harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\theta A+I_d=\begin{pmatrix}1 \theta 0 \hdots 0 \theta\cr\theta 1 \theta 0 \hdots 0\cr 0 \theta 1 \theta 0 \hdots\cr \ddots \ddots \ddots \cr0 \hdots 0 \theta 1 \theta\cr \theta 0 \hdots 0 \theta 1\end{pmatrix}\in \mathcal{S}_d^+.
\exp\left(-f-\sum_{k=2}^{\infty}\frac{(-1)^k P_{k}\mathrm{ad}_{B}^{k}}{k}\right) =\exp\left(-\sum_{k=1}^{\infty}\frac{(-1)^k\mathrm{ad}_{B}^{k}}{kz^k}+H(z)\right)=\left(\frac{\mathrm{ad}_{B}}{z}+1\right)\exp(H(z)),
P \Big(\Big \lvert \frac{1}{n}\sum_{i=1}^n \phi_b(b^0_i, ..., b^{t}_i, w_{i}) - \mathbb{E}\phi_b \Big \lvert \geq \epsilon \Big) &\leq K t^3 K_{t-1}e^{-{\kappa \kappa_{t-1}n \epsilon^2}/{t^7}}.
\zeta_\pm^1(z)=N_1\left(\begin{array}{c}z^{\frac{M}{2}}(1-z)^{-\frac{M}{2}}F(1,-1,\frac{1}{2}+M;z) \pm \frac{i}{\frac{1}{2}+M}z^{\frac{1+M}{2}}(1-z)^{\frac{M}{2}}F(-\frac{1}{2}+M,\frac{3}{2}+M,\frac{3}{2}+M;z) \end{array}\right)
\Delta\phi(e_1) =\left[\frac{\partial^2\bar{\phi}}{\partial r^2}+\frac{N-1}{r}\frac{\partial\bar{\phi}}{\partial r}\right](1,-e_N) &=\left.\left[\frac{d^2}{dr^2}\bigl(e^{-\alpha r^2}-e^{-\alpha}\bigr)+\frac{N-1}{r}\frac{d}{dr}\bigl(e^{-\alpha r^2}-e^{-\alpha}\bigr)\right]\right|_{r=1}&=2\alpha e^{-\alpha}( 2 \alpha-N)>0,
\pi_C^* \biggl( \omega_{D_i}+ \frac 12 \langle (\mu_{\le i})^* \theta , (\mu_{i,1})^* \theta \rangle \biggr) = \omega_{D_i^+}+ \frac 12 \langle (\mu_{\le i}^+)^* \theta , (\mu_{i,1}^+)^* \theta \rangle
E_{2}^{p , q}\: = \: H^{p}\left( S , \underline{{\mathrm{E x t}}}_{{\cal O}_{S}}^{q}({\cal S}_{1},{\cal S}_{2}) \right) \Longrightarrow \mathrm{E x t}_{S}^{p + q}\left({\cal S}_{1},{\cal S}_{2}\right)
\phi(X_1, \tilde Y-(\tilde t-t)X_1,\tilde t) =\min_{\tilde X\in\overline{{B_\epsilon(X)}}}\phi(\tilde X,\tilde Y-(\tilde t-t)\tilde X,\tilde t)=\inf_{\tilde X\in{{B_\epsilon(X)}}}\phi(\tilde X,\tilde Y-(\tilde t-t)\tilde X,\tilde t).
\begin{aligned}N^{-1}\psi(R(2^{2n_1+1}))N = N^{-1}\psi(R(a))N \equiv R(c)\equiv\left( \begin{matrix}1 c 0 1 \end{matrix}\right) \pmod{2^T}, N^{-1}\psi(L(2^{2n_1+1}))N =N^{-1}\psi(L(b))N \equiv L(d) \equiv \left( \begin{matrix}1 0 d 1 \end{matrix}\right) \pmod{2^T}.\end{aligned}
\Gamma \left[ \Phi \right] = \frac{1}{2}\int_{p}\Phi_{p}\Phi_{- p}C^{- 1}( p , \Lambda ) + S \left[ \Phi \right] + \frac{1}{2}{t r}\ln \left. \left( C^{- 1}+ \frac{\delta^{2}S}{\delta \phi \delta \phi}\right) \right|_{\phi = \Phi}+ \cdots
\left|q_{j}(u(r +s\nu(r)))\,\nu_{j}(r)\,\beta(r)\right|\,\leq\begin{cases}M\,\left(\int_{k}^{u(r +s\nu(r))}\,\eta^{\prime}(s)\,ds\right)\beta(r) M\left(\int_{u(r +s\nu(r))}^{k}\,-\eta^{\prime}(s)\,ds\right)\beta(r) \end{cases}
D_0(17,\{1,2,2,3\}) =D(17+8,\{1,2,2,3\}) &=D(22,\{1,2,2,3\})+D(22,\{1,2,2\}) &=D(19,\{1,2,2,3\})+D(19,\{1,2,2\})+25 &=D(16,\{1,2,2,3\})+D(16,\{1,2,2\})+20+25 &=D(13,\{1,2,2,3\})+D(13,\{1,2,2\})+12+20+25 &=D(10,\{1,2,2,3\})+D(10,\{1,2,2\})+9+12+20+25 &=D(7,\{1,2,2,3\})+D(7,\{1,2,2\})+4+9+12+20+25 &=0+2+4+9+12+20+25=72.
\mbox{det}(W(G))=(-1)^{1+d}\prod\limits_{k=1}^{\mu}t_k^{u_k}\prod\limits_{k=1}^{\mu}(1-t_k^{-1})^{n_k}\, \frac{(-1)^n \mbox{det}(F_D^n)}{\theta(x_n)-1}\,.
\int_{\mathbb{R}^n}(v-w) \left(-\frac{1}{2m}\Delta +\mu \right)\phi (x) dx = \int_{\mathbb{R}^n}|v|^{p-2}v(x)\phi (x) dx -\int_{\mathbb{R}^n}|v|^{p-2}v (x)\phi (x) dx=0.
\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}\left(-c^{2}z\right)^{n}}{\left(q^{2};q^{2}\right)_{n}}A_{q^{2}}\left(-c^{2}q^{2n}\right)=\frac{\left(c^{2}q;q^{2}\right)_{\infty}}{\left(c^{2}zq;q^{2}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\binom{n+1}{2}}\prod_{j=0}^{n-1}\left(z-q^{2j}\right)\left(-c^{2}\right)^{n}}{\left(q,cq^{1/2},-cq^{1/2};q\right)_{n}}
G_{n_1,\mu_1,\xi_1}(z) =e^{f_1(z;0)}\,\,,\,\,G_{\Delta n,\Delta\mu,\Delta\xi}(w)=e^{f_2(w;0)}G_{k,\mu_1,\xi_1}(\zeta) =e^{f_1(\zeta;y)}\,\,,\,\,G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)=e^{f_2(\omega;x)}G_{n_2-k,\Delta\mu,\Delta\xi}(w) =e^{f_2(w;y)}\,\,,\,\,G_{\ell-1,\mu_1,\xi_1}(z)=e^{f_1(z;x)}
\Bigl ( \prod_{( j , \kappa ) \in \Lambda^{1}}\sum_{a_{j \kappa}, b_{j \kappa}}\Bigr ) = \underbrace{\sum_{a_{j \kappa}, b_{j \kappa}= 1}^{\dim V_{\tau_{j \kappa}}}\cdots \sum_{a_{j \kappa}, b_{j \kappa}= 1}^{\dim V_{\tau_{j \kappa}}}}_{\mathrm{o n e ~ \sum ~ f o r ~ e a c h ~ l i n k}}.
\gamma_{\psi'}^{-1}(\det c_{b_*h})\mathfrak{h}(b_*h) =\gamma_{\psi'}^{-1}(\det c_{b_*}\det c_{h})(\det c_{b_*},\det c_h)_2\mathfrak{h}(b_*)\mathfrak{h}(h) &=\gamma_{\psi'}^{-1}(\det c_{b_*})\gamma_{\psi'}^{-1}(\det c_{h})\mathfrak{s}(b_*)\mathfrak{h}(h) &=\gamma_{\psi'}^{-1}(\det c_{h})\mathfrak{s}(b_*)\mathfrak{h}(h).
Q^i_\alpha = \phi^i_{Q\alpha}+ \sqrt{2}\theta \psi^i_{Q\alpha}+ \theta^2 F^i_{Q\alpha}, \qquad{\tilde Q}^{\alpha j}= \phi^{\alpha j}_{\tilde Q}+ \sqrt{2}\theta \psi^{\alpha j}_{\tilde Q}+ \theta^2 F^{\alpha j}_{\tilde Q},
F ( \frac{s}{2}+ \frac{5 - m}{4}, \frac{s}{2}+ \frac{3 - m}{4}; 1 ; \frac{k^{4}}{( k^{2}+ 8 M^{2})^{2}}) , \, f o r \, \Re ( s ) > \frac{m - 3}{2}\, ,
d\nu^1_{k,\sum_{i=d-k+1}^{d}x_i}(y_{d-k},...,y_1)=d|Df^{e_{d-k}}_{\sum_{i=1}^{d-k-1}y_i+\sum_{i=d-k+1}^{d}x_i}|(y_{d-k})d\mathcal{H}^1(y_{d-k-1})...d\mathcal{H}^1(y_1), & d\nu^2_{k,\sum_{i=d-k+1}^{d}x_i}(y_{d-k},...,y_1)=\mathbf{1}_{|\sum_{i=1}^{d-k}y_{0i}-\sum_{i=1}^{d-k}y_i|\leq 2\varepsilon}d\nu^1_{k,\sum_{i=d-k+1}^{d}x_i}(y_{d-k},...,y_1).
\{((p_1, q_1), (p_2, q_2))\in(P\times Q)^2\mid (p_1, q_1)\leq (p_2, q_2)\}&=\{(p_1, q_1, p_2, q_2)\in(P\times Q)^2 \mid (p_1<p_2)(p_1=p_2 q_1\leq q_2)\}&\simeq \left(\{(p_1, p_2)\in P^2\mid p_1<p_2\}\times Q^2\right) \sqcup \left( P \times \{(q_1,q_2) \in Q^2\mid q_1\leq q_2\}\right)
\left[\begin{array}{cc}\tau \frac{1}{3}(h_{j+1}-h_j) \frac{1}{3}(h_{j+1}-h_j) \tau\end{array}\right] \left[\begin{array}{c}\zeta_j \xi_j\end{array}\right]= \left[\begin{array}{c}0 0\end{array}\right]\, .
\langle W_0 (\sum h_{j,k}\otimes f_j\otimes f_k), \, W_0 (\sum g_{a,b}\otimes f_a\otimes f_b)\rangle = \sum_{j,k}\langle Q(j,k)h_{j,k}, \, g_{a,b}\rangle = [ \sum h_{j,k}\otimes f_j\otimes f_k, \, \sum h_{a,b}\otimes f_a\otimes f_b ],
\check{x}_{3 1 ; 2}= \tilde{x}_{3 1}+{\frac{4 i}{( 1 2 )}}\theta_{3 1}( \bar{\theta}_{2}^{+}- ( 2 1^{-}) \bar{\theta}_{1}^{+}) \quad \Rightarrow \quad \delta \check{x}_{3 1 ; 2}= 0 .
D_{\phi\phi\phi}^3\Phi(0, \mu^*_{k})[\phi^*_{k},\phi^*_{k},\phi^*_{k}] =\Pi D_\phi F(0,\mu^*_{k})D_{\phi\phi\phi}^3 \psi(0,\mu^*_{k})[\phi^*_{k},\phi^*_{k},\phi^*_{k}] &\quad+3 \Pi D_{\phi\phi}^2 F(0,\mu^*_{k})[\phi^*_{k},D_{\phi\phi}^2 \psi(0,\mu^*_{k})[\phi^*_{k},\phi^*_{k}]].
\frac{[Ext^1_U(j_{*}\hat{T},\mathbb{G}_m)]}{[Ext^1_U(j_{*}\hat{T}',\mathbb{G}_m)]}=\frac{q(\hat{\lambda}(\alpha))}{q(\lambda(O_{K,S}))q(\hat{\lambda}(H^0_{et}))}\prod_{v\in S}[\mathrm{cok}(\hat{\lambda}(H^{0}(K_v))].
\frac{1}{\sqrt{\varepsilon \left( L\frac{1}{t}\right)^{\frac{\varepsilon}{2}}}}\sqrt{\frac{\left( L2^{m+1}\right)^{\frac{\varepsilon}{2}}}{2L_{2}2^{m+1}}}\sqrt{\frac{L2}{2}}\sum_{j=0}^{\infty}2^{-j/2}\sqrt{2+j}.
L_p(r)= \frac{R+R_g^{(p)}-r}{r}(T)^{-\frac{1}{\alpha}}, C(L_p(r),\alpha)=\frac{{}_2 F_1(1,\frac{\alpha-2}{\alpha},2-\frac{2}{\alpha},-\{L_p(r)\}^\alpha)}{\alpha-2}\{L_p(r)\}^{2-\alpha}.
\Phi_{n + 1}= \frac{- 1}{1 - \Phi_{n}^{2}}\left[ \frac{d \Phi_{n}}{d r}- \frac{n}{r}\Phi_{n}\right] = \Phi_{n - 1}- \frac{2}{1 - \Phi_{n}^{2}}\frac{d \Phi_{n}}{d r}, \; n \geq 1 .
\sum_{n=0}^\infty \|T_{n+1}u-T_nu\|_{L^1}\lesssim\sum_{n=0}^\infty 2^{-nQ}\|\tilde d_nu\|_{\ell^1}\lesssim \sum_{n=0}^\infty 2^{-nQ/p}\| du\|_{\ell^{p,\infty}}\lesssim \| du\|_{\ell^{p,\infty}}.
[G]_{i,j}= [ D^{(1)}]_{i,j}+ \sum_{n=2}^\infty \sum_{k=0}^\infty [ A_0]_{i,k}[D^{(n-1)}]_{k,j}+ \sum_{n=3}^\infty \sum_{m=1}^{n-2}\sum_{k=0}^\infty \sum_{l=0}^\infty [ A_1]_{i,k}[^{m}D^{(m)}]_{k,l}[D^{(n-m-1)}]_{l,j}\cr = [ A_{-1}]_{i,j}+ [ A_0 G]_{i,j}+ [ A_1 G^2]_{i,j},
W_{\pi}(g_{t,l,v}) = \begin{cases}\epsilon(\frac{1}{2},\tilde{\pi}) l=0 t=-k, \gamma q^{-\frac{t}{2}}K(\xi^{-1},\Omega^{\frac{t}{f}},v\varpi^{-l}) l=\frac{n}{2}-k \leq t<0, l\neq 0,\frac{n}{2}t= -k, 0 \end{cases}
\lambda = (n-2+p,\: n-1-p, \: \underbrace{2pq,\: \ldots,\: 2pq}_{2p}, \: \underbrace{2p(q-1),\: \dots,\: 2p(q-1)}_{2p}, \: \ldots, \: \underbrace{2p,\: \ldots,\: 2p}_{2p}, \: \underbrace{0,\: \ldots,\: 0}_{p-1})
\mathfrak{G}_p = \Sigma_1\times \left( 1-\frac{1}{p}\right)\prod_{\substack{\hat{\alpha}\neq \alpha \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}}\right) \prod_{\hat{\alpha}\neq \alpha}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}}\right) \prod_{\hat{\beta}\neq \beta}\left(1-\frac{1}{p^{1+\hat{\beta}-\beta}}\right),
\Sigma'_q =\left(\begin{matrix}-\frac{r}{r_2}0 0 \cdots 0 0 -r 0 \vdots 0 0 -\frac{r}{r_4}\vdots \ddots 0 0 0 \cdots -r\end{matrix}\right)+\left(\begin{matrix}1 1 \vdots 1 \end{matrix}\right)\left(\begin{matrix}1 1 \cdots 1 \end{matrix}\right)
\begin{array}{r@{\}l}\left( 1 - \alpha^{3n-3j}\right) \kappa^n_{3n-2j, j, j}= \displaystyle \alpha^{3n-5j-3}\left( \sum_{\nu=0}^{j}\alpha^{3\nu}\right) \kappa^n_{3n-2j-2, j+1, j+1}& = \displaystyle \alpha^{3n-5j-3}(1-\alpha^3)^{-1}(1-\alpha^{3j+3})\kappa^n_{3n-2j-2, j+1, j+1},\end{array}
\hat{\Delta}_{equ}(x - y) = \int d^{\, 3}u \, d^{\, 3}v \sum_{\tau = \pm}\hat{B}_L^{(\tau)}({\bf x}-{\bf u}) \, \hat{\Delta}_{diag}^{(\tau)}({\bf u}-{\bf v}; x_0 - y_0) \, \hat{B}_R^{(\tau)}({\bf v}-{\bf y}) ,
\begin{matrix}{{R}}\\ \end{matrix}{i j}{k \ell}= g^{i m}\begin{matrix}{{\tilde{R}}}\\ \end{matrix}{n j}{m \ell}g_{n k}= g^{j m}\begin{matrix}{{( R^{- 1})}}\\ \end{matrix}{i n}{k m}g_{n \ell},
\lim_{\substack{m\to\infty \frac{\omega_{m}}{m^{\alpha}}\to c}}\frac{d_{n,k}^{m,\omega_{m}}}{d^{m,\omega_{m}}}= V^{3IBP,\alpha,\theta}_{n,k}\Big( \frac{c\alpha\Gamma(\alpha+\theta)}{\Gamma(\theta +1)}\Big).
\lim\limits_{n\to+\infty}\sup_{w\in[1,A]}\frac{1}{n\sin(w\alpha_n/2)}=\lim\limits_{n\to+\infty}\frac{1}{n\sin(\alpha_n/2)}=\lim\limits_{n\to+\infty}\frac{2}{n\alpha_n}=0.
B = - \frac{\vert e \vert H}{( 2 \pi )^{2}}\frac{2 \sqrt{\pi}\vert e \vert H}{\Gamma ( 1 + \delta )}\int_{0^{+}}^{\infty}d \eta \eta^{- 3 / 2 + 1 + \delta}\mathrm{e}^{- ( 4 + \beta^{2}m^{2}) \eta}\mathrm{c s c h}^{2}( \beta^{2}\vert e \vert H \eta )
\|\mathbb P_{\mathbb R^3}G(t)\|_{q,\mathbb R^3}\leq \|G(t)\|_{q,\mathbb R^3}+\|\nabla Q(t)\|_{q,\mathbb R^3}&\leq C\|G(t)\|_{q,\mathbb R^3}=C\|\bar g(t)\|_{q,\mathbb R^3}\leq C\|g(t)\|_q\leq CM_q
{\rm arctanh}_p(x) < \frac x2 \left( 1 - \frac2p\, \log\big(1-x^{\frac p2}\big) + \frac{2^{\frac2p}\,b_{\frac p2}}{(1+x^{\frac p2})^{\frac2p}}\right){\rm arctan}_p(x) < x\left( 1- \frac1{p(1+p)}\, \log(1-x^p) - \frac1p\,\log (1+x^p) \right) =: R_p(x)\, ,
\left(x_1\overline{y_1}+x_2\overline{y_2}\right)\left(y_1\overline{z_1}+y_2\overline{z_2}\right) \left(z_1\overline{x_1}+z_2\overline{x_2}\right) &\equiv\left|x_1\overline{y_1}+x_2\overline{y_2}\right|^2+\left|y_1\overline{z_1}+y_2\overline{z_2}\right|^2+\left|z_1\overline{x_1}+z_2\overline{x_2}\right|^2-4.
\frac{(-1)^j}{j!}(N_-^j\Psi)(b_Z) =(m-j)(m-j-1)(P_{0+}\vec\Phi)(b_Z)_j \qquad-(m-j-1)(j+1)(P_{1+}\vec\Phi)(b_Z)_{j+1}\qquad+(j+2)(j+1)(X_+\vec\Phi)(b_Z)_{j+2}.
W = \int \frac{2 d^{1 0}\tilde{p}_{1}}{\tilde{p}_{1}^{2}\tilde{p}_{2}^{2}}\sum_{( \lambda , \lambda^{\prime})}\hat{A}_{( 0 )}( \{p , \epsilon \}_{( i )}, \tilde{p}_{1}\lambda , \lambda^{\prime}) \hat{A}_{( n - 1 )}( \lambda , \lambda^{\prime}; \tilde{p}_{1}, \{p , \epsilon \}_{( f )}) ,
( \mathrm{m a s s})^{2}={\frac{( \frac{1}{2}m + \Delta a )^{2}}{N}}- \frac{1}{4}{\frac{m^{2}}{N}}+ \frac{Q^{2}}{8}={\frac{\Delta a ( \Delta a + m )}{N}}+ \frac{1}{4 N},
\|(F_M-F)(\gamma^r)\|_{L^2(\mathbb P^H)}^2=E\left[\sum_{\alpha\in \mathcal J}\sum_{\beta\in\mathcal J}c_\alpha^{(M)}c_\beta^{(M)}\tilde{\mathcal H}_{\alpha}(\gamma^r)\tilde{\mathcal H}_{\beta}(\gamma^r)\right]<+\infty.
[_{i}\leq \gamma_{}]= &\Bigg[\Phi_{i}\leq \frac{\gamma_{}}{\gamma_{}\left(\frac{(2\sqrt{\omega}+1)^{-1}}{(\kappa^{2}_{T}(\omega+1)+\omega+1}-1\right)+\frac{1}{(\kappa^{2}_{T}(\omega+1)+\omega+1}}\Bigg],
b_t=\left \lfloor \frac{\mathbb{E}_\pi(\widetilde{N}_t)}{4}\right \rfloor \geq \frac{\mathbb{E}_\pi(\widetilde{N}_t)}{4}-1 \geq \frac{(1-\lambda)^2t -48d^2}{48d(d-1)}\geq \frac{(1-\lambda)^2}{48d^2}t.
\frac{u_{-}-v_{+}+h}{u_{-}-v_{+}}k_{1}^{+}(u)e_{1}^{+}(u)k_{1}^{-}(v)e_{1}^{-}(v)=\frac{u_{+}-v_{-}+h}{u_{+}-v_{-}}k_{1}^{-}(v)e_{1}^{-}(v)k_{1}^{+}(u)e_{1}^{+}(u),
(f_1\circ f_2)\star f_3+(-1)^{(m_2-1)m_3}f_1\star (f_3 \circ f_2)-(-1)^{(m_3-1)(m_2-1)}(f_1\star f_3 )\circ f_2 = (-1)^{(m_1+m_2)(m_3-1)+(m_1-1)(m_2-1)}f_1\star_0(f_2\star_0f_3)+ f_1\star_1(f_2\star_1 f_3).
\gamma(\sigma \times \sigma^\prime, \psi) \sum_{g \in U_n(\mathfrak{f}) \backslash G_n(\mathfrak{f})}W(g)W^\prime(g)\phi(e_n g) &= \sum_{g \in U_n(\mathfrak{f}) \backslash G_n(\mathfrak{f})}W(g)W^\prime(g)\widehat{\phi}(e_1\,^tg^{-1}),
\overline{C}(\alpha_l)=\frac{\Gamma(\frac{D}{2}-1)}{2^D\pi^{\frac{3}{2}D-2}}\sum_{m=0}^M (L_m+2 \delta_{m,l})\sum_{n=1}^\infty \frac{\cos(2n\pi \alpha_m)}{n^{D-2}}.
\mathcal B_\Omega f(x)=\int_{\Omega}f(y)\Big(\frac{x-y}{|x-y|^{n}}\int_{|x-y|}^\infty\omega\bigg(y+\zeta\frac{x-y}{|x-y|}\bigg)\zeta^{n-1}\,d\zeta\Big)\,dy\hbox{for $x \in \Omega$,}
\beta^{(n)}_{e}\,= \, \int_{0}^{nT}\;\langle \nu_{e}(t)|\; U^{-1}(t)\,i \partial_t \,\Big( U(t)\;|\nu_{e}(t)\rangle\Big)\,dt \,= \, \int_{0}^{nT}\langle \tilde{\nu_{e}}(t)|\;i \partial_t|\tilde{\nu_{e}}(t)\rangle \,dt \, =\, 2 \pi \,n\,\sin^{2}\theta \, ,
\begin{array}{ll}Q^S(F,F)= \int_{(w,\sigma)\in{\mathbb R}^3\times S^2}{\Big (}\sum_{i=1}^{2d}\int_0^1(1-\theta)\partial_{z'_i}G\left( v+\theta h,w+\theta 2h\right)d\theta h_i{\Big )}|w|dwd\sigma .\end{array}
d_{\mathcal{S}(\theta)}([u],[v]) \le \sum_{j=0}^n \bigg( 2 \int_X \theta_{V_\theta}^j \wedge \theta_{\max\{u,v\}}^{n-j}- \int_X \theta_{V_\theta}^j \wedge \theta_u^{n-j}-\int_X \theta_{V_\theta}^j \wedge \theta_v^{n-j}\bigg) \le C d_{\mathcal{S}(\theta)}([u],[v]),
P(T, x) = \prod_{v \in V(T)}\mathcal{G}(v, x) = \mathcal{G}(r(T), x) \prod_{j = 1}^{d}\left( \prod_{v \in V(T_j)}\mathcal{G}(v, x) \right) = \mathcal{G}(r(T), x) \prod_{j = 1}^{d}P(T_j, x) .
\begin{dcases}\partial_t \bar \rho + \nabla \cdot (\bar \rho \bar u) = 0 \partial_t \bar u + \bar u \cdot \nabla \bar u + \nabla\big(h_1^\prime(\bar \rho)+h_2^\prime(\bar n) \big) = 0 - \Delta h_2^\prime(\bar n) + \bar n = \bar \rho \end{dcases}
\delta A_{( 3 )}= d \Delta^{( 2 )}+{\textstyle \frac{1}{2}}\vec{\Sigma}_{( 1 )}^{\T}\eta \vec{F}_{( 2 )}-{\textstyle \frac{1}{2}}\vec{\Sigma}_{( 0 )}{}^{T}\eta \left[ \vec{F}_{( 3 )}- A_{( 1 )}\vec{F}_{( 2 )}\right] .
\Theta_\pm(\psi_-,\psi_+) :=\Pi_\pm^{\psi_\pm,T}(D_{\pm,T}\psi_{\pm,T}+ e_{\pm,T}+ \mathcal L_\pm(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_\pm(\psi_{\pm,T}))=0.
\kappa_{n}^{1}=\zeta_{n}^{1}sin\left(\sqrt{CL\left(\omega_{n}^{1}\right)^{2}-\frac{CR^{2}}{4L}}x_{0}\right)+\eta_{n}^{1}cos\left(\sqrt{CL\left(\omega_{n}^{1}\right)^{2}-\frac{CR^{2}}{4L}}x_{0}\right)
\partial_{x}f^{j}\left( x\right) =\left( -1\right)^{j}{\displaystyle\prod\limits_{i=0}^{j-1}}\frac{F_{x}\left( f^{i}\left( x\right) ,f^{i+1}\left( x\right) \right)}{F_{y}\left( f^{i}\left( x\right) ,f^{i+1}\left( x\right) \right)}.
H_\mu \left( \alpha^F \big \vert \beta^F \right) \leq \sum_{\gamma \in F}H_\mu \left(\gamma^a \alpha \big \vert \beta^F \right) \leq \sum_{\gamma \in F}H_\mu (\gamma^a \alpha \vert \gamma^a \beta) = |F| \cdot H_\mu(\alpha|\beta) \leq |F|.
Z(G_{4,1}\vee A^{(1)\vee G_{4,1}})=\begin{cases}Z(C(3,4))=13 (A^{(1)}=C(\dot1,\dot2)) Z(C(5,2))=14 (A^{(1)}=C(\ddot1,2)) Z(C(7))=7 (A^{(1)}=C(\ddot3)). \end{cases}
V_{p,q}=\{v\in W^{1,p}(\Omega):\v|_\Gamma\in L^{q}(\Gamma)\}; L^{p,q}(Q_T) =L^q(0,T;L^p(\Omega)); L^{p,q}(\Sigma_T) =L^q(0,T;L^p(\Gamma)); W^{1,q}(0,T;X,Y) =\{v\in L^q(0,T;X):\v'\in L^q(0,T;Y)\},
(2k_1+1)^{t_1-s_1}\cdots(2k_l+1)^{t_l-s_l}\leq (2k_l+1)^{t_1+\cdots+t_l-(s_1+\cdots+s_l)}&=(2k_l+1)^{t_1+\cdots+t_r-(s_1+\cdots+s_r)}\cdot(2k_l+1)^{s_{l+1}-t_{l+1}+\cdots+s_r-t_r}&\leq (2k_l+1)^{s_{l+1}-t_{l+1}+\cdots+s_r-t_r}&\leq (2k_{l+1}+1)^{s_{l+1}-t_{l+1}}\cdots(2k_r+1)^{s_r-t_r}.
cov[f_{d}(x),f_{d^{\prime}}(x^{\prime})]=\sum_{q=1}^{Q}\sum_{i=1}^{R_{q}}a_{d,q}^{i}a_{d^{\prime},q}^{i}k_{q}(x,x^{\prime})=\sum_{q=1}^{Q}b_{d,d^{\prime}}^{q}k_{q}(x,x^{\prime})
\begin{array}{rcl}\partial^{ci}_{\tau,w}\varphi(\tau,z,w(\cdot)) + \langle\partial^{ci}_{\tau,w}g(\tau,w(\cdot)), \nabla_z \varphi(\tau,z,w(\cdot)) \rangle begin{align*}0.3cm] + H(\tau,z,w(-h),\nabla_z \varphi(\tau,z,w(\cdot))) = 0,\end{array}\ (\tau,z,w(\cdot)) \in \mathbb G_*,
u = \frac{C_1 \theta(t-t_0) + C_0}{(t-t_0)^2}\, \exp\left[-\frac{(x-x_0)^2}{4(t-t_0)}-\frac{3}{(t-t_0)^3}\left(y-y_0 -(t-t_0) \frac{x+x_0}{2}\right)^2 \right].
\Upsilon^{*}(w_{j}-A_j^{[5]}(w_1,\ldots,w_{n-1})w_n)=w_{j}+ A_{j,1}^{[3]}w_{n}^2+A_{j,2}^{[3]}w_n\overline{w}_{n}+ A_{j,3}^{[3]}\overline{w}_n^2,
P(x) \geq \int_{\zeta_{2x}}^1 \exp\Big(-2xt - m(t)\Big) dt \geq \exp\big(-m(\zeta_{2x})\big) \int_{\zeta_{2x}}^1 \exp (-2xt) dt = \exp\Big(-m(\zeta_{2x})\Big)\frac{\exp(-2x\zeta_{2x}) - \exp(-2x)}{2x}= \frac{\exp\big(-k(2x)\big)}{2x}\Big(1-\exp( -2x + 2x\zeta_{2x})\Big) \geq \frac{\exp\big(-k(2x)\big)}{4x}.
\rho^\lambda_a\partial_\lambda E_{\mu\nu}=\widetilde{C}^{kl}_a\rho^\lambda_k\rho^\tau_lE_{\lambda\nu}E_{\mu\tau}-(\partial_\mu\rho^\lambda_a)E_{\lambda\nu}-(\partial_\nu\rho^\lambda_a)E_{\mu\lambda}.
W_{e f f}= - N S ( \log ( S / \Lambda^{2}) - 1 ) + ( 6 g_{4}+ 4 \widetilde{g}_{2}N ) N S^{2}- 2 ( 6 g_{4}+ 4 \widetilde{g}_{2}N )^{2}N S^{3}+ \frac{2 0}{3}( 6 g_{4}+ 4 \widetilde{g}_{2}N )^{3}N S^{4}+ \cdots .
s_{(2,0;1)}s_{(;3)}=s_{(5,0;1)}+s_{(4,0;2)}+s_{(4,0;1,1)}+s_{(3,0;2,1)}+s_{(2,0;3,1)}+s_{(2,0;4)}, s_{(2,0;1)}s_{(3;)}=s_{(4,2,0;0)}+s_{(3,2,0;1)}, s_{(2,0;1)}s_{(;1^2)}=s_{(2,0;1^3)}+s_{(3,0;1^2)}+s_{(2,0;2,1)}+s_{(3,0;2)}+s_{(2,1;2)}, s_{(2,0;1)}s_{(0;1^2)}=s_{(2,1,0;2)}.
\phi_1 := \phi_{10A}(\tau,z) = 2 \zeta^{\pm 1}- 4 + (-4 \zeta^{\pm 2}+ 10 \zeta^{\pm 1}-12) q + (2 \zeta^{\pm 3}-12 \zeta^{\pm 2}+ 28 \zeta^{\pm 1}- 36) q^2 + O(q^3)
A_{nm}^l = \frac{2}{2l+\delta_1+\delta_2}\left[\frac{(l-|m|)(l+|m|+\delta_1+\delta_2)(l+\delta_1)(l+\delta_2)(n-l)(n+l+\delta_1+\delta_2)}{(2l+\delta_1+\delta_2-1)(2l+\delta_1+\delta_2+1)}\right]^{1/2}.
\begin{aligned}I_{j}(x)-\beta_j\frac{x-z_j}{\left|x-z_j\right|^{2}}=\frac{1}{2\pi}\int_{B_{\delta}(z_j)}\left(\frac{x-y}{\left|x-y\right|^{2}}-\frac{x-z_j}{\left|x-z_j\right|^{2}}\right)f_{k}(y)dy &+\frac{1}{2\pi}\int_{B_{\delta}(z_j)}\frac{x-z_j}{\left|x-z_j\right|^{2}}\left|\beta_{k,j}-\beta_{j}\right|\end{aligned}
\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\mu}}}\partial_{\hat{\hat{\mu}}}= \partial_{m}\hspace{. 5 c m}\rightarrow \hspace{. 5 c m}\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\mu}}}\hat{\hat{k}}_{( n )}{}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}= G_{n m}\; ,
f ( D ; l ) = \theta ( 1 / l - | D |^{\kappa}) = \theta ( 1 - | D |^{\kappa}l ) , \quad \Theta ( D_{e}, D_{\bar{e}}) = \theta ( | D_{e}| - | D_{\bar{e}}| ) = \theta ( 1 - \frac{D_{\bar{e}}}{D_{e}}) .
\delta_3= \phi(\phi_{ss}\phi_{zz}-\phi_{sz}^2)+\phi_s(\phi_s\phi_{zz}-\phi_z\phi_{sz})+\phi_z(\phi_{ss}\phi_z-\phi_s\phi_{sz}) = -\det \left(\begin{array}{c c c}-\phi \phi_s \phi_z \phi_s \phi_{ss}\phi_{sz}\phi_z \phi_{sz}\phi_{zz}\end{array}\right),
q^{{n}\choose{2}}C_n\left(\frac{1}{q}\right)=\sum_{\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k),\atop \lambda_1+k\leq n}\prod_{i \geq 1}{\lambda_i-\lambda_{i+2}+1 \choose \lambda_{i+1}-\lambda_{i+2}}q^{{\lambda_1\choose 2}+|\lambda|}.
\sum_{n \geq 1}(-1)^n(q)_{n-1}q^{\binom{n+1}{2}}\beta_n = \sum_{n \geq 1}\frac{(-1)^nq^{\binom{n+1}{2}}}{1-q^n}\alpha_n, \sum_{n \geq 1}\big(q^2;q^2\big)_{n-1}(-q)^n\beta_n = \sum_{n \geq 1}\frac{(-q)^n}{1-q^{2n}}\alpha_n,
P_{V}(\{\omega \in \mathbb{N}_{0}^{\infty}: \forall f \in \mathcal{C}(\mathbb{N}_{0}^{\infty}), \; \underset{n\rightarrow \infty}{\lim}\Pi_{n}f(\omega)=\textstyle\int \!f \, \mathrm{d}P_{V}\}) = 1.
\int_{3}^{q}\frac{\ln4\pi(1-\tau)}{r^{2}}dr=\left(\frac{1}{3}-\frac{1}{q}\right)\ln4\pi(1-\tau), &\int_{3}^{q}\frac{\ln(r-2)}{r^{2}}dr=\left(\frac{1}{2}-\frac{1}{q}\right)\ln(q-2)+\frac{1}{2}\ln\frac{3}{q}, &\int_{3}^{q}\frac{\ln r^{2}}{r^{2}}dr=2\left(\frac{1+\ln3}{3}-\frac{1+\ln q}{q}\right).
\frac{1}{2 \lambda f^{2}}\int d^{4}X \frac{d^{4}q}{\left( 2 \pi \right)^{4}}\left( \varphi ( X ) \right)^{2}\tilde{\pi}_{0}( q ) \left[ \partial_{q}^{2}+ \frac{4 i \lambda}{q^{2}- \Sigma^{2}( q )}\right] \tilde{\pi}_{0}( q ) ,
\dim \mathbf{H}^1({\mathcal F}^{\bullet}_x) = \dim H^0({\mathcal F}^1_x) + \dim H^1({\mathcal F}^0_x) -\dim H^0({\mathcal F}^0_x) - \dim H^1({\mathcal F}^1_x) + 2\dim_{\mathbb{C}}\mathbb{C}= \chi({\mathcal F}^1_x)-\chi({\mathcal F}^0_x) +2.
\begin{array}{rcl}f(x) = \xi(g(x),h(x))\ge \lambda \|g(x)\|^2 -\|c\|\, \|h(x)\| & &= \|g(x)\|\left( \lambda \|g(x)\| -\|c\|\, \dfrac{\|h(x)\|}{\|g(x)\|}\right) & &\ge \|g(x)\|\left(\lambda \|g(x)\| -R\|c\|\right)\,,\end{array}
\left( \begin{array}{l}{X^{I}}{F_{I}}\end{array}\right) \rightarrow \left( \begin{array}{l l}{{\bf a}}{0}{0}{{{\bf a}^{t}}^{- 1}}\end{array}\right) \left( \begin{array}{l}{X^{I}}{F_{I}}\end{array}\right) \ ,
\sigma(A) = \left\{\sqrt{n}^{(1)},-\sqrt{n}^{(1)},0^{(n-1)}\right\}, \sigma(B) = \left\{\sqrt{n}+1^{(1)}, - \sqrt{n}+1^{(1)}, 1^{(n-1)}, \sqrt{n}-1^{(1)}, -\sqrt{n}-1^{(1)}, -1^{(n-1)}\right\}.
b_{1}g_{1}|_{{\textstyle\bigcup_{i=1}^{m}}\Omega_{i}}+\cdots+b_{n}g_{n}|_{{\textstyle\bigcup_{i=1}^{m}}\Omega_{i}}=-b_{n+1}f_{1}|_{{\textstyle\bigcup_{i=1}^{m}}\Omega_{i}}-\cdots-b_{n+m}f_{m}|_{{\textstyle\bigcup_{i=1}^{m}}\Omega_{i}}
\frac{\partial}{\partial z}\Big |_{z=1}(\widetilde{Q}_{r, O_r}(z;q) - \widetilde{E}_{r, L_r}(z;q)) =(-q;q^2)_\infty \sum_{\ell \in L_r \cup O_r}\mathop{\sum_{k=0}^\infty}\frac{q^{2kr+\ell}}{1+q^{2kr+\ell}}.
P_{n}(v', y') = \Phi(y') + \frac{m_3}{6 \sigma^3 \sqrt{n-1}}[1 - (y')^2] \phi(y') - \frac{b(f, v')}{\sigma \sqrt{n-1}}\phi(y') + o \left( \frac{1}{\sqrt{n}}\right) & = \int_{- \infty}^{y'}H_n(v', t) dt + o \left( \frac{1}{\sqrt{n}}\right),
|T(e)''(t)| \leq \frac{3 D_{2}}{\alpha \log(\log(t))t^{2}\log^{b+1}(t)}&\leq \frac{1}{100 \sqrt{\log(\log(t))}t^{2}\log^{b+1}(t)}, e \in \overline{B}_{1}(0) \subset X
{}_2F_1\bigg[\begin{matrix}\alpha \alpha &1\end{matrix}\bigg|\,z\bigg]-z^{1-\alpha}{}_2F_1\bigg[\begin{matrix}\alpha \alpha &1\end{matrix}\bigg|\,\frac1z\bigg]=(1-z)^{1-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12-\frac12\alpha &1\end{matrix}\bigg|\,-\frac{4z}{(1-z)^2}\bigg].
\begin{aligned}\mathcal{R}_{2,a}(t_n,\xi) = \int_0^\xi e^{(\xi - s) \mathcal{L}}f \left(u(t_n+s), \overline u(t_n+s) \right) ds-\xi f(u(t_n), \overline u(t_n)) & \Psi_{\xi}( v,w) = \mathcal{B}\left( F\left( v\right) \cdot e^{\xi \mathcal{A}}\mathcal{C}[G,\mathcal{A}] \left( w\right)\right), \, \xi \in \mathbb{R}.\end{aligned}
\frac{Var_Q\Big( Z^{\beta, u, q}_{N, \tilde{\nu}^q_{(I,J)}}(\Omega_N^{},\theta_{IN,JN}(V))\ \big|\ \mathcal{F}_{I}\Big)}{\Big( E^Q\Big( Z^{\beta, u, q}_{N, \tilde{\nu}^q_{(I,J)}}(\Omega_N^{},\theta_{IN,JN}(V))\ \big|\ \mathcal{F}_{I}\Big)\Big)^2}\leq \frac{\epsilon}{12},.
\max_{0\leq \eta \leq 1}A_{x,x',\eta}\leq \max_{0\leq \eta \leq 1}\frac{1-\eta}{1-\chi_{x,x',\eta}}&=\max_{0\leq \eta \leq 1}\frac{\eta P_{Y_1|X}(y|x)+(1-\eta) P_{Y_1|X}(y|x')}{P_{Y_1|X}(y|x')}&=\frac{P_{Y_1|X}(y|x)}{P_{Y_1|X}(y|x')}.
J = T^2 + y^2[n] -2Ty[n] + 4\frac{c^2[n]}{\tau^2}+ T^2 + y^2[n] &- 4\frac{c[n]}{\tau}y[n] + 4\frac{c[n]}{\tau}T - 2Ty[n] &= 2T^2 + 2w^2[n]\xi^2 - 2Tw[n]\xi + 4\frac{c^2[n]}{\tau^2}- 4\frac{c[n]}{\tau}w[n]\xi &+ 4\frac{c[n]}{\tau}T -2T\xi w[n] + 2w^2[n]\eta^2[n] + 4w^2[n]\eta[n]\xi &- 2Tw[n]\eta[n] 4\frac{c[n]}{\tau}w[n] \eta[n] - 2T\eta[n]w[n]
C_1 \cdot C_{2e}=C_{12e}+C_e, &C_{2e}\cdot C_{e2}=\frac{z-z^{-1}}{q-q^{-1}}C_{2e2}, &C_e\cdot C_{12e}=\frac{q^{2}z-q^{-2}z^{-1}}{q-q^{-1}}C_e.
\protect S = S_{G}+ \lambda S_E = \sum_{i,j,\mu}C_{ij}(X^{\mu}_i - X^{\mu}_j)^2 + \lambda\sum_{\hat{k},\hat{l},\mu}C^{\hat{k}\hat{l}}(1 - n^{\mu}_{\hat{k}}\cdot n^{\mu}_{\hat{l}}) .
\Delta (J^{\pm})\langle v|q^{-J^{0}}|v\rangle=(J^{\pm}\langle v|)|v\rangle +(q^{-J^{0}}\langle v|) J^{\pm}q^{-J^{0}}|v\rangle= -\langle v| J^{\pm}|v\rangle + \langle v|J^{\pm}|v\rangle = 0,
EE(D) - EO(D) = EE(D_1)EE(D_2) + EO(D_1)EO(D_2) - EE(D_1)EO(D_2) - EO(D_1)EE(D_2) &= (EE(D_1) - EO(D_1))(EE(D_2) - EO(D_2)) &\ne 0.
0=(\Delta_Eh,h)_{L^2}=\left\|D_2 h\right\|^2_{L^2}-\mu \left\| h\right\|_{L^2}^2-(h,\mathring{R}h)_{L^2}\geq-\mu\left\| h\right\|_{L^2}^2+\mu\left\| h\right\|_{L^2}^2=0.