harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}|F^{ij}(\nabla_\alpha u)_{ij}| \leq \, |\nabla_\alpha f^{1/k}| + C \sum_{i,j=1}^n F^{ij}u_{ij}+ C |Du| \sum_{i=1}^n F^{ii}\leq \, Cf^{\frac{1}{k}-\frac{1}{2(k-1)}}+ C b_{n-1}\sum_{i=1}^n F^{ii}\leq\, C b_{n-1}^{1/2-1/k}+ C b_{n-1}\sum_{i=1}^n F^{ii}\end{aligned}
{\cal F}_{i j}= \left( \begin{array}{c c c c}{0}&{{\cal F}_{0 1}}&{0}&{0}\\{-{\cal F}_{0 1}}&{0}&{0}&{0}\\{0}&{0}&{0}&{{\cal F}_{2 3}}\\{0}&{0}&{-{\cal F}_{2 3}}&{0}\\ \end{array}\right) .
G\cdot R-\bar G\cdot \bar R-\bar R\cdot \nabla\bar G(\nabla \bar A)^{-1}(A-\bar A)-\bar G \cdot (P-\bar P)=(G-\bar G) \cdot (R-\bar R) &+\bar R \cdot G(U|\bar U;x,t)+\bar G\cdot \left[ A_t-\bar A_t+f_{\alpha,x_\alpha-}\bar f_{\alpha,x_\alpha}\right]\,.
x_k = \frac{\tau_k}{1+\tau_k}z_k + \frac{1}{1+\tau_k}y_k z_{k+1}- z_k = \tau_k \left(x_k - z_k - \frac{1}{\mu}\nabla \varphi(x_k) - \frac{1}{\mu}\nabla \psi(z_{k+1})\right) y_{k+1}= \mathcal{G}(x).
a_{\beta}= \Lambda_{\beta}\left(\frac{W_{0}}{2}\right)^{1/(\beta-2)}, \Lambda_{\beta}:= \frac{\Gamma\left(\frac{\beta-3}{\beta-2}\right)}{\Gamma\left(\frac{\beta-1}{\beta-2}\right)}\left(\frac{2}{\beta-2}\right)^{2/(\beta-2)}.
\frac{\left\langle b \varphi ( x ) e^{a \cdot \varphi ( 0 ) / b}\right\rangle_{b}}{G ( a / b )}= - 2 a \log r + \frac{\left\langle b \varphi ( 0 ) e^{a \cdot \varphi ( 0 ) / b}\right\rangle_{b}}{G ( a / b )}+ O ( r^{\sigma}) .
\frac{d^{2}}{d y^{2}}\Re{f_{M}(x + iy; \theta)}={}- \left.\Re{\frac{d^{2}}{d z^{2}}f_{M}(z; \theta)}\right|_{z = x + iy}={}- \frac{(M+1+x)x(M x - (M+1))+ y^{2}((M+1)(M+1+x) - x)}{((M+1+x)^{2}+ y^{2}) (x^{2}+ y^{2})}.
(1-\dfrac{\Delta t}{2}\{(1-r_1)\delta_{\alpha_1 x}^{S^-}+r_1\delta_{\alpha_1 x}^{S^+}\}) S_{i,j}^{*}=(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{S^-}+r_2\delta_{\alpha_2 y}^{S^+}\})S_{i,j}^{n}-\dfrac{\Delta t}{2}\left \{\mu-\mu S_{i,j}^{n+1/2}-\beta S_{i,j}^{n+1/2}V_{i,j}^{n+1/2}\right\}
\left[ \left( \mathsf{S}_{j}\left( s\right) \varphi\right) \right]_{\operatorname*{D};j}\left( s\right) =0\quad, \left[ \left( \mathsf{S}_{j}\left( s\right) \varphi\right) \right]_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right) =-\varphi.
\varphi_t = \begin{cases}- dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge \omega_t + dr^2 \wedge \Re (\Omega_t) + dr^3 \wedge \Im (\Omega_t) T^3 \times X^4, begin{align*}5pt] \Re (\Omega_t) - dr \wedge \omega_t S^1 \times X^6, \end{cases}
\begin{aligned}\leq C\mathbb{E}\int_0^\infty \big(|y_n-y|^2+|z_n-z|^2+|\tilde{z}_n-\tilde{z}|^2+||\gamma_n-\gamma||^2\big)e^{-\beta s}ds\rightarrow0,\ \n\rightarrow\infty,\end{aligned}
N^{*}(\zeta) =P(UV^{-1}B-A)P+PUV^{-1}L(\zeta I-LV^{-1}L)^{-1}LV^{-1}BP &=P(-A+U(I+(\zeta V-I)^{-1})V^{-1}B)P &=-P(A+U(\zeta^{-1}I-V)^{-1}B)P=-PN(\zeta^{-1})P,
{\cal C}_{\underline{w}_0}=\left\{(\lambda,\mathbf m)\in \Lambda_{\mathbb R}\times{C}_{\underline{w}_0}\mid\begin{array}{l}m_k \le \langle\lambda,\alpha_{i_k}^\vee\rangle -\sum_{\ell=k+1}^N\langle \alpha_{i_\ell},\alpha_{i_k}^\vee \rangle m_\ell, k=1,\ldots,N\end{array}\right\}.
= \frac{1}{\sqrt{2 \pi k i}}({\cal M}^{- 1}- 1 ) \frac{e^{- i [ \Omega ] \theta}}{1 - e^{i \theta}}\left( 1 + e^{i \left( [ \Omega ] + 1 / 2 \right) ( 2 \theta - \pi )}\tan \theta / 2 \right)
\frac d{dt}(a_1+c_1)= a_1^2+2a_2a_3 + c_1^2+2c_2c_3 = a_1(a_1+a_2+a_3)+c_1(c_1+c_2+c_3)+I = (a_1+c_1)\cdot \frac R2 + I = (\kappa+\delta t) \frac{R^2}2 + I
b_{2n+1-k}\sigma_{k-1}^\ell+{}a_{2n+1-k}\sigma_{k}^\ell+c_{2n+1-k}\sigma_{k+1}^\ell &=b_{2n+1-k}\sigma_{2n+2-k}^\ell+a_{2n+1-k}\sigma_{2n+1-k}^\ell+c_{2n+1-k}\sigma_{2n-k}^\ell &=s_\ell( \sigma_{2n+1-k})=s_\ell(-\sigma_{2n+1-k})=s_\ell(\sigma_{k}).
K_{2}=\left(\int_{\wp}\int_{0}^{\infty}s^{Q-1}(\psi_{3}(sy))^{1-p'}\left(\frac{\int_{\wp}\int_{0}^{s}r^{Q-1}h(rw)(\psi_{3}(rw))^{1-p'}drd\sigma(w)}{\int_{\wp}\int_{0}^{s}r^{Q-1}(\psi_{3}(rw))^{1-p'}drd\sigma(w)}\right)^{p}dsd\sigma(y)\right)^{\frac{q-1}{p}}
\begin{aligned}\int_{A_1 \cap A_2}|g - g_{\rho}| \, d \pi \leq \int_{A_1 \cap A_2}M \left[\boldsymbol{d}_{\mathcal{S}_1}(s^\star_1, s_1)^{p_1^{\prime}}+\boldsymbol{d}_{\mathcal{S}_2}(s_2, s^\star_1)^{p_2^{\prime}}\right] d\pi(s_1, s_2) & \leq B (\rho^{p_1^\prime - p_1}+ \rho^{p_2^\prime - p_2}).\end{aligned}
a = \left[\begin{array}{c}1 1 1 \end{array}\right],B = \left[\begin{array}{ccc}-2 -0.35 -0.35 -0.35 -4 -0.35 -0.35 -0.35 -6\end{array}\right],d = \left[\begin{array}{c}0.2 0.2 0.2 \end{array}\right].
w^{\rho_{i_k}}= [-d_1\Delta+I]^{-1}\left\{w^{\rho_{i_k}}+w^{\rho_{i_k}}q(\rho_{i_k}w^{\rho_{i_k}})\left[f(\rho_{i_k}w^{\rho_{i_k}})-v^{\rho_{i_k}}\right]\right\}, v^{\rho_{i_k}}= [-d_2\Delta+I]^{-1}\left\{v^{\rho_{i_k}}+v^{\rho_{i_k}}\left[f(v^{\rho_{i_k}})+q(\rho_{i_k}w^{\rho_{i_k}})w^{\rho_{i_k}}\right]\right\},
\det \left| \begin{array}{c c c c}{{0}}&{{\phi_{x_{1}}}}&{{\phi_{x_{2}}}}&{{\phi_{x_{3}}}}\\{{\phi_{x_{1}}}}&{{\phi_{x_{1}x_{1}}}}&{{\phi_{x_{1}x_{2}}}}&{{\phi_{x_{1}x_{3}}}}\\{{\phi_{x_{2}}}}&{{\phi_{x_{2}x_{1}}}}&{{\phi_{x_{2}x_{2}}}}&{{\phi_{x_{2}x_{3}}}}\\{{\phi_{x_{3}}}}&{{\phi_{x_{3}x_{1}}}}&{{\phi_{x_{3}x_{2}}}}&{{\phi_{x_{3}x_{3}}}}\\ \end{array}\right| \, = \, 0 .
\frac{\partial \mathcal{L}}{\partial \varphi}- \frac{\partial}{\partial t}\left(\frac{\partial \mathcal{L}}{\partial \varphi_t}\right) - \frac{\partial}{\partial x}\left(\frac{\partial \mathcal{L}}{\partial \varphi_x}\right) + \frac{\partial^2}{\partial x^2}\left(\frac{\partial \mathcal{L}}{\partial \varphi_{xx}}\right) = 0.
Q_{\lambda, q}(\Omega) g_q^{q-1}(\varsigma)=\int_{\Omega_q}\frac{g_q(\eta)}{|\eta^{-1}\varsigma|^{Q-\alpha}}d\eta+ \lambda \int_\Omega\frac{f_q(\xi_q)^{1-q}\cdot f_q(\eta)}{|\eta^{-1}\xi|^{Q-\alpha-1}}d\eta
\begin{aligned}(\Delta (u+v))^m\wedge\beta_n^{n-m}=(\Delta u)^m\wedge\beta_n^{n-m}+(\Delta v)^m\wedge\beta_n^{n-m}+\sum\limits_{p=1}^{m-1}\binom{m}{p}(\Delta u)^p\wedge(\Delta v)^{m-p}\wedge\beta_n^{n-m}&\geq (\Delta u)^m\wedge\beta_n^{n-m}+(\Delta v)^m\wedge\beta_n^{n-m}. \end{aligned}
\begin{aligned}G: L^2_{\mathcal{F}}(\Omega \times I; L^2_x) \times U \to L^2_{\mathcal{F}}(\Omega \times I; L^2_x) , (v,u) \mapsto G(v)u := \sum_{j \in \mathbb{N}}g_j\big(\cdot,v(\cdot)\big)( u_j, u)_U.\end{aligned}
f(x)\approx\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{(x-\mu)^{2}}{2\sigma^{2}}][1+\frac{\kappa_{3}}{3!\sigma^{3}}He_{3}(\frac{x-\mu}{\sigma})+\frac{\kappa_{4}}{4!\sigma^{4}}He_{4}(\frac{x-\mu}{\sigma})],
U(y,\Lambda_R)\tilde{a}(\mathbf{x})U(y,\Lambda_R)^{-1}= \frac{iy^{0}}{2\pi^2}\left(\frac{m^2 K_{2}(m\sqrt{\vert \mathbf{x+y}\vert^{2}-(y^{0})^2})}{\vert \mathbf{x+y}\vert^{2}-(y^{0})^2}\right)\ast\tilde{a}(\mathbf{x+y}).
\gamma ( \beta \sqrt{- \triangle}) = \frac{2 \pi^{2}}{\beta \sqrt{- \triangle}}+ \frac{4 \pi{\mathrm i}}{\beta \sqrt{- \triangle}}\ln \left( \frac{\Gamma ( 1 +{\mathrm i}\frac{\beta \sqrt{- \triangle}}{4 \pi})}{\Gamma ( 1 -{\mathrm i}\frac{\beta \sqrt{- \triangle}}{4 \pi})}\right) + 2 \ln \Big ( \frac{\beta \mu}{4 \pi}\Big ) .
\nabla_1h_{11}=-\frac{\frac{\partial G}{\partial \lambda_2}}{\frac{\partial G}{\partial \lambda_1}}\nabla_1h_{22}=\frac{\lambda_1}{\lambda_2}\nabla_1h_{22},~\nabla_2h_{22}=-\frac{\frac{\partial G}{\partial \lambda_1}}{\frac{\partial G}{\partial \lambda_2}}\nabla_2h_{11}=\frac{\lambda_2}{\lambda_1}\nabla_2h_{11}.
A(u,s)=\frac{1}{4}\iint \left \vert \nabla u\right \vert^{2}dxdt+\frac{1}{2}\iint W(u)dxdt+\frac{1}{2}\iint \left( \frac{\partial s}{\partial t}+\frac{1}{2}\left \vert \nabla s\right \vert^{2}\right) u^{2}dxdt
\mathbf{Y}^{T_E}_{[\ell+1:K]}+ \tilde{\mathbf{n}}^{T_E}_{[\ell+1:K]}= \mathbf{H}_3 \begin{bmatrix}\mathbf{X}_{[1:(M-\ell)]}^{T_E}\mathbf{H_1}^{\dagger}\tilde{\mathbf{Y}}^{T_E}_{[1:\ell]}\end{bmatrix}+ \mathbf{n}_{[\ell+1:K]}^{T_E},
\frac{\ddot{b}}{b}= \frac{\beta^{\prime \prime}}{\beta}, ~ \Biggl [ \frac{d}{d t}\left( \frac{\dot{c}}{c}\right) + \frac{\dot{b}}{b}\frac{\dot{c}}{c}\Biggr ] = \Biggl [ \frac{d}{d x}\left( \frac{\kappa^{\prime}}{\kappa}\right) + \frac{\beta^{\prime}}{\beta}\frac{\kappa^{\prime}}{\kappa}\Biggr ] .
\mathcal{R}_{n,\alpha}(\theta(\mathcal{P}_\varepsilon), \Phi \circ \rho) \geq \frac{\Phi(\omega(\varepsilon)/2)}{2}\left(1-\sqrt{4n\alpha^2}\mathrm{TV}(\widetilde{R}_0, \widetilde{R}_1)\right) = \frac{\Phi(\omega(\varepsilon)/2)}{2},
S_{3}={\frac{4 \pi \mu}{\sqrt \lambda}}\int_{0}^{\infty}d r r^{2}\left[{\frac{1}{2}}({\frac{d \varphi}{d r}})^{2}+{\frac{1}{2}}( \varphi^{2}- 1 )^{2}- f \varphi \right] .
\sum_{k=0}^{n-1}\frac{(q^{d+r};q^d)_k^{d-r}(q^r;q^d)_k^{r-1}(q^{r-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}\equiv\frac{(1-q^r)^{r}(1-q^{d-r})(q^d;q^d)_{n-1-(n+r)/d}}{-(-1)^{n-1-(n+r)/d}(q^d;q^d)_{(n+r)/d}^{d-1}}q^{A(d,n,r)},
\|x_n -w\| \leq \sum_{i=0}^{n-1}\|x_{i+1}-x_i\| +\|x_0 -w\| \leq \sigma\sum_{i=0}^{n-1}d_C(x_i) +\|x_0 -w\| &\leq \sigma\sum_{i=0}^{n-1}\rho^i d_C(x_0) +\|x_0 -w\| \leq \big(\sigma\frac{1}{1 -\rho}+1\big)\|x_0 -w\| \leq \delta.
A_{\tilde{\varphi}}\rightarrow L^{- 1}e^{\phi_{0}}\frac{\hat{B}\rho^{2}}{2 \tilde{\Lambda}}, \, B_{\tilde{\varphi}}\rightarrow L e^{- \phi_{0}}\frac{\hat{E}\rho^{2}}{2 \tilde{\Psi}},
\sup_{j\ge 0}\ \|P_{n}T^{\,j}P_{l}\,x\| \le 2^{\,\delta^{(k-1)}-\tau^{(k)}}\,\cdot\,\bigl(\Delta^{(k)}\bigr)^{1-\frac1{p}}\,\cdot\,\Big\|\sum_{i=b_{l}}^{b_{l+1}-1}\Big(\prod_{s=i+1}^{b_{l+1}-1}w_s\Big)\,x_{i}e_{i}\Big\| &\le \dfrac{\beta_{l}}{4}\cdot \Big\|\sum_{i=b_{l}}^{b_{l+1}-1}\Big(\prod_{s=i+1}^{b_{l+1}-1}w_s\Big)\,x_{i}e_{i}\Big\|
\begin{array}{llcl}\inf_{z_1,z_2 \in \mathbb{C}}\hat{f}(z_1,z_2) := 1-\frac{4}{3}|z_1|^2+a|z_1|^4 \hat{g}(z_1,z_2) := 1 - |z_1|^2 - |z_2|^2 = 0.\end{array}
Q_j(x_3) =x_{2}^{2^{j-1}}Q_{j-1}(x_3)+ x_{3}^{2^{j-1}}Q_{j-2}(x_3) &= x_3 \left( \alpha_{j-1,0}x_2^{2^{j-1}}+ \alpha_{j-2,0}x_3^{2^{j-1}}\right),
\begin{aligned}v_1^e(x,t) :=v_1(x,t)-v_1(\sigma_e(x),t),\x\in \overline{B},\t>0, v_2^e(x,t) :=v_2(\sigma_e(x),t)-v_2(x,t),\x\in \overline{B},\t>0, \end{aligned}
\sinh^2\tfrac{1}{2}d_{\xi}z = \sinh^2(\tfrac{1}{2}T_{\xi})\cosh^2d_z\mathcal{A}+\sin^2\theta\sinh^2d_z\mathcal{A}\leq \left( \sinh^2(\tfrac{1}{2}T_{\xi})+\sin^2\theta\right)\cosh^2d_z\mathcal{A},
-\left. \frac{i}{4}\mbox{Tr}\,\log\widehat{\Omega}^2\right|_{div}=\left.\frac{i}{4}\mbox{Tr}\,\log\left(-\Delta+{R\over 4}\hat1\right)\right|_{div}= -{1\over 2\epsilon}\int d^2x\, \sqrt{g}\,{n\over12}R \ .
\left|\psi(\zeta )\right| \leq \sum_{n=0}^{\infty}\left| h\left( f^{\circ n}(\zeta )\right)\right|\leq \mathrm{e}^{-\alpha \Re (\zeta )}\cdot \frac{1}{1- \frac{1}{\mathrm{e}^{\alpha \cdot \rho^-_{\beta , \varepsilon ,k}(R)}}},
x^{(1)}=[\begin{matrix}0&-1/2\\ -5/7&0\\ \end{matrix}][\begin{matrix}1\\ 1\\ \end{matrix}]+[\begin{matrix}11/2\\ 13/7\\ \end{matrix}]=[\begin{matrix}5.0\\ 8/7\\ \end{matrix}]\approx[\begin{matrix}5\\ 1.143\\ \end{matrix}].
\bigcup_{i=2}^{n}\bigcup_{\substack{1\le j_1,j_2\le i j_1\ne j_2 \gcd(i,j_1,j_2)=1}}\left\{\left<E^{p\times q}_{kl}\otimes E^{i\times i}_{j_1j_2}\right>\right\}
\left[ \frac{\mathrm{d}^{2}}{{\mathrm{d}{\rho}^{*}}^{2}}+ \left( \omega - e \frac{Q}{\rho}\right)^{2}- \mu^{2}\frac{\Delta ( \rho )}{R^{2}( \rho )}\right] \chi ({\rho}^{*}) = 0 \; ,
g h_{s}^{j}+ \lambda h_{t}^{j}= g \eta^t_s \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}+\lambda \eta^t_t \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}- \eta^{t}\Lambda \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}+ g^{2}F^{j}+ g \lambda G^{j}+ \lambda^{2}H^{j}.
\lambda(\rho(g_1,g_2),\iota(z_1,z_2))S^{-1}B(z_1,z_2)\left[\begin{array}{cc}\lambda(g_1,z_1) 0 0 \overline{\lambda(g_2,z_2)}\end{array}\right]^{-1}B(g_1z_1,g_2z_2)^{-1}S=1,
(\begin{matrix}12\\ 0\end{matrix})12!-(\begin{matrix}12\\ 1\end{matrix})(12-1)!+(\begin{matrix}12\\ 2\end{matrix})(12-2)!-\cdot\cdot\cdot+(-1)^{12+1}(\begin{matrix}12\\ 12\end{matrix})0!=176214841
S_{1}:= \sum_{j \geq 3}\int_{B}\int_{B}\frac{\left\vert \underset{x}{Hess}\left( \left\Vert x - y \right\Vert^{j}\right) \right\vert^{2}}{\left( (j+1)! \right)^{2}}dy dx \leq \left\vert B \right\vert^{2}\, \sum_{j \geq 3}\frac{j^{2}\, \left( diam(B)^{2}\right)^{(j-2)}}{\left( (j+1)! \right)^{2}}< +\infty,
{\frac{d}{d \tau}}\zeta^{\mu}( \tau , \vec{\sigma}) = \lbrace \zeta^{\mu}( \tau , \vec{\sigma}) , H_{D}\rbrace = - \lambda^{\mu}( \tau , \vec{\sigma}) -{\dot{x}}_{s}^{\mu}( \tau ) -{\dot{b}}_{\check{r}}^{\mu}( \tau ) \sigma^{\check{r}}\approx 0 ,
\mu \widetilde\rho(s) = -\frac{\widetilde f\,^\prime(s|\mu)}{\widetilde f(s|\mu)}\quad{\rm and}\mu\,\widetilde r_\alpha(s) = -\frac{\widetilde m_\alpha\,^\prime(s|\mu)}{\widetilde m_\alpha(s|\mu)}= -\frac{\widetilde f\,^\prime(s^\alpha|\mu)}{\widetilde f(s^\alpha|\mu)}
\eta(x) =p_i(x)+\sum_{0\leq |k|\leq n-1}c_i(k)D^{k}\ln\left(\frac{1}{|x|}\right) &=p_i(x)+c_{0,i}\ln(|x|)+\sum_{1 \leq |k|\leq n-1}c_i(k)D^{k}\ln\left(\frac{1}{|x|}\right) &=p_i(x)+c_{0,i}\ln\left(\frac{1}{|x|}\right)+q_i\left(\frac{x}{|x|^2}\right) B_{r_i}\backslash \{0\},
f=A_{0}(1+\frac{-1}{2}z^{2}+\frac{-1}{8}z^{4}+\frac{-7}{240}z^{6}+\cdot\cdot\cdot)+A_{1}(z+\frac{1}{6}z^{3}+\frac{1}{24}z^{5}+\frac{1}{112}z^{7}+\cdot\cdot\cdot)
\displaystyle \frac{\partial^{j}}{\partial \rho^{j}}\left(H(1,\rho)\right){\big|_{\rho =0}}=j!\int\limits_{-1}^1f(t)G^{j-1}(t)dt=j!\int_{-1}^1f(t)r_j(t)dt=0, \mbox{for each}j=1,2
\xi*\int_{0}^{t}\Gamma_{\alpha\beta}(\tau,t-\tau)d\tau+\int_{B}\left[{\cal L}_{j}^{(\beta)}*\theta_{,j}^{(\alpha)}(t)+{\cal L}_{j}^{(\alpha)}*\theta_{,j}^{(\beta)}\right]dv =\xi*\int_{0}^{t}\Gamma_{\beta\alpha}(\tau,t-\tau)d\tau+\int_{B}\left[{\cal L}_{j}^{(\beta)}*\theta_{,j}^{(\alpha)}(t)+{\cal L}_{j}^{(\alpha)}*\theta_{,j}^{(\beta)}\right]dv,
\Re Tr B_jY_k^{[1]}J \geq \frac{y_k^{[1]}}{(s+t)^2}\cdot \frac{\{\Re b_{j,1}, \Re b_{j,2}\}}{(s_j+t_j)^2}\geq c\frac{y_k^{[1]}}{1+2t^2},k=p,q,
u_\alpha(0)=\pi+\upsilon, \mbox{with}|\upsilon|<2^{-5}, u_\alpha\left(\frac{1}{T}\right)=\pi^{-1}+\delta_{1/T}, \mbox{with}|\delta_{1/T}|<2^{-3}, &u_\alpha\left(\frac{1}{T^2}\right)=\pi^2+\delta_{1/T^2}, \mbox{with}|\delta_{1/T^2}|<2^{-6}.
P\left(X_t^{(h)}(\epsilon) \ge \frac 1 4 \eta x b_t\right)=P\left(e^{\lambda X_t^{(h)}(\epsilon)}\ge e^{\frac \lambda 4 \eta x b_t}\right) \le E\left(e^{\lambda X_t^{(h)}(\epsilon)}\right) e^{-\frac \lambda 4 \eta x b_t}.
{{\frac{1}{2}}{\frac{{\partial}^{2}V}{{\partial x}^{2}}}= P ( x , T , T^{*}) \left[ \widetilde{E}+{\frac{{\left( \alpha + \beta + 2 x \alpha \beta \right)}^{2}- \left( 1 + \alpha + \beta + 2 x \alpha \beta \right)}{1 + \alpha + \beta + 2 x \alpha \beta -{\frac{1}{4 x^{2}}}}}\right]}
F^{C}( \beta ) \sim -{\frac{\pi^{2}}{9 0}}{\frac{c_{0}}{\beta^{4}}}-{\frac{1}{2 4}}{\frac{c_{1}}{\beta^{2}}}-{\frac{1}{1 6 \pi^{2}}}\ln ( \beta \rho ) c_{2}-{\frac{1}{1 6 \pi^{5 / 2}}}\sum_{n = 3}^{\infty}\Gamma \left( n - \frac 3 2 \right) \zeta ( 2 n - 3 ) c_{n}\left({\frac{\beta}{2 \pi}}\right)^{2 n - 4},
T_{\mathcal{C}}= \begin{pmatrix}1 0 0 0 0 0 0 1 0 0 0 0 0 0 \zeta_5^3 0 0 0 0 0 0 \zeta_5^2 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 \end{pmatrix}
F ( m , \mu ) = \frac{2}{3}\int \frac{d^{3}p}{( 2 \pi )^{3}}\frac{4 \mu ( p^{2}/ m^{2}) ( 1 - 2 \mu^{3}p^{2}/ m^{2}) + 3}{f ( \mu , p^{2}) [ p^{2}- m^{2}f^{2}( \mu , p^{2}) ]^{2}}.
A_{k}^{(1)}(m,n) = \int_{-1/2}^{1/2}\int_{-1/2}^{1/2}\frac{\psi(\tau) F_2(\tau)}{(S_4^\ell \psi F_{k'})(\tau)}\cdot \frac{S_4^\ell(z) \psi(z) F_2(z)}{\psi(\tau)-\psi(z)}e^{2\pi i m \tau}e^{-2\pi i n z}du dx.
\Pi_\xi:L^2_{\rm per}([0,T])\to\bigoplus_{j=1}^3{\rm ker}\left(L_\xi-\lambda_j(\xi){\bf I}\right),~~\widetilde{\Pi}_\xi:L^2_{\rm per}([0,T])\to\bigoplus_{j=1}^3{\rm ker}\left(L_\xi^\dag-\overline{\lambda_j(\xi)}{\bf I}\right)
X^{2}= b \left( \begin{array}{c c c c c}{{\ddots}}&{{\ddots}}&{{\ddots}}&{{}}&{{}}\\{{\ddots}}&{{0}}&{{e^{i \omega t}}}&{{\ddots}}&{{}}\\{{\ddots}}&{{e^{- i \omega t}}}&{{0}}&{{e^{i \omega t}}}&{{\ddots}}\\{{\ddots}}&{{\ddots}}&{{e^{- i \omega t}}}&{{0}}&{{\ddots}}\\{{}}&{{}}&{{\ddots}}&{{\ddots}}&{{\ddots}}\\ \end{array}\right)
\gamma=\sqrt{\frac{-c_{gh}-c_M-r +24(|\rho||\bar\rho|-\rho\cdot\bar\rho)}{48|\bar\rho|^2}}-\sqrt{\frac{-c_{gh}-c_M-r -24(|\rho||\bar\rho|+\rho\cdot\bar\rho)}{48|\bar\rho|^2}}~,
\langle|F^\ell(k)|^2\rangle \lesssim \sum_{\stackrel{k'+k''=k}{k',k''\not=0}}d^{-4}(k',0)d^{-1}(k'',0), \langle|\ell\frac{\partial}{\partial \ell}F^\ell(k)|^2\rangle \lesssim\sum_{\stackrel{k'+k''=k}{k',k''\not=0}}{\min}^2 \{1,\ell(d(k',0)+d(k'',0))\}d^{-4}(k',0)d^{-1}(k',0).
\ddot{\Phi}- \frac{2}{a}\, \dot{\Phi}_{,r}+ \frac{1}{a^2}\, \Phi_{,rr}+ H\dot{\Phi}= 8\pi G\, \dot{\varphi}\left( \delta\dot{\varphi}- \frac{1}{a}\, \delta\varphi_{,r}\right),
{\cal Z}=\left[\frac{2}{\beta}I_1(\beta)\right]^{\#{\rm\;of\;plaquettes}}\sum_{J_P,\;j_l}\quad\prod_{{\rm plaquettes}}(2J_P+1)\;T_{J_P}(\beta)\;(-1)^{2J_P}\prod_{{\rm links}}(2j_l+1)
\begin{matrix}{{L = N \sum_{k = 1}^{N}\left( ( \partial_{\mu}\phi_{k})^{2}+ V ( \phi_{k}) \right)}}\\{{+ \sum_{i , j = 1}^{N}( \phi_{i}- \phi_{j})^{2}| B_{\mu}^{i j}|^{2}}}\\ \end{matrix}
g_{\tilde{r}\tilde{r}}(G^{\mu}_{\mu}-2G^{t}_{t})=16\pi G\frac{m}{f}(L_M+g^{tt}\frac{\partial L_M}{\partial g^{tt}}-g^{\tilde{r}\tilde{r}}\frac{\partial L_M}{\partial g^{\tilde{r}\tilde{r}}}-g^{\theta\theta}\frac{\partial L_M}{\partial g^{\theta\theta}}-g^{\varphi\varphi}\frac{\partial L_M}{\partial g^{\varphi\varphi}})
\vert\widehat{\bold{1}_{P}}(x+yi)\vert\leq\frac{4}{\tilde{N}^{2}}\Vert x/\tilde{N}\Vert\cdot\Vert y/\tilde{N}\Vert=\frac{4}{\min\{x,\tilde{N}-x\}\cdot\min\{y,\tilde{N}-y\}}
\begin{array}{c}{{\left( \sum_{i}r_{i}^{( l )}\right)_{L}\times \left( \sum_{i}r_{i}^{( l )}\right)_{R}}}\\{{\times \left[ \begin{array}{l}{{\{\left( 4 4 + 8 4 \right)_{B}+ 1 2 8_{F}\}_{L}}}\\{{\times \{\left( 4 4 + 8 4 \right)_{B}+ 1 2 8_{F}\}_{R}}}\\ \end{array}\right]}}\\ \end{array}
\sum_{m,n \geq 0}\frac{q^{m^2+n^2-mn}x^m y^n}{(q)_m (q)_n}=\sum_{n_1,n_2,n_3 \geq 0}\frac{x^{n_1+n_3}y^{n_2+n_3}q^{n_1^2+n_2^2+n_3^2+ n_1 n_3+n_2 n_3}}{(q)_{n_1}(q)_{n_2}(q)_{n_3}}.
D_{k}(\tau)=\left(\frac{\alpha_{0}}{2C_{k}}(k+1)!-\frac{2\alpha_{1}k(k+1)}{(2\pi)^{2}}+\tilde{C_{k}}\right)(1+\tau)^{k}+\frac{2\alpha_{1}k(1+\tau)^{k-2}}{(2\pi)^{2}}.
\lim_{K\rightarrow +\infty}p(x_K)\sum_{j=0}^{{1\over \delta_K}-1}h_K G\left(\overline{(i_K-j)\delta_K}\, \log K\right) =\lim_{K\rightarrow +\infty}p(x_K)\sum_{\ell=-\lfloor 1/2\delta_K\rfloor}^{1/\delta_K-1-\lfloor 1/2\delta_K\rfloor}h_K G\left(h_K\ell\right) =p(x)\int_{\mathbb{R}}G(y)\,dy=p(x),
S=\begin{pmatrix}-a_{12}0 a_{16}-a_{15}a_{14}-a_{13}0 -a_{12}-a_{13}-a_{14}-a_{15}-a_{16}a_{16}-a_{13}-2 a_{14}0 0 0 -a_{15}-a_{14}0 0 0 0 a_{14}-a_{15}0 0 0 0 -a_{13}-a_{16}0 0 0 -2 a_{14}\end{pmatrix},a_{14}\neq 0.
\sharp \left\{(\hat{x}^a_k,\hat{y}^a_k,\,\hat{p}^a_{kx_k},\,\hat{p}^a_{ky_k}),\,k=1,...,N,\,a=2,3,4\right\}\gg \sharp \left\{\widehat{X}^a,\dot{\widehat{X}^a},\,a=2,3,4\right\}.
\partial_tw_k+(J_kw_k)\cdot\nabla w_k+Sw_k=\Delta w_k-\nabla p_k+f, &\mbox{div $w_k$}=0, &w_k|_{\partial\Omega}=0, &w_k\to 0 \quad\mbox{as $|x|\to\infty$}, &w_k(\cdot,0)=w_0.
ln\gamma_{2}^{\infty}=\frac{\nu_{2}}{RT}[(\lambda_{1}-\lambda_{2})^{2}+\frac{q_{1}^{2}q_{2}^{2}(\tau_{1}^{T}-\tau_{2}^{T})^{2}}{\psi_{1}}+\frac{(\alpha_{1}^{T}-\alpha_{2}^{T})(\beta_{1}^{T}-\beta_{2}^{T})}{\xi_{1}}]+d_{12}
I ( \epsilon ) = \frac{d}l \int_{\epsilon}d \rho \, d^{d}x \, \sqrt{\hat{g}}\, \rho^{- 1 - \frac{d}2}+ \frac 2 l \int d^{d}x \sqrt{\hat{g}}\, \epsilon^{- \frac{d}2}( \epsilon h - 1 ) .
g^{(k)}(x) =\sum_{n}{\binom{k}{n}}(-1)^{k-n}x^{n}\sum_{i}\frac{(k-N_{1})_{i}(k-N_{w})_{i}}{(N_{2}-N_{w}+1)_{i}i!}x^{i}& =\sum_{i}\sum_{n}{\binom{k}{n}}(-1)^{k-n}\frac{(k-N_{1})_{i}(k-N_{w})_{i}}{(N_{2}-N_{w}+1)_{i}i!}x^{i+n}& =(-1)^{k}\sum_{i}\left[ \sum_{n}{\binom{k}{n}}(-1)^{n}\frac{(k-N_{1})_{i-n}(k-N_{w})_{i-n}}{(N_{2}-N_{w}+1)_{i-n}(i-n)!}\right] x^{i}
M =\left((-\infty,0] \times \{-1,1\}\right)\cup \left( (0,\infty) \times \{0\}\right) U_+ = \left((-\infty,0] \times \{1\}\right)\cup \left( (0,\infty) \times \{0\}\right) U_- = \left((-\infty,0] \times \{-1\}\right)\cup \left( (0,\infty) \times \{0\}\right)
(\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2))^{\omega+1}\quad\quad=\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2).
\Gamma_{N}= - i e^{N}\int_{0}^{\infty}\frac{d T}{T}e^{- i m^{2}T}\left( \prod_{i = 1}^{N}\int_{0}^{T}d t_{i}\right){\cal N}\int_{x ( T ) = x ( 0 )}{\cal D}x \: e^{\displaystyle \frac{i}{4}\int_{0}^{T}d \tau x \frac{d^{2}}{d \tau^{2}}x}e^{\displaystyle \int_{0}^{T}J ( \tau ) x ( \tau )}.
T_{C, 2}(b_1, b_2)\ge\lim_{k\rightarrow \infty}c(1-k^{-1})k^{\frac 12}\langle 1_{(0,1)}\rangle_{C, I_k}=\lim_{k\rightarrow \infty}c(1-k^{-1})\frac{k^{\frac 12}}{C^{-1}(k)}=\infty.
h(x^\infty)+\langle\nabla g(x^{\infty}),x^{\infty}\rangle =h(x^*)+\langle\nabla g(x^{\infty}),x^{*}\rangle &=h(x^*)+\langle\nabla g(x^{*}),x^{*}\rangle, \ \forall~x^*, x^{\infty}\in X^*.
\Delta_{0}= \{3, 17, 18, 21\}, \Delta_{1}= \{1, 4, 12, 14, 16, 22, 26, 34\}, \Delta_{2}= \{2, 6, 7, 9, 15, 23, 29, 36\}, \Delta_{3}= \{5, 8, 10, 11, 13, 19, 20, 24, 25, 27, 28, 30, 31, 32, 33, 35\}.
P_{\lambda}^{\omega}(x\in I_{\infty}) =P_{\lambda}^{\omega}(U_{x_i,x_{i+1}}<H_{x_i},\forall i\in[0,n-1]) &=\prod_{i=0}^{n-1}[\frac{\lambda\rho(x_i,\omega)\rho(x_{i+1},\omega)}{1+\lambda\rho(x_i,\omega)\rho(x_{i+1},\omega)}].
c_I({\theta}) = \sum_{k=k_0: k\neq k^\star}^K{\mu^\star - r_k\theta_{c^\star k}\over I(\theta_{c^\star k}, \frac{\mu^\star}{r_k})}+\sum_{c\neq c^\star}\sum_{k=k_0}^K{\mu^\star - r_k\theta_{ck}\over I(\theta_{ck}, \frac{\mu^\star}{r_k})}.
H_{m, n}^+(x) = \frac{4}{\pi}\int_{t = 0}^\infty \tanh(\pi t) \biggl ( \int_{\zeta = -\infty}^\infty \cos(x \cosh \zeta) e\Bigl( \frac{t \zeta}{\pi}\Bigr) \, d \zeta \biggr ) k(t) V(m^2n, t) t \, d t.
\sum_{n=0}^{\infty}\frac{q^{n^2 + sn}}{(q; q)_n}= \prod_{\begin{subarray}{c}n=1 n \equiv \pm (s+1) \, \textit{mod}\, 5 \end{subarray}}^{\infty}\frac{1}{1-q^n}, s = 0, 1,
=\frac{e^{4}}{(k-k^{\prime})^{4}}(\sum_{r^{\prime}}\overline{v}_{k^{\prime}}\gamma^{\mu}(\sum_{r}v_{k}\overline{v}_{k})\gamma^{\nu}v_{k^{\prime}})(\sum_{s}\overline{u}_{p}\gamma_{\mu}(\sum_{s^{\prime}}u_{p^{\prime}}\overline{u}_{p^{\prime}})\gamma_{\nu}u_{p})
\underline{m}^*\circ S_{k,k}=\;\; \underline{m}^*\circ ((Ri)^{\simeq})^{-1}\circ\mathsf{cof}^{\underline{1}}\circ(Li)^{\simeq}=\;\; \underline{m}^*\circ \mathsf{cof}^{\underline{1}}\circ(Li)^{\simeq}\cong\;\; C^{|\underline{m}|}\circ (1_{\underline{m}^\vee})^\ast\circ (Li)^\simeq.
\sigma^{\ast}\mathcal{L}\cong \sigma^{\ast}(\mathcal{O}_{E}(3o_{E}))\cong \mathcal{O}_{E}(\widetilde{\sigma}(3o_{E}))=\mathcal{O}_{E}(3q), &(\sigma^{2})^{\ast}\mathcal{L}\cong (\sigma^{2})^{\ast}(\mathcal{O}_{E}(3o_{E}))\cong \mathcal{O}_{E}(\widetilde{\sigma^{2}}(3o_{E}))=\mathcal{O}_{E}(3r),
-\mu_{1,i}= -1 + \prod_{k=2}^d (1-\mu_{k,i})^{-1}= \sum_{k=2}^d \mu_{k,i}+ \sum_{\kappa=2}\sum_{\|\ell\|_1=\kappa}\prod_{k=2}^d \mu_{k,i}^{\ell_k}= \sum_{k=2}^d \mu_{k,i}+ M_i;
([\nabla_t,D]\xi)(X,X_1,\ldots,X_k) =R^V_{\partial_t,X}(\xi(X_1,\ldots,X_k))-\frac{1}{2}D_{\dot{g}(X)}\xi(X_1,\ldots,X_k) &\qquad+\frac{1}{2}\sum_{i=1}^k\xi(X_1,\ldots,D_X\dot{g}(X_i),\ldots,X_k).
\left\langle \prod_{b = 1}^{N}2 \cosh \left( \; 2 \omega \beta^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \left( 2 \sqrt{\pi}\varphi^{( b )}( x ) \right) :_{Q^{M , S}}\; \right) \; \right\rangle_{Q^{M , S}}\; .
\begin{aligned}-M\begin{bmatrix}1-F_{j,j}1_{i-j}-f_{j,j}-\Gamma_{j}-G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 -G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 1-F_{j,j-1}^{T}x -g_{j,j}-f_{j,j}\end{bmatrix}\end{aligned}
I_2 \le C\|u-\bar{u}\|_{L^t(R_1)}^{as}\|u\|_{L^q(R_0)}^{(1-a)s}\le C\big( \|u-\bar{u}\|_{L^1(R_1)}+\|\nabla(u-\bar{u})\|_{L^1(R_1)}\big)^{as}N_0^{(1-a)s/q}\le CM_1^{as}N_0^{(1-a)s/q}.
\check{\mu}_{\delta}*\mu_{\delta}(B(x,\rho)) =(Q_{\delta}*\mu*\mu*Q_{\delta})(B(x,\rho)) &\leqslant \mu*\mu(B(1,\delta)B(x,\rho)B(1,\delta))\ll\delta^{-\varepsilon}\rho^{\kappa}.
{\cal P}_{3}( y ) \| = \frac{1}{r^{n}}\int_{- \infty}^{u}d{\bar{u}}\, \log \Bigl ( m \frac{u -{\bar{u}}}{2}\Bigr ) \frac{d}{d{\bar{u}}}A ({\bar{u}}) + O \Bigl ( \frac{1}{r^{n + 1}}\Bigr ) \quad .
\mathrm{T r}e^{- ( L_{i}+ m_{i}^{2}g_{\tau \tau}) t}\simeq{\frac{n_{i}}{( 4 \pi t )^{{\frac{D - 1}{2}}}}}\int_{\widetilde{\cal M}^{\prime}}d \tilde{V}^{\prime}e^{- m_{i}^{2}g_{\tau \tau}t},
F_{x_{\perp}\bar{x}_{\perp}}^{2 1}= - \frac{1}{2 \pi | x_{\perp}|}e^{2 i x \cdot \omega}\partial_{\bar{x}_{\perp}}( \bar{x}_{\perp}I ) \sim e^{2 i x \cdot \omega}e^{- 2 | \omega | | x_{\perp}|}{\frac{( \omega_{1}+ i \omega_{2}) \pi^{{\frac{1}{2}}}}{2^{{\frac{3}{2}}}| \omega |^{{\frac{1}{2}}}| x_{\perp}|^{{\frac{3}{2}}}}}
u(x)=\int_{\partial M}\Big[u\cdot\partial_\nu^y\triangle_y\hat{E}(x,y)-\partial_\nu u\cdot\triangle_y\hat{E}(x,y)+\triangle u\cdot\partial_\nu^y \hat{E}(x,y)-\partial_\nu\triangle u\cdot\hat{E}(x,y) \Big]d\sigma(y).