harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F = \left\{\lambda \in \overline{C}\,\, \middle| \,\, \begin{aligned}\langle \lambda + \rho, \alpha^{\vee}\rangle = n_{\alpha}p, \,\, (n_{\alpha}-1)p < \langle \lambda + \rho, \alpha^{\vee}\rangle < n_{\alpha}p, \,\, \end{aligned}\right\}
\delta_{\lambda \mu}=\langle P_{\lambda}(x;\zeta),{Q'}^{(r)}_{\mu}(x;\zeta)\rangle_{0}=b_{\lambda}(\zeta)^{-1}\sum_{\rho\in CP_{r,n}}Q_{\rho}^{\lambda}{Q'}_{\rho}^{\mu}z_{\rho},
\begin{aligned}{1\over q}\varphi(x_q, t_q)^{1-q}(\lambda h_q+(1-\lambda) k_q)+\mu(w_q(x_q, t_q)) &\geq -w_q(x_q, t_q)^{1-q}\left (\lambda u(y_q, t_q)^{q-1}+ (1-\lambda)u(z_q, t_q)^{q-1}\right) \omega(|X_q|).\end{aligned}
V_{eff}(z)=\frac{15}{4\big(|z|+\ell\big)^2}-\frac{3}{\ell}\,\delta(z)\bigg[1-\frac{2\ell\gamma}{9}\left(\partial_z\psi(0_+)+\frac{3}{2\ell}\,\psi(0_+)\right)\bigg]\,,
(f_1(x)+2f_2(x)+2uf_{14}(x))^* = f_1^*(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) = f_1(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) & = (f_1(x)+2f_2(x)+2uf_{14}(x))c \in C,
(\underline{c}_1)_n = \begin{cases}c_n, n>\lfloor\max(p_K(\epsilon),q_T(\epsilon))\rfloor, 0,{\rm otherwise}, \end{cases}(\underline{c}_2)_n = \begin{cases}c_n, n\leq\lfloor\max(p_K(\epsilon),q_T(\epsilon))\rfloor, 0,{\rm otherwise}. \end{cases}
\Omega = \,\displaystyle\sum_{q_2\sim C/q_1}\,\displaystyle\sum_{q'_2\sim C/q_1}\,\,\sum_{m \sim M_1}\,\sum_{m_1 \sim M_1}\frac{\lambda_f(m)\,\lambda_f(m_1)}{(mm_1)^{1/4}}\,\mathcal{H}(...),
\mathbb{E}_0 \left[ \prod_{\omega = 1}^4\left(h\left(\mathbf{R}^{(1 2)}_{\mathbf{i}^\omega}\right) \right)\right] = \prod_{j=1}^2\;\mathbb{E}_0 \left[ \prod_{\omega\in \Omega_j}\left(h\left(\mathbf{R}^{(12)}_{\mathbf{i}^{\omega}}\right) \right)\right] \;=\; (\zeta_d^h)^2.
\begin{array}{r c l}{{}}&{{}}&{{\delta ( i_{h}i_{k}N^{( 8 )}) = 6 \partial ( i_{h}i_{k}\Omega^{( 7 )})}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{\delta ( i_{h}i_{k}{\cal N}^{( 8 )}) = 6 \partial ( i_{h}i_{k}{\tilde{\Omega}}^{( 7 )}) \, .}}\\ \end{array}
J_1(\beta_0):=\frac{1}{i_2(\beta_0)}-\bigg(B^{*}\beta_0^{\nu+1}-\frac{u_c}{M(\beta_0)}\bigg)^{\tfrac{1}{\nu+1}}, J_2(\beta_0):=\frac{1}{i_1(\beta_0)}-B^{*}\beta_0^{\nu+1}.
\|B_{6,M_n,n}\|^2_{L^2(\Omega)}= \sum_{p=M_n+1}^\infty \Big<u(\cdot,t), \phi_p \Big>^2 &= \sum_{p=M_n+1}^\infty p^{-2\beta \alpha}e^{-2p^{2\beta}t}e^{2p^{2\beta}t}p^{2\beta \alpha}\Big<u(\cdot,t), \phi_p \Big>^2\le M_n^{-2\beta \alpha}e^{-2M_n^{2\beta}t}\widetilde P_2.
\{J_{( \mu )}^{j , m}, J_{( \mu^{\prime})}^{j^{\prime}, m^{\prime}}\}( x ) = ( x , [ \mathrm{g r a d}_{x}J_{( \mu )}^{j , m}, \mathrm{g r a d}_{x}J_{( \mu^{\prime})}^{j^{\prime}, m^{\prime}}] )
{}~~~~~~~~~~~~~\times \Bigg\{\sum_{\sigma\in Z_N}\frac{w(x_{13},x_1u,x_{58}|\sigma+j_2+j_3)w(x_2x_{67},x_2v,x_6x_{24}|\sigma)s(\sigma,j_1)}{w(x_5x_{13},\omega x_1u,\omega x_1x_{58}|\sigma+j_2)w(x_{67},x_2v,x_{24}|\sigma+i_3)}\Bigg\}_0 ~~
{}_3F_{2}\begin{pmatrix}-j, 2N-M-j+\frac{1}{2}, -M ; 1 N-M-j, N-M-j+\frac{1}{2}\end{pmatrix}\times \times 4^{-M}(-2j-2M+2N)_{2M}=\sum_{l=0}^\infty(-1)^{l}{M\choose l}(j-l+1)_l\times 4^{l-M}\times (2N-2M-2j+2l)_{2(M-l)}(j-l-2N+M+\frac{3}{2})_{l},
\omega: \bigsqcup_{k=1}^{K}\bigsqcup_{\ell=1}^{m_{k}}\Gamma z_{(k,\ell)}\Gamma \to \Gamma g^{-1}\Gamma\times_{\Gamma}\Gamma h^{-1}\Gamma \gamma z_{(k,\ell)}\delta \mapsto [\gamma g_{i(k,\ell)},h_{j(k,\ell)}\delta],
- g \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}_{s}-\begin{pmatrix}0 -e g -2f\end{pmatrix}\begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}_{t}= \Lambda \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}+ \begin{pmatrix}e H^{j}- g F^{j}2f H^{j}- g G^{j}\end{pmatrix}
M (X_{n-5}') = M(\Delta_n) \setminus \{\sigma \}\cup\{\sigma \setminus \{v^{i_1}, \ldots , v^{i_{n-5}}, v^{p}\}, \sigma \setminus \{v^{i_1}, \ldots , v^{i_{n-5}}, v^{n}\}\}\cup & \{\sigma \setminus \{v^{l_1}, v^{l_2}, \ldots , v^{l_{n-5}}\}: \{l_1, l_2, \ldots, l_{n-5}\}\subseteq \{4, 5, \ldots, n-1\}\setminus \{i_1, \ldots, i_{n-5}\}\}.
\vert \tau_j(y) -x \vert \leq \vert \sigma_{j}(\tau_{j-1}(y)) - \tau_{j-1}(y) \vert + \vert \tau_{j-1}(y) -x\vert \leq (1+36C_1(m)\delta + 24 \delta)r_j + r -(2+36C_1(m)\delta+24 \delta)\sum \limits_{k=j}^\infty r_k \leq r- (2+36C_1(m)\delta+24 \delta)\sum \limits_{k=j+1}^\infty r_k.
\Phi^*(e^{\lambda\mathtt{P}})^*e^{\lambda\mathtt{P}}\Phi =\mathbf{1}+\Phi^*\big[\lambda \mathtt{U}(\mathtt{P})+\lambda^2\mathtt{V}(\mathbf{0},\mathtt{P})\big]\Phi+\lambda^3\mathtt{S}^{(\lambda)}(\Phi,\mathtt{P})\,,
\frac{1}{N^2}\sum\limits_{\mathbf{m}\in \mathbb{Z}^3}\Big( \prod\limits_{j=1}^4 f_j(\Xi(\mathbf{m})_j+ \widetilde{\mathbf{r}}_j) \Big) F(\Xi(\mathbf{m}) + \widetilde{\mathbf{r}})1_{[-10,10]^2}(L(\Xi\mathbf{m}+ \widetilde{\mathbf{r}}))
\widehat{H}= \underbrace{-\frac{\hbar^2}{2M}\Delta_{\boldsymbol{r}}}_{\hbox{$:= \,\widehat{T}_n$}}-\,\underbrace{\frac{\hbar^2}{2m}\Delta_{\boldsymbol{x}}+ V(\boldsymbol{r},\boldsymbol{x})}_{\hbox{$:= \,\widehat{H}_e$}}\, ,
\sum_{\substack{i \in M_P, i \neq i^*}}\sum_{j \in N_i}a_{ij}\tilde x_{ij}+ \sum_{\substack{ij \in P, i \notin M_0, i \neq i^*}}(b-s) \tilde x_{ij}+ a_{i^* j^*}\tilde x_{i^* j^*}\leq s + (b-s) |\bar P| = s + (|M_P - M_0| - 1) (b-s) & = b + (|M_P - M_0| - 2) (b-s).
\left\langle (f_1\otimes g_1)\mathbf{1}_V, f_2\otimes g_2\right\rangle_{\mathcal{H}}=\int_V\left\langle (f_1\otimes g_1)_x, (f_2\otimes g_2)_x\right\rangle_{\mathcal{H}_x}m(dx) &=\int_V G_1(y)G_2(y)\left\langle \nabla F_1(y), Z(x)\nabla F_2(y)\right\rangle_{l_2}m(dx).
{x\zeta_{(p, q)}}{\cdot}\alpha^{\rm op}=\begin{cases}x\zeta_{(\widetilde{p}, e_{t(\alpha)})}-\sum\limits_{\beta\in T(\alpha)}x \zeta_{(\widetilde{p}\beta, \beta)}, \begin{matrix}l(q)=0, p=\widetilde{p}\alpha, \alpha ;\end{matrix}\delta_{s(\alpha),t(q)}x\zeta_{(p, \alpha q)}, .\end{cases}
\sum_{[K_p:\mathbb{Q}_p]=n}\frac{|\mathrm{Disc}(K_p)|_{p}^{1/2}}{|\mathrm{Aut}(K_p)|}\int_{\mathcal{O}_{K_p}}|\mathrm{Disc}(f_\alpha)|_p^{1/2}\psi_{p^{2k}}(f_\alpha) e_{p^{2k}}(-u_1\sigma_1(\pmb{\lambda}) + u_2\sigma_2(\pmb{\lambda})) d\mu(\alpha),
g \cdot g^{\prime}= \left( \begin{array}{r r}{{a \cdot a^{\prime}+ b \cdot c^{\prime}}}&{{a \cdot b^{\prime}+ b \cdot d^{\prime}}}\\{{c \cdot a^{\prime}+ d \cdot c^{\prime}}}&{{c \cdot b^{\prime}+ d \cdot d^{\prime}}}\\ \end{array}\right)
G^{E}( 2 \pi n T , k ) = - \pi C_{\cal{O}}\left[ \psi \left({\frac{1 + n}{2}}-{\frac{i k}{4 \pi T}}\right) + \psi \left({\frac{1 + n}{2}}+{\frac{i k}{4 \pi T}}\right) \right] .
I ( N ) = \sum_{k , l = 1}^{N - 1}\frac{1}{2}[ 1 - ( - 1 )^{k + l}] \frac{1}{4 N^{4}}\frac{\cos^{2}k \pi / 2 N \cos^{2}l \pi / 2 N}{\sin^{2}( l - k ) \pi / 2 N \sin^{2}( k + l ) \pi / 2 N}.
\sum_{n=0}^{\infty}L_{n}x^{n}=\sum_{n=0}^{\infty}(\phi^{n}+\psi^{n})x^{n}=\sum_{n=0}^{\infty}\phi^{n}x^{n}+\sum_{n=0}^{\infty}\psi^{n}x^{n}=\frac{1}{1-\phi x}+\frac{1}{1-\psi x}=\Phi(x)
S_{( i s e n t r p i c )}[ J^{\mu}, \varphi^{'},{\tilde{\alpha}}^{r}, \beta_{r},{}^{4}g^{\mu \nu}] = \int d^{4}x \Big [ - \sqrt{{}^{4}g}\rho ({\frac{| J |}{\sqrt{{}^{4}g}}}) + J^{\mu}( \partial_{\mu}\varphi^{'}+ \beta_{r}\partial_{\mu}{\tilde{\alpha}}^{r}) \Big ]
\left( \sum_{k = 1}^{K}\lambda_{k}\: F_{k}\right)^{*}\; = \; \sum_{k = 1}^{K}\overline{{\lambda_{k}}}\: F_{k}^{*}\; \stackrel{( \ref{e q : 4 7})}{=}\; \sum_{k = 1}^{K}\lambda_{k}\: F_{k}\; .
F_{1}\circ f_{2}^{( 3 )}( z ) = \widehat{F}_{2}\circ f^{( m )}\circ I ( z ) , \quad G_{1}\circ \left( F_{1}\circ f_{3}^{( 3 )}\right) ( z ) = \widehat{G}_{2}\circ f^{( m )}\circ I ( z ) ,
\begin{aligned}\int_\Omega \left[p\left(\frac{|\nabla u_h|}{\lambda_h}\right)^{p-2}+qa(x)\left(\frac{|\nabla u_h|}{\lambda_h}\right)^{q-2}\right]\frac{\nabla u_h}{\lambda_h}\cdot\nabla v \,dx &\,=\lambda_hS(u_h)\int_\Omega\left(p|u_h|^{p-2}+qa(x)|u_h|^{q-2}\right)u_hvdx\quad\mbox{for all}v\in W^{1,\mathcal H}_0(\Omega)\mbox{and}h\ge1,\end{aligned}
\langle I_T \rangle_T = \lim_{M_n \rightarrow 0}\displaystyle \sum \limits_{i = 0}^{n-1}f(t_i)^2 (t_{i+1}- t_i) & = \lim_{M_n \rightarrow 0}\displaystyle \sum \limits_{i = 0}^{n-1}\int^{t_{i+1}}_{t_i}f(u)^2 du & = \int^{T}_0 f(u)^2 du.
\nabla_{p}\varphi\left(z,t\right) =\frac{V\left(0\right)^{-1}p}{2}-z +h\sum_{s_{k}\in\mathcal{S}}\Big(Q_{p}\left(s_{k}\right)E_{p}\left(s_{k}\right)\left(E_{p}\left(s_{k}\right)^{\dagger}p\right) -Q_{e}\left(s_{k}\right)E_{e}\left(s_{k}\right)\left(E_{e}\left(s_{k}\right)^{\dagger}p\right)\Big),
\frac{\tau_{n}(t)}{\sigma_{n,n}(t)}= \alpha_{n+2}(t) + \beta_{n+1}(t) + \frac{\tau_{n-1}(t)}{\sigma_{n-1,n-1}(t)}= \sum_{k=1}^{n+1}[\alpha_{k+1}(t) + \beta_{k}(t)] = \sum_{k=1}^{n+1}\gamma_{k}(t),
u(t,x)=\frac{e^{-x}}{2}E_+(t),~~x>q(t,b), u(t,x)=\frac{e^{x}}{2}E_-(t),~~x<q(t,a), v(t,x)=\frac{e^{-x}}{2}F_+(t),~~x>q(t,b), v(t,x)=\frac{e^{x}}{2}F_-(t),~~x<q(t,a),
\psi(h_3) =\psi(h_4)=\psi(h_5)=\psi(h_6)=\frac{n+1}{4}e \, n\equiv 3 \pmod{4}\,, \psi(h_3) =\psi(h_4)=\psi(h_5)=\psi(h_6)=\frac{3n+1}{4}e n\equiv 1 \pmod{4}\,.
\begin{aligned}X(e_i)X'(e_i) =v^{\varLambda(e_i,\sum\limits_{1\leq l\leq m}[b_{li}]_+ e_l)}X({\sum\limits_{1\leq l\leq m}[b_{li}]_+ e_l})+v^{\varLambda(e_i,\sum\limits_{1\leq l\leq m}[-b_{li}]_+ e_l)}X({\sum\limits_{1\leq l\leq m}[-b_{li}]_+ e_l}). \end{aligned}
I_{g r a v}^{( t )}- 3 ( \frac{6}{\pi^{2}})^{3}( x_{1 2}^{2}x_{3 4}^{2})^{- 4}u^{- 3}\frac{1 6}{2 7}\sum_{n = 0}^{\infty}\frac{u^{n}}{n !}G_{n}( 2 , 2 ; v ) .
A_N:= \sqrt{\tfrac{2}{\tilde\mu}}\sum_{i=1}^N\sqrt{\tfrac{\delta_i}{a_i}}+\left(\|x^*-v_0\|^2+\tfrac{M^2N(r+\frac{\rho}{2}(N+3))}{\tilde\mu}+\tfrac{2}{\tilde\mu}\sum^N_{i=1}\tfrac{\delta_ia_i+2\varepsilon_i}{a_i^2}+\tfrac{2}{\tilde\mu}\left(\sum_{i=1}^N \sqrt{\tfrac{\delta_i}{a_i}}\right)^2\right)^{1/2}.
\Gamma_{[ \mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda ]}= \frac{1}{4 !}\sum_{P}( - 1 )^{\delta_{P}}\Gamma_{\mu_{P ( 1 )}}\Gamma_{\nu_{P ( 2 )}}\Gamma_{\rho_{P ( 3 )}}\Gamma_{\lambda_{P ( 4 )}}
\;\|Au_*\|_{L^2}^2(t) + 2\nu\int_0^t \|A^{1+s/2}u_*\|_{L^2}^2\,d\tau = \;\|Au_0\|_{L^2}^2 - 2\int_0^t \langle A^2 u_*, u_*\cdot \nabla u_*+\mathcal{U}^\alpha(u_*,u_*)\rangle(\tau)\,d\tau.
\kappa_E^{-1}( \mathcal{Z}_E(\mu) ) = \kappa_E^{-1}(\{e_1\cdots e_{i_0-1}e_{i_0}^\infty\}) = \{e_1\cdots e_{i_0-1}\}= \mathcal{Z}_{E_\curlyvee}( e_1 \cdots e_{i_0-1})
\phi ({\vec{X}}) = \dot{\vec{X}}\cdot ({\ddot{\vec{X}}\wedge \dddot{\vec{X}}\wedge \ldots\wedge \mathop{\vec{X}}\limits^{\left( n \right)}}) = \det({\dot{\vec{X}},\ddot{\vec{X}}, \dddot{\vec{X}},\ldots ,\mathop{\vec{X}}\limits^{\left( n \right)}}) = 0
d s^{2}= \frac{X^{1 / 2}}{\rho}e^{2 A ( r )}d x_{/ /}^{2}+ \frac{X^{1 / 2}}{\rho}\left( d r^{2}+ \frac{L^{2}}{\rho^{2}}\left[ d \theta^{2}+ \frac{\sin^{2}\theta}{X}d \phi^{2}+ \frac{\rho^{6}\cos^{2}\theta}{X}d \Omega_{3}^{2}\right] \right) ,
\lim\limits_{x\to -\infty}\frac{1}{|x|^{\frac{1}{\beta}}\ell^{\frac{1}{\beta}}(|x|^{\frac{1}{\beta}})}\eta_K(x)=\int^{\infty}_0 \big(K_{\infty}(-t^{-\beta})-K_{\infty}(0)\big) dt.
- \int\limits_{0}^{\infty}e^{q_1}\, b_1 \, q_1 \, e^{b_1 x}\, e^{-q_1 \, e^{b_1 x}}\, q_1 \, e^{b_1 x}\, dx= - e^{q_1}\, q_1^2 \, \int\limits_{1}^{\infty}y \, e^{-q_1 y}\, dy
q_{k,\tau,\mu}(L)=1-\alpha_{k,\min}\epsilon_{k,\min}\Bigg( 2-2\mu-\frac{H_{k,\tau}(L^2+\epsilon_{k,\max}^2)\alpha_{k,\max}^2}{\alpha_{k,\min}\epsilon_{k,\min}}-\frac{2L\Delta_k}{\alpha_{k,\min}\epsilon_{k,\min}}\Bigg).
\sum_{j=1}^{k}\psi_{k,j}^{\ast}(Q^{\prime})\leq (k-n)\frac{(k+1)n}{k(n+1)}\psi_{n,1}^{\ast}(Q) + \frac{(k+1)n}{k(n+1)}\sum_{j=1}^{n}\psi_{n,j}^{\ast}(Q) + \frac{k-n}{n+1}.
\sin \theta_{n}=\left\Vert P_{T^{\dag}T\left( X_{n}\right)}-P_{T^{\ast}T\left( X_{n}\right)}\right\Vert =\sqrt{1-{\Vert I-Q_{n}\Vert}^{-2}}\in \left[ 0,1\right) ,n\in\mathbb{N}.
\begin{cases}\gamma_{2}+ \gamma_{4}+ \gamma_{14}= \gamma_{3}+ \gamma_{5}+ \gamma_{14}= 0, \gamma_{2}+ \gamma_{4}+ \gamma_{14}= \gamma_{3}+ \gamma_{5}+ \gamma_{14}= 0 , \gamma_{14}= \gamma_{18},\ \gamma_{7}= \gamma_{10},\ \gamma_{8}= \gamma_{11},\ \gamma_{9}= \gamma_{16},\ \gamma_{14}= \gamma_{17}. \end{cases}
P_{n}\stackrel{(d)}{=}\boldsymbol{1}_{\{B_{n}=1\}}\cdot \left(P_{U_{n}}' + P_{n-U_{n}}''\right) + \boldsymbol{1}_{\{B_{n}=0\}}\cdot \left(P_{U_{n}}' \cdot P_{n-U_{n}}''\right), , P_{1}= 1,
A_2 = g^{\mu\nu}g^{\alpha\beta}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}+ b^{\mu\nu\alpha}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}+C^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+ d^{\mu}\nabla_{mu}+ X,
\bigg({}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12\beta &\frac12+\frac12(\alpha+\beta)\end{matrix}\bigg|\,z\bigg]\bigg)^2={}_3F_2\bigg[\begin{matrix}\alpha \beta \frac12(\alpha+\beta) &\alpha+\beta \frac12+\frac12(\alpha+\beta)\end{matrix}\bigg|\,z\bigg].
S_{1}=\sum_{2a_1\le a}\frac{(-2)^{a-2a_1}a!}{a_1!a_1!(a-2a_1)!}E\{\|X_1\|_{2}^{2a_1}\|Y_1\|_{2}^{2a_1}(X_1^{\top}Y_1)^{a-2a_1}\}=\sum_{2a_1\le a}\frac{(-2)^{a-2a_1}a!}{a_1!a_1!(a-2a_1)!}E\{\|Y_1\|_{2}^{2a_1}\|Y_2\|_{2}^{2a_1}(Y_1^{\top}Y_2)^{a-2a_1}\},
V_{e f f}( \sigma ) ={\frac{\sigma^{2}}{2 \lambda}}- \left[ \frac{\Lambda \sigma^{2}}{2 \pi^{3 / 2}}+ \frac{\sqrt{2}}{\pi}( e B )^{3 / 2}\zeta \left( - \frac{1}{2}, 1 + \frac{\sigma^{2}}{2 e B}\right) + \frac{1}{2 \pi}e B \sigma \right]
\left[\hat{x}^\mu_k,\hat{x}^\nu_l\right]|_{t}=\,\left[\hat{y}^\mu_k,\hat{y}^\nu_l\right]|_{t}=\,\left[\hat{x}^\mu_k,\hat{y}^\nu_l\right]|_{t}=0, \quad\,\mu,\nu=1,2,3,4;\, k,l=1,...,N.
\frac{R}{(L+1)(1-R)}\le \frac{1}{L+1}\frac{1}{1-R}\le \frac{\log q}{2 \log d(\chi)}\frac{4\log d(\chi)}{\log q \log (d(\chi) \log d(\chi) /n)}= \frac{2}{\log (d(\chi) \log d(\chi)/n)}\le 1
\begin{aligned}\dot{x}_i = \{x_i, H^\textsc{csm}\}= \hphantom{+}\frac{\partial H^\textsc{csm}}{\partial p_i}= p_i \, , \dot{p}_j = \{p_j, H^\textsc{csm}\}= -\frac{\partial H^\textsc{csm}}{\partial x_j}= -g^2 \, \partial_{x_j}\! \sum_{i (\neq j)}^N \frac{1}{4 \sin^2[(x_i - x_j)/2]}\, .\end{aligned}
\phi_1(z) ={_2F_1}\left(\frac{1}{2},\frac{2-a}{2};\frac{1}{2};-z^2\right) = (1+z^2)^{\frac{a-2}{2}}, \phi_2(z) = z\cdot{_2F_1}\left(1,\frac{3-a}{2};\frac{3}{2};-z^2\right).
\begin{aligned}\xi_\gamma u^{\gamma'}p^{\gamma''}\equiv \frac{a_\gamma}{q}u^{\gamma'}p^{\gamma''}+ \theta_\gamma n^{\gamma'}p^{\gamma''}\pmod 1 &\equiv \frac{a_\gamma}{q}(r')^{\gamma'}(r'')^{\gamma''}+ \theta_\gamma u^{\gamma'}p^{\gamma''}\pmod 1,\end{aligned}
h_{\mathrm{r}_i\mathrm{d}}[k]=\alpha_{\mathrm{r}_i\mathrm{d}}h_{\mathrm{r}_i\mathrm{d}}[k-1]+\sqrt{1-\alpha_{\mathrm{r}_i\mathrm{d}}^2}e_{\mathrm{r}_i\mathrm{d}}[k]
U^{(\theta)}Tg - g = \sum_{\substack{I,I'\in\mathcal{D}_{\leq n}I \neq I'}}a_{I'}\frac{\langle T b_{I'}^{(\theta)}, b_I^{(\theta)}\rangle}{\langle Tb_I^{(\theta)}, b_I^{(\theta)}\rangle}b_I^{(\theta)}.
\quad\ \|\mathcal{V}[v+\eta] - \mathcal{V}[v] - M \eta\|_{L^r(0,T)}\qquad\qquad\qquad\qquad\qquad\le \rho_{v,p,r}(\|\eta\|_{\infty}) (\|\eta'\|_{L^p(0,T)}+ |\eta(0)|) .
\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}=\frac{12(1-\nu^{2})}{Et^{3}}(-N_{x}\frac{\partial^{2}w}{\partial x^{2}})
{1\over{32\omega^2}}\left( -{{32\omega z_0^2}\over l^2}+ 9 +3\sqrt{-64\omega{{z_0^2}\over l^2}+9 -64{{z_0^2}\over l^2}\omega^2}\right) < z_0^2 \varepsilon^2 <{1\over{8\omega +3}}\left(-3-{{5z_0^2}\over l^2}\right) \, .
C_1:=\sqrt{32\, \delta C^2 + 4\, \delta^2 C^4}\,,\ \ \ \ \C_2:= - \max_{\partial \mathcal{D}_{\delta}}\, \big(\widehat{\Phi}^* - \varphi \big)(t,x)\,, \ \ \ \ \ \C_3:=\frac{C_2\, e^{-C_1}(||x_0||_1-n\delta)}{4(||x_0||_1 + n\delta)}
\delta_{H}= 2 \times 1 0^{- 5}\alpha_{0}^{- 1}( \Omega_{\mathrm{N R}})^{c_{1}+ c_{2}\ln \Omega_{\mathrm{N R}}}\exp{[ c_{3}( n - 1 ) + c_{4}( n - 1 )^{2}]}
\mathbb{E}_n \bigg[ \sup_{0 \le t \le T}\bigg| \frac{1}{\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}\sum_{i=0}^{\ell-1}F^\ell_{i,j}(s) \psi_i \nabla^n \varphi^n_j (s) ds \bigg|^2 \bigg]\le C \frac{T}{n^2}\| \partial_x \varphi \|^2_{L^2(\mathbb{R})}.
E(\prod_{x,y}s_{x,y}^{N^{(\frac{1}{2})}_{x,y}}\prod_x e^{-\sum_{x}\chi_{x}\hat{\mathcal{L}}_{\frac{1}{2}}^x})=E(e^{-\frac{1}{2}\sum_{x,y}C_{x,y}( s_{x,y}-1)\varphi^{\mathbb{R}}_x \varphi^{\mathbb{R}}_y-\frac{1}{2}\sum \chi_{x}({\varphi^{\mathbb{R}}_x})^2}).
M=\left (\begin{matrix}1 1 1 \dots 1 1 \psi_1(\sigma) \psi_1(\sigma)^2 \dots \psi_1(\sigma)^{2r-1}\dots \dots \dots \dots \dots 1 \psi_r(\sigma) \psi_r(\sigma)^2 \dots \psi_r(\sigma)^{2r-1}\end{matrix}\right )
I_{Q}(\mu):=\int(H\mu-Q')^{2}d\mu-\left(\int Q'd\mu\right)^{2}H\mu(x)=-p.v. \int \frac{z+w}{z-w}\mu(dw)=p.v.\int \cot\left(\frac{x-y}{2}\right)\bar{\mu}_{u}(dy)
( X , Y )_{\lambda ,{\bf k}}= \frac{1}{2}\int{\mathrm{T r}}_{\bf k}\bigg ({\frac{\delta_{r}X}{\delta{\cal A}}}{\frac{\delta_{l}Y}{\delta{\cal A}_{\lambda}^{*}}}-{\frac{\delta_{r}X}{\delta{\cal A}_{\lambda}^{*}}}{\frac{\delta_{l}Y}{\delta{\cal A}}}\bigg ) ,
\mathbb{P}_X = \mathbb{Q}_Y \iff \mathbb{E}_{\mathbb{P}}[\langle \ell, S(X) \rangle_{\mathcal{H}^1}] = \mathbb{E}_{\mathbb{Q}}[\langle \ell, S(Y) \rangle_{\mathcal{H}^1}], \forall \ell \in T(\mathbb{R}^{d+1}) &\iff \mathbb{E}_{\mathbb{P}}[S(X)] = \mathbb{E}_{\mathbb{Q}}[S(Y)].
f_i^\tau(\mathbf{R}, \mathbf{Q}) = \sum_{j \in N}R_j x_{ij}^{1, \tau}+ \sum_{j \in N}(1-\beta) Q_j x_{ij}^{0, \tau}= \mathbf{x}_i^{1, \tau}\mathbf{R}+ (1-\beta) \mathbf{x}_i^{0, \tau}\mathbf{Q}, g_i^\tau = \sum_{j \in N}x_{ij}^{1, \tau}= \mathbf{x}_i^{1, \tau}\mathbf{1}.
S_{2}^{( 0 )}={\frac{1}{2}}x^{i}\Lambda_{i j}x^{j}+ y_{\alpha a}^{*}D_{2 \beta}^{\alpha}c^{\beta a}+ \bar{y}_{\alpha}D_{4 \beta}^{\alpha}B^{\beta}+ \varepsilon^{a b}c_{\alpha a | b}^{*}D_{2 \gamma}^{- 1 \alpha}D_{4 \beta}^{\gamma}B^{\beta}.
(-q;q^2)_\infty\sum_{k=0}^{\infty}\frac{q^{2kr+\ell}}{1+q^{2kr+\ell}}=\sum_{k=0}^{\infty}\mathop{\prod_{m=0}^\infty}_{2m+1\neq 2kr+\ell}q^{2kr+\ell}(1+q^{2m+1}) \succeq 0 .
T(z) = -\frac{1}{2}:\! \partial X \partial X \! : +\,i\frac{\kappa}{2}\partial^2 X, \qquad\tilde T(\bar z) = -\frac{1}{2}:\! \bar\partial X\bar\partial X \! : -\, i\frac{\kappa}{2}\bar\partial^2 X,
\sum_{\mu=1}^n(-1)^{\mu}\begin{vmatrix}[0]_{\substack{j=1,...,s-1 k=1,...,s-1}}[0]_{j=1,...,s-1}[\lambda_j^{k-1}]_{\substack{j=1,...,s-1 k=1,...,n (k\neq \mu)}}[f_j(\lambda_k)]_{\substack{j=1,...,n k=1,...,s-1}}[h_{j,\mu}]_{j=1,...,n}[h_{j,k}]_{\substack{j=1,...,n k=1,...,n (k\neq \mu)}}\end{vmatrix}.
E \Bigl[ F_I( x^o, y^o) \Bigr] = E \Bigl[ E_q [ F_I( x^o, y^o) ] , Q_{\partial K}\Bigr] + O(n^{-2}) = E \Bigl[ E_q [ F_I( x^q, y^q) ] , Q_{\partial K}\Bigr] + O(n^{-2}),
\left. -\frac{1}{4}e^\varphi \,{\widetilde F}_{\mu\nu}^J{\widetilde F}^{J\mu\nu}-\frac{1}{\sqrt 2}e^{\varphi}{\widetilde F}_{\mu\nu}^J{\widetilde C}^{[\mu}\partial^{\nu]}{\tilde D}^J-\frac{1}{2}e^{\varphi}{\widetilde C}_{[\mu}\partial_{\nu]}{\tilde D}^J{\widetilde C}^{[\mu}\partial^{\nu]}{\tilde D}^J \right.
\left|\left(\frac{zf_2'(z)}{f_2(z)}\right)^2-1\right| =\left|\left(1-\frac{\rho_2}{(1-\rho_2^2)}\left( \frac{u\rho_2^2+4\rho_2+u}{\rho_2^2+u\rho_2+1}+\frac{v\rho_2^2+2\rho_2+v}{v\rho_2+1}+\frac{q \rho_2^2+4\rho_2+q}{\rho_2^2+q \rho_2+1}\right)\right)^2-1\right|=1
{\cal Z}= \sum_{k = 0}^{+ \infty}( - 1 )^{k}\exp ( - \frac{\pi k}{g_{n c}^{2}\theta}) \frac{1}{2 \pi i}\int_{{\cal C}_{0}}\frac{d z}{z^{k + 1}}\exp \left[ z \Phi ( - z ; \frac{3}{2}; 1 ) \frac{A_{n c}}{\sqrt{2 \pi g_{n c}^{2}\theta^{3}}}\, \, \right] ,
Z_L^{(m,n)}(\tau) = Z^{m/2}_{n/2}(\tau)^{16}+Z^{m/2}_{1+n/2}(\tau)^{16}+Z^{1+m/2}_{n/2}(\tau)^{16}+Z^{1+m/2}_{1+n/2}(\tau)^{16}.
N_\infty(\exp(T_L(g_1) + ... + T_L(g_r))) = \max_i |T_L(g_i)|^{1/i}\geq |T_L(g_k)|^{1/k}\overset{\eqref{e:short-layer-vectors}}{\geq}\epsilon |g_k|^{1/k}= \epsilon N_\infty(g).
p_N(n)= \sum_{\substack{0\leqslant h<k \leqslant N (h,k)=1}}\sum_{\ell=1}^{\lfloor N/k \rfloor}C_{hk\ell}(N) \binom{\ell-1+n}{\ell-1}(-1)^\ell e^{-2\pi i h(\ell+n)/k},
\cos\zeta(s) + \cos\theta_c = (s-s^{*}_b)\frac{F^{(1)}_{\pi-\theta_c}(s)F^{(2)}_{\pi-\theta_c}(s)}{F^{(3)}_{\pi-\theta_c}(s)}\frac{\sqrt2 e^{\frac{i3\pi}4}}{\sqrt{k_{+}}}.
\Psi(\omega)=\begin{cases}\frac{1}{4^n \pi^{n+m}}\sqrt{\frac{(2 \pi)^m y_\omega^{m-1}\sin(y_\omega)^3}{2\omega^{m-1}(\sin(y_\omega)-y_\omega\cos(y_\omega))}}, \frac{(3\pi)^{m/2}}{4^n \pi^{n+m}}, \end{cases}
\widetilde{H}(z_{1},z_{2}, z_{3}) = -\displaystyle\frac{k}{2c_{1}}\left(z_{1}^{2}+\displaystyle\frac{c_{1}}{c_{2}}z_{2}^{2}\right)~~\hbox{and}~~\widetilde{C}(z_{1}, z_{2}, z_{3}) = z_{2}-\displaystyle\frac{1}{2 k}z_{3}^{2}
d \Theta_{\underline{{\mu}}}^{2}= v_{\underline{{\mu}}q}^{+}\pi_{q}^{2 -}+ v_{\underline{{\mu}}\dot{q}}^{-}\pi_{\dot{q}}^{2 +}, = e^{- 2}\psi_{- 2 \dot{q}}^{2 +}v_{\underline{{\mu}}\dot{q}}^{-}+ e_{q}^{-}v_{\underline{{\mu}}q}^{+},
\tilde{\mbox{Aux}}\left[J\right]= \begin{pmatrix}\mbox{Aux}\left[J\right] \mbox{O}_{N^{p-1}}\dots \mbox{O}_{N^{p-1}}\mbox{O}_{N^{p-1}}\mbox{Aux}\left[J\right] \dots \mbox{O}_{N^{p-1}}\vdots \vdots \ddots \vdots \mbox{O}_{N^{p-1}}\mbox{O}_{N^{p-1}}\dots \mbox{Aux}\left[J\right] \end{pmatrix}.
\Phi(z_0)[x^{\Gamma}]=\int_{{\bf x}\nolinebreak\begin{picture}(10,3)(0,0)\setlength{\unitlength}{.5pt}\put(1,-7){\line(0,1){14}}\put(2,-7){$\scriptscriptstyle\partial\Sigma$}\end{picture}={\bf x}_0}\phi(z_0)\exp{\left(-S\left[{\bf x}^{\Sigma}\right]\right)}\, d\!{\bf x}^{\Sigma}.
\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(1)_{k}(a+b)_{k}}\frac{H_k(a-1)+H_k(b-1)}{x^k}=\bigg(\log\frac{x}{x-1}\bigg)\sum_{k=0}^{\infty}\frac{(a)_k(1-a)_k}{(1)_{k}(a+b)_{k}}\frac{1}{x^k}.
W ( x_{i}) = - \frac{1}{4}\sum_{i = 1}^{N}x_{i}^{4}+ \alpha \sum_{\stackrel{i , j = 1}{i < j}}^{N}\ln ( x_{i}^{2}- x_{j}^{2}) + \beta \sum_{i = 1}^{N}\ln x_{i}^{2}\, .
\left(y_{\tau_2}^{(i_2;L_2)}\right)^2 \circ y_{\tau_1}^{(i_1;L_1)}\circ \left(y_{\tau_2}^{(i_2;L_2)}\right)^2 \circ y_{\tau_1}^{(i_1;L_1)}= \mathrm{Id}.
\tilde{\cal H}= \frac{1}{2}(\vec{\cal E}^{\,2}+ \vec{\cal B}^{\,2}) + \frac{1}{2}p^{\dagger}p + \frac{1}{2(e\chi)^2}(\vec{\nabla}\vec{\cal E})^2 + \frac{1}{2}(\vec{\nabla}\chi)^2 +\frac{1}{2}(e\chi)^2\vec{\cal A}^{\,2}+ V(\chi^2/2) \ .
h_{R,r}(\eta_j, u_j) \ =\h_{R,r}(u_j z_j, u_j) \ \stackrel{\eqref{ree60}}{\le}\K_b h_{R, r}(2^{r(j)}u_j, u_j) \ \le\K_bd^{r(j)}h_{R,r}(u_j,u_j)\ \lesssim\K_bd^{r(j)}h_{R,r}(k_j,u_j),
F(x_{1},x_{2},x_{3}) =a_{1}f_{1}(x_{1})+a_{2}f_{2}(x_{2})+a_{3}f_{3}(x_{3}) +a_{4}f_{1}(x_{1})f_{2}(x_{2})+a_{5}f_{1}(x_{1})f_{3}(x_{3})+a_{6}f_{2}(x_{2})f_{3}(x_{3})
\times \int_{0}^{\infty}\frac{d \tau}{\tau}~ e^{2 \pi \tau}~ e^{- 2 \pi \tau \alpha^{\prime}k^{2}}~ e^{- \frac{y^{2}\tau}{2 \pi \alpha^{\prime}}}~ \mathrm{e}^{2 \pi \tau}~ T r \left( \prod_{n = 1}^{\infty}e^{- 2 \pi \tau \alpha_{- n}\cdot \alpha_{n}}\right) =
-j r^{-1}((r(\dfrac{\zeta{\pi'_n}^j}{1-\zeta{\pi'_n}^j}\alpha),\pi'_n)_{\rho',L,n}) = -j r^{-1}([r(\dfrac{\zeta{\pi'_n}^j}{1-\zeta{\pi'_n}^j}\alpha),\pi'_n]_{\rho',L,n}).
0 = T^{D}(e_{-},v_{+},w_{+}) = \langle D_{e_{-}}v_{+}- D_{v_{+}}e_{-}- [e_{-}, v_{+}], w_{+}\rangle + \langle v_{+}, D_{w_{+}}e_{-}\rangle & = \langle D_{e_{-}}v_{+}- [e_{-}, v_{+}], w_{+}\rangle,
F_{- \gamma}( p ) \, F_{\gamma}( p ) = \left( \frac{\Gamma ( 1 - \alpha - i \frac{\gamma p}{2 \pi}) \, \Gamma ( 1 - \alpha + i \frac{\gamma p}{2 \pi})}{\Gamma ( 1 + \alpha - i \frac{\gamma p}{2 \pi}) \, \Gamma ( 1 + \alpha + i \frac{\gamma p}{2 \pi})}\right)^{\frac{1}{2}}\, .
\texttt{P}_{3\nu}^t\big[\mathbf{X}_t\big]=\texttt{E}_\emptyset\big[ \boldsymbol{F}_\nu{\mathbf{Z}_\mathrm{f}^\otimes}^\star\mathbf{X}_t\mathbf{Z}_\mathrm{f}^\otimes \boldsymbol{F}_\nu^\star\big],
+ H_{1}^{\frac{1}{4}}H_{2}^{-\frac{1}{4}}(dz_{1}^{2}+dz_{2}^{2}+dz_{3}^{2}+dz_{4}^{2})+ H_{1}^{\frac{1}{4}}H_{2}^{\frac{3}{4}}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}).
D_F = \left( \begin{array}{cc}D_+, -\sqrt{2}e\phi -\sqrt{2}e\phi^*, \partial_-\end{array}\right)\,,\qquad-D_F^\dag = \left( \begin{array}{cc}D_-, \sqrt{2}e\phi \sqrt{2}e\phi^*, \partial_+\end{array}\right)\,.
I(X^n;W,Z^n,F) &\; \overset{(a)}{\leq}n\big(I(X;Z|A,U)+I(X;A,V,Y)-I(X;Y|A,U) \big) &+n\big(I(V;Z|A,X,U)\!-\!I(V;Y|A,X,U)\big) &-n(I(U;X|Z,A)+\!\epsilon^{\prime\prime}) &\; = n\big(I(V,X;Z|A,U)+I(X;A,V,Y)-I(V,X;Y|A,U) \big) &-n(I(U;X|Z,A)+\!\epsilon^{\prime\prime})
F_4=\, a+a^2+a^3+a^4+b+a^2 b+b^2+a b^2+b^3+b^4, F_3=\, 1+a^2+b+a^2 b+b^2+b^3, F_2=\, a^3+a^4+b+b^2+b^3+b^4, F_1=\, a+b+a b+b^3+k+a^2 k+b k+a^2 b k+b^2 k+b^3 k, F_0=\, b+b^2+a b^2+a^2 b^2+a k+b k+a b^2 k+b^3 k+a k^2+a^2 k^2 &+a^3 k^2+a^4 k^2+b k^2+a^2 b k^2+b^2 k^2+a b^2 k^2+b^3 k^2+b^4 k^2.
M_{pl}^2 ={M_X^3}\left({{k_0}\over\displaystyle{k_0^2 - (k_1-k_c)^2}}e^{-2 \sigma(y_0)}+{{k_1}\over\displaystyle{k_1^2 - (k_0+ k_c)^2}}e^{-2 \sigma(y_1)}\right)~.~\,
U(t)=e^{t^{\frac{1}{\alpha}}\Lambda_{1}-t\Lambda^{\alpha}}u_{0}-\int_{0}^{t}\left[e^{t^{\frac{1}{\alpha}}\Lambda_{1}-(t-\tau)\Lambda^{\alpha}}\nabla\cdot \left (e^{-\tau^{\frac{1}{\alpha}}\Lambda_{1}}U\cdot e^{-\tau^{\frac{1}{\alpha}}\Lambda_{1}}\nabla(-\Delta)^{-1}U\right)\right](\tau)d\tau.