harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathcal{L}_{j}}{\left\vert{\chi_{n}\left( \pm{u,t}\right) dt}\right\vert}\leq 2\int_{\mathcal{L}_{j}}{\left\vert{F_{n}\left( t\right) dt}\right\vert}+\sum\limits_{s=1}^{n-1}{\frac{1}{\left\vert u\right\vert^{s}}\int_{\mathcal{L}_{j}}{\left\vert{G_{n,s}\left( t\right) dt}\right\vert ,}}
H'_0={e^2 \over 2\pi}\int_{k>0}{dk \over 4\pi}\Biggl[\int_{p>0}dp \biggl({1 \over (k-p)^2}-{1 \over (k+p)^2}\biggr)\Biggr] \bigl(b_k^\dagger b_k+d_k^\dagger d_k\bigr) \;.
{\hat x}^i(\varepsilon)=e^{i{\hat H}\varepsilon}{\hat x}^i (0)e^{-i{\hat H}\varepsilon}\simeq{\hat x}^i(0)+{{\hat \pi}^i(0)\over m}\varepsilon +{e\over 4m^2}(2{\hat F}^{ik}{\hat \pi}_k(0)-i\partial_k{\hat F}^{ik})\varepsilon^2\,,
\mu(T_f^{-k}[a_L, \ldots, a_R]_L^R \cap [a_{L'}, \ldots, a_{R'}]_{L'}^{R'}) = \mu(T_f^{-k}[a_L, \ldots, a_R]_L^R) \mu([a_{L'}, \ldots, a_{R'}]_{L'}^{R'}) = \mu([a_L, \ldots, a_R]_L^R) \mu([a_{L'}, \ldots, a_{R'}]_{L'}^{R'})
\bigg| \sum_{k=2}^{D_xS_A}\big(\Delta f_B(S_A+k-1)-\Delta f_B(S_A)\big) \bigg| \leq 2\|\Delta f_B\|_\infty \max\{0, D_xS_A-1\}& \leq 2\varepsilon_{1,A}\max\{0, D_xS_A-1\}
C = \frac{4 (\sqrt{M}- 1)}{M \log_2 M}\left[ \frac{(\sqrt{M}- 1) \,_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}- m; \frac{3}{2}; \frac{1}{2}\right)}{\sqrt{2}\: \pi}+ \frac{\; \Gamma( m + \frac{1}{2})}{2 \sqrt{\pi}m\Gamma(m)}\right].
u_{z}=\frac{u_{z}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}},u_{z}^{\prime}=\frac{u_{z}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u_{x}},
(1-u)(1-rs) = \underbrace{(1-u)(1- e_1 r e_1 s e_1)}_{=1}- (1-\underbrace{u}_{=ue_1})(\sum_{i>1}e_i r e_1 s e_1) = 1 - \sum_{i > 1}e_i r e_1 s e_1.
(\partial X^j(z))^2 \cdot \partial_n X^i(w) \sim G^{ii}\frac{1}{(z-w)^2}\partial_n X^i \sim \frac{t^{-2p}}{(z-w)^2}\partial_n X^i, \qquad ({\rm no}~~{\rm sum}~~{\rm over}~~i)
dy_{u}^{n,s,t}=\sum_{k=1}^{\left[ p\right]}\left( F\left( f\left( \cdot+\pi_{1}\left( y_{s_{0}}^{n}\right) \right) \right) \right)^{\circ k}\pi_{k}\left( \log X_{s,t}\right) \left( Id_{L^{\left[ p\right]}\left( \mathcal{U}\right)}\right) \left( y_{u}^{n,s,t}\right) du,u\in \left[ 0,1\right] ,y_{0}^{n,s,t}=y_{s_{0},s}^{n}
\alpha(B_{n,3,3,3}^{(n-5)},x)=x^{n}-(n+1)x^{n-2}+(3n-9)x^{n-4}; \alpha(B_{n,3,3}^{(n-5)},x)=x^{n}-(n+1)x^{n-2}+(2n-5)x^{n-4}-(n-5)x^{n-6}.
\begin{aligned}b_1 =\{n_1(n_1^2-n_2^2)a_1-n_3(n_2^2-n_3^2)a_3\}/\{n_1(n_1^2-n_2^2)\}, b_2 =\{n_2(n_1^2-n_2^2)a_2+n_3(n_1^2-n_3^2)a_3\}/\{n_2(n_1^2-n_2^2)\}.\end{aligned}
\tau_C\sum^{N}_{s=1}\|(R)^{sk}\| = \tau_C\sum^{N}_{s=1}\|(R1)^{sk}+(R2)^{sk}+(R3)^{sk}+(R4)^{sk}\| \leq \tau_C\sum^{N}_{s=1}\left(\|(R1)^{sk}\|+\|(R2)^{sk}\|+\|(R3)^{sk}\|+\|(R4)^{sk}\|\right) \leq \bar{C}(T)(\tau_C^2+h_1^2+h_2^2).
\left(\left\Vert x\right\Vert +\mu\right)\left|\left\Vert x\right\Vert +\mu\right|\left(1-\frac{\lambda^{2}}{c_{2}^{2}}\right)=\mu\left|\mu\right|\left(1-\frac{\lambda^{2}}{c_{1}^{2}}\right),
h_{m f_2^-}(x) = h_{m f_1^-}(x) + g_{m(f_2^--f_1^-)}(x + mf_1^- \alpha) - \sum_{i=mf_2^-}^{mf_1^- - 1}\eta_i h_{m f_2^+}(x) = h_{m f_1^+}(x) + g_{m(f_2^+-f_1^+)}(x + mf_1^+ \alpha) + \sum_{i=mf_1^+}^{mf_2^+ - 1}\eta_i
\widetilde V_{1}( \psi , z ) = : \prod_{a = 1}^{M}{\frac{i}{( m_{a}- 1 ) !}}{\frac{d^{m_{a}}}{d z^{m_{a}}}}C_{+}^{i_{a}}( z ) \prod_{b = 1}^{N}{\frac{i}{( n_{b}- 1 ) !}}{\frac{d^{n_{b}}}{d z^{n_{b}}}}C_{-}^{j_{b}}( z ) e^{i{\frac{\bar{\lambda}}{\alpha}}\cdot C_{+}( z )}e^{i{\frac{\lambda}{\alpha}}\cdot C_{-}( z )}: \gamma_{\lambda}\, ,
A_{n}\,:=\biggl\{\Delta X_{s}\left( \Big(\frac{c}{2}\,2^{-n},\frac{3c}{2}\,2^{-n}\Big)\right) \geq2^{-(\bar{\eta}_{\mathrm{c}}+1)n}\,n^{1/(1+\beta)}s\in(t-2^{-2 n},t-2^{-2(n+1)})\biggr\}.
\mathcal{G}_1(r(T), x) = x \displaystyle\prod_{j=1}^{k}\mathcal{G}_1(r(T_j), x) - \displaystyle\sum_{j=1}^{k}\left( \alpha_j \ \mathcal{G}_2(r(T_j), x) \displaystyle\prod_{i = 1, i \neq j}^{k}\mathcal{G}_1(r(T_i), x) \right) .
\widehat{\alpha}_{(\gamma_{b_1}, b_1^{-1}\phi(b_1))}\widehat{\alpha}_{(1, \phi(b_1)^{-1}b_1a_1)}=\widehat{\alpha}_{(\gamma_{b_2}, b_2^{-1}\phi(b_2))}\widehat{\alpha}_{(1, \phi(b_2)^{-1}b_2a_2)}\Rightarrow\widehat{\alpha}_{(1, a_3)}=\widehat{\alpha}_{(\gamma_{b_1^{-1}b_2}, (b_1^{-1}b_2)^{-1}\phi(b_1^{-1}b_2))}
r_{\ell}^{\vee}( \bar{z}) = \mbox{aplim}(r_{\ell}, ( - \infty, \bar{z}), \bar{z}) \mbox{and}r_{\ell}^{\wedge}(\bar{z}) =\mbox{aplim}(r_{\ell}, (\bar{z}, + \infty), \bar{z})=0.
\sum_{n=1}^{\infty}\frac{q^n}{(q^{n};q)_{n+1}(q^{2n+2};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{q^{2n^2+2n+1}}{(q;q^2)_{n+1}}, \sum_{n=0}^{\infty}q^n (-q^{n+1};q)_n (-q^{2n+2};q^2)_{\infty}=\sum_{n=0}^{\infty}\frac{q^{n^2+n}}{(q;q^2)_{n+1}}.
J_{B}(g,h)(t,\mathbf{x},\mathbf{v})=\int\limits_{\mathbb{R}^{3}\times \mathbb{S}^{2}}b(|\mathbf{v}-\mathbf{v}_{\ast}|,\boldsymbol{\omega})(g^{\prime}h_{\ast}^{\prime}-gh_{\ast})d\mathbf{v}_{\ast}d\boldsymbol{\omega}.
\begin{array}{l}Y_t=Y_0+\int_0^t\Phi(r(s),Y_{r(s)},v(r(s),Y_{\cdot \wedge r(s)}))dW_s+\int_0^t g(r(s),Y_{r(s)},v(r(s),Y_{\cdot \wedge r(s)}))ds\ ,\textrm{for any}\t\in [0,T] \ , \end{array}
D_{\bar{v}_i}E (\bar{x}) = k \, D^+_{v_i}(d_{x_{i-1}}^2 + d_{x_{i+1}}^2 )x_i &= 2k \, d(x_{i-1},x_i) \, D^+_{v_i}(d_{x_{i-1}})x_i + 2k \, d(x_i , x_{i+1}) \, D^+_{v_i}(d_{x_{i+1}})x_i &= c_1 \, \min{\{- \langle v_i,\xi \rangle : \xi \in \widehat{x_i x_{i-1}}}\}+ c_2 \, \min{\{- \langle v_i,\eta \rangle : \eta \in \widehat{x_i x_{i+1}}}\}
\hat{H}_1=\phi^{7_3 7_3}_{(\tilde{u}+2,\bar{\tilde{u}})}\phi^{7_37_3}_{(\tilde{u},\overline{\tilde{u}+1})}\phi^{3 7_3}_{(1,\bar{\tilde{u}})}\phi^{7_3 3}_{(\tilde{u},\bar{2})}M_s^{-3}
\begin{aligned}\limsup_{N\rightarrow \infty}\left\Vert \left( f_N^{(s)}(0) -\int_{\mathcal{P}\left( \mathbb{R}^{2d}\right)}h_0^{\otimes s}d\pi(h_0) \right) \mathbf{1}_{Z_s \in \mathcal{K}_s\cap \mathcal{U}_s^{\eta(\varepsilon)}}\mathbf{1}_{E_s (Z_s)\leq R^2}\right\Vert_{L^\infty_{Z_s}}= 0\end{aligned}
p_{X^{\rho m}|S^m}(x^{\rho m}|s^m)=\mathbb{I}(x^{\rho m}=f^{(m,{\rho m})}(s^m)), &p_{\hat{S}^m|Y^{\rho m}}(\hat{s}^m|y^{\rho m})=\mathbb{I}(\hat{s}^m=g^{({\rho m},m)}(y^{\rho m})),
h^1_{1,-\beta,x}(y)=\frac{\partial f_1^{-\beta}}{\partial x}(0,y)=-6\beta ty^{1+3k}(6^3y^{3(1+3k)}+1)^{-\beta-1}, h^2_{1,-\beta,y}(x)=\frac{\partial f_2^{-\beta}}{\partial y}(x,0)=-6\beta tx^{3(1+2k)}(6^3+x^{(1+3k)3})^{-\beta-1}.
\left\{\sigma =\left(\begin{matrix}\sigma_{1,1}\sigma_{2,2}\ddots \sigma_{2g,2g}\end{matrix}\right) \in \operatorname{GL}_{2g}\left(\mathbb{Z}/\ell^n\mathbb{Z}\right) \bigm\vert \sigma \cdot \left( \begin{matrix}1 \vdots 1 \end{matrix}\right)=\left( \begin{matrix}1 \vdots 1 \end{matrix}\right) \right\},
L ( x , \dot{x}, \{q_{n}, \dot{q}_{n}\}, t ) = \frac{M}{2}\dot{x}^{2}- V ( x , t ) + \sum_{n}\left\{\frac{m}{2}( \dot{q}_{n}^{2}- \omega_{n}^{2}q_{n}^{2}) + c_{n}q_{n}F_{n}( x , t ) \right\},
\frac{1}{2(h-t)}\int_{U_{h-t}(\Gamma_\infty)}j_1(p_n(y),\nabla b_{\Omega_n}(p_n(y)),H_{\Gamma_n}(p_n(y))) \, \det(dT_\infty) \, dy = \frac{1}{2(h-t)}\int_{U_{h-t}(\Gamma_\infty)}j_1(p_n(y),\nabla b_{\Omega_n}(p_n(y)),\operatorname{Tr}\nabla^2 b_{\Omega_n}(y )) \, \det(dT_\infty) \, dy+ \operatorname{O}(h).
\begin{aligned}T^{-+}(x) = \max \left(0, \sum_{\substack{y \in S y \neq x}}\left( T^{+}(x,y) - T^{-}(x,y) \right) \right), T^{+-}(x) = \max \left(0, \sum_{\substack{y \in S y \neq x}}\left(T^{-}(x,y) - T^{+}(x,y)\right) \right). \end{aligned}
\begin{aligned}\Delta_{k+1}=\xi^{k+1}\Delta_0 + \Delta_0 \zeta \sum_{i=0}^{k}\beta^i \xi^{k-i}&\leq \xi^{k+1}\Delta_0(1+\zeta \frac{1-(\frac{\beta}{\xi})^{k+1}}{\xi-\beta}) &\leq \xi^{k+1}\Delta_0(1+\frac{\zeta}{\xi-\beta})\end{aligned}
( - 1)^m \sum_{p = 0}^n{\left\{{\binom{m+p-1}p\frac{{\mu^{m + 1}}}{{\nu^{m + p}(\nu + \mu )^{n - p + 1}}}}\right\}}& &= \frac{\mu}{{(\nu + \mu )^{n + 1}}}+ \sum_{i = 1}^m{\left\{{( - 1)^i \binom{i+n-1}n\frac{{\mu^i}}{{\nu^{i + n}}}}\right\}}\,,
\frac{\partial}{\partial_a}(t-r\psi_0(a))^{a_j}r^{b_j}+ (t-r\psi_0(a))^{a_j}r^{b_j}\cdot (t-r\psi_0(r)) = ~- a_j \psi_1(a)(t-r\psi_0(a))^{a_j-1}r^{b_j+1}+(t-r\psi_0(a))^{a_j+1}r^{b_j}.
A(z_1,z_2,z_3,z_4) = \frac{\zeta(1+z_1 + z_3)\zeta(1+z_1 + z_4)\zeta(1+z_2 + z_3)\zeta(1+z_2 + z_4)}{\zeta(2+z_1 + z_2+ z_3 + z_4)}.
\ell\left( \sum_{i=1}^s \left \lfloor \frac{n_i+t\lambda_i}{m}\right\rfloor Q_i + \sum_{i=s+1}^r \left \lfloor \frac{t\lambda_i}{m}\right\rfloor Q_i \right) &-\ell\left( \sum_{i=1}^s \left \lfloor \frac{n_i-c_i+t\lambda_i}{m}\right\rfloor Q_i + \sum_{i=s+1}^r \left \lfloor \frac{t\lambda_i}{m}\right\rfloor Q_i\right)=0
G(s,x;t,y) = \frac{1}{4\pi^2a}\int_{|\theta| \leq \pi}d\theta &\times \int_{|\lambda| \leq \pi}d\lambda e^{i(\theta x+\lambda y)/a + (t-s)(1-\cos(\lambda /a))}\sum_{j \in \mathbb{Z}}e^{-i(\theta+\lambda)j}e^{s\{\cos(\theta /a)-\cos(\lambda /a) \}},
\mathcal{R}_{1,P}(R_1')= \left\{\begin{aligned}(R_2,R_3): R_2,R_3\geq 0,\R_2\leq I(X_1;Y_2|U_1,U_2),\R_3\leq I(X_2;Y_3), R_2+R_3\leq I(X_1,U_2;Y_2|U_1)+I(X_2;Y_3|U_2) \end{aligned}\right\}.
Z_{\Phi}(\mathfrak{p},\chi,s,f)=q^{-n}\sum_{\substack{I\subset T, \forall i\in I: d\mid N_{i}}}c_{I,\Phi,\chi}\prod_{i\in I}\frac{(q-1)q^{-N_{i}s-\nu_{i}}}{1-q^{-N_{i}s-\nu_{i}}},
m_{r_{j+1}}\circ p_N^{-1}\big((a,b)\big) = \sum^N_{i=0}m_{r_{j+1}}\bigg(\Big(\frac{a+i}{N}, \frac{b+i}{N}\Big)\bigg) = \sum^N_{i=0}\int_{\frac{a+i}{N}}^{\frac{b+i}{N}}W_{r_{j+1}}(t)\,dt.
{\mathbf T}^{\rm d'A.}_{\rm int}= -\frac{\hbar^{2}}{2I}{\mathbf D}_{l}+ \frac{1}{8I}\sum_{a,b}\frac{\left({\mathbf M}^{a}{}_{b}\right)^{2}}{\left(Q^{a}-Q^{b}\right)^{2}}+\frac{1}{8I}\sum_{a,b}\frac{\left({\mathbf N}^{a}{}_{b}\right)^{2}}{\left(Q^{a}+Q^{b}\right)^{2}},
\int_{V}|g|^l \, d\nu =l \int_0^\infty t^{l-1}\nu\big(\{V: |g| >t\}\big)\, dt\geq l \sum_{i=0}^\infty\int_{\alpha_0 M^{i-1}}^{\alpha_0 M^i}t^{l-1}\nu\big(\{V: |g|>t\}\big)\, dt &\geq \sum_{i=0}^\infty \big[(\alpha_0 M^i)^l - (\alpha_0 M^{i-1})^l\big]\, \nu\big(\{V: |g|> \alpha_0 M^i\}\big).
\int_0^t |u_t|_{x=0}|^2d\tau\leq\int_0^t\|u_tr^{\frac{1}{2}}\|_{L^\infty}dx\leq\int_0^t\int_0^k\rho(r^2u_{tx})^2dxd\tau+2\int_0^t\int_0^k\rho^{-1}\frac{u_t^2}{r^2}dxd\tau\leq C(T).
f\left( \left( M+m \right){{1}_{K}}-\sum\limits_{i=1}^{n}{{{\Phi}_{i}}\left({{A}_{i}}\right)}\right)\le \left( f\left( M \right)+f\left( m \right) \right){{1}_{K}}-\sum\limits_{i=1}^{n}{{{\Phi}_{i}}\left( f\left({{A}_{i}}\right) \right)}.
X_3\psi =-X_3GV\psi +X_3Gv=-X_3GVX_1\psi+X_3Gv+\O{}(h^\infty)\psi &=-X_3GG_{\Delta}^{-1/2}\varphi+X_3Gv+\O{}(h^\infty)\psi=-X_3GG_{\Delta}^{-1/2}X_2\varphi+X_3Gv+\O{}(h^\infty)\psi.
\begin{aligned}H_r(\tilde M)/ TH_r(\tilde M)= H^{N}_r(M;\xi) \to H^{BM}_r(\tilde M)/ TH^{BM}_r(\tilde M) \to \to (H_{n-r}(\tilde M))^\ast / T(H_{n-r}(\tilde M)^\ast)\xleftarrow{=}(H^N_{n-r}(M;\xi))^\ast\end{aligned}
\ \ = 2 a^{2N-1}\cdot \frac{a}{2}\big[ (1+a)^{2N-1}- (1-a)^{2N-1}\big]+ 2 a^{2N}\cdot \frac{1}{2}\big[ (1+a)^{2N-1}+ (1-a)^{2N-1}\big] &= a^{2N}\cdot \big[ (1+a)^{2N-1}- (1-a)^{2N-1}\big]+ a^{2N}\cdot \big[ (1+a)^{2N-1}+ (1-a)^{2N-1}\big] &= 2 a^{2N}\cdot (1+a)^{2N-1}.
\eta(1,2)=(\begin{matrix}-1&2\\ 0&1\end{matrix}),\eta(1,3)=(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&-1\end{matrix}),\eta(2,3)=(\begin{matrix}0&-2\\ -\frac{1}{2}&0\end{matrix}).
\overset{\cdot \cdot}{\vec{r}}_{n}=-\left( \overset{\cdot}{\vec{r}}_{n}\cdot \overset{\cdot}{\vec{r}}_{n}\right) ~\vec{r}_{n}+g^{2}~\hat{z}\wedge \vec{r}_{n}~\sum_{\ell =1,~\ell \neq n}^{N}\left\{\frac{\vec{r}_{n}\cdot \vec{r}_{\ell}}{\left[ \left( \vec{r}_{\ell}\wedge \vec{r}_{n}\right) \cdot \hat{z}\right]^{3}}\right\}~.
\sup_{B_{R_{k}}(x_k)}u > C^{-1}\delta^{-\gamma}\left(u(x_k) - u(x_k)/2\right) &=C^{-1}\delta^{-\gamma}u(x_k)/2 &>C^{-1}\delta^{-\gamma}M_{k-1}/2\left(\frac{1}{M_{k_0}}\sup_{B_3}u + \inf_{B_1}u+\rho\right) &=M_k\left(\frac{1}{M_{k_0}}\sup_{B_3}u + \inf_{B_1}u+\rho\right),
\begin{pmatrix}\dot{y}_{1}\dot{y}_{2}\dot{z}\end{pmatrix}= \begin{pmatrix}m(l_u) 1 0 0 -[n-2-m(l_u)] 0 0 0 \varpi \end{pmatrix}\begin{pmatrix}y_1 y_2 z \end{pmatrix}+ \begin{pmatrix}0 - g(y_1,\frac{\ln(z)}{\varpi};l_u) 0 \end{pmatrix}
\sum_{m = 1}^{\infty}\frac{\sin^{2}( \pi m y )}{m^{2}- n^{2}/ y^{2}}\frac{\sqrt{x + \sqrt{x^{2}+ m^{2}}}}{\sqrt{x^{2}+ m^{2}}}\approx - \frac{\pi y^{3 / 2}}{4 n}\frac{\sqrt{\sqrt{n^{2}+ x^{2}y^{2}}- x y}}{\sqrt{n^{2}+ x^{2}y^{2}}}.
\delta \psi_{A \mu}= \mathcal{D}_{\mu}\, \epsilon_{A}- \frac{1}{3}\, \mathcal{T}_{A B}^{\rho \sigma}\left( g_{\mu \rho}\, \gamma_{\sigma}- \frac{1}{8}\, \epsilon_{\mu \rho \sigma \lambda \nu}\, \gamma^{\lambda \nu}\right) \, \epsilon^{B}
\delta^{2}_{hom}\delta^{1}_{hom}f(x_1,x_2,x_3) = (a1)+(b1)+(a2)+(b2)+(a3)+(b3)+(a4)+(b4)+(a5)+(a6)+(a7)+(a8) = C1+C2+C3+C4+C5+C6+C7+C8=0.
\phi_{a}= \left( \begin{array}{c}{{2 ( x_{1}x_{3}+ x_{2}x_{4})}}\\{{2 ( x_{2}x_{3}- x_{1}x_{4})}}\\{{x_{1}^{2}+ x_{2}^{2}- x_{3}^{2}- x_{4}^{2}}}\\ \end{array}\right) , \quad | \phi | = r^{2}, \quad n = \phi / r^{2}\equiv n_{\mathrm{H}}.
\delta E = \tilde{\epsilon}(\lambda_p) + \sum_{ia}\tilde{\epsilon}(\lambda_{ia})[N_{ia}-n^{\rm imp}_{ia}] + \frac{2\pi}{L}\sum_{ia}s_a\tilde{v}_{ia}\left[n_{ia}+ \frac{1}{2}\left(\sum_{jb}U_{jb,ia}[N_{jb}-n^{\rm imp}_{jb}] \right)^2\right]
f(x,y,n,z)^{-\alpha}=(x^2+y^2+z^2)^{-\alpha}\sum_{j=0}^\infty{\alpha+j-1 \choose j}\Big(\sum_{k=0}^\infty d_k\frac{(x^{2k+4}+y^{2k+4})}{(x^2+y^2+z^2)}n^{-2k-2}\Big)^{j}.
\begin{array}{ll}S_{H}^{(2)}=\int [d^2z]{\pounds}_H^{(2)}= \eta_{0}\int \left[ d^{2}z\right] \,\left[ p_{0}\left( \left|\psi \right|^{2}+\left| \varphi \right|^{2}\right) + \right. & \left. + 2p\left( \psi_{R,0}\psi_{R}+\psi_{I,0}\psi_{I}+\varphi_{R,0}\varphi_{R}+\varphi_{I,0}\varphi_{I}\right) \right] \quad .\end{array}
W ( x_{B}, Q ;{\mathcal T}) = \left( \begin{array}{c c c c}{W^{0 0}}&{0}&{W^{0 2}}&{0}\\{0}&{W^{1 1}}&{0}&{0}\\{W^{2 0}}&{0}&{W^{2 2}}&{0}\\{0}&{0}&{0}&{0}\\ \end{array}\right)
(\sigma_1;s_2-s_1-r_1)(\pi_1;s_1)=(\sigma_1\pi_1;s_2-r_1+\langle\sigma_1,\pi_1\rangle)=(\pi_2;s_2), (\sigma_2;s_1-s_2-r_2)(\pi_2;s_2)=(\sigma_2\pi_2;s_1-r_2+\langle\sigma_2,\pi_2\rangle)=(\pi_1;s_1).
|w_2| \lesssim \bar w_2 = T^{\frac{1}{2}{\sf c}_1}(t-t)^{\frac{\gamma}{2}+J+\frac{1}{2}{\sf c}_1}e_{J+1}(z) \lesssim T^{\frac{1}{2}{\sf c}_1}(t-t)^{\frac{\gamma}{2}+J+\frac{1}{2}{\sf c}_1}|z|^\gamma |z|<{\sf r},\t\in(0,T).
\frac{d}{dt}\Re \langle Q_{in}w, w \rangle = \ i \langle Q_{in}(P+B) w, w \rangle - i \langle Q_{in}w, (P+B) w \rangle + 2 \Re \langle Q_{in}w, - i f \rangle = \ \langle i ([Q_{in},P] + Q_{in}B - B^* Q_{in}) w, w \rangle + 2 \Re \langle Q_{in}w, - i f \rangle : = \ - \langle C w, w \rangle + 2 \Re \langle Q_{in}w, - i f \rangle .
\frac{1}{N_{F}}\sum^{N_{F}}_{j=1}\tilde{X}_{C}^{j, n}=\sum^{N_{F}}_{j=1}\sum^{N_{F}}_{i=1}\sum^{N_{F}}_{k=1}\frac{1}{N_{F}}\tilde{T}_{i,j}N_{F}D_{k,i}\big(w_{F}^{i, n}\big)^{-1}\hat{X}_{C}^{k, n}&=\sum^{N_{F}}_{i=1}\sum^{N_{F}}_{k=1}D_{k,i}\hat{X}_{C}^{k, n}=\sum^{N_{F}}_{k=1}\hat{w}_{C}^{k, n}\hat{X}_{C}^{k, n},
\int_{\partial\Omega}\{(\nu_1+i\nu_2)\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})+(\nu_1-i\nu_2)\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})\}d\sigma=0,
E^{tot}(x_{j},t)=\sum_{n\ne j}\frac{E_{n}^{ret}(x_{j},t)+E_{n}^{adv}(x_{j},t)}{2}+\sum_{n}\frac{E_{n}^{ret}(x_{j},t)-E_{n}^{adv}(x_{j},t)}{2}
\lVert AB-A_0 B_0 \rVert^2 \sim \sum_{i=1}^{M-1}\left[ x_i^2 + \sum_{j=2}^{N}\{x_i-(a_i b_1-a_i b_j)\}^2 \right] &\sim \sum_{i=1}^{M-1}\left[ x_i^2 + \sum_{j=2}^{N}(a_i b_1-a_i b_j)^2 \right] &= \sum_{i=1}^{M-1}\left[ x_i^2 + \sum_{j=2}^{N}a_i^2(b_j-b_1)^2 \right].
\|E_{y_{0}}\pi - \widehat{\mathfrak I}^{B^x_1}_{y_{-1}}(E_{y_{0}}\pi) \|_{L^\infty(B_1^x)}\leq C_T \widehat q^m \|\pi\|_{L^\infty(B_0^x)}\pi \in{\mathcal Q}_m, C_T := \frac{2d}{\rho-1}(1+\Lambda_m)^d \exp\left(\gamma \left(\frac{\rho + 1/\rho}{2}+1\right)\right).
\sum_{c}\sum_{Q^{\prime \prime}}\Gamma_{S}^{Q^{\prime}b Q^{\prime \prime}c}( p ) \left[ \delta_{Q^{\prime \prime}Q}\delta^{c a}- N_{Q^{\prime \prime}}^{c a}( p ) g_{Q^{\prime \prime}Q}{( - 1 )}^{a + 1}\right] = \frac{i}{2}g_{Q^{\prime}Q}\delta^{b a}{( - 1 )}^{a + 1}, \ \Q^{\prime}, Q = U , D , \ \b , a = 1 , 2 ,
{\tilde f}^{(0)}= \left ( \begin{array}{cccccc}0 -1 0 0 0 q\partial_y 1 0 0 0 0 q 0 0 0 0 0 0 0 0 0 0 -1 q^2 0 0 0 1 0 \partial_y - q\partial_x - q 0 - q^2 - \partial_x 0 \end{array}\right )\delta(x-y)
\widetilde{a}_+^{\widetilde{\mathbb P}}(\tau_{1,s}(\vartheta \widetilde{\varphi}))= \int_{B_1^+}\widetilde{\mathbb P}\nabla (\tau_{1,s}(\vartheta \widetilde{\varphi})) \cdot \nabla (\tau_{1,s}(\vartheta \widetilde{\varphi})),
J_n(x) =\left( \frac{2}{\pi x}\right)^{1/2}\cos(x- n \pi /2-\pi/4) \sum_{k=0}^\infty (-1)^k (n,2k) \; (2 x)^{-2 k}&\;\;-\left( \frac{2}{\pi x}\right)^{1/2}\sin(x-n \pi/2 -\pi/4)\sum_{k=0}^\infty (-1)^k (n,2k+1)\; (2 x)^{-2 k-1},
\begin{gathered}J_{t,0}(i,j)=\delta_{i,j}+\sum_{r=1}^d\bigg[\int_0^t \frac{\partial a_i}{\partial x_r}(s,X_s) J_{s,0}(r,j)\d s +\sum_{k=1}^m \int_0^t \frac{\partial b_{i,k}}{\partial x_r}(s,X_s)J_{s,0}(r,j)\d W^{k}_s+ \sum_{q=1}^l \int_0^t \frac{\partial c_{i,q}}{\partial x_r}(s,X_s)J_{s,0}(r,j) \d B^{H,q}_s\bigg].\end{gathered}
\left. I_{\mathrm{\scriptsize g r a v}}\right|_{\mathrm{\scriptsize s i n g}}= \frac{2^{1 0}}{3 5 \pi^{6}}\frac{1}{x_{1 3}^{8}x_{2 4}^{8}}\left[ s ( 7 t^{2}+ 6 t^{4}) + s^{2}( - 7 + 3 t^{2}) - 8 s^{3}\right] .
e^{ikR_3(\tau,\mathbf{v}_1,\mathbf{v}_2)}=\sum_{l=0}^N\frac{k^l}{l!}\,R_{3l}(\tau,\mathbf{v}_1,\mathbf{v}_2)+ O\left(k^{N+1}\,R_{3(N+1)}(\tau,\mathbf{v}_1,\mathbf{v}_2)\right).
\exp(B(F_p(z), \pi(\phi(\overrightarrow{z\eta})), f(\eta))) \leq \exp(B(F(z), \pi(\phi(\overrightarrow{z\eta})), f(\eta))) (1 + \epsilon) = \frac{df_*\rho_z}{d\rho_{F(z)}}(f(\eta)) (1 + \epsilon) \leq M(1 - \epsilon^2)
\begin{aligned}\frac{\partial F}{\partial\theta}=\left(0,\frac{1}{2}\sinh\rho\cos\theta\sin\phi,-\frac{1}{2}\sinh\rho\sin\theta\sin\phi\right) \frac{\partial F}{\partial\phi}=\left(-\frac{1}{2}\cosh\rho\sin\phi,\frac{1}{2}\sinh\rho\sin\theta\cos\phi,\frac{1}{2}\sinh\rho \cos\theta \cos\phi\right).\end{aligned}
S_{0}= \frac{1}{2}\int_{0}^{1}d \tau [ \frac{\dot{x}^{2}}{e}+ e m^{2}+ \chi \psi_{a}\dot{x}^{a}/ e - \psi_{a}\dot{\psi}^{a}+ \psi_{S}\dot{\psi_{S}}- m \psi_{S}\chi ] ,
H^{\Lambda,\omega}(x_\Lambda) := \sum_{i \in \Lambda}\phi(x_i) + \sum_{i,j \in \Lambda,\, j \sim i}J_{ij}V(x_i, x_j)+ \sum_{i\in \Lambda,j \in \partial \Lambda,\, j \sim i}J_{ij}V(x_i, \omega_j),
e^{SO_{red}}(T)\le \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+r M_{12}+M_{22}+s M_{24}\le \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+M_{12}+M_{22}+ M_{24}< \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+2M_{13}+M_{34}= 2f(1,3)+f(3,4)+\frac{2n-5}{3}f(1,4)+\frac{n-7}{3}f(4,4).
\Psi_i^+ = \lbrace \mu \in \Psi_i \mid a_\mu > 0 \rbrace, \Psi_i^- = \lbrace \mu \in \Psi_i \mid a_\mu < 0 \rbrace, \Psi_j^+ = \lbrace \nu \in \Psi_j \mid b_\nu > 0 \rbrace, \Psi_j^- = \lbrace \nu \in \Psi_j \mid b_\nu < 0 \rbrace
\left. \exp \left\{- \frac{i}{\hbar}\sum_{j = 1}^{N}\epsilon H_{\Delta}\left( \frac{\hbar}{i \epsilon}\frac{\partial}{\partial K_{j}}, \frac{\hbar}{i \epsilon}\frac{\partial}{\partial J_{j}}\right) \right\}W_{f i}[ K , J ] \right|_{K , J = 0}.
\begin{aligned}\widetilde{a}_j = - \overline{A}_{z_j}- \overline{B}_{z_j}, & \widetilde{b}= \overline{A}_{z_1}+ \overline{B}_{z_1}, & \widetilde{c}= \widetilde{a}_j (A_{z_1}+ B_{z_1}) + \widetilde{b}(A_{z_j}+ B_{z_j}) ,\end{aligned}
H_{k+1}^{(s,t)}\left[\begin{array}{c}1 1\end{array}\right]H_{k+1}^{(s,t)}\left[\begin{array}{c}k+1 k+1\end{array}\right]=H_{k+1}^{(s,t)}\left[\begin{array}{c}1 k+1\end{array}\right]H_{k+1}^{(s,t)}\left[\begin{array}{c}k+1 1\end{array}\right]+H_{k+1}^{(s,t)}H_{k+1}^{(n,t)}\left[\begin{array}{cc}1 k+1 1 k+1\end{array}\right].
L_{ij}=p_i\delta_{ij}+\frac{(1-\delta_{ij})}{(x_{i}-x_{j})}, B_{ij}=\delta_{ij}\sum_{l=1}^n\frac{(1-\delta_{jl})}{(x_j-x_l)^2}-\frac{(1-\delta_{ij})}{(x_i-x_j)^2},
\begin{array}{l}\overline{Scal}=e^{-2u}\cos{2v}\Scal-e^{-2u}\sin{2v}\Scal^*+2n\left(1-e^{-2u}\cos{2v}\right), begin{align*}4pt]\overline{Scal}^*=e^{-2u}\sin{2v}\Scal+e^{-2u}\cos{2v}\Scal^*-2n\e^{-2u}\sin{2v}.\end{array}
R_{S'}(\lambda)=\begin{pmatrix}0 1/\lambda^2 -1 -2/\lambda\end{pmatrix}, \det(R_{S'}(\lambda)-\lambda I)=\frac{(\lambda^2+1)^2}{\lambda^2}, \sigma(R_{S'}(\lambda))=\{i,i,-i,-i\}; R_{S'}(i)=\begin{pmatrix}0 -1 -1 2i\end{pmatrix}, \det(R_{S'}(i)-\lambda I)=(\lambda-i)^2,\sigma(R_{S'}(i))=\{i,i\};
\begin{array}{c}\epsilon_{c}=k^{ab}M_{ab}+ b^a P_a + f^a K_a +f D; M_{ab}=x_a p_b -x_b p_a\;;\; P_a=p_a\;;\;K_a=x^2p_a-2(xp)x_a\;;\; D=(xp).\end{array}
{\Phi}=\frac{1}{2}\left(\begin{array}{c}-A_{Ij}E^{j}_{\nu}D_{ij}E^{j}_{\nu}E^{i}_{\nu}\end{array}\right)={\frac{1}{2}}\left(\begin{array}{ccc}{\cal E}_{I}^{M}0 0 0 T_{i}^{\mu}0 0 0 T_{i}^{\mu}\end{array}\right){\hat \Phi}
\Big{(}\mathcal{D}_tu_H^{\frac12},v_H\Big{)}-\gamma(\nabla \sigma_H^{\frac12},\nabla v_H)+(\nabla u_H^{\frac12},\nabla v_H)+(f^{\frac12}(u_H),v_H)= (g^{\frac12},v_H),~\forall v_H\in L_H,
DS^{\mathrm{soft}}(\kappa, \pi;u,v) =DS^{(\mathrm{soft})}(\kappa;u,v) +\sum_{p=1}^{m}\Big( \phi_{1,p-1}(\kappa, \pi;u) \phi_{1,p}(\kappa, \pi;v)-\phi_{1,p}(\kappa, \pi;u)\phi_{1,p-1}(\kappa, \pi;v) \Big),
P_{L+1, \, 1}(t) = P'_{L, \, 1}(t), & P_{L+1, \, j}(t) = P'_{L, \, j}+ P_{L, \, j-1}DR, 2 \leq j \leq L-1, & P_{L+1, \, L}(t) = P_{L, \, L-1}DR- L(DR)^{L-1}D^2R,
(i)\mu_{n+1}\otimes\mu_{n+1}\simeq\Lambda^{2}\mu_{n+1}+S^{2}\mu_{n+1}:(g,P)\rightarrow gPg^{T}(ii)\mu_{n+1}\otimes\check{\mu}_{n+1}\simeq Ad_{U(n+1)}^{\mathbb{C}}=_{\mathbb{R}}2Ad_{U(n+1)}:(g,P)\rightarrow gPg^{-1}
\frac N{n_1+n_2}(||x(1)||^2 + ||x(2)||^2) = \sum\limits_{i=1}^N (\beta \langle x(1),x_i(1)\rangle + \gamma \langle x(2),x_i(2)\rangle)^2 = \beta^2 \frac N{n_1}||x(1)||^2 + \gamma^2 \frac N{n_2}||x(2)||^2 +2\beta\gamma \sum\limits_{i=1}^N \langle x(1),x_i(1)\rangle \langle x(2),x_i(2)\rangle.
\begin{aligned}\int_{t_1}^{T_{x, \xi}}p\left( f\left(\xi(s)\right)-u\left(\xi(s)\right)\right) ds \leq \int_{t_1}^{T_{x, \xi}}p(1-c_0) f\left(\xi(s)\right) ds \leq \left(T_{x, \xi}-t_1\right)p(1-c_0) f(\xi(t_1))\end{aligned}
\sum_{j_1=0}^{N_1}\sum_{j_2=0}^{N_2}\big\langle \psi_{n_1(k_1')}^{n_2(k_2')},\phi_{m_1(j_1)}\big\rangle_{V_1}\big\langle \phi_{m_1(j_1)},\psi_{n_1(k_1)}^{n_2(k_2)}\big\rangle_{V_1}\big\langle \psi_{n_2(k_2')}, \phi_{m_2(j_2)}^{m_1(j_1)}\big\rangle_{V_2}\big\langle \phi_{m_2(j_2)}^{m_1(j_1)},\psi_{n_2(k_2)}\big\rangle_{V_2}.
\Pi(y)= y^{2}[\beta_{0}+\beta_{1}\omega y+\ \beta_{2}\omega^{2}y^{2}+\ \beta_{3}\omega^{3}y^{3}+\ \beta_{4}\omega^{4}y^{4}] & +h[\alpha_{0}+\alpha_{1}\omega y+\alpha_{2}\omega^{2}y^{2}]=0
\ell_1 :=F(1,1)=(a_0+a_2)+(a_1+a_3) +(b_0+b_2)+(b_1+b_3), \ell_2 :=F(1,-1)=(a_0+a_2)+(a_1+a_3) - (b_0+b_2)- (b_1+b_3), \ell_3 :=F(-1,1)=(a_0+a_2)-(a_1+a_3) + (b_0+b_2)-(b_1+b_3), \ell_3 :=F(-1,-1)=(a_0+a_2)-(a_1+a_3) -(b_0-b_2)+(b_1+b_3).
\int_{Y^{\ast}}\vert\nabla_{y}v_{\varepsilon, k}\vert^{2}dy = \sum_{l\in{\mathbb{Z}}^{n}}\int_{\varepsilon(lz^{0}+Z^{\ast}) \cap Y^{\ast}}\vert\nabla_{y}v_{\varepsilon, k}\vert^{2}dx.
\left( \bar{a}_{\alpha}^{n^{\prime}}\bar{b}^{m^{\prime}\beta}- \frac{1}{C}\delta_{\alpha}^{\beta}\bar{a}^{n^{\prime}}\cdot \bar{b}^{m^{\prime}}\right) | 0 > , \quad \bar{a}_{\alpha_{1}}^{n_{1}^{\prime}}\cdots \bar{a}_{\alpha_{k}}^{n_{k}^{\prime}}| 0 > , \quad e t c .
P^l_i\left(x_1\right):=\int_{{\hat{G}}_{{\mathbb{C}}_i}}{{\Phi}_1({\alpha}^{-l}_i\left(\omega_i\right))\omega_i\left(x_1\right)d\omega_i}\, &P^l_j\left(x_2\right):=\int_{{\hat{G}}_{{\mathbb{C}}_j}}{{\Phi}_2({\alpha}^{-l}_j\left(\omega_j\right))\omega_j\left(x_2\right)d\omega_j},
\left\langle \prod_{b = 1}^{N}2 \cosh \left( 2 \omega \beta^{( b )}\int_{\Lambda}d^{2}x t ( x ) : \cos \left( 2 \sqrt{\pi}\varphi^{( b )}( x ) \right) :_{Q^{M , S}}\right) \right\rangle_{Q^{M , S}}.
B=\left[\begin{array}{cccc}\vec{B}_{00}\vec{B}_{01}\cdots \vec{B}_{0(n-1)}\vec{B}_{10}\vec{B}_{11}\cdots \vec{B}_{1(n-1)}\vdots \vdots \ddots \vdots \vec{B}_{(m-1)0}\vec{B}_{(m-1)1}\cdots \vec{B}_{(m-1)(n-1)}\end{array}\right],
U_{1}( \phi_{0}, a_{1}, . . . , a_{d}) = \mu^{D}\sqrt{a_{1}\cdots a_{d}}\sum_{s = 1}^{\infty}f ( D , d , s ) g^{s}\phi_{0}^{2 s}A_{d}^{c^{2}}( s - \frac{D - d}{2}; a_{1}, . . . , a_{d}) ,
C_1 := 2 Md^2 +2d\sqrt{d}M^d, C_2 := 2d\sqrt{d}\frac{K_a}{\theta}+ 2d^2 K_\zeta, C_3 := \frac{2d^2}{\theta}, C_4 := K_2 + 2d M_V K_\zeta + 2\sqrt{d}M_V \frac{K_a}{\theta}+ K_2 \frac{K_a}{\theta}, C_5 := 2 M_V \frac{\sqrt{d}}{\theta}+\frac{K_2}{\theta}+ \sqrt{d}(M_\zeta +M).
L=\int d^{2}{\bf r}({\kappa \over 4}\epsilon^{\alpha\beta\gamma}A_{\alpha}F_{\beta\gamma}+ i \psi^{\ast}D_t \psi -{1 \over 2}|{\bf D}\psi |^{2}+{1 \over{2 \kappa}}( \psi^{\ast}\psi )^{2})