harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\rho_{n,x,y}\stackrel{\rm def}{=}{\bf E\,}T_{\eta}(x)\,T_{\eta}(y)={\bf \,E\,}Y_{n}(x)\,Y_{n}(y)=\frac{{\bf E\,}\left[ K\left( \frac{x-X}{h_{n}}\right) \,K\left( \frac{y-X}{h_{n}}\right) \right]}{\sqrt{{\bf E\,}K^{2}\left( \frac{x-X}{h_{n}}\right){\bf \,E\,}K^{2}\left( \frac{y-X}{h_{n}}\right)}}.
w^2 = ( y_{2^k-1}\cdots y_1y_0 )^2 = (y_{2^k-1}\cdots y_1)^2y_0^2[y_0,y_{2^{k}-1}]\ldots [y_0,y_{2^{k-1}+1}][y_0,y_{2^{k-1}}] [y_0,y_{2^{k-1}-1}]\ldots[y_0,y_1] \,\,\vdots = y_{2^k-1}^{\, 2}\cdots y_1^{\, 2}y_0^{\, 2}= y^{2^{k+1}}= 1.
\frac{\bar{B}( p^{2})}{m}= P P \Big [{\frac{e^{-{\frac{m^{2}}{2 \mu^{2}}}}}{p^{2}}}+ \int_{0}^{\infty}d \lambda^{2}{\frac{\bar{\sigma}( \lambda^{2})}{p^{2}- \lambda^{2}}}\Big ] \; .
\bigl \langle \partial_t \bigl(u_{\delta,\lambda}(t)-u_0 \bigr), F^{-1}\bigl(u_{\delta,\lambda}(t)-u_0 \bigr) \bigr\rangle_{V_0^*,V_0}+ \int_\Omega \nabla \mu_{\delta,\lambda}(t) \cdot \nabla F^{-1}\bigl( u_{\delta,\lambda}(t)-u_0 \bigr)\,dx =0,
\frac{N-1}{c_{k}t}-\frac{M\log t-M}{t^2}-\frac{N-1}{c_{k}t+L\log t}&=\left[-M+\frac{L(N-1)}{c_{k}^2}+o(1)\right]\frac{\log t}{t^2}&\leq\left[-\frac M2+\frac{L(N-1)}{c_{k}^2}\right]\frac{\log t}{t^2}
\sum_{k=j}^{n-1}\binom{k}{j}\frac{1}{n-k}= \sum_{k=j}^{n-1}[z^{k}]\frac{z^{j}}{(1-z)^{j+1}}[z^{n-k}]\log\Bigl(\frac{1}{1-z}\Bigr) &=[z^{n}]\frac{z^{j}}{(1-z)^{j+1}}\log\Bigl(\frac{1}{1-z}\Bigr) &= [z^{n-j}]\frac{1}{(1-z)^{j+1}}\log\Bigl(\frac{1}{1-z}\Bigr)= \binom{n}{j}(H_n-H_j)
G_{\beta}(t,y)=\frac{t^{\beta - 1}}{(2\pi)^{d}}\int_{0}^{\infty}x^{-1/\beta}w(x^{-1/\beta}, \beta, 1)\left(\frac{\sqrt{\pi}}{\sqrt{t^{\beta}x}}\right)^{d}\exp\left\{\frac{-y^{2}}{4at^{\beta}x}\right\}dx
\frac{1}{|H_n|}\log_2 \gamma^{\geq\underline{s}}(n) =\frac{1}{|H_n|}\log_2\gamma^{\geq \underline{s}_{L(n)}}_{L(n)}(n) &\geqslant\frac{1}{|H_n|}\frac{1}{m^d}\log_2\gamma^{\geq \underline{s}_{L(n)}}_{L(n)}(nm).
\mathop{\mathrm{sgn}}(\boldsymbol{\pi})(-1)^{f_0(\boldsymbol{\pi})}=\mathop{\mathrm{sgn}}(\boldsymbol{\pi}_\sigma)\mathop{\mathrm{sgn}}(\tilde{\boldsymbol{\pi}})(-1)^{f_0(\tilde{\boldsymbol{\pi}})}=\mathop{\mathrm{sgn}}(\boldsymbol{\sigma})(-1)^{N-n}.
\MoveEqLeft e^{itH}e^{-itH_0}e^{itH_0}[P_\delta(\tfrac{x-y}{t})]e^{-itH_0}e^{itH_0}[P_\delta(\tfrac{x-y}{t})]e^{-itH_0}= e^{itH}[P_\delta(\tfrac{x-y}{t})]e^{-itH}e^{itH}e^{-itH_0}e^{itH_0}[P_\delta(\tfrac{x-y}{t})]e^{-itH_0},
\frac{{{m_w}^{{m_w}}}}{{\Gamma \left({{m_w}}\right)}}\sum\limits_{j = 0}^\infty{\frac{\Gamma^{-1}\!\left({j\! +\! 1}\right){{K_w}^j{\alpha_{wj}}{I_E}}}{{j!{{\left({2{C_{1w}}\sigma_w^2}\right)}^{j + 1}}}}}>{\left({{\varepsilon_{th}}\!- \!{\kappa^2}}\right)^{q +\!{m_{fw}}- 2}},
\inf_{x\in\mathbb{R}}f\left(x\right)+g\left(x\right) =\alpha =\inf_{x\in\mathbb{R}}f\left(x\right)+\psi\left(x\right)\leq \sup_{\psi\in\psi}\inf_{x\in\mathbb{R}}f\left(x\right)+\psi\left(x\right),
\det(A_n) = \prod_{\substack{d_1, d_2 \mid n 1 \leqslant d_1 < d_2 \leqslant n}}\mathcal{R}(\Phi_{d_2}, \Phi_{d_1}) = (-1)^{n-1}\prod_{i=1}^r p_i^{\frac{n(1-p_i^{-\alpha_i})}{(p_i-1)}},
\Gamma = \Gamma(\sigma,S)= \Gamma_1 +\Gamma_2 \Gamma_1 = \{\{s,(s,c_{s,1}),(c_{s,1},c_{s,2}),\cdots,(c_{s,k_s 1},c_{s,k_s})\}; s \in \sigma \}\Gamma_2 = \{\{(t,c_{t,1}),(c_{t,1},c_{t,2}),\cdots,(c_{t,k_t-1},c_{t,k_t})\}; t \in S_\setminus (S_+ + \sigma)\}.
O_{F}= \left( \begin{array}{c c}{{- \frac{d^{2}}{d z^{2}}+ \frac{\partial^{2}}{\partial \phi^{2}}V}}&{{\frac{\partial^{2}}{\partial \chi \partial \phi}V}}\\{{\frac{\partial^{2}}{\partial \phi \partial \chi}V}}&{{- \frac{d^{2}}{d z^{2}}+ \frac{\partial^{2}}{\partial \chi^{2}}V}}\\ \end{array}\right)_{\vert \phi = \phi ( z ) , \chi = \chi ( z )}
H_{\rm eff}=\frac{1}{2}p^2_r+\frac{1}{2r^2}p^2_\theta+\frac{1}{2}p^2_z+\frac{1}{2}\omega^2r^2+\xi\Big[p_z+gp_\theta\Big]+F(\xi)p_\xi-F'(\xi){\cal P}_1\eta^1-{\cal P}_2\eta^1\ \ \ ,
\displaystyle ||h||_{H_{x,v}^{s,r}}^2 = \sum_{|m|\leq r}\, ||\partial^m h||_{H_{x,v}^s}^2\,, \qquad||h||_{H_{\Lambda}^{s,r}}^2 = \sum_{|m|\leq r}\, ||\partial^m h||_{H_{\Lambda}^s}^2\,, begin{align*}2pt] \displaystyle ||h||_{H_{x}^{s,r}L_v^2}^2 = \sum_{|m|\leq r}\, ||\partial^m h||_{H_{x}^s L_v^2}^2\,.
S_{\rm dil1}={1 \over{2 \kappa_4^2}}\int d^4 x \sqrt{-g}\left [ R - (\partial \tilde d)^2 - \frac{32k^2(\Delta +4)}{\Delta^2}\left (2e^{-a \sqrt{4/3}\tilde d}- e^{-2a \sqrt{4/3}\tilde d}- 1\right ) \right].
\alpha_k = 2 \rho_{\rm r}\left( E\bigg[ \left\| \frac{\boldsymbol{w}_k}{\epsilon_{\rm r}+ |\boldsymbol{w}_{k-1}|}\right\|_{1}\bigg] \!\!-\!\! E \bigg[\left\| \frac{\boldsymbol{w}}{\epsilon_{\rm r}+ |\boldsymbol{w}_{k-1}|}\right\|_{1}\bigg] \right) .
\beta_n^i=\begin{cases}\varepsilon_n\kappa_n+\kappa_{n-1}i=1, \varepsilon_n\omega_n\kappa_{n-1}+\beta_{n-1}^1 i=2, \varepsilon_n\beta_{n-1}^{i-2}+\beta_{n-1}^{i-1}i=3,\ldots,2n-1, \varepsilon_n\beta_{n-1}^{2n-2}i=2n.\end{cases}
\overline{F}_{\,1}^{\,\alpha}\left ( t \right )= \mathbb{P}\left ( X_{\alpha}^{\left ( 1 \right )}>t \right )= \frac{\Gamma \left ( \alpha +1 \right )}{\mathbb{E}\left [ X^{\alpha}\right ]}I_{-}^{\alpha}\overline{F}\left ( t \right ),t\geq 0,
\alpha_{- n}^{\mu}M_{\mu \nu}\widetilde \alpha_{- n}^{\nu}\rightarrow \alpha_{- n}^{\alpha}M_{\alpha \beta}\widetilde \alpha_{- n}^{\beta}- \alpha_{- n}^{\alpha}M_{\alpha j}\widetilde \alpha_{- n}^{j}+ \alpha_{- n}^{j}M_{j \alpha}\widetilde \alpha_{- n}^{\alpha}- \alpha_{- n}^{i}M_{i j}\widetilde \alpha_{- n}^{j}
{\tau}_{2}({\pi}( f_{0}) ,{\pi}( f_{1}) ,{\pi}( f_{2}) ) = - \int_{S^{2}}T r{\Gamma}D^{- 1}[ D , f_{0}] D^{- 1}[ D , f_{1}] D^{- 1}[ D , f_{2}] d^{2}x ,
f^{+}(\tau) =q - 64\, q^2 - 1836\, q^3 + 4096\, q^4 + 3990\, q^5 + 117504\, q^6 +\cdots f^{-}(\tau) =q + 64\, q^2 + 1236\, q^3 + 4096\, q^4 - 57450\, q^5 + 79104\, q^6 +\cdots \, ,
u(x_0, 0) \geq \left. \int_{M}u \left(4\pi K^{-2}r^2 \right)^{-\frac{m}{2}}e^{-l}d\mu \right|_{t=-K^{-2}r^2}= \left. \int_{M}\varphi \left(4\pi K^{-2}r^2 \right)^{-\frac{m}{2}}e^{-l}d\mu \right|_{t=-K^{-2}r^2}.
\Big| \int_{I}e^{i\lambda x \cdot \gamma(t)}a(x) f(t) dt \Big|= \lambda^{d-1}\Big| \int_{|\mathbf v|\lesssim \lambda^{-1}}\int_{I}e^{i \lambda x \cdot (\gamma(t) + (0,\mathbf{v}))}a(x) f(t) dt \, e^{-i \lambda x \cdot (0,\mathbf{v})}d\mathbf{v}\Big|.
\langle x_1, y_1, \ldots, x_n, y_n, c| [x_i, x_j] = [y_i, y_j]= [x_i, y_j]=1\ \i\neq j, & [x_i, c]= [y_i, c]=1,\ [x_i, y_i]=c,\x_i^p= y_i^p= c^p=1\ \ 1\le i\le n\rangle.
\bar{\alpha}^{-}| \Phi_{s}\rangle = \Bigl ( - \frac{\partial^{I}}{\partial^{+}}\bar{\alpha}^{I}+ \frac{s + d - 2}{\hat{\partial}^{+}}\bar{\alpha}^{z}- \frac{2 ( \hat{\partial}^{+}- \alpha^{+}\bar{\alpha}^{z})}{\hat{\partial}^{+}( \hat{\partial}^{+}- 2 \alpha^{+}\bar{\alpha}^{z})}\bar{\alpha}^{z}\Bigr ) | \Phi_{s}\rangle \, .
G_c:=-2[\bar{\mathfrak{F}}, \mathcal{H}\frac{1}{\zeta_{\alpha}}+\bar{\mathcal{H}}\frac{1}{\bar{\zeta}_{\alpha}}]\bar{\mathfrak{F}}_{\alpha}+\frac{1}{\pi i}\int \Big(\frac{D_t\zeta(\alpha,t)-D_t\zeta(\beta,t)}{\zeta(\alpha,t)-\zeta(\beta,t)}\Big)^2(\zeta-\bar{\zeta})_{\beta}d\beta :=G_{c1}+G_{c2}.
\#(\psi^{-}_{\varepsilon})^{-1}(n)=\# \{m\in\mathbb{N}:\tilde{\psi}^{-}_{\varepsilon}(m)\in[n,n+1)\}\gg |m(n+1)-m(n)| &\geq ((\tilde{\psi}^{-}_{\varepsilon})'(m(n)))^{-1}&\gg \frac{m(n)}{C_2}.
\sum_{a_1,a_2:(*)_{2,b}}(a_1a_2)^{-2}= \sum_{a=1}^{b-1}\left( a^{-1}(b-a)^{-1}\right)^{2}=\sum_{a=1}^{b-1}\left((a^{-1}+(b-a)^{-1})b^{-1}\right)^2 &= b^{-2}\sum_{a=1}^{b-1}\left(a^{-1}+(b-a)^{-1}\right)^2 \leq 2 b^{-2}\sum_{a=1}^{b-1}\left( a^{-2}+ (b-a)^{-2}\right) \leq b^{-2}C.
S_{{\rm boson}}=\frac{8\pi}{e^{2}}\int dt dr \sqrt{A(r)B(r)}\;\:r^{2}\: \: \left \{\frac{1}{\sqrt{B(r)}}\frac{\partial a_{1}(r, t)}{\partial t}-\frac{1}{\sqrt{A(r)}}\frac{\partial a_{0}(r, t)}{\partial r}\right \}^{2}
\int_{\Sigma_t}\sigma_k \,\div(V)\,d\mu = A\int_{\Sigma_t}\sigma_k\,d\mu + \sum_{i=1}^{n+1}B^i\int_{\Sigma_t}\sigma_k x_i\,d\mu & = \left(A + \sum_{i=1}^{n+1}B^i\right)\int_{\Sigma_t}\sigma_k \,d\mu.
lim_{h\rightarrow0}(\frac{1}{h}\cdot ln(\frac{x+h}{x}))=ln(lim_{h\rightarrow0}(\frac{x+h}{x})^{1/h})=ln(lim_{k\rightarrow0}(1+k)^{1/k})^{1/x}=ln(e)^{1/x}=1/x
W_{0}=\frac{{\partial}_{2}({\partial}_{1}c /{\partial}_{2}c) \cdot{\partial}_{1}c -{\partial}_{1}({\partial}_{1}c /{\partial}_{2}c ) \cdot{\partial}_{2}c}{({\partial}_{1}c)^{2}+({\partial}_{2}c)^{2}}\cdot{\partial}_{2}c,
{\cal{V}}= - \frac{\pi^{2}}{V}\frac{s ( N - s )}{N}+ \frac{V}{2 \pi^{2}}\left( \frac{4 \pi}{V}\sum_{i = 1}^{s}\sum_{j = s + 1}^{N}| A_{+}^{( j i )}|^{2}+ \frac{1}{2 \pi^{2}}\sum_{k}( B^{k})^{2}+ \frac{1}{\pi^{2}}\sum_{i}\sum_{j > i}| X^{( i j )}|^{2}\right) .
{\omega^c}_{ab}\partial_c = \nabla_a \partial_b &= \nabla_a \big( (\partial_b \Theta^j) \partial_j) &= (\partial_a \partial_b \Theta^j) \partial_j + (\partial_b \Theta^j) (\partial_a \Theta^j) (\nabla_i\partial_j) &= (\partial_a \partial_b \Theta^j) \partial_j + (\partial_b \Theta^j) (\partial_a \Theta^j){\omega^k}_{ij}\partial_k,
d\alpha(P,\zeta)= (dp\wedge dq-dH\wedge dt)(P,\zeta)=\sum_{i=1}^n dp_i(P)dq_i(\zeta)-dH(P)dt(\zeta) = \sum_{i=1}^n (F_i+R_i)\big(\xi_i-\tau \frac{\partial H}{\partial p_i}\big)=\sum_{i=1}^n (F_i+R_i)(\xi_i-\dot q_i\tau),
d_p(u) =\inf\left\{\left\|(x_1,\ldots,x_n)\right\|_{\ell^{n,w}_{p^*}(E)}\left\|(y_1,\ldots, y_n)\right\|_{\ell^n_p(F)}\right\}, g_p(u) =\inf\left\{\left\|(x_1,\ldots,x_n)\right\|_{\ell^n_p(E)}\left\|(y_1,\ldots, y_n)\right\|_{\ell^{n,w}_{p^*}(F)}\right\},
\left[ - \frac{1}{\rho}\frac{d}{d \rho}\left( \rho \frac{d}{d \rho}\right) + \frac{k^{2}}{\rho^{2}}+ \rho^{2}\right] f_{k}( \rho ) = \left( \frac{2}{e B}\right) \left[ E^{2}- m^{2}+ e B ( k + 1 ) \right] f_{k}( \rho ) ,
{\mathcal D}_{0}^{- 1}\simeq \widetilde{\phi}_{1}\phi_{1}^{\dagger}\lambda_{1}P_{L}- \widetilde{\phi}_{1}\widetilde{\phi}_{1}^{\dagger}\widehat{/ \! \! \! \partial}P_{R}+ \phi_{1}\widetilde{\phi}_{1}^{\dagger}\lambda_{1}P_{R}- \phi_{1}\phi_{1}^{\dagger}\widehat{/ \! \! \! \partial}P_{L}- \phi_{0}\phi_{0}^{\dagger}\widehat{/ \! \! \! \partial}P_{L}.
\int\limits_t^s\phi_j(\tau)d\tau=\frac{T-t}{2}\left(\frac{\phi_{j+1}(s)}{\sqrt{(2j+1)(2j+3)}}-\frac{\phi_{j-1}(s)}{\sqrt{4j^2-1}}\right)\ \ \ \hbox{for}\ \ \j\ge 1.
f_n(e)= \epsilon_n2^n\frac{\displaystyle \prod_{1\leq i<j\leq m}(e_i-e_j) \prod_{m<k< l\leq n}(e_k-e_l)}{\displaystyle\prod_{1\leq i\leq m}e_i \prod_{\scriptsize\begin{array}{c}1\leq i\leq m m<k\leq n\end{array}}(e_i+e_k)}
(1-q_1q_3^{-1})F^{(1)}_{0|1,l}=\sum_{j\ge0}q_1^{-j-l}\Bigl(F_{1,-j}F_{0,l+j}-(q_1^{-1}+q_3^{-1})F_{1,-j-1}F_{0,l+j+1}+q_1^{-1}q_3^{-1}F_{1,-j-2}F_{0,l+j+2}\Bigr) &+\sum_{j\ge1}q_1^{j-l}\Bigl(q_1^{-1}q_3^{-1}F_{0,l-j}F_{1,j}-(q_1^{-1}+q_3^{-1})F_{0,l-j+1}F_{1,j-1}+F_{0,l-j+2}F_{1,j-2}\Bigr)\,,
\sum_{k=0}^n (3k+1){2k\choose k}^3 16^{n-k}\equiv 0 \mod{4(2n+1){2n\choose n}}, begin{align*}5pt]\sum_{k=0}^n (3k+1){2k\choose k}^3 (-8)^{n-k}\equiv 0 \mod{4(2n+1){2n\choose n}},
\int_0^1 f(x) h (x) \d x = \int_0^1 Tf (x) h(x) \d x = \int_0^1 \frac{\mathcal L (fg) (x)}{\lambda g(x)}h(x) \d x= \int_0^1 f(x) \frac{g(x)}{\lambda}\mathcal P\left( \frac{h}{g}\right) \d x.
\begin{aligned}(\mbox{id}\otimes \Delta_L) \circ \Delta_L=(\Delta_{\mathcal{A}}\otimes \mbox{id}) \circ \Delta_L, \textsl{m}\circ(\varepsilon_{\mathcal{A}}\otimes \mbox{id})\circ \Delta_L = \mbox{id}. \end{aligned}
\alpha : = \left( \begin{array}{c c c c c c c c}{{\gamma_{1}}}&{{0}}&{{\gamma_{3}}}&{{\gamma_{4}}}&{{\ldots}}&{{\gamma_{N - 2}}}&{{\gamma_{N - 1}}}&{{\gamma_{N}}}\\{{0}}&{{\beta_{2}}}&{{\gamma_{3}}}&{{c_{4}\gamma_{4}}}&{{\ldots}}&{{c_{N - 2}\gamma_{N - 2}}}&{{\beta_{N - 1}}}&{{\beta_{N}}}\\ \end{array}\right)
\mathcal{L}_{xx}\left( x,a_{0}\right) >\mathcal{L}_{xx}\left( x,\tfrac{1}{2}\right) =-\tfrac{4}{49}\tfrac{x^{2}+x-1/3}{\left( x^{2}+x+5/6\right)^{2}}-\tfrac{45}{49}\tfrac{x^{2}+x+19/90}{\left( x^{2}+x+13/45\right)^{2}},
\hat{U}_{\mathrm{B R S T}}( t_{f}, t_{i}) = \left[ e^{- \frac{i}{\hbar}\epsilon \hat{H}_{\mathrm{e f f}}}\right]^{N}\ , e p s i l o n = \frac{\Delta t}{N}= \frac{t_{f}- t_{i}}{N}\ .
\mathfrak{M}= [\mathfrak{S}_{1}, \mathfrak{S}_{2}] = \mathfrak{S}_{1}\cdot \mathfrak{S}_{2}&= ((x - 1)(y - 1)x^{3}, (y - 1)x^{2}y, (y - 1)xy^{2}, (x - 1)z, y(y-1)x^{3}, yz),
\sum_{t=1}^{+\infty}\beta^t D_1 \phi(\hat{x}_t, \hat{u}_t) \delta x'_t = \sum_{t=1}^{+\infty}\langle q_{t-1}-q_t \circ D_1 f(\hat{x}_t, \hat{u}_t), \delta x'_t \rangle.
\sum_{\alpha = 1}^A a_\alpha^p \delta_{\alpha+1}= \epsilon^{p/k}\sum_{\alpha=1}^A \frac{\epsilon^{1/k}\delta_{\alpha+1}}{\delta_\alpha a_\alpha}= \epsilon^{p/k}\sum_{\alpha=1}^A \zeta_\alpha^2 =: \epsilon^{p/k}A_u,
O_{\beta \mu_{1}\ldots \mu_{n}}= O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 2}}+ O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 3}}+ O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 4}}+ \ldots \; ,
\int_U\frac{\partial \bar{q}}{\partial t}\bar q\phi dx =\int_U \nabla \cdot(K(x,|\nabla p |)\nabla p)_t \bar q dx-\int_U \Psi_{tt}\phi)\bar q dx &=-\int_U (K(x,|\nabla p |)\nabla p)_t\cdot\nabla\bar qdx-\int_U\Psi_{tt}\bar q \phi dx.
\prod_{j = 1}^{k}\frac{z - \cos ( y + 2 \gamma ( s - j ) )}{z - \cos ( y - 2 \gamma ( s + j ) )}= 1 + \sum_{m = 0}^{k - 1}\prod_{l = 0}^{m}\frac{2 v_{n}v_{s}\sin \gamma ( 2 s - l )}{\sin \gamma ( l + 1 )}\frac{\sin \gamma ( n - k + l ) \sin \gamma ( k - l )}{z - v_{n}v_{s}\cos \gamma ( n + 1 - 2 k + 2 l - 2 s )}
\frac{2}{k^+k^{'+}}\left[\left(m^2+k_{\perp}k'_{\perp}e^{+i\lambda(\varphi-\varphi')}\right)\delta^{\lambda}_{\lambda'}-m\lambda\left(k'_{\perp}e^{+i\lambda \varphi'}-k_{\perp}e^{+i\lambda\varphi}\right)\delta^{\lambda}_{-\lambda'}\right]
\phi_{t}^{n}(\boldsymbol{x}) =\left(\phi_{t}^{n}(x_{1}),\phi_{t}^{n}(x_{2}),\dots\phi_{t}^{n}(x_{m})\right)\,, X_{n}(\boldsymbol{y},t) =\left(X_{n}(y_{1},t),X_{n}(y_{2},t),\dots,X_{n}(y_{p},t)\right)\,,
\lambda^{\alpha}= - \frac{i}{4}{\bar{D}}^{2}L^{\alpha}\ , \qquad \Omega_{\alpha \beta}= \frac{i}{8}{\bar{D}}^{2}D_{( \alpha}L_{\beta )}\ , \qquad \sigma = - \frac{i}{2 4}{\bar{D}}^{2}D^{\alpha}L_{\alpha}\ .
Z = \int{\cal D}A^{\mu}{\cal D}\pi_{\mu}{\cal D}\rho{\cal D}\pi_{\rho}\prod_{i , j = 1}^{2}\delta ( \tilde{\Omega}_{i}) \delta ( \Gamma_{j}) d e t \mid \{\tilde{\Omega}_{i}, \Gamma_{j}\}\mid e^{i S},
\int_{\Omega_1\setminus S}| U_*|^{2^*}= \int_{\Omega_1}u_0^{2^*}- 2^*\int_{\Omega_1\setminus S}u_0^{2^*-1}V+ O(\frac{1}{\mu^{\frac{N-2}{2}(1+\delta)}}).
\star \equiv \exp \left({\frac{i \hbar}{2}}\int d x ~ \left( \stackrel{\leftarrow}{\frac{\delta}{\delta \phi ( x )}}\stackrel{\rightarrow}{\frac{\delta}{\delta \pi ( x )}}- \stackrel{\leftarrow}{\frac{\delta}{\delta \pi ( x )}}\stackrel{\rightarrow}{\frac{\delta}{\delta \phi ( x )}}\right) \right) ,
\frac{\partial}{\partial x_1}[z^n] N_1 \Big|_{x_1 = x_2= 1}= \frac{1}{n}[t^{n-1}] \left(1+(3n-7)t-(9n-8)t^2\right) (1-t)^{1-4n}&= \frac{1}{n}\left[ \binom{5n-3}{n-1}+(3n-7) \binom{5n-4}{n-2}-(9n-8)\binom{5n-5}{n-3}\right] &= \frac{2(3n^2-n-1)}{(n-1)(n-2)}\binom{5n-5}{n-3}.
\lbrace Q=0\rbrace \cap S \cap B_\delta(x_2) = \lbrace x_2 \rbrace, Q(x)=s(x)\left( n_0 \otimes n_0 -\frac 13 \mathbf I \right) \forall x\in [x_2,x_0]\cap B_\delta (x_2), Q(x)=s(x)\left( n_1 \otimes n_1 -\frac 13 \mathbf I \right) \forall x\in [x_2,x_1]\cap B_\delta (x_2).
x^{|\alpha^3|}y^{|\alpha^3|}z^{|\alpha^3|}= \alpha^{\frac{(|\alpha^3|-1) |\alpha^3|}{2}}\left(xyz + (1-\alpha^3)^{-1}x^3\right)^{|\alpha^3|}- \alpha^{\frac{(|\alpha^3|-1) |\alpha^3|}{2}}(1-\alpha^3)^{-|\alpha^3|}x^{|\alpha|}.
E_k\big(\epsilon_t(\xi)\big) =\sum_{n\geq0}\binom{t}{n}E_kF_n(\xi) &=\sum_{n\geq0}\binom{t}{n}\sum_{\alpha_1,\ldots,\alpha_k>0}{n\brack\alpha}F_\alpha(\xi) =\sum_{\alpha_1,\ldots,\alpha_k>0}\binom{t}{\alpha_1}\ldots\binom{t}{\alpha_k}F_\alpha(\xi)
a_1^{x_1}a_2^{x_2}\cdots a_{m-1}^{x_{m-1}}a_m^{x_m}\cdots a_{2m-2}^{x_{2m-2}}\cdots a_{n-2}^{x_{n-2}}a_{n-1}^{x_{n-1}}a_n^{x_n}&= s_{1,2}^{x_{1,2}}s_{2,3}^{x_{23}}\cdots s_{m-1,m}^{x_{m-1,m}}s_{1,3}^{x_{1,3}}\cdots s_{1,m-1}^{x_{1,m-1}}s_{2,m}^{x_{2,m}}s_{1,m}^{x_{1,m}}.
I(b): = \frac{1}{n-2s}\left[\sum_{j=1}^kb_j^{n-2s}H(q_j, q_j) - \sum_{i\neq j}b_j^{\frac{n-2s}{2}}b_i^{\frac{n-2s}{2}}G(q_j, q_i)-\sum_{j=1}^kb_j^{2s}\right].
m+o(1) =I(u_k) &\geq\frac{1}{bp^2}M_{u_k}^p-\frac{1}{\mu}\int_0^Tf(t,u_k(t))u_k(t)dt-cT-\frac{a^p}{bp^2}&=\frac{1}{bp^2}M_{u_k}^p-\frac{1}{\mu}M_{u_k}^{p-1}\|u_k\|_{E^{\alpha,p}}^p-cT-\frac{a^p}{bp^2}&=M_{u_k}^{p-1}\left(\left(\frac{1}{p^2}-\frac{1}{\mu}\right)\|u_k\|_{E^{\alpha,p}}^p+\frac{a}{bp^2}\right)-cT-\frac{a^p}{bp^2}.
u_{n\ell}(z)=N_{n\ell}z^{\xi_{n\ell}}(1-z)^{\frac{1}{2}[1+\sqrt{1+4\beta_2}]}\_2F_1\left(-n,n+2(\xi_{n\ell}+\frac{1}{2}[1+\sqrt{1+4\beta_2}]);2\xi_{n\ell}+1,z\right),
\int_{B_{\rho}}r^{2-n}|\nabla u|^2 \d x \leq C \int_{B_{8\rho}\setminus B_{\rho}}r^{2-n}|\nabla u|^2 \d x + C \varepsilon \int_{B_{\rho}}r^{2-n}|\nabla u|^2 \d x & \rho \leq 1/8 \varepsilon \leq \varepsilon_0.
T(a(1+q), b(1+q)) = \frac{1-(a-b)(1+q) - \sqrt{1-2(1+q)(a+b)+(1+q)^2(a-b)^2}\,}2 = \frac{1-(a-b)-(a-b)q}2 \qquad- \frac{\sqrt{1-2(a+b) +(a-b)^2 - 2q(a+b-(a-b)^2)+q^2(a-b)^2}\,}2.
\frac{\partial}{\partial \theta}(-\log \mathbf P_{\delta_x}[e^{-\theta X_T(f)}]) = \frac{\mathbf P_{\delta_x}[X_T(f)e^{-\theta X_T(f)}]}{\mathbf P_{\delta_x}[e^{-\theta X_T(f)}]}= P^\beta_T f(x) \dot{\mathbf P}_x^{(T,f)}[e^{-\theta Y_T(f)}].
\begin{aligned}\sup_{|\alpha| \le 10}\sup_{\psi_1}\int_{\Sigma_0}(\partial_t \psi) (\partial_y \partial^\alpha \phi) + (\partial_y \psi) (\partial_t \partial^\alpha \phi) d x \le{C \over (1 + s)^2}\chi_{|\tau| \le 5}\Vert \phi \Vert_{H^{11}(\Sigma_0)}. \end{aligned}
\frac{\sum_{0\not=x\in \mathfrak{a}_{d-1-l}}a^{1-q^{2}}}{\left(\sum_{0\not=x\in \mathfrak{a}_{d-1-l}}a^{1-q}\right)^{1+q}}= \tilde{J}(\mathfrak{a}_{d-1-l}),
C(\rho e^{-\eta}) = \Theta(h(\eta^{-1}) \eta^{-\alpha}), C'(\rho e^{-\eta}) = \Theta(h(\eta^{-1})\eta^{-(\alpha+1)}), C''(\rho e^{-\eta}) = \Theta(h(\eta^{-1})\eta^{-(\alpha+2)}).
d ( z ) \! \! = \! \! \left( \! \! q^{2 j - 1}x_{+}z \! - \! 2 x_{0}s k q^{j}\phantom{\frac{q^{j}}{s}}\! \! \! \! \! \! - \! x_{-}k^{2}z^{- 1}\! \right) \! \! \left( - q^{2 j}y_{+}z \! + \! \frac{2 y_{0}}{t k}q^{j}\! \! + q y_{-}k^{- 2}z^{- 1}\! \right) ,
E_{o_+\cdot n^{-1}}(nwn^{-1}) = E_{o_-\cdot w_{P'}}(nwn^{-1}) & = j_{P'}^+(E^{P'}_{o_{-,P'}\cdot w_{P'}}(nwn^{-1})) & = j_{P'}^+(E^{P'}_{o_{+,P'}}(nwn^{-1}))
S^{\prime}\left[ \xi^{\prime}\right] \left( z_{j}\right) =\xi_{j}+\left( \xi_{0}-S\xi\left( z_{0}\right) \right) \left\{\varphi_{z_{0}}\left( z_{j}\right) -S\left[ \varphi_{z_{0}}\mid_{Z}\right]\left( z_{j}\right) \right\}& =\xi_{j}+\left( \xi_{0}-S\xi\left( z_{0}\right) \right) \left\{0\right\}=\xi_{j},
G_{ij}= \frac{{(|D w|^2 )_{ij}}}{{|D w|^2}}- \frac{{(|D w|^2 )_i}}{{|D w|^2}}\frac{{(|D w|^2 )_j}}{{|D w|^2}}+ h'u_{ij}+ h''u_i u_j + \alpha_0 d_{ij},
\aligned Y_{t}=x_0+\int_{0}^tb(s,Y_{s})ds+\int_{0}^t1_{s>t_1}h_{s}^\varepsilon ds+\int_{0}^t\sigma(s,Y_{s})dB_s &\quad+\int_{0}^{t+}\int_Uf_1(s,Y_{s-},u)\tilde{N}_k(ds,du)+\int_{0}^{t+}\int_Uf_2(s,Y_{s-},u)N_k(ds,du).\endaligned
\bar\nabla_{\dot\gamma}\bar\nabla_{J_t}^k X = \sum_{i=0}^{k-1}\bar\nabla_{J_t}^i (\bar R(J_t, \dot\gamma) \bar\nabla_{J_t}^{k-1-i}X) = \sum_{j+i+l+m = k-1}( \bar\nabla_{J_t}^j \bar R) * (\bar\nabla_{J_t}^iJ_t) *( \bar\nabla_{J_t}^l\dot\gamma )* (\bar\nabla_{J_t}^{m}X).
{\cal H}_{B}={\frac{1}{2}}p^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}( g_{| \rho |}- 1 ) | \rho |^{2}V ( \rho \cdot q ) + ({\frac{\omega^{2}}{2}}q^{2}) -{\cal E}_{0},
\phi((W_{1})) = \phi\left(\frac{\mu(m_{1}m_{2})}{\langle m_{1}, m_{2}\rangle_{B_{1}}}\right) = \frac{\mu\phi(m_{1}m_{2})}{\langle m_{1}, m_{2}\rangle_{B_{1}}}= \frac{\mu\phi(m_{1}m_{2})}{\langle \phi(m_{1}), \phi(m_{2})\rangle_{B_{2}}}.
d(e^{c_1\tilde{\gamma}t}\overline{V}_t)= e^{c_1\tilde{\gamma}t}d \overline{V}_t+c_1\tilde{\gamma}e^{c_1\tilde{\gamma}t}\overline{V}_t dt \le c_2\tilde{\gamma}e^{c_1\tilde{\gamma}t}dt+ \frac{\tau}{N}\tilde{\gamma}e^{c_1\tilde{\gamma}t}\sum_{i=1}^N{dB_{i,t}}^Te_{i,t}.
\left<B_{1\beta}^{k},B_{1\beta}^{k'}\right>>0,\quad\quad\mbox{for all $k,k',\beta=1,\dots,q$.}& \left<B_{1\beta}^{k},B_{1\alpha}^{k'}\right>=0 ,\quad\quad\mbox{for all $k,k',\alpha,\beta=1,\dots,q$ with $\alpha\neq\beta$,}
\tau_\theta\Big( U_\theta\Big(\sum_{\ell\in I}(\tilde{\chi}_\ell^\delta \tilde{g})*_\theta(\chi_\ell^\delta g)\Big)^{2}\Big) =\Big\|\sum_{\ell\in I}(\tilde{\chi}_\ell^\delta \tilde{g})*_\theta(\chi_\ell^\delta g)\Big\|_{L_2(\mathbb{R}^2)}^2 \leq\Big\|\sum_{\ell\in I}|\tilde{\chi}_\ell^\delta \tilde{g}|*|\chi_\ell^\delta g|\Big\|_{L_2(\mathbb{R}^2)}^2.
\sigma=\int{d^{2}k\over (2\pi )^{2}}[\not\! k+\lambda^{2}\gamma^{5}\tilde{\sigma}+\lambda^{2}\tilde{\sigma}]^{-1}_{\nu\nu}=\int{d^{2}k\over (2\pi )^{2}}[{4i\lambda^{2}\sigma\over k^{2}-4\lambda^{4}(\sigma^{2}+\tilde{\sigma}^{2})}]
I(z,s,c,d,r_{1},r_{2}) =\int\limits_{C}u^{s-1}\frac{\exp(-\left((c\mu+j-\left\{R_{1}\right\})/ck\right) (cz+d)ku)}{\exp(-ku(cz+d))-1}\frac{\exp((\left( v+\left\{(dj+\rho)/c\right\}\right) /k)ku)}{\exp(ku)-1}du.
\frac{\lambda'}{6}\left( \vec{\varphi}^{\, (0)}(X_0) \right)^2 \varphi_i^{(0)}(X_0) \to \frac{\lambda'}{6}\langle \left( \vec{\varphi}^{\, (0)}(X_0) \right)^2 \rangle_\zeta \varphi_i^{(0)}(X_0) + \frac{\lambda'}{3}\langle \varphi_j^{\, (0)}(X_0) \varphi_i^{\, (0)}(X_0) \rangle_\zeta \varphi_j^{(0)}(X_0)
|N_{F}(s)-N_{H_{1}}(v_{t})| =|N_{F}(s)\cap N_{F}(v_{t})|+|N_{F}(s)\cap N_{H_{2}}(v_{t})| &\geq |N_{F}(s)\cap N_{F}(v_{t})|+|X|+|N_{H_{2}}(s)\cap N_{F}(v_{t})| &\geq |N_{F}(s)|-|N_{H_{1}}(s)\cap N_{F}(v_{t})|+|X| &=r-|N_{H_{1}}(s)\cap N_{F}(v_{t})|+|X|
a_ia_j=0\,\,{\rm if}\,\, i\leq j &a_ia_ja_j^\dag= \alpha_{i, j}a_i &a^\dag_ia_i=a_{i-1}a^\dag_{i-1}- a_ia^\dag_i &a_ia^\dag_i a_j=a_j-\alpha_{i, j}\sum_{j<k\leq i}a^\dag_ka_ka_j\,\,
-{\cal L}= \frac{1}{2}\left[2\sum_{i =1}^{2}\partial_+\phi^{i}\partial_- \phi^{i}- \tanh^2 \left ({\phi^{1}- \phi^{2}\over 2}\right)\partial_+\phi^{3}\partial_- \phi^{3}\right] + \sum_{i =1}^{2}\exp (2\phi^i)
\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)] = \mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi)]e^{-i\mu e^t g}&=e^tp(0,e^tg)e^{-i\mu e^t g}.
\vec{\gamma}= \left( \begin{array}{c c}{0}&{\vec{\sigma}}\\{- \vec{\sigma}}&{0}\\ \end{array}\right) , \gamma_{5}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , \gamma_{0}= \left( \begin{array}{c c}{0}&{- 1}\\{- 1}&{0}\\ \end{array}\right)
\frac{1}{|Q|}\int_Q{\big| \mathfrak{S}_{[a]}^{near}f(x)\big|}dx \lesssim \frac{1}{|Q|}\int_Q{\sum_{k=0}^{\infty}{2^{km}\mathfrak{M}_{\sigma,2^k}\Pi^*_kf(x)}}dx &\lesssim \sup_{0\leq k\leq 2}{\big\Vert 2^{km}\mathfrak{M}_{\sigma,2^k}\Pi^*_kf \big\Vert_{L^{\infty}}}+\frac{1}{|Q|}\int_Q{\sum_{k=3}^{\infty}{2^{km}\mathfrak{M}_{\sigma,2^k}\Pi^*_kf(x)}}dx
\min \Big( \lambda \varepsilon_K + \frac{1}{K}\sum_{k = 1}^K \sum_{a \in \mathcal{A}}\Big(u_a - \lambda (u_a - \hat{c}_a^{\mbox{\tiny \upshape (\itshape k\upshape)}})\Big)x_a \Big) \mbox{s.t.}0 \leq \lambda \leq 1.
\begin{array}{r r r l l}{{{\cal B}_{0}}}&{{:}}&{{f_{0}- 1}}&{{=}}&{{0 \; ,}}\\{{{\cal B}_{1}}}&{{:}}&{{f_{1}- f_{2}}}&{{=}}&{{0}}\\{{\mathrm{~ a n d ~}\tilde{\cal B}_{1 2}}}&{{:}}&{{F ( f_{0}^{- 1 / 2}f_{1}, f_{0}^{- 1 / 2}f_{2})}}&{{=}}&{{0}}\\ \end{array}
\left(\begin{array}[c]{ccc}\varphi\left( \mathrm{i}s-0\right) , \psi\left( \mathrm{i}s-0\right)\end{array}\right) & =\left(\begin{array}[c]{ccc}\varphi\left( \mathrm{i}s+0\right) -2\mathrm{i}\pi\delta\left( s,t\right)\psi\left( \mathrm{i}s+0\right) , \psi\left( \mathrm{i}s+0\right)\end{array}\right) ,
\lim_{n\rightarrow\infty}\left( Y_{l_{1}}^{m_{1}}\star_{\vec{b}}^{n}Y_{l_{2}}^{m_{2}}\right) \ =\ \lim_{n\rightarrow\infty}\left( Y_{l_{1}}^{m_{1}}\star_{1}^{n}Y_{l_{2}}^{m_{2}}\right)
\sigma\in O_{2,0},\gamma\in O_{2,0},p_1\in O_{2,0},p_0=O_{2,-1}, &\omega_x-\phi_x\in O_{2,0},|\partial_x|\omega-|\partial_x|\phi\in O_{2,0},V-\phi_x\in O_{2,-1/2},B-G(h)\phi\in O_{2,-1/2}.
\langle \, \Lambda \, \vert \Lambda \, \rangle \, = 1 \, , \, \, \langle \, \Phi \vert X \Psi \, \rangle \, = \, \langle \,{^t X}\Phi \vert \Psi \, \rangle \, , \, \, \langle \, \Phi \vert \Psi \, \rangle = \langle \, \Psi \vert \Phi \, \rangle \, .
A_k\cap A_m=\varnothing \;\; (k\neq m), q_k<h_k\leq|f(x)| \;\; (k\in\mathbb{N}, \;\; x\in A_k), \lim_{k\to\infty}q_k=\infty, 0<|A_k|\leq c\Big(\frac{h_k}{q_k}\Big) \;\; (k\in\mathbb{N}), \sum_{k=1}^\infty \Phi\Big(\frac{h_k}{q_k}\Big)|A_k|=\infty.
\left\lbrace\begin{aligned}X_{n+1}^{\epsilon,\Delta t}=\Phi(\frac{\Delta tm_{n+1}^{\epsilon,\Delta t}}{\epsilon},X_n^{\epsilon,\Delta t}) m_{n+1}^{\epsilon,\Delta t}=m_n^{\epsilon,\Delta t}-\frac{\Delta t}{\epsilon^2}m_{n+1}^{\epsilon,\Delta t}+\frac{\Delta\beta_n}{\epsilon},\end{aligned}\right.