harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_1 = \begin{bmatrix}0 1 0 -1 0 0 0 0 0\end{bmatrix}, F_2 = \begin{bmatrix}0 0 1 0 0 0 -1 0 0\end{bmatrix}, F_3 = \begin{bmatrix}0 0 0 0 0 1 0 -1 0\end{bmatrix}.
\alpha = \frac{5}{14}+ \epsilon, \sigma_0 = 0, \beta = \frac{3}{2}+ \epsilon, \gamma_{1}= 1, R_1 \sim \langle f, f\rangle, \sigma_1 = \frac{6}{7}+ \epsilon, \eta_{1}= \frac{9}{14}+ \epsilon, \gamma_{2}= 1, R_2 \sim \langle f, \overline{f}\rangle, \sigma_2 = \frac{6}{7}+ \epsilon, \eta_{2}= \frac{9}{14}+ \epsilon.
T_{\overline{L},p}(r) = \int (\log \|s\| + mg_{D_r,p})(\delta_p -\pi_{r,p}) + \int g_{D_r,p}\delta_{\div(s)-m[p]}=\int (\log \|s\| + mg_{D_r,p})\delta_p - \int \log\|s\| \pi_{r,p}+ \int g_{D_r,p}\delta_{\div(s)-m[p]}
V_{\pm}= \frac{a ( a \mp \frac{\mu}{\sqrt{2}}) f^{2}+ ( a ( b \mp \frac{\nu}{\sqrt{2}}) + b ( a \mp \frac{\mu}{\sqrt{2}}) ) f + b ( b \mp \frac{\nu}{\sqrt{2}})}{( f^{\prime})^{2}}\pm \frac{a}{\sqrt{2}}.
q^{\beta^{2}/2}\left(q^{\alpha+\beta+n+1};q\right)_{\infty}L_{n}^{(\alpha+\beta)}\left(x;q\right)=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\left(xq^{\alpha+1/2}e^{iy};q\right)_{n}}{\left(q;q\right)_{n}}\frac{\exp\left(\frac{y^{2}}{\log q^{2}}+i\beta y\right)}{\left(-q^{\alpha+1/2}e^{iy};q\right)_{\infty}}dy,
\Gamma_1 =\xi^\top[\Lambda\otimes (PS+S^\top P) ]\xi-2\Theta \xi^\top (\Lambda^2 \otimes Pdd^\top P)\xi- 2\bar x^\top [H(E-\Theta I_N)H\otimes Pdd^\top P ]\bar x &=\sum_i\lambda_i\xi_i^\top[PS+S^\top P-2\Theta\lambda_i Pdd^\top P)\xi_i-2\sum_i (\theta_i-\Theta)\hat x_{vi}^\top Pdd^\top P\hat x_{vi}
z^\top\nabla^2f(x) z = \sum_{i,j=1}^n z_iz_j\partial_{ij}f(x) = \sum_{i=1}^n z_i^2V''(x_i) + \sum_{i,j=1}^n \big(z_i^2 - z_iz_j\big) J_{ij}K''(x_i-x_j).
\langle|F^\ell(k)|^2\rangle=\sum_{k'+k''=k}\big((G_\ell(k')\tilde G_\ell(k''))^2+(G_\ell(k')\tilde G_\ell(k''))(G_\ell(k'')\tilde G_\ell(k'))\big) &\times\langle|\xi(k')|^2|\xi(k'')|^2\rangle.
\overline{L}_{-n}\phi_+(z)=\phi_+'(z)\sum\limits_{m=0}^{\infty}\left(\left(\frac{\phi_+(z)^{-n+1}}{\phi_+'(z)}\right)_{-m+1}\right)^*z^{m+1}-\frac12z\phi_+'(z)\left(\left(\frac{\phi_+(z)^{-n+1}}{\phi_+'(z)}\right)_1\right)^*,
T_{MN}= \frac{8}{3}\partial_M\Phi\partial_N\Phi-\frac{4}{3}\left(\partial \Phi\right)^2 G_{MN}-V\!\left(\Phi\right)G_{MN}-\sum_i f_i\,\delta\!\left( y -y_i\right) g_{\mu\nu}\delta_M^\mu \delta_N^\nu ,
(MFG)\begin{cases}-\partial_t u +H(x,Du)=F(x,m) \ \ \ \mbox{in}\ [0,T]\times{\Omega}, \partial_t m-div(mD_pH(x,Du))=0\ \ \ \mbox{in}\ [0,T]\times{\Omega}, m(0)=m_0 \ \ \ \u(x,T)=G(x,m(T))\end{cases}
W=\int_{0}^{V}dV\int_{0}^{V}dV^{\prime}C(V^{\prime})=\int_{0}^{V}dV^{\prime}\int_{V^{\prime}}^{V}dVC(V^{\prime})=\int_{0}^{V}dV^{\prime}(V-V^{\prime})C(V^{\prime}),
\lim_{\lambda\in\Lambda}\frac{g_{n,k}(\tau)}{|\tau|}=\begin{cases}\frac{1}{|\tau||\widetilde{F}_{1}^{(\rho)}(\tau)|}\mbox{if}\,\,k=0, begin{align*}0.5em]\frac{1}{|\widetilde{F}_{k-1}^{(\rho)}(\tau)||\widetilde{F}_{k+1}^{(\rho)}(\tau)|}\mbox{if}\,\,k\geq 1,\end{cases}\tau\in[a_{k},b_{k}],
\MoveEqLeft A(t)e^{-tA(t)}x - A(0)e^{-tA(0)}x = \frac{1}{2\pi i}\int_{\Gamma}ze^{-tz}[R(z,A(t)) - R(z,A(0))] x \d z & = \frac{1}{2\pi i}\int_{\Gamma}ze^{-tz}R(z,A_{-1}(t)) [A_{-1}(t) - A(0)] R(z,A(0))x \d z,
E_j^{(b){\mathrm{tot}}}=-\left( \frac{k_Dz_j}{z_2-z_1}\right)^{D}\frac{2^{-D-1}\pi^{-D/2}}{\Gamma (D/2+1)}{\mathrm{p.v.}}\int_0^\infty dt \, t^D\frac{d}{dt}\ln \left[1-\frac{(B_1t+1)(B_2t+1)}{(B_1t-1)(B_2t-1)}e^{-2t}\right] .
\begin{array}{l}H \equiv P_+ = \int dx^- \big[ (1/8){\rm tr}~E^2 -(m/2)\big(\psi^\dagger_-(x)\psi_+(x) + \psi^\dagger_+(x) \psi_-(x) \big) \big] \left[A_-(x^-,x^+), E(y^-, x^+) \right] = i \delta (x^- - y^-)\end{array}
f(x) = \sum_{\lambda \in \S}x^{\ell(\lambda')}q^{|\lambda'| + 3\frac{\ell(\lambda')(\ell(\lambda') - 1)}{2}}=\sum_{m, k \geq 0}\frac{(-1)^k x^{m + 2k}q^{m + \frac{6k(k+1)}{2}}q^{3 \frac{(m+2k)(m+2k-1)}{2}}}{(q;q)_m (q^6; q^6)_k}.
k(z,\zeta)= C_{\eta}(z,\zeta)\epsilon_N^*\wedge \cdots \wedge \epsilon_1^* \lrcorner\sum_{k=0}^n \hat{H}_{p+k}^0 \omega_k(\zeta) \wedge (\hat{g}\wedge \hat{B})_{n-k, n-k-1}
\theta_{i}(z) = \frac{\mu_{i,i+1}\mathcal K_i(z) \,d\bar{z}}{\big (\frac{\|t_{i+1}(z)\|^2}{\|t_{i}(z)\|^2}- |\mu_{i,i+1}|^2\mathcal K_i(z)\big )^{1/2}}.
|\tilde v(t)|^2 = |h(0)|^2+ 2\sum_{i=1}^m \int_{(0,t]}\langle v_i^{\ast}(s), v^{(2)}_n(s) \rangle\,dA(s) & + 2 \int_{(0,t]}(\tilde v_n(s), dh(s)) + \sum_{k=0}^{j-1}|h(\tau^n_{k+1})-h(\tau^n_k)|^2-K_n(t),
u_\varepsilon=\gamma_\varepsilon-\frac{t_\varepsilon}{\gamma_\varepsilon}+\frac{S_{0,\varepsilon}}{\gamma_\varepsilon^3}+\frac{S_{1,\varepsilon}}{\gamma_\varepsilon^5}+(A(\gamma_\varepsilon)-2\xi_\varepsilon)\frac{S_{2,\varepsilon}}{\gamma_\varepsilon}+o\left(t_\varepsilon\frac{\zeta_\varepsilon}{\gamma_\varepsilon}\right)\,,
\int_{\mathfrak{D}_{R\mu^k_\alpha}(x^k_\alpha)\setminus\cup^s_{j=1}\mathfrak{D}_{R'\lambda^j_\alpha}(y^j_\alpha)}|u^l_\alpha|^{2^*}\,d\sigma_{\hat{h}}=o(1), \ \ \ \hbox{as}\ \alpha\rightarrow+\infty.
1 + \left(\frac{w_i'w_i}{d}-1\right)- \sum_{j=0}^k \frac{w_i'N_{(i-1)}}{d}\left( I_{i-1}- \frac{N_{(i-1)}'N_{(i-1)}}{d}\right)^j\frac{N_{(i-1)}'w_i}{d}
H\,=\,{{g}_{4}}\frac{\partial}{\partial t}+{{g}_{5}}\frac{\partial}{\partial r}+\left({{c}_{1}}\sin \varphi -{{c}_{2}}\cos \varphi \right)\frac{\partial}{\partial \vartheta}+\left( \cot \vartheta \left({{c}_{1}}\cos \varphi +{{c}_{2}}\sin \varphi \right)+{{c}_{3}}\right)\frac{\partial}{\partial \varphi}.
\ell_{a,d}-\ell_{a+1,1}= \frac{(n-2d)p^{a+1}}{2}- \frac{(n-2)p^a}{2}= \frac{p^a}{2}\big((n-2d)p - (n-2)\big) & = \frac{p^a}{2}\big((n-(n-\frac{n-n_0}{p}- \delta))p - n + 2\big) = \frac{p^a}{2}(\delta p - n_0 + 2).
\mathcal{L}= e^{5 f}G^{- 1}\left( \frac{\dot{G}^{2}}{2}+ 2 5 \dot{f}^{2}G^{2}+ 1 0 G \dot{G}\dot{f}- \dot{l}^{2}G^{2}- 2 \dot{k}^{2}G^{2}- \frac{G}{2}( \dot{a}^{2}+ \dot{\tilde{a}}) \cosh 2 k - G \dot{a}\dot{\tilde{a}}\sinh 2 k \right.
[(x-s_n)^2+(y+\alpha-t_n)^2][(x-s_n)^2+(y+\alpha+t_n)^2] & - [(x-s_n)^2+(y-\alpha-t_n)^2][(x-s_n)^2+(y-\alpha+t_n)^2] & = 8\alpha y\big((x-s_n)^2+y^2-t_n^2+\alpha^2\big).
\frac{df(x,y(x))}{dx}(x_{0})=\frac{\partial f}{\partial x}(x_{0},y(x_{0}))\cdot\frac{\partial x}{\partial x}(x_{0})+\frac{\partial f}{\partial y}(x_{0},y(x_{0}))\cdot\frac{\partial y}{\partial x}(x_{0}).
\begin{array}{l l}{[ h_{i}, h_{j}] = 0 ,}&{[ e_{i}, f_{j}] = \delta_{i j}h_{i},}\\{\mathrm{[}h_{i}, e_{j}\mathrm{]}= A_{i j}e_{j},}&{\mathrm{[}h_{i}, f_{j}\mathrm{]}= - A_{i j}f_{j},}\\ \end{array}
[M^c\circ \tau]_t - [M^c\circ \tau]_s = [M^c]_{\tau_t}- [M^c]_{\tau_s}\leq [M^c]_{\tau_t}- [M^c]_{\tau_s}+ (\tau_t - \tau_s) = ([M^c]_{\tau_t}+\tau_t) - ([M^c]_{\tau_s}+ \tau_s) = A_{\tau_t}- A_{\tau_s}=t-s.
\Phi_{ij}=\begin{cases}\int_{0}^{\infty}{d t \over \hbar}\; K_t(a_i,a_i) \left( e^{-t \mu_{i}^2/\hbar}-e^{t E/\hbar}\right) \textrm{if $i = j$}- \int_{0}^{\infty}{d t \over \hbar}\; K_t(a_i,a_j) e^{t E/\hbar}\textrm{if $i \neq j$}.\end{cases}\;.
|z_n|^{2(1-\beta)}\frac{\partial^2}{\partial z_n\partial \bar z_n}= \frac{\partial^2}{\partial (r_n)^2}+ \frac{1}{r_n}\frac{\partial}{\partial r_n}+ \frac{1}{\beta^2 (r_n)^2}\frac{\partial^2}{\partial (\theta_n)^2}.
\rho_{p + 1}={\frac{1}{( p + 1 ) !}}( \tau_{3}^{( p - 1 ) / 2}\tau_{1}) \psi^{p + 1}+{\frac{1}{( p - 1 ) !}}( \tau_{3}^{( p + 1 ) / 2}\tau_{1}){\cal{F}}\psi^{p - 1}\quad .
\sum_{i,j=d+1}^n(\partial_{\varphi_{ij}}u)(\partial_{\varphi_{ij}}v) =\sum_{i,j=d+1}^n\Big \{\frac{t_i^2}{|t|^2}(\partial_{t_j}u)(\partial_{t_j}v) -2\frac{t_it_j}{|t|^2}(\partial_{t_i}u)(\partial_{t_j}v)+\frac{t^2_j}{|t|^2}(\partial_{t_i}u)(\partial_{t_i}v)\Big \}.
=\hat{a}_{j}^{\dagger}\hat{a}_{l}^{\dagger}\hat{a}_{i}\hat{a}_{k}+\delta_{il}\hat{a}_{j}^{\dagger}\hat{a}_{k}+\delta_{kl}\hat{a}_{j}^{\dagger}\hat{a}_{i}+\delta_{ij}\hat{a}_{l}^{\dagger}\hat{a}_{k}+\delta_{ij}\delta_{kl}
|r^{1/2}u_t + r^{1/2}u_r|_{(r,t)=(r_0,t_0)}\leq |(\sqrt{r}u)_t + (\sqrt{r}u)_r|_{(r,t)=(r_0,t_0)}+ (1/2) r_0^{-1/2}|u(r_0,t_0)| \lesssim_\delta r_0^{-1}.
R ={\frac{1 - \tau}{1 + \tau}}= \left({\frac{\rho_{0}( 1 + t^{2}) + 2 \tilde{\rho}_{0}t - 4 t \rho^{+}}{\rho_{0}( 1 + t^{2}) + 2 \tilde{\rho}_{0}t + 4 t \rho^{-}}}\right)^{1 / 2}.
V[\hat{h}(X,Y)] =V[g(X,\bar{Y})] =V[g](X,\bar{Y}) + g(V[X],\bar{Y}) + g(X,\overline{V[Y]}) =g(V[X],\bar{Y}) + g(X,\overline{V[Y]}) =h(\hat{\nabla}_VX,Y) + h(X,\hat{\nabla}_VY),
\psi(p)\Bigr|_{U_+}= \begin{pmatrix}\omega(p) 1\end{pmatrix}\varphi_+(p)\delta(p^2), \qquad\qquad \psi(p)\Bigr|_{U_-}= \begin{pmatrix}1 \tilde\omega(p)\end{pmatrix}\varphi_-(p)\delta(p^2),
\sum_{i=d}^{s-1}p^{s-1-i}(T_{i}(1+S_{i}))^{p^{i-d}}=\sum_{i=d}^{s-1}p^{s-1-i}T_{i}^{p^{i-d}}+\sum_{i=d}^{s-1}p^{s-1-i}Q_{i}^{p^{i-d}}.
\Delta = \left( \begin{array}{c c}{{\sqrt{\zeta}}}&{{0}}\\{{\bar{z}_{2}}}&{{- z_{1}}}\\{{\bar{z}_{1}}}&{{z_{2}}}\\ \end{array}\right) , \quad \Delta^{\dagger}= \left( \begin{array}{c c c}{{\sqrt{\zeta}}}&{{z_{2}}}&{{z_{1}}}\\{{0}}&{{- \bar{z}_{1}}}&{{\bar{z}_{2}}}\\ \end{array}\right) ,
\begin{cases}u_t+ \alpha u_{xxx}=f, \,\, (0,L)\times (0,T), u(0,t) = h_0(t),\,\,u(L,t) = h_1(t),\,\,u_{x}(L,t) = h_2(t), \,\, (0,T), u(x,0)=u^0(x), \,\, (0,L).\end{cases}
|B^2_{N_1}|=\frac{\pi^{d/2}N_1^d}{\Gamma(d/2+1)}=|B^2_N|\bigg(1+\frac{d}{4N^2}\bigg)^{d/2}\le|B^2_N|\bigg(1+\frac{1}{4CN}\bigg)^{d/2}\le e^{{1}/{(8C^2)}}|B^2_N|,
\Big [ \begin{array}{c c}{a_{1}}&{a_{2}}\\{b_{1}}&{b_{2}}\\ \end{array}\Big ] \sim \Big [ \begin{array}{c}{a_{1}}\\{b_{1}}\\ \end{array}\Big ] \times \Big [ \begin{array}{c}{a_{2}}\\{b_{2}}\\ \end{array}\Big ] .
\begin{cases}H_n = n, H_{(L-m-1) + n}= L - m - 1 + n + \frac{n(n+1)(n+2)}{6}, begin{align*}1.5mm] G_n = n, G_{L-m + n}= L - m + n + \frac{n(n+1)(n+2)}{6}, .\end{cases}
\begin{array}{r l}{\tan \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right) =}&{( - 1 )^{n}\left( \frac{E + m}{E - m}\right)^{1 / 2}\left( \frac{k}{2 m}\right)^{2 \nu + 1}}\\{}&{\times \frac{\Gamma ( - \nu )}{\Gamma ( \nu + 1 )}\tan \mu .}\\ \end{array}
T_1 f = \sum_i f(x_i) \sigma(x,x^{-1}x_i)\psi_i\# W^\rho_u(u) T_2 f = \sum_i \lambda_i(f) \ell_{x_i}^\sigma W^\rho_u(u) T_3 f = \sum_i c_i f(x_i) \ell_{x_i}^\sigma W^\rho_u(u)
\begin{array}{ccc}{\cal E}=\int_{M_3}<(H_{[\mu \nu}\otimes I_E-I_E\otimes K_{[\mu \nu}) \wedge (D_{\rho ]}^A\Phi_A\otimes I_E-I_E\otimes D_{\rho ]}^B\Phi_B)> -\int_{M_3}<(D_{[\mu}^A\Phi_A\otimes I_E)\wedge (H_{\nu \rho ]}\otimes I_E)> + (A\leftrightarrow B,\Phi_A\leftrightarrow \Phi_B).\end{array}
\sum_{i \in \mathbb{Z}}L_i \left[ \psi(\cdot,\mu)\right](\nu) = \sum_{i \in \mathbb{Z}}\Big( t \nu_i(1-\nu_{i+1}) + (1-\nu_i)\nu_{i+1}\Big)\Big[\psi(s_i \nu, \mu) - \psi(\nu,\mu) \Big],
B[N_G]\cap\bigl(B^{\Sigma}+B^{\Sigma'}+B^{G}\bigr) = B[N_G]\cap\bigl(I_GB +B^{\Sigma}+B^{G}\bigr) &=I_GB + \bigl(B[N_G]\cap\bigl(B^{\Sigma}+B^{G}\bigr)\bigr)
J_{\vec{a}\vec{b}}^{\vec{\alpha}}(w)=\frac{\omega^{-\alpha_2 (\alpha_1+1)}\sigma_{2\vec{\alpha}-\vec{a}+\vec{b}}(0)}{(\sigma_{\vec{\alpha}}\sigma_{\vec{a}-\vec{\alpha}}\sigma_{\vec{a}-\vec{b}-\vec{\alpha}}\sigma_{\vec{b}+\vec{\alpha}})(w/n)}.
h_m(\rho) := \begin{dcases}-f(2-\rho) + \sum_{n = 0}^{(m-1)/2}\frac{2(\rho-1)^{2n}}{(2n)!}f^{(2n)}(1) m f(2-\rho) + \sum_{n = 1}^{m/2}\frac{2(\rho-1)^{2n-1}}{(2n-1)!}f^{(2n-1)}(1) m \end{dcases}.
\lim_{x\to\infty}\frac{1}{x}\left(\frac{m(x)}{M(k^t(x),l^t(x))}\right)^{\frac{1}{1-t}}=\lim_{x\to\infty}\frac{1}{x}\left(\frac{xm\left(\frac{1}{x}\right)}{k^t(x)m\left(\frac{l^t(x)}{k^t(x)}\right)}\right)^{\frac{1}{1-t}}&=\lim_{x\to\infty}\left(\frac{x}{k(x)}\right)^{\frac{t}{1-t}}>1.
\begin{array}[c]{l}Ps_{n}^{m}\left({x,\gamma^{2}}\right) \sim\times\left\{{\left({x^{2}-1}\right) \left({x^{2}-\sigma^{2}}\right)}\right\}^{-1/4}\times\left\{{\cos\left({\gamma\xi-\frac{1}{2}m\pi-\frac{1}{4}\pi}\right)+{O}\left( \xi^{-1}\right)}\right\}\quad\left({\eta=\xi^{2}\rightarrow\infty}\right) .\end{array}
rank\left(\mathbf{I}- \mathbf{B}_{j}\right) = L ~~\forall~1 \leq j \leq \omega, &rank\left(\mathbf{B}_{\omega+1}+ \mathbf{M}_{\omega}\mathbf{M}_{\omega-1}\cdots\mathbf{M}_1\right) = L &~~~~~~~~~~~~~~~~~~\forall~\mathbf{M}_j\in \left\{\mathbf{I}, \mathbf{B}_{j}\right\}, 1 \leq j \leq \omega.
\lim_{\tau \to \infty}\frac{\mathrm{Tr}\left(\Theta_\tau(\xi)e^{-H^{\prime,c}_\tau}\right)}{\mathrm{Tr}\left(e^{-H^{\prime,c}_\tau}\right)}= \frac{\int d\mu \, \Theta(\xi)e^{-H^c}}{\int d\mu \, e^{-H^c}}\,.
\Phi(\omega;\lambda)\simeq{2 \omega^{D-2}\over (4\pi)^{(D-1)/2}}\sum_{n=0}^\infty\left[{a_n(\lambda)\over \Gamma\left({D-1 \over 2}-n\right)}\omega^{-2n}+{b_n (\lambda)\over \Gamma\left({D-2 \over 2}-n\right)}\omega^{-(2n+1)}\right]~~~.
p_{d}( N ) = \mathrm{d i m}\, V_{N + \frac{1}{2}{\alpha}^{2}}^{\alpha}= \sum_{M = 0}^{N}p_{d - 2}( M ) p_{2}( N - M ) - p_{d - 2}( N ) + \mathrm{d i m}\, T_{N + \frac{1}{2}{\alpha}^{2}}^{\alpha}\, .
\dot{\Delta}_{j}u =\varphi\left( 2^{-j}D\right) u=2^{jn}\int_{\mathbb{R}^{n}}h\left( 2^{j}y\right) u\left( x-y\right) dy \dot{S}_{j}u =\chi\left( 2^{-j}D\right) u=2^{jn}\int_{\mathbb{R}^{n}}\tilde{h}\left( 2^{j}y\right) u\left( x-y\right) dy
\left[ \Psi_{M}^{-}( x ) , \overline{{{\Psi}}}_{N}^{+}( y ) \right] = - ( 2 \pi )^{- 3}\int \frac{\left( m + i \widehat{p}\right)_{M N}}{2 p_{0}}e^{- i p ( x - y )}d^{3}p
\int d^{2}\theta \, \theta^{2}\Bigr \{\frac{\delta W_{m}}{\delta \bar{\Phi}_{A}( \theta )}+ \Phi^{A}( \theta ) \Bigr \}= 0 \qquad \Longleftrightarrow \qquad \frac{\delta W_{m}}{\delta \eta_{A}}= \phi^{A},
\mathbb{E}^V \theta [\frac{N}{\pi}\int_I \tilde{G}_{\mu \nu}(E+i\eta)q(\operatorname{Tr}\mathcal{X}_E (Q_1))]-\mathbb{E}^V\theta[\frac{N}{\pi}\int_I \tilde{G}_{\mu \nu}(E+i\eta)q(\operatorname{Tr}f_E(Q_1))dE]=o(1).
{\cal L}_{b r a n e}= - 2 g \, \left( \delta ( x^{5}) - \delta ( x^{5}- \tilde{x}^{5}) \right) \left( \sqrt g_{( 4 )}3 W ( \phi ) +{\frac{1}{4 !}}\varepsilon^{\mu \nu \rho \sigma}A_{\mu \nu \rho \sigma}\right) .
\gamma = \gamma_{\underline{{{a}}}}^{1}I^{\underline{{{a}}}}+ \gamma_{\underline{{{a}}}\underline{{{b}}}}^{1}I^{\underline{{{a}}}}I^{\underline{{{b}}}}+ . . . , \mathrm{~ a n d ~}Q_{\mu}= q_{\mu , \underline{{{a}}}}^{1}I^{\underline{{{a}}}}+ q_{\mu , \underline{{{a}}}\underline{{{b}}}}^{2}I^{\underline{{{a}}}}I^{\underline{{{b}}}}+ . . .
\begin{aligned}\frac{d}{dt}\int |\nabla^3 B|^2 dx+\frac{5}{1+t}\int |\nabla^3 B|^2 dx &\lesssim (1+t)^{-2}\|\nabla^2 B\|_{L^2}^2+(1+t)^{-6}&\lesssim (1+t)^{-2}(1+t)^{-\frac{7}{2}}+(1+t)^{-6}&\lesssim (1+t)^{-\frac{11}{2}}.\end{aligned}
\mathbb{L}'(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})({\mathbf U}^{\pm},\Psi^{\pm}) &=L(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm}){\mathbf U}^{\pm}+C(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm}){\mathbf U}^{\pm}-\{L(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})\Psi^{\pm}\}\frac{\partial_1\hat{{\mathbf U}}^{\pm}}{\partial_1\hat{\Phi}^{\pm}},
\begin{cases}\mu_{-}(D_N\hat u_{-m}+ i\xi_m\hat u_{-N})|_-- \mu_{+}(D_N\hat u_{-m}+ i\xi_m\hat u_{-N})|_+= 0, (2\mu_{-}D_N\hat u_{-N}- \hat \pi_-)|_- = -\sigma_-A^2\hat H(0), (2\mu_{+}D_N\hat u_{-N}+ (\nu_{+}-\mu_{+})(i\xi'\cdot\hat u'_+ + D_N\hat u_{+N})|_+ = -\sigma_+A^2\hat H(0), \hat u_{-m}|_- - \hat u_{+m}|_+ = 0 \end{cases}
[ (a_i), (f_i), \sigma_f ][ (b_i), (g_i), \sigma_g ] = ( \sigma_f*( (b_i), (g_i) ) + (a_i, f_i), \sigma_g\circ\sigma_f ) = [ ( (b_{\sigma_f(i)}), (g_{\sigma_f(i)}))( (a_i), (f_i) ), \sigma_g\circ\sigma_f] = [ ( b_{\sigma_f(i)}+ g_{\sigma_f(i)}a_i), (g_{\sigma_f(i)}f_i), \sigma_g\circ\sigma_f ].
u_1 = v_1 w_1 = v_1 u_2 = -\lambda_3v_2 w_2 = (\lambda_2 + 1)u_3 + u_2 = (\lambda_3\lambda_2 + \lambda_2 + 1)v_2 + (\lambda_2 + 1) v_3 - \lambda_3v_2 u_3 = (\lambda_3 + 1)v_2 + v_3
g^{ii}= \frac{3{u^i}^2+{u^j}^2+{u^k}^2 -3u^iu^j-3u^iu^k+u^ju^k}{(u^i-u^j)^2(u^i-u^k)^2}= \frac{(u^i-u^j)^2 + (u^i-u^k)^2 + (u^j-u^k)^2 +4(u^i-u^j)(u^i-u^k)}{(u^i-u^j)^2(u^i-u^k)^2}, g^{ij}= \frac{-1}{(u^i-u^j)^2},
\mathcal{A}_{i}=\frac{\partial f}{\partial \eta_i}(\eta), \, \, 1\leq i\leq n, \, \mbox{where}\, \, f(t\eta)=t^{p}f(\eta)\, \, \mbox{when}\, \, t>0,\,\,\eta\in\mathbb{R}^{n}\setminus \{0\}
( C_{i j}^{n}( q ) ) = ( C_{j i}^{n}( q ) ) = \frac{n}{[ n ]_{q}}\, \frac{[ 2 ]_{q}^{2}}{q^{2 n}+ 1 + q^{- 2 n}}\, \left( \begin{array}{c c}{{q^{n}+ q^{- n}}}&{{( - 1 )^{n}}}\\{{( - 1 )^{n}}}&{{q^{n}+ q^{- n}}}\\ \end{array}\right) \, ;
\Big\{x\in \Lambda: x \textrm{has no universal coding}\Big\}\subseteq \bigcup_{k=0}^{\infty}\bigcup_{N=0}^{\infty}\bigcup_{(\epsilon_{i})\in\{1,\ldots, n\}^{N}}f_{\epsilon_{1}}\cdots f_{\epsilon_{N}}(U_{B_{k}}).
{\displaystyle V^{(1)}_{\underline\alpha}=V_{\underline\alpha}-\frac{i{\cal P}_{\underline{\alpha\dot\beta}}\bar V^{\underline{\dot\beta}}}{(m-ieW/2)};\ bar V^{(1)}_{\underline{\dot\alpha}}=\bar V_{\underline{\dot\alpha}}-\frac{i{\cal P}_{\underline{\beta\dot\alpha}}V^{\underline\beta}}{(m+ie\bar W/2)},}
\mathbf{E}S_{3(t-\tau_{n})}X_{t}\,(x_{n})=\mathbf{EE}\bigl[S_{3(t-\tau_{n})}X_{t}\,(x_{n})\,\bigl|\mathcal{F}_{\tau_{n}}\bigr] begin{align*}2pt] =\mathbf{E}S_{3(t-\tau_{n})}S_{(t-\tau_{n})}X_{\tau_{n}}(x_{n})=\mathbf{E}S_{4(t-\tau_{n})}X_{\tau_{n}}(x_{n}).
\left<\alpha,v\right> =\int_\Lambda\,\iota_\Lambda^*\delta g\,\kappa &+\int_\Lambda\iota_\Lambda^*(\left\{\tilde{p}_\mu,\delta g\right\})\,\mu+\int_\Lambda \iota_\Lambda^*\tilde{p}_\mu\,\delta\mu.
F\left(\{x_{i,r}\}\right)=0 \ \ \mathrm{once}\ \x_{i,2}=q^2 x_{i,1}\ \mathrm{and}\x_{i+\epsilon,1}=qd^{-\epsilon}x_{i,1}\ \ \mathrm{for}\ \i\in [n] \,,\ \epsilon=\pm 1 \,.
\bigg|\int_{B(g_*, \eta r)}K_j(g_1, g_2) dg_2\bigg| =\int_{B(g_*, \eta r)}\big|K_j(g_1, g_2)\big| dg_2 &\geq \int_{B(g_*, \eta r)}C(j,n) r^{-Q}dg_2 &=C(j,n) r^{-Q}\eta^Q r^Q &=C(j,n) \eta^Q,
\rho_{1}[ \beta ; \phi , \phi ] = - \frac{\lambda}{4 !}\rho_{0}\left[ \int_{1}\widehat \varphi^{4}( 1 ) + 6 \int_{1}G_{\eta}( 1 ; 1 ) \widehat \varphi^{2}( 1 ) + \int_{1}\langle \eta^{4}( 1 ) \rangle \right] .
A_{11}^T \Sigma_1^{-1}+\Sigma_1^{-1}A_{11}+\sum_{i=1}^m N_{i, 11}^T \Sigma_1^{-1}N_{i, 11}+\Sigma_1^{-1}k^2 \leq -\Sigma_1^{-1}B_1B_1^T\Sigma_1^{-1}, A_{11}^T \Sigma_1+\Sigma_1 A_{11}+\sum_{i=1}^m N_{i, 11}^T \Sigma_1 N_{i, 11}+\Sigma_1 k^2 \leq -C_1^TC_1.
(1-\delta)\binom{N}{2}=(1-\delta)(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2}\leq k \left(k-\alpha-\frac{1}{4}\right)\frac{n^2}{2}(1-\delta)\alpha^2-(1-2\delta)k\alpha+\frac{k}{4}-(1-\delta)\frac{k-\alpha}{n}-\delta k^2 \leq 0\,.
Q >-\frac{3-\nu}{\nu-1}\frac{2}{\nu-1}r_1\theta(r_1)+\frac{3-\nu}{\nu-1}\frac{r_1^3}{3}\theta^{\nu}(r_1) &=-\frac{3-\nu}{\nu-1}\frac{2}{\nu-1}r_1\theta(r_1)+\frac{3-\nu}{\nu-1}\frac{r_1^3}{3}\theta(r_1)g(r_1).
\mathcal T_{\delta}= \left\{\gamma = \gamma_1 \cup \cdots \cup \gamma_n\mid \begin{array}{ll}n \in \mathbb N, \gamma_j \in \mathcal S_{\delta}\gamma X \setminus \{p_0, p_{\infty}\}\text{to a circle around $p_0$in}U_{\delta}(p_0) \end{array}\right\}.
\varepsilon^{\Phi\Phi}_{ppqqj}= \Bar{\delta}^{\Phi\Phi}_{ppq(q+1)j}\delta^{\Phi\Phi}_{pp(q+1)qj}+ \Bar{\delta}^{\Phi\Phi}_{p(p-1)qqj}\delta^{\Phi\Phi}_{(p-1)pqqj}, \varepsilon^{\Psi\Psi}_{ppqqj}= \delta^{\Psi\Psi}_{p(p+1)qqj}\Bar{\delta}^{\Psi\Psi}_{(p+1)pqqj}+ \delta^{\Psi\Psi}_{ppq(q-1)j}\Bar{\delta}^{\Psi\Psi}_{pp(q-1)qj}
( n + 2 )^{2}b_{n + 2}= \frac{m_{0}^{2}}{8 b^{2}}b_{n + 1}- \sum_{k = 0}^{n}( k + 1 ) ( - 1 )^{n - k}( \frac{1}{2^{n - k + 1}}- 1 ) b_{k + 1}+ \frac{m_{0}^{2}}{8}\sum_{k = 0}^{n}( - 1 )^{n - k}b_{k}( \frac{1}{2^{n - k + 1}}- 2 ) .
(M_{t})_{i,j}=\begin{cases}G(a_1,\dots,a_{J+1})^t, i-1=\sum_{\ell=1}^J a_{\ell+1}10^{\ell-1},\, j-1=\sum_{\ell=1}^J a_\ell 10^{\ell-1}&a_1,\dots,a_{J+1}\in\{0,\dots 9\}, 0, \end{cases}
\left( \begin{array}{c c}{\partial_{r}-{\frac{a + j - 1 / 2}{r}}}&{- \sqrt{2}e v f}\\{- \sqrt{2}e v f}&{\partial_{r}+{\frac{j + 1 / 2}{r}}}\\ \end{array}\right) \left( \begin{array}{c}{h_{2}( r )}\\{h_{3}( r )}\\ \end{array}\right) = 0 ,
\sup_{\beta\in\mathbb{R}}\sum_{\substack{q\le Q d|q}}\sum_{\substack{0<a<q (a,q)=1}}\sup_{|\eta|<\delta}F_{Y}\Bigl(\frac{a}{q}+\beta+\eta\Bigr) \ll \Bigl(1+\frac{\delta Q^2}{d}\Bigr) \Bigl(\Bigl(\frac{Q^2}{d}\Bigr)^{27/77}+\frac{Q^2}{d Y^{50/77}}\Bigr).
g(A)= 2^{k-1}(2^k-1)m^2+\frac{1}{2}(d-1)(2^k-1)m+(2^{2k-1}+k2^{k-1}-2^{k+1})m &\ \ +2^{2k-3}+(d+k)2^{k-2}-5\cdot 2^{k-2}-d+1.
\sum_z \sum^{d}_{j=1}\{z_j - t_j (\tau) \lambda_j \}p (z(\tau) \mid \lambda) \log p_{\pi}(z(\tau)) = \sum_z \sum^{d}_{j=1}t_j (\tau) \lambda_j p ( z (\tau) \mid \lambda )\log \frac{p_\pi ( z (\tau) + \delta_j )}{p_\pi ( z (\tau) )}.
[\begin{matrix}N_{11}\\N_{22}\\N_{12}\end{matrix}]=\frac{2Eh}{1-\nu^{2}}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}u_{1,1}^{0}\\u_{2,2}^{0}\\ \frac{1}{2}(u_{1,2}^{0}+u_{2,1}^{0})\end{matrix}],
\{z_i , z_k \}= \begin{cases}\langle \gamma^i(\alpha_i), \, \gamma^k(\alpha_k) \rangle z_iz_k, \mbox{if}\gamma_i=e -\langle \gamma^i(\alpha_i), \, \gamma^k(\alpha_k) \rangle z_iz_k -\langle \alpha_i, \alpha_i \rangle \sigma_i(z_k) \mbox{if}\gamma_i=s_i\end{cases},1\leq i < k \leq n,
\frac{1}{N(\lambda)}\sum_{|\xi|\leq \lambda}\sum_{i,j=1}^{d_\xi}d_\xi(A\xi_{ij},\xi_{ij})= \frac{1}{N(\lambda)}\sum_{|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx,
\frac{2}{d-2}\times \frac{d-1}{d+1}\cdot \frac{d-2}{2}= \frac{d-1}{d+1}\rightarrow 1, \frac{d-2}{2}- \frac{d-1}{d+1}\cdot \frac{d-2}{2}= \frac{d-2}{d+1}\rightarrow 1 .
\int_0^1\int_{0}^{\infty}\left\{\rho \partial_t \phi + J \partial_x \phi\right\}\,dxdt + \int_0^1 \rho_0(x) \phi(x,0)\,dx =0 & \int_0^1\int_{0}^\infty \left\{J \partial_t \phi + \rho \partial_x \phi - 2k(x) g(J)\right\}\,dxdt + \int_0^1 J_0(x) \phi(x,0)\,dx =0
\begin{array}{lll}L_{IJ}^{(+)}L_{JK}^{(+)}- (-1)^{P_J(P_I+P_K)}L_{JK}^{(+)}L_{IJ}^{(+)}= (q^{-1}-q)(-1)^{P_IP_J}L_{JJ}^{(+)}L_{IK}^{(+)}, (I<J<K)\end{array}
\varphi_\lambda(\exp H)=\frac{\pi(\rho)}{|W_0|\,\pi(i\lambda)\,\pi_0(\rho_0)}\,\frac{\sum_{s\in W}\det (s)\,\pi_0(is\lambda)\,e^{is\lambda(H)}}{\prod_{\alpha\in\Delta^+\setminus\Delta_0^+}\left(e^{\alpha(H)}-e^{-\alpha(H)}\right)}
\begin{array}{l l l}{{\alpha_{i k}}}&{{=}}&{{1 \otimes \cdots \otimes 1 \otimes \alpha_{i}\otimes 1 \otimes \cdots \otimes 1 \in H^{1}( ( M )^{N},{\bf{Z}})}}\\{{\beta_{k}}}&{{=}}&{{1 \otimes \cdots \otimes 1 \otimes \beta \otimes 1 \otimes \cdots \otimes 1 \in H^{2}( ( M )^{N},{\bf{Z}}) ,}}\\ \end{array}
\begin{gathered}|Du(t,x)\cdot h|\le C(T,\varphi) t^{-\alpha}|(-A)^{-\alpha}h|, |D^2u(t,x)\cdot (h,k)| \le C(T,\varphi) t^{-(\beta+\gamma)}|(-A)^{-\beta}h||(-A)^{-\gamma}k|,\end{gathered}
\int_0^{\rho_L(\theta)}h(f(r, \theta))G(r, \theta)\,dr &\quad\geq \left(\int_0^{\rho_L(\theta)}G(r, \theta)\, dr\right) \cdot \left( (\alpha+1) \int_0^{1}h\Big(f(0, \theta)\tau\Big)(1-\tau)^{\alpha}d\tau\right) \quad=\beta_{\alpha}(\theta)\left(\int_0^{\rho_L(\theta)}G(r, \theta)\, dr\right)=\beta_{\alpha}(\theta)\mathbb{G}(\rho_L(\theta), \theta);
\begin{cases}\Psi_{2n}(X,Y) = \Psi_n(X,Y) \left ( \Psi_{n+2}(X,Y)\Psi_{n-1}^2(X,Y) - \Psi_{n-2}(X,Y)\Psi_{n+1}^2(X,Y) \right)/2Y; \Psi_{2n+1}(X,Y) = \Psi_{n+2}(X,Y)\Psi_n^3(X,Y) - \Psi_{n+1}^3(X,Y)\Psi_{n-1}(X,Y). \end{cases}
\chi_{\psi}^n = \left\{(e^{j (\pi - \angle \varphi_n^+ )}, e^{j (\frac{3}{2}\pi - \angle \varphi_n^+ )}), (e^{j (\pi - \angle \varphi_n^- )}, e^{j (\frac{1}{2}\pi - \angle \varphi_n^- )}) \right\}.
\|w\|_{L^\infty_zL^{2p}_{xy}}\le C\|w\|_{W^{1,p}_zL^{2p}_{xy}}\le C\|\partial_zw\|_{L^p_zL^{2p}_{xy}}= C\|\mathrm{div}_H\,v\|_{L^p_zL^{2p}_{xy}}&\le C\|v\|_{L^p_zW^{1,2p}_{xy}}\le C\|v\|_{L^p_zH^{1+1/p,p}_{xy}}\le C\|v\|_{H^{1+1/p,p}(\Omega)},
M_{H,1}(Z_0,J,\rho_1 \lambda_2) =\|f_0\|^2_{\partial D_1\times M_1,\rho_1 \lambda_2}=\|F_2\|^2_{\partial D_1\times M_1,\rho_1 \lambda_2}+\|f_1f_2\|^2_{\partial D_1\times M_1,\rho_1 \lambda_2}\ge M_{\partial D_1}\times M_{M_1}.
m^2_{\pm}=\frac{1}{2 Z^2}\left(m^2_{r}+(1+ 6 \xi \gamma^2) m^2_h \pm ((m^2_{r}-m^2_h(1+ 6 \xi \gamma^2 ))^2+ 144 \gamma^2 \xi^2 m^2_{r}m^2_h )^{1/2}\right)
S^{(2)}=\frac{(ie)^{2}}{2!}\int d^{4}xd^{4}x^{\prime}T\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)\overline{\psi}(x^{\prime})\gamma^{\nu}\psi(x^{\prime})A_{\nu}(x^{\prime}).
S = S^{( 1 )}+ S^{( 2 )}+ . . . = - \int_{0}^{\infty}\! \! d l \ \eta ( l ) - \frac{1}{2}\int_{0}^{\infty}\! \! d l \int_{0}^{l}\! \! d l^{\prime}\ [ \eta ( l ) , \eta ( l^{\prime}) ] + . . . .
1 - \| \rho \|\, \epsilon < \theta \circ T^n (q) =\|\rho\| \,\big( \frac{1}{\|\rho\|}\rho \circ T^n(q) \big)+ \| \theta-\rho \| \,\big( \frac{1}{\|\theta-\rho\|}(\theta-\rho) \circ T^n(q)\big).
\tilde{g_{1}}(\lambda^{2},t)=\int_{0}^{t}e^{\lambda^{2}\tau}u_{x}(0,\tau)d\tau,\tilde{g_{0}}(\lambda^{2},t)=\int_{0}^{t}e^{\lambda^{2}\tau}u(0,\tau)d\tau,\lambda\in\mathbb{C}.