harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B_3 = 2[8b_2(b_1-b_2) + 3b_3(4a-4b_1-2l-d-5)], B_4 = 8[2b_2^2 + 3b_3(b_1-2b_2)], B_5 = 12b_3(4b_2-3b_3), B_6 = 36b_3^2,
\|1_{|k|\neq 1}(Z, Q)(t)\| \leq C_{\alpha}\exp(10) \sqrt{\frac{t}{\xi}}\|(Z, Q)(2\xi)\|, \|1_{|k|=1}(Z, Q)(t)\| \leq C_{\alpha, \gamma}\left(\frac{t}{\xi}\right)^{\gamma}\exp(10) \sqrt{\frac{t}{\xi}}\|(Z, Q)(2\xi)\|,
\mathbb{E}_{f_0,T}\| \tilde{f}(j)-\mathbb{E}_{f_0,T}\tilde{f}(j)\|_2^2 \lesssim\sum_{\ell\leq j-1}\sum_{k=0}^{2^{\ell}-1}\big( \mathbb{E}_{f_0,T}Z_{\ell k}^2+ \mathbb{E}_{f_0,T}W_{\ell k}^2\big) &\lesssim \sum_{\ell\leq j-1}\sum_{k=0}^{2^{\ell}-1}n_\ell^{-1}\leq 2^j/n_j.
\left[g_{ij}\right]=\left[\begin{array}{cc}R^2 0 0 (L+R\cos\theta)^2 \end{array}\right],\left[g^{ij}\right]=\left[\begin{array}{cc}\frac{1}{R^2}0 0 \frac{1}{(L+R\cos\theta)^2}\end{array}\right],
(j^{-1}\circ \exp_r(a))(rec(u)) = -(\pi_L^{{-r}}- 1)ar \Psi_{CW}^r(u) = \tfrac{1}{(r-1)!}(1 - \pi_L^{{-r}})a \partial_{\mathrm{inv}}^r \log g_{u,t_0}(Z)_{|Z=0}t_L^r \ .
\begin{aligned}\Psi^{t_k}_{t_{k-1}, 1}=\int_{t_{k-1}}^{t_k}T_{q}(t_k-s)BB^{\ast}T_{q}^{\ast}(t_k-s)ds,k=1,\ldots,m+1, \Psi^{t_k}_{t_{k-1}, 2}=S_{q}(t_k-t_{k-1})DD^{\ast}S_{q}^{\ast}(t_k-t_{k-1}),k=2,\ldots,m+1,\end{aligned}
s=\min\left\{r \ : \begin{array}{l}p_{k_1},\ldots,p_{k_r}\in\{p_1,\ldots,p_m \}p_{k_i}\neq p_{k_j}, i\neq j, \frac{1}{2}\leq \frac{1}{p_{k_1}}+\cdots+\frac{1}{p_{k_r}}<1\end{array}\right\}
\int\frac{|\widehat{\phi\phi_z}(t,\xi)|^2}{|\xi|^{2\epsilon'}}\mathrm{d}\xi\lesssim \|D^{-\epsilon'}(\phi\phi_z)\|_2^2\mathop{\,\lesssim_{\epsilon'}\,}_{\mathclap{\substack{}}}\|\phi\phi_z\|_{2-\epsilon''}^2\leq \|\phi\|_{4-2\epsilon''}^4\epsilon''=\frac{4\epsilon'}{3+2\epsilon'}
e_\lambda(x, y)~ =~ \begin{cases}\displaystyle\frac{2}{(2 \pi)^{\frac{d+1}{2}}}\left( \frac{-1}{\sinh(r)}\partial_r \right)^{\frac{d-1}{2}}\frac{\sin(r \cdot \sqrt{\lambda-b_d})}{r}\lambda \ge b_d, begin{align*}3ex] 0 . \end{cases}
\aligned\nu(d_M)= (m_{11},\ldots,m_{n1},m_{12},\ldots,m_{n2},\ldots,m_{1n},\ldots,m_{nn}), \nu(d_M^{-1})= (m_{11},\ldots,m_{1n},m_{21},\ldots,m_{2n},\ldots,m_{n1},\ldots,m_{nn}).\endaligned
\int_{\Sigma_{1,M,B}}\left(\frac12 \left|(\partial_t - \partial_r) u\right|^2 + \frac12 \left|\frac{\partial_\omega u}{r}\right|^2 - \frac{d-2}{2d}|u|^{\frac{2d}{d-2}}\right) \mathrm{d}t\mathrm{d}S = \varepsilon(M).
W = - \, \frac{1}{1 6}\, n \: \Gamma \left( \frac{7 - p}{2}\right) \, \left[ 2 \sum_{i}\omega_{i}^{4}- \left( \sum_{i}\omega_{i}^{2}\right)^{2}\right] \prod_{i}\omega_{i}^{- 1}\left( \frac{2}{b}\right)^{7 - p}+ O \left( \frac{1}{b^{9 - p}}\right) .
\begin{aligned}I_{\alpha}^+ \le 2 \|\phi\|_X \frac{\Big(\frac{|x|^2}{t}\Big)^{\nu/2}\exp\Big( \frac{-|x|^2}{32t}\Big)}{|x|^{\nu}}\le \frac{2D_1\|\phi\|_X}{w(x)}\end{aligned}
i=1: -2(H_{q_1}-E_{9,13})-2(H_{p_1}-E_{1,7})-E_{1-2}-E_{7-8}-2E_{9-10}-2E_{10-11}&-E_{11-12}-E_{13-14}=-2H_{q_1}-2H_{p_1}+ E_{1,2,7,8,11,12,13,14}(i=1), i=2: q_1\leftrightarrow q_2, p_1\leftrightarrow p_2, E_j \leftrightarrow E_{j+8}\ (j=1,\dots,8) \mbox{in the above}.
\mathcal{I}_{+}^{(m,\mathbf{k})}=\int_{\frac{\mu^2_{}\Gamma_0}{\rho}}^{\infty}\exp\Big(-\frac{\Big({\sqrt{\frac{\rho x}{\Gamma_0}}-\mu_{\textrm{D}}}\Big)^2}{2(\sigma^{(m,\mathbf{k})}_{})^2}\Big) f_{\Gamma_{}^{(m)}}(x)dx.
I(F_,F_;\widetilde{X}^n|A^n,X^n,Z^n) &\; \!=\! I(F_;\widetilde{X}^n|A^n,X^n,Z^n)\!+\!I(F_;\widetilde{X}^n|A^n,X^n,Z^n,F_)\leq 2\epsilon_n
|(2\alpha-1)f_k(\alpha) \alpha^{n-1}-(2\beta-1)f_{\ell}(\beta)\beta^{m-1}| & =|((2\alpha-1)f_k(\alpha)\alpha^{n-1}-L_n^{(k)})+(L_m^{(\ell)}-(2\beta-1)f_{\ell}(\beta)\beta^{m-1})|< 3.
\frac{1}{d_{N}^{2}}\frac{\partial^{2}}{\partial z_{1}\partial z_{2}}\left[s(z_{1})s(z_{2})N^{-2}\sum_{k=1}^{N}\sum_{i<k}\mathbb{E}_{k}(G^{(1)}_{k})_{ii}\mathbb{E}_{k}(G^{(2)}_{k})_{ii}\beta_{ik}\right] \to 0, N \to \infty
V ( r , u ) = - ( 8 \pi^{2}\alpha^{\prime})^{- 1 / 2}\int_{0}^{\infty}d l ~ e^{- r^{2}l / 2 \pi \alpha^{\prime}}{\frac{l^{1 / 2}\mathrm{t a n h}( u )}{\mathrm{S i n}( u l )}}\left[ 1 2 + 4 \mathrm{C o s}( 2 u l ) - 1 6 \mathrm{C o s}( u l ) \right] \quad .
\lim_{\delta\searrow 0}2\,\int_0^{1-\delta}\left(1-\varrho^{N-s\,p-\beta\,(p-1)}\right)\,\varrho^{s\,p-1}\,(1-\varrho^{\beta})^{p-1}\,\Phi(\varrho)\,d\varrho=C(\beta),
d_\lambda=\frac{\prod_{\alpha\in\Phi^+, \alpha|_\mathfrak{a}\neq 0}(\lambda+\rho', \alpha)}{\prod_{\alpha\in\Phi^+, \alpha|_\mathfrak{a}\neq 0}(\rho', \alpha)}, \ \rho'=\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha.
\frac1T \sum \limits_{k=0}^{T-1}\langle \hat{g}_{k+1}, x_k -x^* \rangle \leq \frac12 \frac{R_0^{2}}{\nu T}+ \frac{\nu}{2}\frac{1}{T}\sum \limits_{k=0}^{T-1}||\hat{g}_{k+1}||^{2}_q.
\Delta\mathcal{C}^{2}\overset{}{=}\left[ \mathcal{C}^{}\right]^{2}-\left[ \mathcal{C}^{}\right]^{2}\mathcal{C}_{}^{2}\overset{}{=}\left[\mathcal{C}^{}\right]^{2}+\left[ \mathcal{C}^{}\right]^{2}
\int_{D\cap V^c}M_{D_n}(x,y)\eta_n(dy)\le \int_{D_k}M_{D_n}(x,y)\eta_n(dy) &\qquad\qquad\qquad=\int\limits_{D_k}G_{D_n}(x,v)\int\limits_{D\setminus D_n}j(|v-y|)f(y)dydv &\qquad\qquad\qquad\le c_k\left(\int\limits_{D_k}G_D(x,v)dv\right)\left(\int\limits_{D\setminus D_n}f(y)(1\wedge j(|y|))dy\right)\overset{n\to\infty}{\longrightarrow}0,
\theta_{t}(t, x) - a \triangle \theta(t, x) = \bar{g}(t, x)\phantom{0}(t, x) \in (0, T) \times \Omega, \theta(t, x) = 0\phantom{\bar{g}(t, x)}(t, x) \in [0, T] \times \partial \Omega, \theta(0, x) = \theta^{0}(x)\phantom{0}\;\, x \in \Omega.
J^{(d)}_{\nu_1 \nu_2 \nu_3}=\frac{1}{\pi^d}\int \!\! \int{d^d k_1~d^d k_2}~P_{k_1,m_1}^{\nu_1}P_{k_1-k_2,m_2}^{\nu_2}P_{k_2,m_3}^{\nu_3}.
D^{s}_{r}\left(p\|q\right)=\frac{1}{s-1}\left\{\left[\sum_{i}\left(MG\right)_{i}^{1-r}\left(MA\right)_{i}^{r}\right]^{\frac{s-1}{r-1}}-\left[\sum_{i}\left(MA\right)_{i}\right]^{\frac{s-1}{r-1}}\right\}
\|N^{k,1}\|_{\widetilde S_q^p}+ \|N^{k,2}\|_{\widetilde D_{q, q}^p}+ \|N^{k,3}\|_{\widetilde D_{p, q}^p}< \frac{1}{2^k},\;\;\;\;\;\; k\geq 1, \|N^{0,1}\|_{\widetilde S_q^p}+ \|N^{0,2}\|_{\widetilde D_{q, q}^p}+ \|N^{0,3}\|_{\widetilde D_{p, q}^p}\leq 2\|M^{n_0}\|_{\mathcal A_{p,q}}.
\sum_{k\in I_0^c}s_k^{(m)}=(\sum_{k\in I_0}s_k^{(m)})-\sum_{k\in I_0}s_k^{(m)}{}>\frac{1}{2}rd(d-1)-\frac{1}{2}r|I_0|(|I_0|-1){}=r\sum_{n=|I_0|}^{d-1}n{}=r|I_0^c|(d-1-\frac{1}{2}(|I_0^c|-1)).
\ll_\eta \sum_{\substack{n\in \mathcal{A}n<X^{1-\delta}}}1+\frac{\#\mathcal{A}}{\#\mathcal{B}}\sum_{n<X^{1-\delta}}1\ll \#\mathcal{A}^{1-\delta}+\frac{\#\mathcal{A}}{X^\delta}=o_\eta\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr).
\|r_Q(u,\theta)\|_2 \leq \int_{\mathbb T^2}h_Q(y)\|u(\cdot-y)-u(\cdot)\|_2\|\theta(\cdot-y)-\theta(\cdot)\|_\infty\,dy &\lesssim \int_{\mathbb T^2}h_Q(y)\left( \sum_{p\leq Q}|y|\lambda_p\|u_p\|_2+\sum_{p> Q}\|u_p\|_2 \right)\, dy &\lesssim \sum_{p\leq Q}\lambda_Q^{-1}\lambda_p\|u_p\|_2+\sum_{p> Q}\|u_p\|_2.
o_\lambda(x^{\pm})=2\frac{\det(x^{\lambda_j+(l-j)}_i+\delta_{j\neq l}\delta_{\lambda_l\neq 0}x^{-\lambda_j-(l-j)}_i)^l_{i,j=1}}{\det(x^{l-j}_i+x^{-(l-j)}_i)^l_{i,j=1}}.
\int_\Omega |\nabla \varphi|^2=\int_\Omega |U_*+\varphi|^{2^{*}-2}(U_*+\varphi)\varphi-\int_\Omega\Bigl( u_0^{2^-1}-\sum_{j=1}^k U_j^{2^*-1}\Bigr)\varphi.
|\mathbf{v}_1|^2 \equiv \rho^2 \, \cos \phi \,,\qquad |\mathbf{v}_2|^2 \equiv \rho^2 \, \sin \phi \,,\qquad \vec{\mathbf{v}}_1 \, \cdot \,\vec{\mathbf{v}}_2 \equiv \rho^2 \sin \phi \cos \phi \, \cos\theta\,.
\partial_a E_{m,n}^{a,b}[g(t_1)]=\mathrm{Cov}^{a,b}_{m,n}\left( g(t_1),\sum_{i=1}^{t_1}L_i(a) \right)=\sum_{i=1}^m L_i(a) \cdot \mathrm{Cov}^{a,b}_{m,n}\left( g(t_1), 1_{\{t_1\ge i\}}\right)\ge 0.
g'_1(\tilde{x},\tilde{t})= \frac{X_x(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))}{\kappa\tilde{t}^{\alpha-1}(\mathcal{T}_t^\alpha Y)(Y^{-1}(\tilde{t}))}\big[2h(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))(r_2r_{1x}-r_1r_{2x})-f_1r_1r_2 &+g_1r_1^2+f_2r_1r_2-g_2r_2^2\big],
\nu_j(N,\Sigma^{\ast})= \sigma^{\ast}_j(N_j)b^{n-j-1}+\dots+\sigma^{\ast}_{n-2}(N_{n-2})b+\sigma^{\ast}_{n-1}(N_{n-1}) = (b-1-\sigma_j(N_j))b^{n-j-1}+\dots+(b-1-\sigma_{n-2}(N_{n-2}))b +(b-1-\sigma_{n-1}(N_{n-1})) = (b-1)(b^{n-j-1}+\dots+b+1)-\nu_j(N,\Sigma)=b^{n-j}-1-\nu_j(N,\Sigma).
\left\langle a \right| N \left( \psi_{L}^{\dagger}\psi_{L}\psi_{R}^{\dagger}\psi_{R}\right) \left| B , a \right\rangle = \frac{1}{4 \pi^{2}\left| z \right|^{2}}\left( \frac{\sinh^{2}\left( \frac{\alpha}{2}\log \left| z \right| \right)}{\sinh^{2}\left( \log \left| z \right| \right)}- \frac{\alpha^{2}}{4}\right) .
\partial_{\mu}q^{u}\to{D}_{\mu}q^{u}= \partial_{\mu}q^{u}- k_{I}^{u}A_{\mu}^{I}\ , \qquad \partial_{\mu}t^{i}\to{D}_{\mu}t^{i}= \partial_{\mu}t^{i}- k_{I}^{i}A_{\mu}^{I}\ .
\omega=-\displaystyle\frac{1}{C_K}\left(R_1\,dR_2\wedge dL+R_2\,dL\wedge dR_1+\displaystyle\frac{(\lambda -8E)L+4\lambda L^3}{(\lambda -8E)+8\lambda L^2}dR_1\wedge dR_2\right) + \sum_{i=1}^3 d \bar{C}_i\wedge dC^i~,
{\rm n}^{\Gamma,+}_{v}(\zeta) = \frac{1}{M(v)}{\rm n}_v^{+}\left(\zeta{\rm e}^{p|\epsilon_\zeta|}M(\nu_\zeta); \zeta< \texttt{k}^\Omega\right)\leq C\frac{1}{M(v)}{\rm n}_v^{+}\left(\zeta{\rm e}^{p|\epsilon_\zeta|}\right)
\mathbb{E}\left[ f(\underbar{X}) \big | g(\underbar{X}) \right] = \sum\limits_{i=1}^n \mathbb{E}\left[ f_i(X_i) \big | g(\underbar{X}) \right] & > \sum\limits_{i=1}^n \mathbb{E}\left[ f_i(X_i) \right] = \mathbb{E}\left[ f(\underbar{X}) \right].
\begin{aligned}\mathfrak{S}_{\delta, \flat}^{\mu_2}(L) \ll \sum_{h\sim H}\frac{1}{h}\sum_{(D/2L)^{1/2}<d\le (2D/L)^{1/2}}\bigg(\Big(\frac{xh}{d^2L^2}\Big)^{1/6}L^{4/6}+ \Big(\frac{xh}{d^2L^2}\Big)^{-1}\bigg) \noalign{\vskip 1mm}\ll \big((x D^2 H)^{1/6}+ x^{-1}D^2\big) x^{\varepsilon}.\end{aligned}
\sum_{k_{1}\geq0}...\sum_{k_{n}\geq0}(\prod_{j=1}^{n}\rho_{j}^{k_{j}})\cos(\beta+\sum_{j=1}^{n}ik_{j}\alpha_{j}) =\frac{\exp(i\beta)\prod_{j=1}^{n}(1-\rho_{j}\exp(-i\alpha_{j}))+\exp(-i\beta)\prod_{j=1}^{n}(1-\rho_{j}\exp(i\alpha_{j}))}{2\prod_{j=1}^{n}(1+\rho_{j}^{2}-2\rho_{j}\cos(\alpha_{j}))}.
\Bar{\Theta}^{d,T}(\frac{t+1}{T},\frac{i}{d})-\Bar{\Theta}^{d,T}(\frac{t}{T},\frac{i}{d})=-\eta \sum_{j=1}^d W(\frac{t}{T},\frac{i}{d})W(\frac{t}{T},\frac{j}{d})\Bar{\Theta}^{d,T}(\frac{t}{T},\frac{j}{d}) +\eta W(\frac{t}{T},\frac{i}{d})\varepsilon^t.
f_{\alpha}(z) = f_{\alpha}\left(\theta \right) + \frac{\sigma_{\alpha}^2}{2}(z-\theta)^2 + \mathcal{O}\left((z-\theta)^3\right), f_{\alpha}(z) = f_{\alpha}\left(-\theta\right) - \frac{\sigma_{\alpha}^2}{2}(z+\theta)^2 + \mathcal{O}\left((z+\theta)^3\right).
\langle \varphi^{2}( x ) \rangle = \int d^{3}p \left[ | \psi_{0}( z_{0}) |^{2}| \chi_{0}|^{2}+ \int_{- \frac{H}{\Delta + 2}}^{\infty}d m | \psi_{m}( z_{0}) |^{2}| \chi_{m}|^{2}\right] .
\left\{\L_{R}\frac{\partial}{\partial \L_{R}}+ \beta \frac{\partial}{\partial g}- ( n + \bar{n}) \gamma_{\Phi}\right\}\Gamma_{n , \bar{n}}^{\L}\left[ \Phi , \bar{\Phi}\right] = 2 \gamma_{\Phi}{\cal T}_{n , \bar{n}}^{\L}\left[ \Phi , \bar{\Phi}\right]
\vec{\vec{\Delta}}\hat{W}=\vec{\nabla}\otimes\vec{\nabla}\hat{W}=\sum_{a=1}^2\sum_{b=1}^2\vec{e}_a\otimes\vec{e}_b\frac{\partial^2\hat{W}}{\partial\phi^a\partial\phi^b}\quad,\Delta\hat{W}=\vec{\nabla}\vec{\nabla}\hat{W}=\sum_{a=1}^2\frac{\partial^2\hat{W}}{\partial\phi^a\partial\phi^a}
\sum_{(1,j_1),\dots,(k,j_k)}\sum_{\gamma,\gamma' \in \Gamma}w_\gamma w_{\gamma'}\prod_{l=1}^k \frac{1_{p_l = p'_{j_l}}1_{p_l | p_1 + \dots + p_{l-1}- p'_1 - \dots - p'_{j_l-1}}}{p_l}
J_{\nu}(t)=\Big(\frac{t}{2}\Big)^{\nu}\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+1)\Gamma(k+\nu+1)}\Big(\frac{t}{2}\Big)^{2k}=\frac{1}{\sqrt{\pi}}\Big(\frac{t}{2}\Big)^{\nu}\sum_{k=0}^\infty \frac{(-1)^k\Gamma(k+\frac{1}{2})}{\Gamma(k+\nu+1)}\frac{t^{2k}}{(2k)!}\,,
\frac{1}{\sqrt{1-x^2}}= \sum_{j=0}^\infty \frac{\Gamma\bigl(\frac{1}{2}\bigr)}{j!\Gamma\bigl(\frac{1}{2}-j\bigr)}\bigl(-x^2\bigr)^j = \sum_{j=0}^\infty \binom{2j}{j}\Bigl(\frac{x}{2}\Bigr)^{2j}
[ * * * ]^{2}= [ \frac{1}{z - w}k - \bar{\beta}k \frac{1}{z}]^{2}= \frac{k^{2}}{( z - w )^{2}}- 2 \frac{k \bar{\beta}k}{z - w}\frac{1}{z}+ \frac{k \bar{\beta}^{2}k}{z^{2}},
\{\Psi_{2}^{2}, \Psi_{3}^{2}\}\stackrel{\Sigma}{=}0 , ~ \{\Psi_{2}^{2}, \Psi_{2}^{4}\}\stackrel{\Sigma}{=}0 , ~ \{\Psi_{2}^{3}, \Psi_{3}^{1}\}\stackrel{\Sigma}{=}0
0 = c_{d + m - r}begin{align*}2mm] = f_{d}g_{m - r}+ \Biggl( \, \sum_{t = 1}^{\min \{d , r \}}f_{d - t}g_{m - r + t}\Biggr) begin{align*}2mm]\shortintertext{(recall that we must have $d - t \geq 0$ and $m - r + t \leq m$)}begin{align*}-3mm] = f_{d}g_{m - r}+ A .
|W_{i+1}| \geq \delta_0 |V_{i+1}|=\delta_0 (1-i \delta_1)n \geq \delta_0\left(1- \frac{1- \varepsilon^2/10}{\delta_1}\delta_1\right)n =\frac{\varepsilon^2 \delta_0 n}{10}=\delta_1 n.
F_{n,\underline{k}}(f_{\underline{k}})(x)=(2\pi)^{n+1}\int_{\mathbb{R}^{n+2}}f_{\underline{k}}(x-\rho)\prod_{l=1}^{n+1}P'_{k_l}h(x-\rho-y_l)\cdot \prod_{l=1}^{n+1}K_{k_l}(y_l,\rho)\,d\rho d\underline{y}.
u^{\varepsilon,\delta}(t,x)=\sup_{t'\in[0,T]}\{u^\varepsilon(t',x)-\delta^{-1}|t-t'|^2\}\quad &v_{\varepsilon,\delta}(t,x)=\inf_{t'\in[0,T]}\{v_\varepsilon(t',x)+\delta^{-1}|t-t'|^2\}
E = - \frac 1{2 a}\sum_{l = 1}^{\infty}( 2 l + 1 ) \int_{- \infty}^{\infty}\frac{d y}{2 \pi}\, e^{i \delta y}x \frac d{d x}\ln \left[ 1 - \xi^{2}\lambda_{l}^{2}( x ) \right] .
\sqrt{n_i}\frac{\mu - F^{-1}(p)}{\sigma}= - \sqrt{n_i}G^{-1}(p) = \sqrt{\frac{n_i}{m}}\sqrt{m}[G_m^{-1}(p) - G^{-1}(p) ] - \sqrt{n_i}G_m^{-1}(p),
I(f,g,h)(\xi) = \frac{1}{2}\int_\eta e^{-3i \Phi(\xi,\eta)}\bar h(\eta- \xi) \left( \int_\nu e^{\frac{3i}4 \eta \nu^2}f \left( \frac{\eta + \nu}{2}\right) g \left( \frac{\eta - \nu}{2}\right) d\nu \right) d\eta \Phi(\xi,\eta) = \eta \xi^2 - \xi \eta^2 + \frac 1 4 \eta^3,
3 2 \times 2 0 = 6 4 0 1 5 \times 3 0 = 4 5 0 3 2 0 \times 2 0 = 6 4 0 0 1 5 0 \times 3 0 = 4 5 0 0 1 2 \times 4 0 = 4 8 0 1 6 \times 5 0
\left( \begin{matrix}{\sigma_{0}^{( \mathrm{o p )}}}\\{\pi_{0}^{( \mathrm{o p )}}}\\ \end{matrix}\right) = -{\frac{\mu^{2}}{N}}\left( m_{\mathrm{Z M}}^{2}- \partial_{\bot}^{2}\right)^{- 1}\left[ : \bar{\Psi}_{M}^{a}\left( \begin{matrix}{1}\\{i \gamma_{5}}\\ \end{matrix}\right) \Psi_{M}^{a}: \right]_{0},
\mathrm{P}^o_{i,k}= \sum^{k-1}_{p=0}{k-1 \choose p}\frac{(-1)^p K!\left[\frac{\gamma\left(N-M+1,\phi_{i,k}\right)}{(N-M)!}\right]^{K-k+p+1}}{(K-k)!(k-1)!(K-k+p+1)},
A_j^*A_j = I - 2\gamma a_{i_j}a_{i_j}^* + \gamma^2 a_{i_j}a_{i_j}^* \|a_{i_j}\|^2 \frac{1}{n}\sum_{j=1}^{n}A_j^*A_j = (1 - 2\gamma \sigma + \gamma^2 \sigma \mu) \cdot I := \varphi \cdot I \sup \|A_j\|^2 \leq \max\{|1- \gamma \mu|,1\}
\mathcal{M}^{a b}= \left( \begin{array}{l l}{{f^{a b i}\varphi^{i}}}&{{\delta^{a b}\partial^{2}}}\\{{- \delta^{a b}\partial^{2}}}&{{- f^{a b i}\overline{{{\varphi}}}^{i}}}\\ \end{array}\right) \; .
\Gamma_{l}:=\left\{t:\frac{cn^{j+1/2}}{(l+1)^j}\leq t<\frac{cn^{j+1/2}}{l^j}\right\}\textrm{for}l=1,\dots, n-1, \Gamma_{n}:=\left\{t:0\leq t< c\sqrt{n}\right\}, \Gamma_{0}:=\{t:t\geq cn^{j+1/2}\}.
r = \begin{bmatrix}1 0 0 0 -1 0 0 0 1\end{bmatrix},t = \begin{bmatrix}1 0 1 0 1 0 0 0 1\end{bmatrix},\textrm{and}s = \begin{bmatrix}0 1 1 1 0 0 0 0 1\end{bmatrix}.
\gamma^{i}\gamma^{[ a_{1}. . . a_{n}]}\otimes \gamma^{[ a_{n}. . . a_{1}]}= ( - 1 )^{n}\gamma^{[ a_{1}. . . a_{n}i ]}\otimes \gamma^{[ a_{n}. . . a_{1}]}+ n \gamma^{[ a_{2}. . . a_{n}]}\otimes \gamma^{[ a_{n}. . . a_{2}i ]}
(\delta r)_{\vec{k}}\cong\frac{e}{mc^{2}k^{2}}E_{\vec{k}}=\frac{e}{mc^{2}k^{2}}E_{\vec{k}}(a_{\vec{k}}e^{-i\nu t+i\vec{k}\cdot\vec{r}}+h.c.)withE_{\vec{k}}=(\frac{\hbar ck/2}{\epsilon_{0}\Omega})^{1/2},
\frac{\partial u_i^j}{\partial \xi_k}(x_1,x_2,x_3) = \frac{\partial{K_i^j}}{\partial \xi_k}(x_1,x_2,x_3) + \frac{\partial{A_i^j}}{\partial \xi_k}(x_1,x_2,x_3) + \frac{\partial{B_i^j}}{\partial \xi_k}(x_1,x_2,x_3) + x_3 \frac{\partial{C_i^j}}{\partial \xi_k}(x_1,x_2,x_3) ,
L(f)(x,y) =\left(fL(e_{xx})\right)(x,y)-\left(L(e_{xx})f\right)(x,y) &+f(x,y)L(e_{xy})(x,y)-\left(fL(e_{yy})\right)(x,y)+L(e_{yy})(x,x)f(x,y) &f(x,x)L(e_{yy})(x,y)+f(y,x)L(e_{yx})(x,y).
\nu_0(\varepsilon, B)\sim\begin{cases}\varepsilon^{-\frac{n}{m}}\sigma>\frac{m}{n}\varepsilon^{-\frac{n}{m}}|\log\varepsilon| \sigma=\frac{m}{n}\varepsilon^{-\frac{1}{\sigma}}0<\sigma<\frac{m}{n}.\end{cases}
\sum_{\vec\imath_{\sigma}\in \mathfrak C(T)}(-1)^{|\sigma|}\Big(\prod_{j=1}^{3}f(i_{\sigma(j)}, i_{\sigma(j+1)})\cdot v\Big)\Big(\prod_{j=1}^{3}f(i_{\sigma^{*}(j)}, i_{\sigma^{*}(j+1)})\cdot w\Big)=0,
\begin{aligned}\begin{cases}W(x_k, \phi^\circ_k, u^\circ_k, k)=H(\phi^\circ_k) \ \ \ \ +\mathbb{E}^{u^\circ}\left[ W(b(x_k,u^\circ_k)+w_k, \phi^\circ_{k+1}, u^\circ_{k+1}, k+1) \right], W(x_K, \phi^\circ_K,u^\circ_K,K)= (\mu \Gamma(x_K)). \end{cases}\end{aligned}
\sum_{n_1,n_2\in\mathbb{Z}}\prod_{j=1,2}(1+|n_j|)^{-N}\sum_{2^{k_1-1}\leq|q|\leq2^{k_1+1}}\int_{I_{q,k_1}}\!H_1(t,x,y)H_2(t,x,y)\,\mathrm{d}t,
A=b_{n}\int\int K(u)K(v)(f_{2}(x_{j}+ub_{n},x_{p}+vb_{n})-f(x_{j}+ub_{n})f(x_{p}+vb_{n}))dudv \leq b_{n}\int\int K(u)K(v)(f_{2}(x_{j}+ub_{n},x_{p}+vb_{n})dudv.
G_{u,v}-G_\rightarrow =|u\rangle\langle v|, G_{u,v}-G_\leftarrow =|v\rangle\langle u| , G_\rightarrow -G_\leftarrow =|v\rangle\langle u|-|u\rangle\langle v|=G_\leftrightarrow, G_{u,v}-G_{u_1,v_1}=|u\rangle\langle v|-|u_1\rangle\langle v_1|,
\displaystyle\lim_{n\rightarrow\infty}I_\infty(\omega_n) =\displaystyle\lim_{n\rightarrow\infty}\frac{\alpha}{N}\|\omega_n\|_X^2 &=\displaystyle\lim_{n\rightarrow\infty}\frac{\alpha}{N}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx &=\frac{\alpha}{N}S^\frac{N}{2\alpha}+o_n(1).
I'_{h,k,n}(x) = \frac{1}{2 \pi n^{1/3}}\,\int_{- \frac{2 \pi n^{1/3}}{m+m'}}^{\frac{2 \pi n^{1/3}}{m+m''}}\,\exp\left[n^{2/3}\left(\frac{L_m(x)^3}{2 ( \Re L_m(x) - iz)^2}+ ( \Re L_m(x) - iz)\right)\right] \, dz .
\sum_{n=0}^{\infty}p(n)x^{n}=(x;x)_{\infty}^{-1}=\vartheta_{00}(x)^{-1/6}\vartheta_{01}(x)^{-2/3}\{\frac{1}{16x}[\vartheta_{00}(x)^{4}-\vartheta_{01}(x)^{4}]\}^{-1/24}.
S_5 = -\int d^5x\sqrt{-g}\, \left[i \bar\Psi_M\gamma^{MNP}D_N\Psi_{P}+i\frac{3}{2}\frac{a^\prime}{a}\bar\Psi_M\gamma^{MN}\Psi_{N}+\left(i\bar J^M \Psi_{M}+h.c.\right)\right]\, .
\Big|g_2(\log C(t,\pi))-g_2(\log C(t-\nu,\pi))\Big| \ll \frac{\mu_{d,\sigma}}{(1 - \sigma)}\dfrac{|\log C(t,\pi)-\log C(t-\nu,\pi)|}{(\log \log C(t,\pi))^2}& \ll \dfrac{d\,\mu_{d,\sigma}\,(\log C(t,\pi))^{2-2\sigma}}{(1-\sigma)^2(\log\log C(t,\pi))^{3}}.
E_{0}^{(1)}=\lambda(\frac{\alpha}{\pi})^{\frac{1}{2}}\int e^{-\alpha x^{2}/2}x^{4}e^{-\alpha x^{2}/2}dx=\lambda(\frac{\alpha}{\pi})^{\frac{1}{2}}\frac{\partial^{2}}{\partial\alpha^{2}}\int e^{-\alpha x^{2}}dx,
\mathcal B_\delta^{(1)}:=\left\{\omega\in\mathcal A_r\,\Big|\,\begin{array}{c}\forall z,w\in \mathbb B_r\textrm{with}\Vert z-w\Vert<\delta : \sup_n\Vert f^n_\omega(z)-f^n_\omega(w)\Vert<\varepsilon_1\end{array}\right\},
\int_\Omega f \, \overline \varphi \, \pi_i= \int_\Omega \sum_{l=1}^d w_l \, \overline{\partial_l \varphi}\, \pi_i + \sum_{k=1}^d \hat c_{ki}\int_\Omega (\partial_k u) \, \overline \varphi + \lambda \int_\Omega u \, \overline \varphi \, \pi_i.
\Delta_0 (\lambda ) = \Delta_0 = - \left[{\begin{array}{*{20}c}{u_1}{u_2}\end{array}}\right]\left[{\begin{array}{*{20}c}{\sigma_1}0 0{\sigma_2}\end{array}}\right]\left[{\begin{array}{*{20}c}{v_1}{v_2}\end{array}}\right]^\dag,
{\sf a}=\sqrt{|{\kappa}|\left(\frac{\sqrt{4{\ell}({\ell}+1)+1}-1}{2}\right)\;}\;\;\;\;\;\mbox{and}\;\;\;\;{\sf b}=\sqrt{|{\kappa}|\left(\frac{\sqrt{4{\ell}({\ell}+1)+1}+1}{2}\right)\;}\;\;\;.
\eta^{1/2}\breve T_2-{\textstyle{1\over 6}}\breve \kappa_3\eta T_1^2-\breve \kappa_1+{\textstyle{1\over 6}}\breve \kappa_3=\eta^{1/2}T_2-{\textstyle{1\over 6}}\kappa_3\eta T_1^2-\kappa_1+{\textstyle{1\over 6}}\kappa_3
\frac{d}{dt}F = \ - \int_M \left( 1 - \frac{\bar{v}}{v_m}\right)^2v_m dV_u = \ - \int_M \left[1 - \frac{2 \bar{v}}{v_m}+ \frac{\bar{v}^2}{v_m^2}\right] v_m dV_u = \ - v + 2 v - \bar{v}^2 \int_M \frac{1}{v_m}dV_u.
1.E_j(x_2, \ldots, x_n)=(-1)^j x_{i+1}^j +(-1)^{j-1}x_{i+1}^{j-1}E_1(x_1, \ldots, x_n)+ \cdots + (-1)^0 x_{i+1}^0 E_j(x_1, \ldots, x_n).
\frac{\ell+1}{\sin^8 \phi}\ell^2 \sum_{v=0}^4{_4}\omega_{2,v}(\cos \phi) p_{1,\ell+v}(\phi) = \frac{\ell^3}{\sin^8 \phi}\Big[ \sin^3 \phi f_b(\phi) s_{1,0}(\ell,\phi) \Big]+\phi^{-8-1/2}O(\ell^{-1/2})+\phi^{-5}O(\ell),
[H(t), \Lambda(t)] = [(H_1 + H_2 + W_{12}) + H_3 + H_4 + \sum\, '\,W_{ij}, & (H_1 + H_2 + W_{12}) + L_3 + L_4] = [H_3,L_3] + [H_4,L_4] + & \left[\sum\, '\,W_{ij}, H_1 + H_2 + L_3 + L_4\right].
D_{i,j}:=\sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+1}~\sigma_{i}^2~\sigma_{i+1}\cdots\sigma_{j-2}\sigma_{j-1}=\prod_{i \le a < j}g_{a,j}\in P_{n}, F_{i,j}:=\sigma_{n-j}\sigma_{n-j+1}\cdots \sigma_{n-i-1}\sigma_{n-i}^{2}\sigma_{n-i-1}\cdots \sigma_{n-j+1}\sigma_{n-j}=\prod_{a=j}^{i+1}g_{n-a,n-i}\in P_n,
\lim_{k \rightarrow \infty}P ( F_{\ell^{k}}; J \times \mathbb{R}^{n-1}) = \lim_{k \rightarrow \infty}\left( \int_{J}\mathcal{H}^{n-2}\left( ( \partial^{*}F_{\ell^{k}})_{z}\right) \dz + |D^{c}\ell^{k}|(J) \right) &= \int_{J}\mathcal{H}^{n-2}( (\partial^{*}F_{\ell})_{z}\dz + |D^{c}\ell|(J) &= P (F_\ell ; J \times \mathbb{R}^{n-1}),
\sum_{i=m+1}^{n-1}z^{i}4^{i-m-1}\int_{-1}^1 \bigg((t^2-1)\phi_m(t)+4a_{m+1}\psi_m(t)+16\: a_{m+1}^2 \phi_{m-1}(t)\bigg)\bigg(\sum_{j\geq i+1}\ldots\bigg),
q_{-}^{0}( \tau ) = - \frac{\Omega \vartheta_{1}^{\prime}( 0 )}{2 \pi}~ \frac{e^{\Omega \tau \frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}( \Omega y )}\vartheta_{1}( \Omega ( y - \tau ) ) + e^{- \Omega \tau \frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}( \Omega y )}\vartheta_{1}( \Omega ( y + \tau ) )}{\vartheta_{1}( \Omega \tau ) \vartheta_{1}( \Omega y )},
\partial_t^2 \phi - \Delta \phi + h(\phi)\Delta \phi = F(\partial \phi) (t,x)\in (0,\infty)\times{\mathbb R}^4 &\phi(0,x)=f(x) \in H^{s}_{{\rm rad}}({\mathbb R}^4), \partial_{t}\phi(0, x)=g(x) \in H^{s-1}_{{\rm rad}}({\mathbb R}^4)
y_n =\frac{1}{n+1}\sum_{k=1}^{n}\binom{n+k}{k}\frac{k!}{n!}B_{n,k}(0,2!c_2,3!c_3\dots) =\frac{1}{n+1}\sum_{k=1}^{n-1}\binom{n+k}{k}\frac{k!}{(n-k)!}B_{n-k,k}(1!c_2,2!c_3\dots),
\displaystyle \begin{array}{rcl}\varphi_{1}(x) \!\!\!\!\!=4(\pi\!-\!2)(\pi^{3}\!-\!60\pi\!+\!120)x^{2}\!-\!3\left(\pi^{6}\!-\!100\pi^{4}\!+\!200\pi^{3}\!-\!480(\pi\!-\!2)^{2}\right) \end{array}
\|u\|_{(\kappa_n+1)p^*}\leq M_{31}^{\frac{1}{\kappa_n+1}}M_{30}^{\frac{1}{\sqrt{\kappa_n+1}}}\left\|u\right\|_{(\kappa_n+1)\tilde{q}_1}=M_{31}^{\frac{1}{\kappa_n+1}}M_{30}^{\frac{1}{\sqrt{\kappa_n+1}}}\left\|u\right\|_{(\kappa_{n-1}+1)p^*}
\int_{-\infty}^{+\infty}d\kappa \, \rho_{\textsf{fin}}(\kappa)\beta(\kappa) =\lim_{L\to\infty}\left[\sum_{n=1}^{L}\frac{1}{4n}- \sum_{n=1}^{2L}\frac{n}{4n^{2}-1}\right] =\frac{1}{4}-\frac{3\log 2}{4}\, .
\left| 2 - i y - \frac{1}{n-1}\sum_{i=1}^{n-1}t_i\right| = \sqrt{(2-\frac{1}{n-1}\operatorname{Re}\sum_{i=1}^{n-1}t_i)^2 + (y+\frac{1}{n-1}\operatorname{Im}\sum_{i=1}^{n-1}t_i)^2}\geq \sqrt{1+(|y|-1)^2},
RL(b,c) =f_c\left(\binom{b}{2}+b\right)-f_c\left(\binom{b}{2}\right)+b\Delta_c(1) =\sum_{i=1}^b\left[f_c\left(\binom{b}{2}+i\right)-f_c\left(\binom{b}{2}+i-1\right)\right]+b\Delta_c(1) =\sum_{i=1}^b\Delta_c\left(\binom{b}{2}+i\right)+b\Delta_c(1),
(\delta_\alpha H)(a_1, \ldots, a_{m+n+p-1}) ~+ \sum_{j= m}^{m + p -1}(-1)^{-(m-1) + (n-1)(j-1)}h_{0,j}&~+ \sum_{i=0}^{p-1}(-1)^{(m-1)(i-1) + (n-1)(m+i-1) + (m-1)}h'_{i+1, m+i}&~ + \sum_{i=0}^{p-1}(-1)^{(m-1)(i-1) + (n-1)(m+p-2) + (m-1) + (n-1)}h''_{i+1, m+p}= 0.
N_{1,2n+2}=\sum_{p' \in P_n}w(\rho_1(p')) + \sum_{p' \in B_n}w(\rho_2(p')) + 2 w(\mathfrak{p}_{n}) &= q_0(q_1-1)N_{2n}+ \frac{(q_1-1)q_0}{(q_0-1)}N_{2,2n}+ 2q_0^{k+1}q_1^{k}(q_1-1)
b_{\pm}( \eta , z ) ~ = ~ \bar{b}~ \exp \left[ i k ( \eta - z ) ~ \pm ~ i \left[ \frac{h_{r}}\eta ~ + ~ \frac{p^{2}}{2}\left( \frac{h_{0}\eta^{2}}{3}~ - ~ h_{r}\eta ~ - ~ h_{0}z^{2}\right) ~ + ~ O ( p^{4}) \right] ~ + ~ O \left( \frac 1 k \right) \right]
\begin{cases}K_1 > \frac{6}{5}\gamma \alpha_r \leq \frac{4}{3}\frac{6}{5}\gamma < K_1 < \frac{3\alpha_r - 2}{3\alpha_r - 4}\gamma \alpha_r > \frac{4}{3}\end{cases},3 < K_2 < 3\beta ,K_3 > \frac{5 K_1 + 6\gamma}{5 K_1 - 6\gamma}.
\sum_{\lambda = m+i}^{j-1}= \sum_{\lambda = m+i}^{j-1}(-1)^\lambda~ h (a^{m+n-1}_{1, i-1}, ~f^n,~ a^{m+n-1}_{i+m, \lambda -1},~ \alpha^{m+n-2}(a_\lambda \cdot a_{\lambda +1}),~ a^{m+n-1}_{\lambda +2, j},~ g^m,~ a^{m+n-1}_{j+n+1, m+n+p-1})
\|y\|_\mathcal{B}:=\max_{t\in [0,T]}\|y(t,\cdot)\|_{L^2(0,L)}+\left(\int_0^T\|y(t,\cdot)\|_{H^1(0,L)}^2dt\right)^{1/2}&\leq C(T)\|y_0\|_{L^2(0,L)}.
\hat{G}_{c}( t , x ) ={\frac{| t |}{2 L}}\Big [ \theta ( t ) \sum_{n = 1}^{\infty}e x p \big ({\frac{- i \pi n}{L}}( t + x ) \big ) + \theta ( - t ) \sum_{n = 1}^{\infty}e x p \big ({\frac{i \pi n}{L}}( t + x ) \big ) \Big ] .