harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{M}(x_{1},x_{2})=-\sum_{(n,m)\in \mathbb{Z}^{2}\setminus \lbrace\textbf{0}\rbrace}\log\left\lbrace \dfrac{(x_{1}-2na)^{2}+(x_{2}-2ma)^{2}}{4n^{2}a^{2}+4m^{2}a^{2}}\right\rbrace \phantom{XXXXXX}\phantom{XXXXXXX}+ \sum_{(n,m)\in \mathbb{Z}^{2}}\log\left\lbrace \dfrac{(x_{1}-(2n+1)a)^{2}+(x_{2}-(2m+1)a)^{2}}{(2n+1)^{2}a^{2}+(2m+1)^{2}a^{2}}\right\rbrace .
z^{0}\!=\!k_{\mbox{}_H}^{-1}\sqrt{1\!-\!2m/r\!+\! r^2/R^2}\sinh (k_{\mbox{}_H}t)\; , \;\; z^{1}\!=\!k_{\mbox{}_H}^{-1}\sqrt{1\!-\!2m/r\!+\!r^2/R^2}\cosh (k_{\mbox{}_H}t)
\begin{aligned}10^{-2}\delta_3\tau_0^2 \|\underline{\beta}_{s-1},\underline{y}\|_{L^{2}(N\cap B_{4\rho_{s-1}})}\leq \|\underline{f}_{s-1}\|(2\rho_{s-1}, \tau_0) \|\underline{\beta}_{s-1}, \underline{y}_{s-1}\|^2_{L^{2}(N_{s-1}\cap B_{4\rho_{s-1}})}+ \|\underline{f}_{s-1}\|^2_{L^{2}(2\rho_{s-1}, \tau_0)}\leq{\rm VolEx}_{N}(2\rho_{s-1},2^{1-b}\rho_{s-1})\end{aligned}
w = \frac{\varphi}{t}= \frac{\frac{\pi}{2}}{\sqrt{\frac{\frac{1}{2}g}{h}}}= \frac{\frac{\sqrt{h}\pi}{2}}{\sqrt{\frac{1}{2}g}}= \frac{\frac{\sqrt{h}}{2}\pi}{\frac{\sqrt{2}}{2}g}= \frac{\sqrt{h}\pi}{\sqrt{2}g}
\phi(e^{tS}x-\psi) = \phi(x) + \sum_{i=1}^{d}\left(e^{tS}x-x-\psi\right)_i D_i\phi(x) +\sum_{i=1}^{d}\sum_{j=1}^{d}\frac{1}{2}\left(e^{tS}x-x-\psi\right)_i\left(e^{tS}x-x-\psi\right)_j D_jD_i\phi(x) +R_{x,2}\left(e^{tS}x-x-\psi\right)
\delta \tilde{H}^{\alpha}= \delta H_{c}^{\alpha}+ \sum_{n = 1}^{\infty}\left( \delta G_{( n )}^{\alpha}( q_{i}, p_{i}) \theta_{\alpha}^{n}+ n G_{( n )}^{\alpha}( q_{i}, p_{i}) \theta_{\alpha}^{( n - 1 )}\delta \theta_{\alpha}\right) = 0 ,
-\kappa_5\frac{(\eta\cdot\zeta)^2}{2P_+}\frac{\partial}{\partial\theta_\eta}\psi = -\frac{\kappa_5}{2P_+}(\eta\cdot\zeta)^2\chi'_\eta - \kappa_5 \theta'_\eta - \frac{\theta'_\eta}{2P_+}(\eta\cdot\zeta)^2 B
\mathcal{V}^{+}(D) = \bigoplus_{\lambda >0}\mathcal{V}^{\lambda}(D), \mathcal{V}^{-}(D) = \bigoplus_{\lambda <0}\mathcal{V}^{\lambda}(D), \mathcal{V}^{\pm0}(D) = \mathcal{V}^{\pm}(D) + \mathcal{V}^0(D).
c(k) \leq m_1:=\min_{1\leq w\leq c(k)}w\left(d-\sqrt{(2-\frac{1}{r})k}-(2-\frac{1}{r})^{\frac{1}{4}}k^{\frac{1}{4}}\right)-w^2, c(k) \leq m_2:=\min_{1\leq w\leq c(k)}(\frac{1}{r}-1)k-w\left(\sqrt{(\frac{1}{r}-1)k}+(\frac{1}{r}-1)^\frac{1}{4}k^\frac{1}{4}\right).
\mathbb{P}\left[ \exists B, \left| \Gamma_B \cap \mathfrak{A}\right| < \frac{1}{2}q\ell \right] \leq \binom{n}{b}2e^{-q\ell/6}\leq 2e^{b \log n - c^2 \log^2 n/48}< \frac{1}{2}
G_{MN}dx^Mdx^N=H^{-{{4(n-1)}\over{(p+n)\Delta_p}}}_p\left[\eta_{\mu\nu}+h_{\mu\nu}\right]dx^{\mu}dx^{\nu}+H^{{4(p+1)}\over{(p+n)\Delta_p}}_pd{\bf y}\cdot d{\bf y}
b_1+h\frac{q}{a}=\overline{c_1}q_0\ell_0 + b\overline{c_2}q_1\ell_1+b\overline{c_2}(1-\overline{c_2}c_2) q_1\equiv \overline{c_1}q_0\ell_0 + b\overline{c_2}q_1\ell_1\pmod{q}.
A=\sum_{j_{1}=1}^{n_{1}}\sum_{j_{2}=1}^{n_{2}}\cdot\cdot\cdot\sum_{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},...,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes\cdot\cdot\cdot\otimes e_{j_{d}}^{d}
\mathbf{w}_{k}^\star = \left(\sum_{p\in\mathcal{K}}\bar{\mathbf{h}}_{p}\bar{\mathbf{h}}_{p}^H + \lambda^\star\mathbf{I}_N\right)^{-1}\times\sqrt{1 + \iota_{k}}\bar{\mathbf{h}}_{k}, \forall k \in \mathcal{K},
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right) -{\frac{3}{n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right) +{\frac{\ddot{a}}{a}}\right] -{\frac{3 k}{a^{2}}}= \kappa_{5}^{2}\left[{\frac{4}{3}}n^{- 2}\dot{\phi}^{2}+{\frac{4}{3}}b^{- 2}\phi^{\prime \, 2}- e^{- 2 \alpha \phi}\Lambda \right] ,
( A_{\mu}, \psi_{\mu}, \bar{\psi}_{\mu}, \phi , V_{\mu}, T_{\mu \nu}^{\pm}, \lambda_{\mu \nu}^{\pm}, Y_{\mu}^{\pm}, \xi_{\mu}^{\pm}) \rightarrow ( - A_{\mu}, i \bar{\psi}_{\mu}, - i \psi_{\mu}, - \phi , - V_{\mu}, T_{\mu \nu}^{\mp}, \mp \lambda_{\mu \nu}^{\mp}, Y_{\mu}^{\mp}, \mp \xi_{\mu}^{\mp}) ,
\Delta '=\frac{\Delta}{(R^{2n},{\cal Z})_{E}}sdet\Bigl([\kappa^{(l)}\tilde{W_0}D^{(2l+1)}(R^{st}) +\kappa^{(l) \prime}W_0 D^{(2l)}(R^{st})]W^{-1}_0\Bigr).
\Lambda_{M_j}^{r,s}(g_{jk}(\xi_k,t),t)=\sum_{p,q=1}^n\Lambda_{M_k}^{p,q}(\xi_k,t)\frac{\partial g_{jk}^r}{\partial \xi_k^p}\frac{\partial g_{jk}^s}{\partial \xi_k^q}.
F(\vec{q})=\sum_{k=1}^{N}n_{k}[\int d^{3}\vec{v}\rho_{k}(\vec{v})e^{-i\vec{q}\cdot\vec{v}}][\int d^{3}\vec{u}p_{k}(\vec{u})e^{-i\vec{q}\cdot\vec{u}}]e^{-i\vec{q}\cdot\vec{x}_{k0}}
G_{\mu\nu}= \left(\frac{\partial G_{\mu\nu}}{\partial F_{\rho\sigma}}(-F_{\rho\sigma}+B^N_{\rho\sigma})+2\chi\frac{\partial G_{\mu\nu}}{\partial\chi}+2\chi\frac{\partial G_{\mu\nu}}{\partial\phi}\right) + *B^{R}_{\mu\nu}.
g_t =\begin{cases}g_{0}M \setminus B(p_0, 2 t^{1/4}) (1 - \chi_{t}(r) )\phi_{t}^* g + \chi_{t}(r) t h_{t}(1/2) t^{-1/4}< r < 2 t^{-1/4}t h_{t}r < (1/2) t^{-1/4}\end{cases}.
\zeta_2=\frac{2^{2s-2}\mu^{-1}\Gamma(s+\frac{1}{2})\Gamma(-s)}{\pi\Gamma(s)\Gamma(\frac12-s)}\int dk\ \tilde S(k)\tilde S(-k)\left(\frac{k^2}{\mu^2}\right)^{-s-\frac{1}{2}}\,.
\mbox{\;}& (\varphi \Rightarrow \psi)^{\Box}= \Box (\varphi^{\Box}\Rightarrow \psi^{\Box}) & (\exists x \; \varphi)^{\Box}= \exists x \; \varphi^{\Box}& (\forall x \; \varphi)^{\Box}= \Box (\forall x \; \varphi^{\Box})
\Omega=\Omega_S =\Omega_{S,(X,\langle\cdot, \cdot\rangle)}= \{(x_1,x_2) \in X^2: \begin{bmatrix}\langle x_1,x_1 \rangle \langle x_1, x_2\rangle \langle x_1 , x_2 \rangle \langle x_2 , x_2\rangle \end{bmatrix}= S \}.
\delta_{t}(\varepsilon_C)^{k}_{ij}-\mu\Delta_{h}(\varepsilon_C)^{k}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(\varepsilon_C)^{k}_{ij}=\frac{1}{2}\left[g((U_C)^k_{ij})-g((\tilde{U}_C)^k_{ij})\right] +\frac{1}{2}\left[g((U_C)^0_{ij})-g((\tilde{U}_C)^0_{ij})\right] ,(x_i,y_j)\in\Omega_{h},
\nabla \Pi = \sqrt{- \gamma}\; \Pi{}^{\prime}= -{\frac{1}{\sqrt{- \gamma}}}\left[ \left({\frac{L_{2}}{\kappa_{1}}}\right)^{\prime}\; \eta_{2}+{\frac{L_{2}\kappa_{2}}{\kappa_{1}}}\; \eta_{1}-{\frac{L_{2}\kappa_{3}}{\kappa_{1}}}\; \eta_{3}\right] \, .
\frac{1}{\lambda}\tilde{R}_1(u_n+z_nr_n) =\frac{2r_n^{-2}}{\lambda}\int_0^\infty \phi^{(\eta)}(z_n+zr_n^{-1})\psi^{(\eta)}(z_n+zr_n^{-1})dz=\frac{2r_n^{-1}}{\lambda}\int_{z_n}^\infty \phi^{(\eta)}(z)\psi^{(\eta)}(z)dz.
D\ge n - 2\lfloor n/4 \rfloor -1=\begin{cases}n/2-1 \textrm{if $n = 0 \pmod 4$,}(n-1)/2 \textrm{if $n = 1 \pmod 4$,}n/2 \textrm{if $n = 2 \pmod 4$,}(n+1)/2 \textrm{if $n = 3 \pmod 4$.}\end{cases}
p^\kappa(c_p\mathcal T^T_\phi-\mathcal T^T_T)=-a_4(\mathcal T^p)^\kappa,p^\kappa(c_p\mathcal T^\phi_\phi-\mathcal T^\phi_T)=a_4c_p(\mathcal T^p)^\kappa,c_p\mathcal T^i_\phi-\mathcal T^i_T=0, \i\ne\phi,T.
\pi^n(|m^n(x) - m_0| \geq n^{\delta - \gamma}) \leq \pi^n(|m - \tanh \beta(m+h)| \geq (1-\beta) n^{\delta - \gamma}) = \pi^n\left(|m - \tanh \beta(m+h)| \geq \frac{\beta}{n}+ \frac{t_n}{\sqrt{n}}\right) \leq 2 \exp\left(-\frac{t_n^2}{4(1+\beta)}\right)
d_n =\sum\limits_{i=1}^{n}{\left( -1 \right)^i\left(\sum\limits_{\begin{array}{c}0 <k_1,k_2, \ldots ,k_i \leq n k_1+k_2+\ldots+k_i=n\end{array}}\frac{1}{\left[k_1\right]!\left[k_2\right]!\ldots \left[k_i\right]!}\right)}
\left|{\frac{d x}{d \lambda}}\right| = H_{p}^{-{\frac{2 ( p + 1 )}{( D - 3 ) \Delta_{p}}}}\sqrt{E^{2}H_{p}^{\frac{4 ( D - p - 4 )}{( D - 3 ) \Delta_{p}}}-{\frac{{\cal J}^{2}}{x^{2}H_{p}^{\frac{4 ( p + 1 )}{( D - 3 ) \Delta_{p}}}}}- \epsilon},
\vartheta_K^{(N)}=\min_{u\in \mathbb{R}}\left\{u+\frac{1}{\alpha}\left[ \frac{1}{N}\sum_{i=1}^N \int (\max (0,x-u))^q\frac{1}{h}K(\frac{x-X_i}{h})dx\right]^{1/q}\right\}\vartheta_w^{(N)}=\min_{u\in \mathbb{R}}\left\{u+\frac{1}{\alpha}\left[ \int (\max (0,x-u))^q\tilde{d}_{N,j}(x)dx\right]^{1/q}\right\}.
p_*(\psi^2) = \kappa_1 = 12 \lambda -\delta_0 - \delta_1, p_*(\psi\lambda) = 4\lambda, p_*(\psi\delta_0) = 4\delta_0, p_*(\kappa_2) = p_*(p^*(\kappa_2)+\psi^2) = \kappa_1, p_*(\psi\delta_{1,1}) = \delta_1, p_*(\psi\delta_{2,1}) = 3\delta_1, p_*(\delta_{1,1}^2) = -\delta_1, p_*(\delta_{1,1}\delta_{2,1}) = \delta_1, p_*(\delta_{2,1}^2) = -\delta_1.
\psi_Y (w):=\begin{cases}\displaystyle -\frac{1}{L}(Y=H) \displaystyle \frac{v_{n+1}w^{n+1}}{(-H)^{n+2}}(Y=V) \displaystyle \frac{a_n w^{n}\sqrt{1-U(w)}}{(-H)^{n+1}}(Y=A) \end{cases}.
\lim_{t \to 0}\; (\mathbf{u}, \mathbf{H})^{T}(t, \cdot) =\big(\mathbf{0}, \mathbf{F}(\mathbf{0})\big)^{T}L^{2}(G, \mathbb{R}^{k}) \times L^{2}\big(G, \mathbb{R}^{(k \times d) \times (k \times d)}\big).
R_{n \beta}( \theta_{1}, \ldots , \theta_{n}) ={\frac{N !}{( N - n ) !}}\int_{0}^{2 \pi}\cdots \int_{0}^{2 \pi}P_{N \beta}( \theta_{1}, \ldots , \theta_{N}) \, d \theta_{n + 1}\cdots d \theta_{N}\, .
\Gamma^{Ai}_1=\left\{\zeta\in\mathbb{C}\vert \textnormal{arg}\zeta=2\pi/3\right\}, \Gamma^{Ai}_2=\left\{\zeta\in\mathbb{C}\vert \textnormal{arg}\zeta=\pi\right\}, \Gamma^{Ai}_3=\left\{\zeta\in\mathbb{C}\vert \textnormal{arg}\zeta=-2\pi/3\right\}, \Gamma^{Ai}_4=\left\{\zeta\in\mathbb{C}\vert \textnormal{arg}\zeta=0\right\},
\left({\frac{\dot{a}_{R}}{a_{R}}}\right)^{2}+ 2 m_{0}\dot{R}{\frac{\dot{a}_{R}}{a_{R}}}+ m_{0}^{2}\dot{R}^{2}+{\frac{k c^{2}\Omega_{R}^{2}}{a_{R}^{2}}}={\frac{8 \pi m_{0}G_{5}}{3 ( 1 - \Omega_{R}^{2})}}\left[{\frac{\tilde{\varrho}_{1}\Omega_{R}+ \varrho_{2}\Omega_{R}^{5}}{\sqrt{\Omega_{R}^{2}- \dot{R}^{2}/ c^{2}}}}+ V_{r}/ c^{2}\right] .
\sigma_{e}=\frac{1}{\rho_{e}}=\alpha_{ee},k_{e}=\frac{\alpha_{tt}-\alpha_{te}\alpha_{ee}^{-1}\alpha_{et}}{T^{2}},and\alpha_{S}=\frac{\alpha_{et}\alpha_{ee}^{-1}}{T^{2}}(\alpha_{S}=\alpha_{P}T).
\left \{\frac{d^{2}}{dx^{2}}+\frac{1}{2}\left( \frac{1}{x-e_{1}}+\frac{1}{x-e_{2}}+\frac{1}{x-e_{3}}\right) \frac{d}{dx}-\frac{1}{4\prod_{j=1}^{3}(x-e_{j})}\right. & \; \left. \cdot \left( \tilde{C}+l_{0}(l_{0}+1)x+\sum_{i=1}^{3}l_{i}(l_{i}+1)\frac{(e_{i}-e_{i^{\prime}})(e_{i}-e_{i^{\prime \prime}})}{x-e_{i}}\right) \right \}\tilde{f}(x)=0,
b_{2}= - \zeta \cdot \frac{4 y_{2}( y_{1}y_{2}+ ( y_{1}^{2}+ y_{2}^{2}) \operatorname{a r c t a n}{y_{2}/ y_{1}})}{y_{1}( y_{1}^{2}+ y_{2}^{2})}+ \zeta \cdot f ( y_{1}) y_{2},
\biggl(\prod_{i=1}^{N}\frac{\theta_i^{m(e^\mathsf{T}A)_i}}{(\theta_i; \zeta_i)_{e_i}^m}\biggr) \cdot \tau(a_\zeta(\theta)^m) = \biggl(\sum_{k \in (\mathbb{Z}/m\mathbb{Z})^N}\zeta^{\overline{Q(k)}}\prod_{i=1}^{N}\frac{\theta_i^{((k+e)^\mathsf{T}A)_i}}{(\theta_i ; \zeta_i)_{k_i + e_i + 1}}\biggr)^m = a_\zeta(\theta)^m.
\phi_{\pi^{(p^k+1)/2}\alpha,-\pi\beta}(x_2)=(\pi^{\frac{p^k+1}{2}}\alpha)^{p^k}x_2^{p^{2k}}-2(\pi\beta)^{p^k}x_2^{p^k}+\pi^{\frac{p^k+1}{2}}\alpha x_2=0.
\int_T^{2T}\left|R_Y(\sigma+it) \right|^{2k}dt \leq 9^k \bigg(\int_T^{2T}\bigg|\sum_{p \leq k \log k}\frac{1}{p^{\sigma+it}}\bigg|^{2k}dt+\int_T^{2T}\bigg|\sum_{k \log k \leq p \leq Y}\frac{1}{p^{\sigma+it}}\bigg|^{2k}dt & +O(T \log^{2k}\zeta(2\sigma))\bigg).
\mathcal{F}_{F}^{\alpha}(S_{F}^{\alpha}(s))=\mathcal{L}_{F}^{\alpha}[f(x)]= \int_{S_{F}^{\alpha}(0)}^{S_{F}^{\alpha}(\infty)}f(x)e^{-S_{F}^{\alpha}(s)S_{F}^{\alpha}(x)}d_{F}^{\alpha}x,
-\frac{\widetilde f\,^\prime(s\vert t)}{\widetilde f(s\vert t)}= t\, \psi\,^{\prime}(s) = t \, \widetilde \rho(s) \; -\widetilde f\,^\prime(s\vert t) = t \, \widetilde \rho(s)\times \widetilde f(s\vert t) \ \implies x f(x\vert t) = t \, \{\rho\star f(\cdot\vert t)\}(x)
P = \frac{1}{2 \alpha^{\prime}\dot{t}}\frac{\dot{R}^{2}- R^{2}}{( 1 + \frac{K}{4}R^{2})^{2}}= \frac{1}{2 \sqrt{b}\alpha^{2}}[ \frac{\dot{r}^{2}}{1 - K r^{2}}- r^{2}] = \frac{b \alpha^{2}- 2 r^{2}}{2 \sqrt{b}\alpha^{2}}.
\begin{cases}du(t, x) =\Delta u(t, x)dt + \displaystyle\int^0_{-r}\alpha(\theta)\Delta u(t+\theta, x)d\theta dt + \Sigma(u(t-r, x)) dW(t, x), \hskip 250pt\,\,\,\,(t, x)\in [0, \infty)\times{\cal O}, u(0)=\phi_0,\,\,\,\,u(\theta)=\phi_1(\theta),\,\,\,\theta\in [-r, 0],\,\,\,r>0,\end{cases}
K_{w_6}^{++}(y; \mu) = \frac{1}{12\pi^2}\frac{\cos\left(\frac{3}{2}\pi \nu_1\right) \cos\left(\frac{3}{2}\pi \nu_2\right)}{\cos\left(\frac{3}{2}\pi \nu_3\right)}\mathcal{J}_{5}(y;\mu);
{{\tilde{\psi}}}_{\alpha,s,\boldsymbol{\xi}_s,\ell_s}={{\tilde{\psi}}}_{\alpha,s}\bigcirc_{w=1}^{\ell_s}(\theta_{X_s}+\xi_{s,w}) \ \ 0\le s \le r-1\enspace, &\Psi_{\alpha,\boldsymbol{\Xi}_{\boldsymbol{\ell}}}=\bigcirc_{s=0}^{r-1}{{\tilde{\psi}}}_{\alpha,s,\boldsymbol{\xi}_s,\ell_s}\enspace,
\sum_{n=0}^\infty \sum_{m=1}^{n}(-1)^{n}z^{n-m+1}q^{\frac{1}{2}(2n^2-m^2) + \frac{1}{2}(2n+m)}&\qquad=\sum_{n=0}^\infty \sum_{m=1}^{n}(-1)^{n}z^{n-(m-1)}q^{\frac{1}{2}(2n^2-(m-1)^2) + \frac{1}{2}(2n-(m-1))}&\qquad=\sum_{n=0}^\infty \sum_{m=0}^{n-1}(-1)^{n}z^{n-m}q^{\frac{1}{2}(2n^2-m^2) + \frac{1}{2}(2n-m))}.
F_N(1,\xi_n)= e^{-i\xi_n}\begin{pmatrix}1 \alpha_N(0,\xi_n)\end{pmatrix}\exp\Big(i \int_0^1\varphi_1(t)\alpha_N(t,\xi_n),dt\Big)+\frac{R_N(1,\xi_n)}{(2i\xi_n)^N}
\bar m_{n,j}(\theta) \equiv \textstyle{n^{-1}\sum_{i=1}^n m_j(X_i,\theta)},~~j=1,\dots, J_1+J_2 \hat{\sigma}_{n,j}\equiv \textstyle{(n^{-1}\sum_{i=1}^n [m_j(X_i,\theta)]^2-[\bar m_{n,j}(\theta)]^2)^{1/2}},~~j=1,\dots, J_1+J_2
\rho_{\beta *}(E_{\theta})\cdot (\rho_{\beta}(g)\cdot v_{\beta}) ={\frac{d}{dt}}_{|t=0}\left(\rho_{\beta}(gg^{-1}\exp(t\,E_{\theta})g)\cdot v_{\beta}\right) =\rho_{\beta}(g)\cdot\rho_{\beta *}(\mathrm{Ad}(g^{-1})\cdot E_{\theta})\cdot v_{\beta}=\rho_{\beta}(g)\cdot 0=0\,.
\int_0^{1/2}u\varphi \,dx\ge\Phi\left(\frac{1}{2}\right)\left(\frac{1}{2}u\left(\frac{1}{2}\right)+\int_0^{1/2}u(x)\,dx\right)\ge\Phi\left(\frac{1}{2}\right)\int_0^1u(x)\,dx=0.
\sum_{j,k,l=1}^3 c_j^{kl}(A)\,\partial_{jkl}^3 u = -\frac{1}{128\pi}\partial_{111}^3 u -\frac{1}{128\pi}\partial_{122}^3 u +\frac{1}{64\pi}\partial_{133}^3 u \equiv -\frac{15}{32\pi}.
||-\frac{(\cos(2(v_{1}+v_{2}+v_{3}+v_{4}))-1)}{r^{3}}\left(\sin(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\sin(2Q_{\frac{1}{\lambda(t)}})\right)\vert_{r=R\lambda(t)}||_{L^{2}(R dR)}&\leq C ||\frac{v_{1}+v_{2}+v_{3}+v_{4}}{r}||_{L^{\infty}}^{2}||\frac{v_{5}}{r}\vert_{r=R\lambda(t)}||_{L^{2}(R dR)}&\leq \frac{C \log^{6+b}(t)}{t^{5}}
\Pi^{n}(f_{1},f_{2})=\sum_{k=0}^{n}(-1)^{k}(\begin{matrix}n\\k\end{matrix})(\frac{\partial^{k}}{\partial p^{k}}\frac{\partial^{n-k}}{\partial q^{n-k}}f_{1})\times(\frac{\partial^{n-k}}{\partial p^{n-k}}\frac{\partial^{k}}{\partial q^{k}}f_{2})
L_1^2.\,c_1(X_1) = \{(\mu^*L)^2 - 2\mu^*L.\,E +E^2\}.(\mu^*c_1(X) - E) &=L^2.\,c_1(X) - 2L.C + E^2.\,\mu^*c_1(X) - E^3\quad({}\eqref{zero1}, \eqref{int3}).
V \ \approx \ 2 | a_{D}|^{2}\left( | a_{M}|^{2}+ | a_{\tilde{M}}|^{2}\right) \ - \ \pi^{2}\{\frac{\left( | a_{M}|^{2}- | a_{\tilde{M}}|^{2}\right)^{2}\ + \ 1 6 | a_{M}a_{\tilde{M}}|^{2}}{2 \ln{| a_{D}/ 1 6 \Lambda |}}}\ .
|\partial_{t}^{n}D_{x}^{m}q(t,x)| \leq N t^{-n+\alpha-1}\sum_{k=1}^{n}\left|D_{x}^{m}\left(|x|^{-d}\textnormal{H}_{k,0}^{q}(R)\right)\right| &\leq N|x|^{-d-m}t^{-n+\alpha-1}\sum_{k=0}^{n}\sum_{l=1}^{m}\left|d\textnormal{H}_{k,l-1}^{q}(R)+2\textnormal{H}_{k,l}^{q}(R)\right|.
i_{j+1}=\begin{cases}i_j+d^G(z^{j-1}_{i_{j-1}}, z^j_{i_j}) \textnormal{if}i_j\leq r, i_j-d^G(z^{j-1}_{i_{j-1}}, z^j_{i_j}) \textnormal{if}i_j> r.\end{cases}
g_2(z) = \log z^{\frac{1}{2}}+ \int_0^q \log\left(1+ \frac{\sqrt{t}}{z^{\frac{1}{2}}}\right) d\mu^*(t) & = \frac{1}{2}\log z +\int_0^q \left(\frac{\sqrt{t}}{z^{\frac{1}{2}}}- \frac{t}{2 z}+ \mathcal{O}\left(z^{-\frac{3}{2}}\right)\right) d\mu^*(t),
{\hat F}[P^3]~=~{\alpha}^{-1}~~\sum_{m,\ell,R}{(-)^{6 \epsilon_{\ell}- 3\epsilon_R}{S_{0 \Lambda_{\ell}}~S_{0 \Lambda_R}~S_{\Lambda_{\ell}\Lambda_m}~ S_{\Lambda_{\ell}\Lambda_m}~~S^*_{\Lambda_R \Lambda_m}~q^{-5C_{\ell}+{3 \over 2}C_R}}\over{S_{00}~S_{0 \Lambda_m}}}~.
K_p(q,\bar{q})W^{2g}_{2a+2}(q,\bar{q},K) &=K_p(q,\bar{q})K_q(q_1,\bar{q}_1)\left(W^{g}_{a+2}(q_1,\bar{q},L)W^{g}_{a+1}(\bar{q}_1,M)\right. &\left. +W^{g}_{a+1}(q_1,L)W^{g}_{a+2}(\bar{q}_1,\bar{q},M)+\dots\right)
P = P_0(l) \left( \frac{2}{\lambda}\right)^{2l+3}\frac{(l+1)!^2\Gamma^4(1+l/4)}{\pi}= P_0(l) \left( \frac{2\pi T}{\omega}\right)^{2l+3}\frac{(l+1)!^2\Gamma^4 (1+l/4)}{\pi}\,,
M_0 = C(\frac{1}{h_{\min}}, \frac{1}{h_{b, \min}}, |\zeta|_{H^{t_0}},|b|_{H^{t_0}}) M(s) = C(M_0, |\zeta|_{H^{\max\{t_0 +2, s\}}},|b|_{H^{\max\{t_0 +2, s\}}}) N(s) = C(M(s),|\nabla_X \psi |_{H^{s}}).
\int_\Omega|\mathbf{w}_x|^2\omega^2 dx+\mu \iint_{Q_t}|\mathbf{w}_{xx}|^2\omega^2 dxds\leq C+C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}, \iint_{Q_t}\big(|\mathbf{w}_t|^2+u^2|\mathbf{w}_x|^2\big)dxdt\leq C+C \iint_{Q_t}u_{xx}^2 dxds.
\tau_2 := Dv (Dv^T Dv)^{-1}\nabla \Phi_2\, , \nu_2 := \frac{\partial_{x_1}v \times \partial_{x_2}v}{|\partial_{x_1}v\times \partial_{x_2}v|}\,, t_2 := \frac{\tau_2}{|\tau_2|^2}\, , n_2 := \frac{\nu_2}{|\tau_2|}\, .
cN^2(\xi_1^2+t\xi_2^2) \le 2c(N\xi_1-M\xi_2)^2+2cM^2\xi_2^2+cN^2t\xi_2^2 &=2c(N\xi_1-M\xi_2)^2+\big(2cM^2+cN^2t\big)\xi_2^2 &\le (N\xi_1-M\xi_2)^2+KB^2\xi_2^2.
\left ({{\Delta X^{\mu}}\over{d \tau}}\right ) \left ({{\partial}\over{\partial \xi^{c}}}|{{\Delta X^{\mu}}\over{\partial \xi^{c}}}|^{-1}\left [{{\Delta X^{\nu}}\over{d \tau}}\right ] \right ) +\left ({{\Delta X^{\mu}}\over{d \tau}}\right )\Gamma^{\nu}_{\mu\lambda}\left ({{\Delta X^{\lambda}}\over{d \tau}}\right )= 0 \quad .
\mathcal P(x_k = a_m\vert{\bf y}) = \frac{p({\bf y}|x_k = a_m)P(x_k = a_m)}{\sum_{m=1}^{M}p({\bf y}|x_k = a_m)P(x_k = a_m)}\approx \frac{\varphi_m(x_k)}{\sum_{m=1}^{M}\varphi_m(x_k)}.
\begin{aligned}G-c_3 ={}\{v_1:[v_2, v_4, v_2, v_5], v_2:[v_1, v_3, v_1, v_4], v_3:[v_2, v_5, v_4, v_5], v_4:[v_1, v_3, v_5, v_2], v_5:[v_1, v_3, v_4, v_3]\}. \end{aligned}
i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^{2}}{2m}[2(1-q)l_{0}^{2}(1+(1-q)x)\frac{\partial\psi}{\partial x}+(1+(1-q)x)^{2}l_{0}^{2}\frac{\partial^{2}\psi}{\partial x^{2}}]+V\psi.
\sum_{k = 1}^{\infty}T_{2 n , 2 k - 1}\frac{v_{2 k - 1}^{( \kappa )}}{\sqrt{2 k - 1}}= - \frac{v_{2 n}^{( \kappa )}}{\sqrt{2 n}}\tanh \left( \frac{\kappa \pi}{4}\right) \, ,
\frac{d}{dt}\|\Lambda^{m}u\|_{2}^{2}+\|\Lambda^{m+\alpha}u\|_{2}^{2}\leq C(t+1)^{-\frac{3}{4\alpha-2}}\|\Lambda^{m}u\|_{2}^{2}\leq C(t+1)^{-1}\|\Lambda^{m}u\|_{2}^{2}
m''_3(\xi,\eta,\rho) = O \Big( \frac{|\xi-\rho|^{3/2}+ |\rho-\eta|^{3/2}}{(|\xi|+|\eta|)^{1/2}}\mathbf{1}_{[2^2,\infty)}(|\xi|/|\xi-\rho|) \mathbf{1}_{[2^2,\infty)}(|\eta|/|\rho-\eta|) \Big) .
\sum_{(p,q,\Upsilon) \in I_j \times Y_j}\frac{\dim \Upsilon'_{p'q'j}}{\dim \Upsilon_{pqj}}\, \Bar{\delta}_{pp'qq'j}^{\Upsilon\Upsilon'}= \sum_{(p,q,\Upsilon) \in I_j \times Y_j}\frac{\dim \Upsilon'_{p'q'j}}{\dim \Upsilon_{pqj}}\, \delta_{pp'qq'j}^{\Upsilon\Upsilon'}= 1.
S = \int d^{2}x \mathrm{t r}\left( - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}D_{\mu}\phi D^{\mu}\phi + i \bar{\Psi}\gamma^{\mu}D_{\mu}\Psi - 2 i g \phi \bar{\Psi}\gamma_{5}\Psi \right) ,
T(\phi_{1}(f)\delta_{1}-\phi_{1}(g)f) =T(f\ast g-\phi_{1}(g)f) &=f\cdot T(g-\phi_{1}(g)e^{\sharp}) &=f\ast T(g-\phi_{1}(g)e^{\sharp}) &=\phi_{1}(f)T(\delta_{1}-\phi_{1}(\delta_{1})e^{\sharp}) &=\phi_{1}(f)T(\delta_{1}-e^{\sharp}).
U^{{\rm gh}\dagger}_2 = \left( \prod_{\scriptstyle i=1\atop \scriptstyle i\neq F}^{N_2}{}_i\langle 0|\right) \int \prod_{r=-1}^1d\beta_r\int \prod_{\scriptstyle i=1\atop \scriptstyle i\neq d,g,h}^{N_2}d{\cal M}^i \exp{\phi_2}\exp \left( \sum_{n=2}^\infty A_n^2b_{-n}^F\right) \exp \left( \sum_{n=-1}^\infty c_{-n}^FB_n^2\right) |0\rangle_F
Z_{k\ell}[a]:=\left(\begin{array}{ccccc}a_1 a_2 \ldots a_{\ell-1}-a_1 - a_2 - \ldots - a_{\ell-1}\ldots \ldots \ldots \ldots \ldots a_1 a_2 \ldots a_{\ell-1}-a_1 - a_2 - \ldots - a_{\ell-1}\end{array}\right),{\rm if\}\ell\geq 2,
0 \leq \frac{1}{\log(2+n)}\sum_{k=0}^n \mu(k,A) + \frac{1}{\log(2+n)}\sum_{k=0}^n \mu(k,B) - \frac{1}{\log(2+n)}\sum_{k=0}^n \mu(k,A+B) \leq O(\frac{1}{\log(2+n)}), n\to\infty.
U_0^{(\kappa)}(u)=\emptyset \mbox{for $\kappa>0$}Q_{0,i}^0(u)=\{X_i>u\}\mbox{and}Q_{0,i}^{\kappa}(u)=\emptyset \mbox{for $\kappa>0$}&U_{0,i}^0(u,x)=\{X_i>u,m_u(\{X_i\})>x\}\mbox{and}U_{0,i}^{\kappa}(u,x)=\emptyset \mbox{for $\kappa>0$}&R_{0,i}(u,x)=U_{0,i}(u,x)
\langle{\bf p}\, \vert{\bf 0},0 \rangle =\langle{\bf p}\, \vert U(B(p) ) \vert{\bf 0},0 \rangle = \sqrt{{m \over \omega_m ({\bf p}\,)}}\langle{\bf p}={\bf 0}\vert{\bf x}={\bf 0},t=0 \rangle =N \sqrt{{m \over \omega_m ({\bf p}\,)}}
9\sum^{T-1}_{t=0}\left(F_t+\frac{D_y^2}{2\lambda_t}\right) =\sum^{T-1}_{t=0}\frac{1}{(t+2)^2}+9\sum^{T-1}_{t=0}\frac{D_y^2}{2(t+2)}\leq\frac{\pi^2}{6}+\ln(T+2)\frac{9D_y^2}{2}.
\hat{\mu}= \frac{1}{n}\sum_{i=1}^n X_i, \hat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n (X_i-\hat{\mu})^2, \bar{Z}= \frac{1}{n}\sum_{i=1}^n{Z_i}, Y_i = \hat{\mu}+ \hat{\sigma}(Z_i-\bar{Z}).
\frac{|z|^{2 (n - 1)+1}}{1-|z|^{n-1}}=\frac{|z|^{2n - 1}}{1-|z|^{n-1}}~<~ \left| G_{n}(z) \right|\mbox{for}~z \in C_{j,n},j = 1, 2, \ldots, J_n
I \left( y \right) = \int_{0}^{\infty}\frac{Q d Q}{Q^{2}+ \left( 2 \pi T y \right)^{2}}\frac 1{\exp \left( Q / T \right) - 1}= \frac 1 2 \Theta [ \mathrm{R e}( y ) ] \left( \ln y - \frac 1{2 y}- \psi \left( y \right) \right) + ( y \leftrightarrow - y ) ,
\sum_{\alpha = 0}^{N}Y_{\alpha}^{N}( \frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}) Y_{\alpha}^{N}( \frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}) = \sum_{\alpha = 0}^{N}Y_{\alpha}^{N}({\eta}) Y_{\alpha}^{N}({\eta}^{\prime})
\left( \begin{array}{c c}{\hat{\alpha}( p )}\\{\hat{\beta}( p )}\\ \end{array}\right) = \left( \begin{array}{c c}{a ( p )}\\{1}\\ \end{array}\right) \hat{\xi}( p ) + \left( \begin{array}{c c}{b ( p )}\\{1}\\ \end{array}\right) \hat{\eta}( p )
\int_0^\infty\sin(2\pi nt)e^{-2^k\pi(m+1)t}dt =\dfrac{1}{2i}\left(\int_0^\infty e^{2\pi i nt}e^{-2^k\pi(m+1)t}dt-\int_0^\infty e^{-2\pi i nt}e^{-2^k\pi(m+1)t}dt\right) &=-\dfrac{1}{2i}\left(\dfrac{1}{2\pi(i n-2^{k-1}(m+1))}+\dfrac{1}{2\pi(i n+2^{k-1}(m+1))}\right) &=\dfrac{1}{2\pi}\left(\dfrac{n}{n^2+2^k(m+1)^2}\right).
\left. \begin{array}{c}{\theta_{\beta \bar{\gamma}}= i u_{\beta \bar{\gamma}}^{\bar{\alpha}}}{\theta_{\alpha \delta}= i u_{\alpha \delta}^{\gamma}}\end{array}\right\}\Leftrightarrow \left\{\begin{array}{c}{\theta_{\alpha \beta}= i u_{\alpha \beta}^{\gamma}}{\theta_{\beta}= \theta_{\delta}}\end{array}\right.
\mathcal A_{p\mu}\;(\delta p,\delta\mu,\delta\b u):= \mathcal A_{p\mu\rho}(\delta p,\delta\mu,0,\delta \b u)= \begin{pmatrix}\mathcal L_{p\mu}\;(\delta p,\delta\mu,\delta\b u) \mathcal B \;(\delta\b u)|_{\partial\Omega}\end{pmatrix}=\begin{pmatrix}\mathcal S 0\end{pmatrix}
\int_{-\infty}^\infty dv\thinspace\left[{\partial^2\over \partial v^2}\thinspace\psi_{s,F}(-v)\right]\psi_{s+p,G}(v)=\int_{-\infty}^\infty dv\thinspace\psi_{s,F}(-v){\partial^2\over \partial v^2}\thinspace\psi_{s+p,G}(v),
\phi_{0}^{( - 1 )}={\frac{1}{{J_{H}^{- 1}}^{\prime}}}, \qquad \; \phi_{1}^{( - 1 )}={\frac{J_{H}^{- 1}}{{J_{H}^{- 1}}^{\prime}}}, \qquad \; \phi_{2}^{( - 1 )}={\frac{\left( J_{H}^{- 1}\right)^{2}}{{J_{H}^{- 1}}^{\prime}}}.
\tilde{\Theta}_-B\varphi =\tilde{\Theta}_-S(1-\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=S\tilde{\Theta}_-(1-\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=\frac{1}{2}S\tilde{\Theta}_-(-D_a-UD_aU)\varphi &=-\frac{1}{2}S(-D_a-UD_aU)\tilde{\Theta}_-\varphi=-B\tilde{\Theta}_-\varphi,
{\cal D}D_{n,\mp 1/2}^{(j)}D_{.,\pm 1/2}^{(1/2)}=-[D^{(j)}_{n,n'}(J^{(j)}_\alpha)_{n',\mp 1/2}]D^{(1/2)}_{.,\mu'}[\frac{1}{2}\sigma_3,[\frac{1}{2}\sigma_3,\sigma_\alpha]]_{\mu',\pm 1/2}
E= -E_n \, , \; n \geq 1 \; \; \psi^{(-)}_{n,p}= \sqrt{\frac{E_n-m}{2 E_n}}\frac{e^{i p x_2}}{\sqrt{2 \pi}}e^{-\frac{1}{2}{\zeta}^2}\left( \begin{array}{c}N_n H_n(\zeta) - \frac{\sqrt{E_n^2-m^2}}{E_n-m}N_{n-1}H_{n-1}(\zeta)\end{array}\right)
s := \begin{pmatrix}0 0 0 1 0 0 0 1 0 \end{pmatrix}, a := \begin{pmatrix}1 0 0 0 \tfrac12 \sqrt2 0 0 0 \tfrac12 \sqrt2 \end{pmatrix}, a_+ := \begin{pmatrix}\tfrac12 \sqrt2 0 0 0 \tfrac12 \sqrt2 0 0 0 1\end{pmatrix}.
\phi_1^+(y)=\mathbf{1}_{[\alpha_1,+\infty)}(y),\quad\phi_n^-(y)=\mathbf{1}_{(-\infty,\alpha_n]}(y), &\phi_i^-(y)=\psi\bigg(\frac{3(y-\alpha_i)}{\alpha_i-\alpha_{i+1}}\bigg)\mathbf{1}_{[\alpha_{i+1},\alpha_i]}(y),\forall i=1,\ldots,n-1, &\phi_i^+(y)=[1-\phi_{i-1}^-(y)]\mathbf{1}_{[\alpha_i,\alpha_{i-1}]}(y),\forall i=2,\ldots,n.
\phi_{i,k}^\gamma(z_{i+1}, \ldots, z_{k-1}) \,\stackrel{{\rm def}}{=}\, \phi_{\alpha_i}^{(\gamma_{i+1}, \ldots, \gamma_n)}(z_{i+1}, \ldots, z_{k-1}), \psi_{i,k}^\gamma(z_{i+1}, \ldots, z_{k-1}) \,\stackrel{{\rm def}}{\,=\,}\, \psi_{\alpha_i}^{(\gamma_{i+1}, \ldots, \gamma_n)}(z_{i+1}, \ldots, z_{k-1})
\mathrm{tr}(\pi'^{S'})(b_{E/F'}\Sigma^{S_0}_{\phi}(X))=\mathrm{Res}_{s=1}\left(\widetilde{\phi}(s)X^sL(s,\Pi \times \Pi^{\tau\vee S_0})\right)+O_{E,F',n,S_0,\varepsilon}(C_{\Pi \times \Pi^{\tau \vee}}(0)^{\tfrac{1}{2}+\varepsilon})