}\left( \exp\left(-\frac{(x_n-\mu_X)^2}{2\sigma_X^2}-\frac{(y_n-\mu_Y)^2}{2\sigma_Y^2}\right) + o(1) \right),
(\mathrm{\mathrm{H}}^{s,p}(\Omega))' = \mathrm{H}^{-s,p'}_0(\Omega)\, (\mathrm{B}^{s}_{p,q}(\Omega))'=\mathrm{B}^{-s}_{p',q',0}(\Omega) (\mathrm{B}^{s}_{p,q,0}(\Omega))'=\mathrm{B}^{-s}_{p',q'}(\Omega)
\Phi(A,B) \leqq C_1 \sum_{j=1}^N \sum_{i=1}^{M-1}\left\{\sum_{k=1}^{H_0-1}(a_{ik}b_{kj}-a^0_{ik}b^0_{kj}) + c_i \right\}^2{+}C_2 \sum_{k=H_0}^{H-1}\left( \sum_{j=1}^N b_{kj}^2 \right) \left( \sum_{i=1}^{M-1}a_{ik}^2 \right),