harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\partial_t\boldsymbol u + \boldsymbol u\cdot \nabla \boldsymbol u + \nabla\pi = \theta \boldsymbol e_2 + S\boldsymbol e_2,\boldsymbol e_2=(0,1),{\mathrm{div}\,}\boldsymbol u=0, \partial_t \theta - \Delta\theta + \boldsymbol u\cdot\nabla\theta +\boldsymbol u\cdot\nabla S = \Delta S, \end{cases}
(\begin{matrix}N\\x_{1},\cdot\cdot\cdot x_{n}\end{matrix})p_{1}^{x_{1}}\cdot\cdot\cdot p_{n}^{x_{n}}\approx e^{-\frac{N}{2}\sum_{i}\frac{(\hat{p}_{i}-p_{i})^{2}}{p_{i}}}
\|\tilde \phi_1[T]- (p, 0)\|_{H^1_x\times L^2_x}= \|\left(\phi_0[T]- (p, 0)\right)^p\|_{H^1_x\times L^2_x}\leq 9 \varepsilon \|\phi_0[T]- (p, 0)\|_{H^1_x\times L^2_x},
\frac{C_{2,n}}{(k-4)!}- \frac{C_{1,n}}{(k-3)!}e_1 = 2 [(2n-3) a_k + b_{k-1}] e_2 + \Bigl[\frac{2}{k}(n-1)(n-k) a_{k-1}+ \frac{1}{k-1}(n-k+1) b_{k-2}\Bigr] e_1 - \frac{2n(n-1)}{k(k-1)}(2k-3) a_{k-2}- \frac{2n}{k-1}b_{k-3}.
\psi_0^S(x) ={}\max \bigg\{-\frac{\theta^{Q_x}(t_0^S,s)}{\theta_u' (t_0^S+\langle s,u\rangle)}\mid s \in S, t_0^S=t_0^s\bigg\}\psi_1^S(x) ={}\min \bigg\{-\frac{\theta^{Q_x}(t_1^S,s)}{\theta_u '(t_1^S+\langle s,u\rangle )}\mid s \in S, t_1^S=t_1^s\bigg\}.
\begin{array}{r l}\mathcal{L}_\xi(Z_\alpha-V_{\alpha}-R_\alpha)_{ij}(x)= \int_{B_0(s_\alpha)}\mathcal{H}_{ij,\alpha}(x,y)_p\overrightarrow{\Delta}_\xi (Z_\alpha-V_{\alpha}-R_\alpha)^p(y)\,dy &+\int_{\partial B_0(s_\alpha)}\mathcal{H}_{ij,\alpha}(z_\alpha,y)_p\nu_p\mathcal{L}_\xi (Z_\alpha-V_{\alpha}-R_\alpha)^{pq}(y)\,d\sigma.\end{array}
2\sum_{t=1}^{\chi-1}\left(\sum_{j=t}^{\chi-1}\frac{1}{j}\right)^2\le \,\sum_{t=1}^{\chi-1}2\left(\frac{1}{t}+\log\frac{\chi-1}{t}\right)^2\le\sum_{t=1}^{\chi-1}\left(\frac{1}{t^2}+\log^2\frac{\chi-1}{t}\right) \le \,\sum_{t=1}^{\infty}\frac{1}{t^2}+(\chi-1)\int_0^1\log^2 x\,dx=O(\chi).
(b) \!=\!\mathrm{C}_{K\!+\!1}r^{\!-\!K\!-\!1}\sum\limits_{i=0}^K\mathrm{C}_i r^{\!-\!i}\!-\!\mathrm{C}_K r^{\!-\!K}\sum\limits_{i=1}^{K\!+\!1}\mathrm{C}_i r^{\!-\!i}\!-\!\mathrm{C}_K r^{\!-\!K}\!<\! \sum\limits_{i=0}^K{\left(\mathrm{C}_{K+1}\mathrm{C}_i-\mathrm{C}_K\mathrm{C}_{i+1}\right) r^{-k-i-1}}<0.
\displaystyle \mathcal{L}_{\sigma}(w\otimes \widetilde{v},w'\otimes \widetilde{v}')=\int_{P_{Z'}(F)}\int_{H_\natural(F)}\left(\widetilde{v},\widetilde{\sigma}(ph_\natural)\widetilde{v}'\right) (w,\sigma_{1,V}(p)w')\delta_{H_{\overline{Q}}}(p)^{1/2}dh_\natural d_Lp
Z_{L}^{( n )}( \{q_{N_{s}}\}) = \tilde{Z}^{( n )}( \{q_{N_{s}}\}, L ) H ( \{q_{N_{s}}\}) \prod_{s = 1}^{n}\frac{( - 1 )^{2 l_{1 s}+ 2 l_{2 s}- 1}1 6^{2 l_{1 s}}Z^{( 1 )}( k_{s}; l_{1 s}, l_{2 s})}{k_{s}^{( 3 - 2 l_{1 s}) / 2}}
=\left[\frac{N^2 +224 N +5344}{3600}-\frac{(N-1)(N+42)}{18(N+82)}\left(\frac{B}{\phi B^{\prime}}\right) +\frac{(N-1)(N+42)^2}{18(N+82)^2}\left(\frac{B}{\phi B^{\prime}}\right)^2\right.
{\cal{L}}_N W^N=\frac{1}{2N}\sum_{i\neq j}\int \,dv_i' \,dv_j'\, K(v_i,v_j| v_i',v_j')\delta( v_i+v_j-v_i'-v_j') \delta( v_i^2+v_j^2-v_i'^2-v_j'^2 ) &\{W^N(v_1,\ldots, v_{i}',\ldots, v_{j}',\ldots, v_N)-W^N(v_1,\ldots, v_N)\},
g_a =\bar{g}+ (V+a^2)\sum_{i=1}^3(dx_i)^2 + \frac{1}{V+a^2}\eta^2, I_i^a =I_i-\frac{a^2V}{a^2+V}\theta\otimes I_iT + a^2dx_i\otimes T, \omega_i^a =\omega_i + a^2dx_j\wedge dx_k,
\begin{aligned}\|(1+3x)^{\frac{2}{3}}(\tilde{u}_k)_{xt}\|_{L^2_tL^2_x}\leq \|(1+3x)^{\frac{2}{3}}(u_k)_{xt}\phi_k\|_{L^2_tL^2_x}+\|(1+3x)^{\frac{2}{3}}(\phi_k)_x(u_k)_t\|_{L^2_tL^2_x}\leq C(T)\|r_k^2(u_k)_{xt}\|_{L^2_tL^2_x(Q_{k,T})}+C(T)\|\frac{(u_k)_t}{r_k}\|_{L^2_tL^2_x(Q_{k,T})}\leq C(T).\end{aligned}
\begin{cases}U''-{\Delta}U+AU=0 \hbox{in}(0,+\infty) \times \Omega, U= 0 \hbox{on}(0,+\infty) \times \Gamma_0,\cr\partial_\nu U = DH \hbox{on}(0,+\infty) \times \Gamma_1 \end{cases}
\begin{matrix}\begin{aligned}W_{1}W_{2}=UVz, W_{1}W_{3}=UVy, W_{2}W_{3}=UVx, W_{1}x =Uv W_{2}y =Uv W_{3}z =Uv W_{1}x =Vu, W_{2}y =Vu, W_{3}z =Vu\end{aligned}\end{matrix}
d s_{M 2}^{2}\approx{\frac{r^{4}}{R^{4}}}( - d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}) + R^{2}\frac{d r^{2}}{r^{2}}+ R^{2}\frac{\Lambda}{6}d s_{X_{7}}^{2}.
\int_{|M| > 3 |E|}|M+E|^K dU = \int_{|M| > 3 |E|}(|M|^2 + |E|^2 + M\overline{E}+ \overline{M}E)^{K/2}dU = \int_{|M| > 3 |E|}(A + B)^{K/2}dU = \int_{|M| > 3 |E|}|M|^K \left(1+\frac{B}{A}\right)^{K/2}dU = \int_{|M| > 3 |E|}|M|^K \left(1+\frac{K}{2}\frac{B}{A}+ O_K\left(\frac{B^2}{A^2}\right)\right) dU,
S_{F}( p ) = \sqrt{2}\left( \frac{p \! \! \! \slash + m}{p^{2}- m^{2}+ i \epsilon}- \frac{\gamma^{+}}{2 p^{+}}\right) = \sqrt{2}\frac{\bar{p}\! \! \! \slash + m}{p^{2}- m^{2}+ i \epsilon}
s^c_{(3,0;4,1)}=m_{(3,0;4,1)}+m_{(3,0;3,2)}+2 m_{(3,0;3,1,1)}+2 m_{(3,0;2,2,1)}+3m_{(3,0;2,1,1,1)}\phantom{s^c_{(3,0;4,1)}=}{}+4m_{(3,0;1,1,1,1,1)}+m_{(3,1;3,1)}+m_{(3,1;2,2)}+2 m_{(3,1;2,1,1)}+3 m_{(3,1;1,1,1,1)}\phantom{s^c_{(3,0;4,1)}=}{}+m_{(3,2;2,1)}+2 m_{(3,2;1,1,1)}.
\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!}=\frac{1}{k!}\big(e_{\lambda}(t)-1\big)^{k}=\frac{1}{k!}\sum_{m=0}^{k}\binom{k}{m}(-1)^{k-m}e_{\lambda}^{m}(t) &=\sum_{n=0}^{\infty}\bigg(\frac{1}{k!}\sum_{m=0}^{k}\binom{k}{m}(-1)^{k-m}(m)_{n,\lambda}\bigg)\frac{t^{n}}{n!}.
l^{x_1}\cdots l^{x_n}= \frac{1}{n}\sum\limits_{j=1}^{n}\sum\limits_{\sigma\in\mathfrak{S}_n}\int\limits_{0<s_1<\cdots<s_n<|l|}\prod\limits_{i=1}^{n}1_{\{l(s_i)=x_{\sigma(i+j)}\}}\,ds_i = \frac{1}{n}\sum\limits_{\sigma\in\mathfrak{S}_n}l^{x_{\sigma(1)},\ldots,x_{\sigma(n)}}.
R_{s,k}=\log_{2}\bigg(1+\frac{\mathbf{h}_{s,k}^{H}\mathbf{q}\mathbf{q}^{H}\mathbf{h}_{s,k}}{\mathbf{h}_{s,k}^{H}\mathbf{V}\mathbf{h}_{s,k}+\sigma_{sa,k}^{2}+\frac{\sigma_{sp,k}^{2}}{\rho_{s,k}}}\bigg)
v_{0}= u - w_{1}a^{- \gamma}- ( \mathrm{~ \frac{1}{1 2}~}u^{\prime \prime}+ u^{2}) a^{- 2 \gamma}+ 2 ( \mathrm{~ \frac{1}{6}~}w_{1}^{\prime \prime}+ u w_{1}) a^{- 3 \gamma}+ \mathrm{O}( a^{- 4 \gamma}) .
\| \nabla u_{x^*, \rho_0}\|_{L^{n,\infty}(B_{\theta_0})}=\| \nabla u\|_{L^{n,\infty}(B_{\theta_0 \rho_0}(x^*))}\leq \frac{1}{2}\| \nabla u_{x^*, \rho_0}\|_{L^{n,\infty}(B_1)}=\frac{1}{2}\| \nabla u\|_{L^{n,\infty}(B_{\rho_0}(x^*))}.
\begin{cases}-\Delta p-k^2p=2(\eta-\eta_0)-2\Delta\eta+2\nabla\cdot A\,\,\,\,\Omega\backslash K begin{align*}0.3cm] \frac{\partial p}{\partial n}=0\,\,\,\,\partial K begin{align*}0.3cm] p=0\,\,\,\,\partial\Omega \end{cases}
V^\mathrm{total}(N_1,N_0,n_1,n_0) =\mbox{var}\big(\widehat{\theta}\,|\,N_1,N_0\big)=\frac{S^2_1}{N_1}+\frac{S^2_0}{N_0}-\frac{S^2_{\theta}}{n_0+n_1},
\delta_{k,\ell}({\lambda_{\ell}}) = \left[1-\frac{N_k-1}{P_{k,\ell}\alpha_{k,\ell}N_k \log 2}\lambda_{\ell}\right]_+ &-\sum_{k\in \mathcal{K}_{\ell}}(N_k-1)\log_2(1-\delta_{k,\ell}(\lambda_{\ell})) = B_{\ell}.
\bar{C}= ( \bar{\rho}_{0}^{2}- 1 ) - 2 \frac{( - 1 - \bar{\rho}_{0}\bar{\rho}_{\frac 1 2}) \sinh ( k b ) + ( \bar{\rho}_{0}+ \bar{\rho}_{\frac 1 2}) \cosh ( k b )}{\sinh ( k b ) - \bar{\rho}_{\frac 1 2}( \cosh ( k b ) - 1 )}.
f_{\lambda}(t) = \frac{g_{\lambda}(t)}{\lambda^{n}}= \sum_{j=1}^{n}\frac{A_{j}(t)}{\lambda^{j}}, f_{\lambda}'(t) = \frac{g_{\lambda}'(t)}{\lambda^{m}}- \sum_{j=1}^{m}\frac{A_{j}'(t)}{\lambda^{j}}.
T ( z ) = -{\frac{1}{2}}~ : \partial_{z}X ( z ) \partial_{z}X ( z ) : ~ +{\frac{2}{3}}q^{2}~ : \psi_{1}( z ) \partial_{z}\psi_{2}( z ) : ~ -{\frac{1}{3}}q^{2}~ : \partial_{z}\psi_{1}( z ) \psi_{2}( z ) :
L =\frac{1}{4}F_{\sigma\tau}R^{\tau\sigma}-\frac{4\pi}{c}j^{\sigma}A_{\sigma}& =\frac{1}{4}F_{\sigma\tau}\epsilon^{\tau\sigma\lambda\rho}F_{\lambda\rho}-\frac{4\pi}{c}j^{\sigma}A_{\sigma},
\delta(t)=\delta(t_0)+\frac{k}{6H^2a^2(t_0)}+\frac{\dot{\delta}(t_0)}{3H}-\frac{k}{2H^2a^2(t_0)}e^{-2H(t-t_0)}+\left[\frac{k}{3H^2a^2(t_0)}-\frac{\dot{\delta}(t_0)}{3H}\right]e^{-3H(t-t_0)}
\mathcal{F}\left( u,\Omega \right) =\int\limits_{\Omega}\sum\limits_{\alpha=1}^{m}\left\vert \nabla u^{\alpha}\right\vert^{p}+G\left( x,u,\left\vert\nabla u^{1}\right\vert ,...,\left\vert \nabla u^{m}\right\vert \right) \,dx
|\beta| = \left| \left [\Delta_h \, (\mathcal{P}_h \,u - u^I), u^I - \mathcal{P}_h \,u\right]_{\mathcal{C}_h}\right| \leq \|\Delta_h\| \|u^I - \mathcal{P}_h \,u\|_{\mathcal{C}_h}^2 & \leq C \, h^{-2}\, h^4 \, |u|^2_{H^2(\Omega)}= C\, h^2 \, |u|^2_{H^2(\Omega)}.
\frac{\partial}{\partial t}f_\sigma - \Delta f_\sigma - 2 \, (1-\sigma) \, \Big \langle \frac{\nabla H}{H},\nabla f_\sigma \Big \rangle &+ 2 \, \sum_{i=1}^n \frac{H^{\sigma-1}}{\mu-\lambda_i}\, (D_i \mu)^2 - \sigma \, |A|^2 \, f_\sigma \leq 0
\partial_t a \mu^{1/2}+ (\partial_t b + \nabla_x a)\cdot v \mu^{1/2}+ \sum_i (\partial_t c+\partial_i b_i) \vert v_i \vert^2 \mu^{1/2}&\quad+\sum_{i< j}(\partial_j b_i +\partial_i b_j) v_i v_j \mu^{1/2}+ \nabla_x c \cdot \vert v \vert^2 v \mu^{1/2}=-\partial_t g_2 + R_1+ R_2.
\begin{aligned}g_1\left(x_1,x_2,y_1 \right) = 0, & \frac{\partial g_1\left(x_1,x_2,y_1 \right)}{\partial y_1}= 0, & \frac{\partial g_1\left(x_1,x_2,y_1 \right)}{\partial x_1}f_1\left(x_1, x_2, y_1 \right) + \frac{\partial g_1\left(x_1,x_2,y_1 \right)}{\partial x_2}f_2\left(x_1,x_2,y_1 \right) = 0.\end{aligned}
{1 \over (b+1)q_k}>{a_{k+1}-b\over b+1}\|q_{k}\theta\|+{b\|q_{k}\theta\|+\|q_{k+1}\theta\| \over b+1}&>{a_{k+1}-b\over b+1}\|q_{k}\theta\|+\|q_{k+1}\theta\|.
h_{E}=h_{H}\circ f-\left<A \vec{c},{\bf h}_{\vec{D}}\right> &{\bf h}_{\vec{E}}=\left(\begin{array}{c}h_{E_{1}}h_{E_{2}}\vdots h_{E_{r}}\end{array}\right)={\bf h}_{\vec{D}}\circ f -{}^{\rm t}A{\bf h}_{\vec{D}}\ .
{2\over \alpha}k_\rho \tilde\Pi_{\mu \nu, \rho \sigma}=-{k_\sigma \over 2}\left(\eta_{\mu \nu}-{k_\mu k_\nu \over k^2}\right){1\over 16\pi^2}\hskip .1truecm{\rm ln}\hskip .05truecm{\Lambda^2\over \mu^2}\neq 0.
0\leq \varliminf_{n_i\to \infty}A_s^{n_i,\infty}(\omega, \theta_t \omega')=\varliminf_{n_i\to \infty}(A_{s+t}^{n_i,\infty}(\omega, \omega')-A_t^{n_i,\infty}(\omega, \omega'))=\tilde{A}_{t,s+t}(\omega, \omega').
d\xi_1 =-(C_1^1\omega_1+C_2^1\omega_2)Y-\mu(\omega_1Y_2+\omega_2Y_1)+\theta_{12}\xi_2, d\xi_2 =-(C_2^1\omega_1-C_1^1\omega_2)Y-\mu(\omega_1Y_1-\omega_2Y_2)-\theta_{12}\xi_1.
\mathbb{P}\left[\left| H_{1}- \exp \left\{\sum_{i=1}^{k}\frac{2\mu_{i}H_{i+1}- \mu_{i}^2}{2 \sigma_{i}^2}\right\}\right|\ge \frac{\tilde{\epsilon}}{2}\right]\le \frac{\delta \tilde{\epsilon}^2}{25 \tilde{\epsilon}^2}=\frac{\delta}{25}.
g_{n}(m)=\frac{1}{2}\sum_{k=0}^{m}(\begin{matrix}m\\k\end{matrix})(\begin{matrix}n-1+m-k\\m-1\end{matrix})=\frac{1}{2}\sum_{k=0}^{min(n,m)}\frac{m}{n+m-k}(\begin{matrix}n+m-k\\k,n-k,m-k\end{matrix})
\Omega_{\pm}=\frac 1{\sqrt{2}}\left(\begin{array}{cc}\exp \pm \frac{i\phi}2 -\exp \mp \frac{i\phi}2 \exp \pm \frac{i\phi}2 \hphantom{-}\exp \mp \frac{i\phi}2\end{array}\right) ;\qquad \Omega_{\pm}^{-1}=\Omega_{\pm}^{\dagger};
P(\tau=j,R_j=k) =P(X_i\notin S_i,\;i=1,\dots,j-1,\;X_j\in S_j,R_j=k) &=\left[\prod_{i=1}^{j-1}P(X_i\notin S_i)\right] P(X_j\in S_j, R_j=k) &\le\left[\prod_{i=1}^{j-1}P(X_i\notin S_i')\right] P(X_j=1,R_j=1) &=P(\tau'=j,R_j=1),
\sum_{n,m\ge0}\bar{A}_{k,m}(n)a^mq^n=\frac{(q)_kq^{1+1+2+2+\cdots +(k-1)+(k-1)+k}}{(1-aq^{k+1})(1-aq^{k+2})\cdots(1-aq^{2k})}=\frac{(q)_kq^{k(k+1)/2+k(k-1)/2}}{(aq^{k+1})_{k}}.
\limsup_{T \to \infty}\frac{1}{T}\left( \sum_{t=1}^T \ell \bigl( \hat Y_t,Y_t \bigr) \right) \leq \inf_{f \in \mathcal{L}_{L}^d}\ \limsup_{T \to \infty}\left( \frac{1}{T}\sum_{t=1}^T \ell\bigl(f(Y_{t-d}^{t-1}),Y_t\bigr) \right) \,.
\widetilde\phi_1(u_1)=2^{-k_1n_1}s_{k_1}^{b_1}(2^{-k}u_1+y_1,y_1) \widetilde\phi_3(v_1)=2^{-k_1n_1}d_{k_1}^{\tilde b_1}(x_1,2^{-k}v_1+x_1)
\pi_{k}^{a}(x^{0},{\bf x}) \equiv \frac{\delta L'}{\delta \dot{A}_{a}^{k}(x^{0},{\bf x})}= \left (\nabla_{k}(A) A_{0}\left \{{\bf A}\right \}\right )^{a}- \dot{A}_{k}^{a}- \left ( \nabla_{k}(A) C \right )^{a},
I(M_2; Y_{2,0}^T) = \lim_{n \to \infty}I(M_2; Y^{(n)}_{2}(\Delta_n)) & = \lim_{n \to \infty}I(M_2; \Delta Y^{(n)}_{2}(\Delta_n)) & = \lim_{n \to \infty}h(\Delta Y^{(n)}_{2}(\Delta_n))-h(\Delta Y^{(n)}_{2}(\Delta_n)|M_2),
b_{0}^{\ast}( \overline{{{K}}}, t ) b_{0}^{\ast}( \overline{{{P}}}, t ) | 0 \rangle = \int \int \overline{{{\kappa}}}_{\overline{{{K}}}, \overline{{{P}}}}^{( + )}( K , P ) b^{\ast}( K , t ) b^{\ast}( P , t ) | 0 \rangle d K d P
D_m\Big\{\big[v(t),w(t)\big];\big[\bar{v}(t),\bar{w}(t)\big]\Big\}&=\sqrt{\left(\begin{array}{c}v(t)-\bar{v}(t) w(t)-\bar{w}(t)\end{array}\right)^{\top}\Theta_{ij}^{+}(t)\left(\begin{array}{c}v(t)-\bar{v}(t) w(t)-\bar{w}(t)\end{array}\right)}.
\lim_{| x | \to \infty}\langle J_{\mu}{}^{a}( x ) J_{\nu}{}^{b}( 0 ) \rangle = \delta^{a b}f_{\pi}^{2}\partial_{\mu}\partial_{\nu}\left( \frac{1}{| x |^{2}}\right) +{\cal O}\left( \frac{1}{| x |^{6}}\right) \quad .
\begin{array}{ll}\displaystyle\int_{R^n}\frac{u^{q_1+1}(x)}{|x|^t}dx=\int_{R^n}\frac{u^{q_1}(x)}{|x|^t}\int_{R^{n}}\frac{v^{q_2}(y)}{|y|^{t}|x-y|^{n-2l}}dydx begin{align*}3mm] =\displaystyle\int_{R^n}\frac{v^{q_2}(y)}{|y|^{t}}\int_{R^{n}}\frac{u^{q_1}(x)}{|x-y|^{n-2l}}dx dy=\int_{R^n}\frac{v^{q_2+1}(y)}{|y|^t}dy.\end{array}
\lim_{r \rightarrow 0}\frac{\overline{v}(r)}{r \langle \log(r) \rangle}= \int_{0}^{\infty}\overline{y}(\xi) \lim_{r\rightarrow 0}\left(\frac{\phi(r,\xi)}{r^{3/2}\langle \log(r) \rangle}\right) \rho(\xi) d\xi=0
F \rightarrow F+\frac{\Gamma(-1/2)}{(4\pi)^{3/2}R}[\int_{B^{3}}(\frac{\lambda}{2}\hat{\phi}^{2})^{3/2}dv+\frac{2}{3R}\int_{S^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1/2}ds]
(1+\frac{1}{n})^{n}=\sum_{k=0}^{n}(\begin{matrix}n\\k\end{matrix})\frac{1}{n^{k}}=\sum_{k=0}^{n}\frac{1}{k!}\times\frac{n}{n}\times\frac{n-1}{n}\times\cdot\cdot\cdot\times\frac{n-k+1}{n},
\begin{array}{r c l}{Q}{=}{\int_{0}^{L}d x \, \psi^{\dagger}\psi = - \lambda ( p_{+}- p_{-}) \ ,}{Q_{5}}{=}{\int_{0}^{L}d x \psi^{\dagger}\gamma_{5}\psi = - \frac{i \lambda \sqrt{2 \mu}}{2 \sqrt{\pi}\sqrt{L}}\, \left[ \varphi_{0}- \varphi_{0}^{\dagger}\right] \ .}\end{array}
\int_{B_R(\xi)}V_{\delta,\xi}^q \leq C\delta^{n-q\frac{n-4}{2}}\left(1+\int_1^\frac{R}{\delta}r^{n-1-q(n-3)}\, dr \right)= \begin{cases}O(\delta^{q\frac{n-2}{2}}) q<\frac{n}{n-3}O(\delta^\frac{n(n-2)}{2(n-3)}|\log \delta|) q=\frac{n}{n-3}\end{cases}
\sum\limits_{q\le Q}\sum\limits_{\substack{a=1 (a,q)=1}}^{q^k}\left| \sum\limits_{M<n\le M+N}a_n e\left(n\cdot \frac{a}{q^k}\right) \right|^2\ll_{\varepsilon}(QN)^{\varepsilon}\left(N+Q^{k+1}\right)\cdot \sum\limits_{M<n\le M+N}|a_n|^2
A^s_0 := \left(\begin{matrix}0 0 1 1 1 1 -1 0 1 -1\end{matrix}\right)A^s_1 := \left(\begin{matrix}0 0 1 1 1 -1 -2 0 1 -1\end{matrix}\right)A^s_2 := \left(\begin{matrix}0 0 1 1 1 1 2 0 1 -1\end{matrix}\right)
S_{\scriptscriptstyle Y M}= \int d^{4}x \; T r \left\{- \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}- \frac{1}{2}\bar{\lambda^{a}}\hat{D}\lambda^{a}+ \frac{1}{2}( D^{a})^{2}\right\},
\frac{1}{1+z}e^{-\Phi_z^{+}}m_{-4}e^{\Phi_z^{+}}(1+z) = m_0+4m_{-1}+6m_{-2}+4m_{-3}+m_{-4}+\frac{1}{8}z^{-4}+\frac{1}{3}z^{-3}+\frac{1}{4}z^{-2}-\frac{1}{24}-\frac{1}{8}\eta^{-4}.
P^{\mathrm{\scriptsize{p}}}\: P^{\mathrm{\scriptsize{s e a}}}= 0 = P^{\mathrm{\scriptsize{s e a}}}\: P^{\mathrm{\scriptsize{p}}}{\mathrm{a n d}}P^{\mathrm{\scriptsize{a}}}\: P^{\mathrm{\scriptsize{s e a}}}= P^{\mathrm{\scriptsize{a}}}= P^{\mathrm{\scriptsize{s e a}}}\: P^{\mathrm{\scriptsize{a}}}.
\mathrm{U}_{\mathrm{e}}(\xi,t) = -\mathcal{S}\Big[\gamma^{\textbf{ext}}_{1}\mathrm{U}_{\mathrm{e}}\Big](\xi,t) + \mathcal{D}\Big[\gamma^{\textbf{ext}}_{0}\mathrm{U}_{\mathrm{e}}\Big](\xi,t), \ (\xi,t) \in \mathbb{R}^2\setminus\overline{\Omega}\times (0,\mathrm{T}),
\begin{aligned}\phi(f_h(x)) \approx \sum_{l=0}^R \frac{\phi^{(l)}(\hat{f}_{h,2}(x))}{l!}(f_h(x)-\hat{f}_{h,2}(x))^l &=\sum_{l=0}^R\frac{\phi^{(l)}(\hat{f}_{h,2}(x))}{l!}\sum_{j=0}^l \binom{l}{j}f_h(x)^j(-\hat{f}_{h,2}(x))^{l-j}.\end{aligned}
\MoveEqLeft O(|\eta|^{-3/2}) + \int_{|\mu|>|\eta|^{3/2}/2}\frac{1}{\mu\sqrt{\eta}}|\mu/\sqrt{\eta}|^{-k-1}+ \frac{1}{\mu\sqrt{\eta}}|\mu/\sqrt{\eta}|^{-k}d\mu + \left(\frac{1}{\mu}\left|z\left(\eta+\frac{\mu}{\sqrt{\eta}}\right)\right|\right)\Bigg|_{\mu=|\eta|^{3/2}}= O(|\eta|^{-3/2}) + O(|\eta|^{-3/2-k}) + O(|\eta|^{-1/2-k})
g^* h_{\alpha \beta \gamma \delta}\: = \: \left( h_{\alpha \beta \gamma\delta}\right) \, \left( \nu^g_{\beta \gamma \delta}\right) \, \left( \nu^g_{\alpha \gamma \delta}\right)^{-1}\,\left( \nu^g_{\alpha \beta \delta}\right) \,\left( \nu^g_{\alpha \beta \gamma}\right)^{-1}
A_{q}(-q^{m})=\sum_{n=0}^{\infty}\frac{q^{n^{2}+mn}}{(q;q)_{n}}=\frac{(-1)^{m}q^{-\binom{m}{2}}a_{m}(q)}{(q,q^{4};q^{5})_{\infty}}-\frac{(-1)^{m}q^{-\binom{m}{2}}b_{m}(q)}{(q^{2},q^{3};q^{5})_{\infty}}.
{\cal H}= \frac{1}{2}\left[ \vec{E}_{\bot}^{2}+ \vec{B}\,^{2}\right] + \frac{1}{2}{\cal E}_{3}^{2}+ \frac{1}{2 L^{2}}e_{3}^{2}\left( \vec{x}_{\perp}\right) + \frac{1}{2}\vec{\eta}\,^{2}\left( \vec{x}_{\perp}\right) \ .
\Lambda_1 = x+(\zeta^5+\zeta)y+(\zeta^5+\zeta^4+\zeta^2+1)z, \Lambda_2 = x-(\zeta^5+\zeta^4+\zeta^3+\zeta+1)y+(\zeta^5+\zeta^3+\zeta^2+1)z, \Lambda_3 = x+(\zeta^5+1)y-(\zeta^3+\zeta^2+\zeta)z, \Lambda_4 = x+(\zeta^2+1)y+(\zeta^3+\zeta^2+\zeta+1)z.
X_1 = \partial_{x_1}+ \frac{x_3}{2}\partial_{t_2}- \frac{x_2}{2}\partial_{t_3}, X_2 = \partial_{x_2}+ \frac{x_1}{2}\partial_{t_3}-\frac{x_3}{2}\partial_{t_1}, X_3 = \partial_{x_3}+ \frac{x_2}{2}\partial_{t_1}- \frac{x_1}{2}\partial_{t_2}.
1 \rightarrow 3 \rightarrow 5 \rightarrow 7 \rightarrow \overline{9}=1, 2 \rightarrow 4 \rightarrow 6 \rightarrow 8 \rightarrow \overline{10}=2, 1 \rightarrow 4 \rightarrow 7 \rightarrow \overline{10}=2 \rightarrow 5\rightarrow 8 \rightarrow \overline{11}=3 \rightarrow 6 \rightarrow \overline{9}=1.
p(t)=\frac{1}{2}[\begin{matrix}1&t&t^{2}&t^{3}\end{matrix}][\begin{matrix}0&2&0&0\\ -1&0&1&0\\ 2&-5&4&-1\\ -1&3&-3&1\end{matrix}][\begin{matrix}f_{-1}\\f_{0}\\f_{1}\\f_{2}\end{matrix}]
\langle u(t) , \vartheta(t) \rangle_{(H^1)',H^1}- \int_\Omega u^{in}\vartheta(0)\ \mathrm{d}x = \int_0^t \int_\Omega u(s)\gamma(v(s)) \Delta\vartheta(s)\ \mathrm{d}x\mathrm{d}s & + \int_0^t \langle u(s) , \partial_t \vartheta(s) \rangle_{(H^1)',H^1}\ \mathrm{d}s
e_{j}\; = \; - \frac 1 2 \, \Bigl ( \sigma_{j}^{x}\sigma_{j + 1}^{x}\; + \; \sigma_{j}^{y}\sigma_{j + 1}^{y}\; + \; \frac{q + q^{- 1}}2 \, ( \sigma_{j}^{z}\sigma_{j + 1}^{z}- 1 ) \; + \; \frac{q - q^{- 1}}2 \, ( \sigma_{j}^{z}- \sigma_{j + 1}^{z}) \Bigr ) \, .
W^s_{\epsilon}(x) := \left\{y\in M : \d(f^n(x),f^n(y))\leq\epsilon ,\ \forall n \geq 0\right\}, W^u_{\epsilon}(x) := \left\{y\in M : \d(f^n(x),f^n(y))\leq\epsilon ,\ \forall n \leq 0\right\},
i \Delta^{\left( + \right)}( \xi ) = \frac{1}{4 \pi}\epsilon ( \xi_{0}) \delta ( \xi^{2}) + \frac{m}{\sqrt{- \xi^{2}}}\frac{i}{8 \pi}\left( J_{1}( m \sqrt{\xi^{2}}) \epsilon ( \xi_{0}) + i N_{1}( m \sqrt{\xi^{2}}) \right) , \, \, \xi^{2}\geq 0
Q_{\kappa'\kappa}(t) := \sum_{n=0}^\infty \int_0^t\int_0^{t_1}\cdots \int_0^{t_{n-1}}T^{(n)}_{\kappa'\kappa}(t,t_1, t_2 , \dots, t_n) d t_n d t_{n-1}\cdots d t_1.
{\rm Det}^{\star}_2\begin{pmatrix}a b c d \end{pmatrix}=[a\star d-a\star c\star a^{-\star}\star b]_{\overline{\mathbb{F}}^{\star}}={\rm Det}^{\star}_2\begin{pmatrix}a b 0 d- c\star a^{-\star}\star b\end{pmatrix}.
\ell_T(b_0 + h/\sqrt{T}) - \ell_T(b_0) = \frac{1}{\sqrt{T}}\int_0^T h(X_t).dX_t - \frac{1}{\sqrt{T}}\int_0^T b_0(X_t).h(X_t)dt & - \frac{1}{2T}\int_0^T\| h(X_t)\|^2 dt & = \frac{1}{\sqrt{T}}\int_0^T h(X_t).dW_t - \frac{1}{2T}\int_0^T\| h(X_t)\|^2 dt .
f_\mathbf{Y}(\mathbf{y}) = \int_{\mathbb{R}^{d-1}}f_{\mathbf{Y}|\mathbf{X}}(\mathbf{y}|\mathbf{x}) f_\mathbf{X}(\mathbf{x}) \mathrm{d}\mathbf{x}= \int_{\mathbb{R}^{d-1}}f_\mathbf{N}(\mathbf{n}) f_\mathbf{X}(\mathbf{y}-\mathbf{n}) \mathrm{d}\mathbf{n}.
\begin{array}{l}\displaystyle\langle P\phi,\rho\rangle_{{L_e^2}}=\int\langle K_P(x,\tilde x)\phi(x),\rho(\tilde x)\rangle d\nu_xd\nu_{\tilde x}, \displaystyle K_P(x,\tilde x):=(2\pi)^{-m}\int e^{-\sqrt{-1}(x-\tilde x)\cdot\xi}p(x,\xi)d\nu_\xi\,.\end{array}
\psi_0(r,z) = \sum_{\sqrt M \le j \le M}2^{-3j}a_3(2^j(r,z)), & \omega_0 (r,z) = \omega_0^{\theta}e_{\theta}, & \omega_0^{\theta}(r,z) = -\sum_{\sqrt M \le j \le M}b_3 (2^j(r,z)),
\beta C_{n,1}+ \sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\stackrel{d}{\to}N\left( \beta^2+\frac{1}{4}\log(1-4\beta^2), -\beta^2 - \frac{1}{2}\log(1-4\beta^2) \right).
L_t(\widehat{\Phi}_0,\widehat{\Phi}_1)= \langle\!\langle (\widehat{U}_1,-\widehat{U}_0),( \widehat{\Phi}_0, \widehat{\Phi}_1)\rangle\!\rangle+\int_0^t \int_{\Gamma}(DH(\tau), \Phi(\tau)) dxdt.
\big( \mathcal{L}^S_{t_1, t_2}\varphi \big)( x, y, z ) = \varphi'( S_{t_1, t_2}\, x ) \, S_{t_1, t_2}\, y + \tfrac{1}{2}\sum_{u \in \mathbb{U}}\varphi''( S_{t_1, t_2}\, x ) ( S_{t_1, t_2}\, z \, u, S_{t_1, t_2}\, z \, u ) .
\mathbb{\tilde{E}}_{t_{n}}^{X^{n}}[\eta]:=\sup \limits_{\sigma \in \left \{\underline{\sigma},\overline{\sigma}\right \}}\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[ \eta \right] :=\sup_{\sigma \in \left \{\underline{\sigma},\overline{\sigma}\right \}}\mathbb{E}_{P^{\sigma}}\left[ \left.\eta \right \vert X_{t_{n}}=X^{n}\right] ,
0 = x^2(u) = (\alpha_{11}^2+\alpha_{21}\alpha_{12})u + \alpha_{21}(\alpha_{11}+\alpha_{22})v = (\alpha_{11}^2+\alpha_{21}\alpha_{12})u, 0 = x^2(v) = \alpha_{12}(\alpha_{11}+\alpha_{22})u + (\alpha_{22}^2+\alpha_{21}\alpha_{12})v = (\alpha_{22}^2+\alpha_{21}\alpha_{12})v.
{(v\circ \widehat F^n)}_+(x)-{(v\circ \widehat F^n)}_-(x) =\sup_z v\circ \widehat F^n(x,z)- \inf_z v\circ \widehat F^n(x,z) \le \sup_{z\in \widehat{F}^n\pi^{-1}(x)}v - \inf_{z\in \widehat{F}^n\pi^{-1}(x)}v \to0.
\begin{array}{r c l}{{< r | \hat{U}^{- 1}( \theta_{0}) | A_{\bar{z}}( z , \bar{z}) >}}&{{=}}&{{< r | A_{\bar{z}}( z , \bar{z}) > \ \ \ ,}}\\{{}}&{{}}&{{}}\\{{< r | \hat{U}^{- 1}( k_{1}, k_{2}) | A_{\bar{z}}( z , \bar{z}) >}}&{{=}}&{{< r | A_{\bar{z}}( z , \bar{z}) > \ \ \ ,}}\\ \end{array}
{n+m \brack m}_{q}= \sum_{k=\ell}^n{n-\ell \brack k-\ell}_{q}{k+m-(k-\ell) \brack m-(k-\ell)}_{q}q^{k (k-\ell)}= \sum_{k=0}^{n-\ell}{n-\ell\brack k}_{q}{m+\ell \brack m-k}_{q}q^{k(k+\ell)},
\sum_{n=1}^{\infty}\left( -1\right)^{n}\chi\left( n\right) \sum_{m=2}^{\infty}\frac{1}{m}\left( -\frac{s}{n}\right)^{m}=\sum_{n=1}^{\infty}\left( -1\right)^{n}\chi\left( n\right) \left[ \frac{s}{n}-\log\left( 1+\frac{s}{n}\right) \right] ,
\left(x\alpha_{1}\right)\left(y\alpha_{2}\alpha_{3}^{a}\right) =\rho^{y_{2}}xy\alpha_{1}\alpha_{2}\alpha_{3}^{a}\ \left(y\alpha_{2}\alpha_{3}^{a}\right)\left(x\alpha_{1}\right) =\rho^{ax_{3}-x_{3}y_{2}}xy\alpha_{1}\alpha_{2}\alpha_{3}^{a},
\left(\vphantom{z_i^b}D_i\cdot z_i^a\right)\left(D_i\cdot z_i^b\right) = \left(z_i^{a+b}+4z_i^{q+b+1}+6z_i^{q+b+2}+4z_i^{q+b+3}+z_i^{q+b+4}\right)\left(z_i\frac{\partial}{\partial z_i}z_i^a\right)\left(z_i\frac{\partial}{\partial z_i}z_i^b\right).
\delta{\cal L}_{t o t a l}= \omega \left[ J^{\mu , a}( D_{\mu}c^{a}) - i \overline{{\eta}}^{a}( -{\frac{1}{2}}f^{a b c}c^{b}c^{c}) + i F^{a}\eta^{a}+ H^{\mu}{\frac{\partial{\cal L}_{t o t a l}}{\partial n^{\mu}}}\right]
A_{dr,2}f(\eta )=\int_{{\Bbb U}_{dr,2}}\Bigg(E\big[\prod_{(x,u)\in\eta}(\xi_{x,u}\hat{g}(x^{*},z,u)+(1-\xi_{x,u})g(x,u))\big]-f(\eta )\Bigg)\mu_{dr,2}(dz),
M(r)=\frac{r^{2}d\Phi}{Gdr}=\int_{0}^{r}dr\int_{0}^{\pi}(rd\theta)\int_{0}^{2\pi}(rsin\theta d\varphi)\rho_{0}H(r_{0}-r)=M_{0}x^{3}|_{x=\frac{r}{r_{0}}}.
\int_{-\infty}^{x}e^{\frac{y}{\alpha}}(\alpha^2 u_x^2+ (u+k)^2)(y)dy \geq 2\alpha \int_{-\infty}^{x}e^{\frac{y}{\alpha}}(u+k)u_x(y)dy &= \alpha e^{\frac{x}{\alpha}}(u+k)^2 - \int_{-\infty}^{x}e^{\frac{y}{\alpha}}(u+k)^2(y)dy
\sqrt{N}\int_0^{T^*}\gamma ( Z_s^N ) - \gamma(z(s)) ds = \sqrt{N}\int_0^{T^*}\gamma'(z(s)) ( Z_s^N - z(s) ) + \tilde R_N(s) ds = \int_0^{T^*}\gamma'(z(s)) W_s^N ds + \sqrt{N}\int_0^{T^*}\tilde R_N(s) ds
(WW^\top)_{ij}=\sum_{1\leq \ell \leq m}(N^{c_{i\ell}}C_2)(N^{c_{j\ell}}C_2)^\top &=\sum_{1\leq \ell \leq m}N^{c_{i\ell}}C_2C_2^\top N^{-c_{j\ell}}&=k\sum_{1\leq \ell \leq m}N^{c_{i\ell}-c_{j\ell}}&=\begin{cases}m kI_n O i\neq j, \end{cases}
+\infty>\int_{\mathbb{R}^N}\frac{u^\theta}{|x|^\alpha}\,dx =N\,\omega_N\, \int_0^{+\infty}u(\varrho)^\theta\,\varrho^{N-1-\alpha}\, d\varrho &\ge N\,\omega_N\, \int_0^{R}u(\varrho)^\theta\, \varrho^{N-1-\alpha}\,d\varrho\ge N\,\omega_N\, u(R)^\theta\, \frac{R^{N-\alpha}}{N-\alpha},
w_1:=[153]=ws_{\alpha_1}\textrm{with}\ \alpha_1=e_1-e_3 &w_2:=[241]=ws_{\alpha_2}\textrm{with}\ \alpha_2=e_1+e_2 &w_3:=[315]=ws_{\alpha_3}\textrm{with}\ \alpha_3=e_2-e_3 &w_4:=[321]=ws_{\alpha_4}\textrm{with}\ \alpha_4=2e_2
\left( \begin{array}{c c}{\partial_{1}}{\partial_{2}}{\partial_{4}}{\partial_{3}}\end{array}\right) \mapsto \left( \begin{array}{c c}{\partial_{1}q^{\alpha L_{6}}}{\partial_{2}}{\partial_{4}}{\partial_{3}q^{- \alpha L_{6}}}\end{array}\right) \equiv \tilde{\partial}\quad ,