harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\mathrm{C}}= \frac{1}{e^{2}}\int d^{4}x \, \sqrt{g}\, \Bigr \{( \nabla^{\mu}T_{\mu \rho}^{-}- i T_{\mu \rho}^{-}V^{\mu}) ( \nabla_{\nu}T_{+}^{\nu \rho}- i V_{\nu}T_{+}^{\nu \rho}) \Bigr \},
K_{a,+}(\lambda |\tau_{\epsilon},\kappa_{\epsilon},\zeta_{\epsilon})=\left( \sigma_{a}^{x}\right)^{\mathsf{x}}K_{a,+}^{sG}(\lambda |\tau_{-}^{sG},\kappa_{-}^{sG},\zeta_{-}^{sG})\left( -\sigma_{a}^{x}\right)^{\mathsf{x}},
\tau_{a b}= \frac{U_{0}^{7 - p}\Omega_{8 - p}}{( 2 \pi )^{7}g_{\mathrm{Y M}}^{4}}\; \mathrm{d i a g}\left[ \frac{9 - p}{2}, \frac{5 - p}{2}, \cdots , \frac{5 - p}{2}\right] ,
\int_{0}^{|I|}\frac{I_{1}^{2}}{t^{4-p-2\beta}}dt \lesssim\int_{0}^{|I|}t^{p+2\beta-2}\left(\int_{4I}\Big|\frac{\partial\hat{f}}{\partial n}((1-t)e^{is})\Big|^{2}ds \right)dt &\lesssim \int_{S(4I)}|\nabla\hat{f}(z)|^{2}(1-|z|)^{p+2\beta-2}dA(z),
{}^{( 5 )}g_{A B}= \left( \begin{array}{c c c}{{- n^{2}}}&{{- a^{2}S_{i}}}&{{0 \nonumber}}\\{{- a^{2}S_{j}}}&{{a^{2}\left[ \delta_{i j}+ F_{i | j}+ F_{j | i}\right]}}&{{- a^{2}S_{y i}\nonumber}}\\{{0}}&{{- a^{2}S_{y i}}}&{{1 \nonumber}}\\ \end{array}\right) \, ,
-{1\over r^2}\; \left(1-{r_0\over r}\right){d\over dr}\left( r(r-r_0)\;{dR_\ell\over dr}\right) +{\ell(\ell+1)\over r^2}\; \left(1-{r_0\over r}\right)\;R_\ell =k^2 \; R_\ell
|S_{m,N}|^{\frac{1}{q}-\frac{1}{p}}\gamma \left|\bigcap_{i=M}^{\infty}\{k\in S_{m,N}:|x^{(i)}_{k}-x^{(j)}_{k}|>\gamma\}\right|^{\frac{1}{p}}&\leq|S_{m,N}|^{\frac{1}{q}-\frac{1}{p}}\gamma\left|\{k\in S_{m,N}:|x^{(M)}_{k}-x^{(j)}_{k}|>\gamma\}\right|^{\frac{1}{p}}&\leq||x^{(M)}-x^{(j)}||_{w\ell_{q}^{p}}.
\left( \begin{array}{c}{{\phi}}\\{{\chi}}\\ \end{array}\right) = \frac{\sqrt{1 2}( P ( z ) Q^{\prime}( z ) - Q ( z ) P^{\prime}( z ) )}{| q | ( | P ( z ) |^{2}+ | Q ( z ) |^{2})^{3 / 2}}\left( \begin{array}{c}{{P ( z )}}\\{{Q ( z )}}\\ \end{array}\right) \, ,
\omega_{\psi}(g, z) \varphi(x) = z \cdot \int_{F^n / \ker(c)}\psi \left( \frac{1}{2}(xa, xb) + (xb, yc) + \frac{1}{2}(yc, yd) \right) \varphi(xa + yc) \, d \mu_g(y),
\rho_\zeta^\varepsilon(x,t) = \bar{\zeta}(x,t)-k_d^{-1}d(x,t)\Bigl(\overline{V_\Gamma\zeta}\Bigr)(x,t)+\frac{1}{2}d(x,t)^2\bar{\zeta}_2(x,t), (x,t)\in\overline{Q_{\varepsilon,T}},
\frac{b_6x+b_8}{b^2}= \frac{x+a_2}{a_1^2}+ \frac{x^3+a_2x^2+a_4x+a_6}{(a_1x+a_3)^2}+ \frac{a_4}{b}+\left(\frac{a_4}{b}\right)^2.
F(\zeta e^t) = \sum_{n=0}^\infty \frac{b_n(\zeta)t^n}{n!}= e^{t/24}\sum_{n=0}^\infty \frac{c_n(\zeta)t^n}{24^nn!}= \sum_{M=0}^\infty \frac{t^M}{24^MM!}\sum_{n=0}^M \binom{M}{n}c_n(\zeta),
|u(x,t)|\leq C\left( \left( \frac{|\sigma |(\Omega )+|\mu |(\Omega_{T})}{d^{N}}\right)^{m_{2}}+1+\left( \mathbb{I}_{2}^{2d}[|\sigma |\otimes \delta_{\{t=0\}}+|\mu |](x,t)\right)^{\frac{2}{2-N(1-m)}}\right) ,
| \lambda_{-}|^{2}\: \overline{{{\lambda_{-}}}}\: T_{[ 0 ]}^{( - 1 )}\; = \; \left( T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}\: \overline{{{T_{[ 0 ]}^{( - 1 )}}}}\right)^{2}\overline{{{T_{[ 0 ]}^{( 0 )}}}}\; .
\mathrm{E}_{\theta}T_1=\mathrm{P}_{\theta}(A^c_M) = \mathrm{P}\Big(\sum_{i=1}^{m}\xi_{[i]}^2 > \tfrac{(1+B)}{\beta}m \log(\tfrac{en}{m}) +\tfrac{M}{8}\Big) \le \exp\big\{-\beta M/8\big\}.
h_{n,k}(z)=\begin{cases}\varepsilon_{n,k-1}\,z\int_{a_{k-1}}^{b_{k-1}}\frac{p_{n,k-1}^{2}(\tau)}{z-\tau}\,d|\nu_{n,k-1}|(\tau) \ell<k, begin{align*}1em]\varepsilon_{n,k-1}\int_{a_{k-1}}^{b_{k-1}}\frac{p_{n,k-1}^{2}(\tau)}{z-\tau}\,d|\nu_{n,k-1}|(\tau) k\leq \ell.\end{cases}
\begin{aligned}\left \|f\right \|_{h_{\omega}^{p}}=\left \|T_{N}^{-1}\circ T_{N}(f)\right \|_{h_{\omega}^{p}}&\leq C\left \|T_{N}(f)\right \|_{h_{\omega}^{p}}&\leq C\left \|\left\{\sum\limits_{j\in\mathbb N}\sum\limits_{Q\in\Pi_{j+N}}\vert(\psi_{j}\ast f)(u_{Q})\vert^{2}\chi_{Q}\right\}^{\frac{1}{2}}\right \|_{L_{\omega}^{p}}\end{aligned}
\dot{z}(t)= G_{\bar{t}_{0}}(t,z(t),y_{\tau}(t),u_{\tau}(t)) := F(t-\tau,z(t),y_{\tau}(t),u_{\tau}(t))+\phi_{R}(t)P_{R}^{-1}(t)H'(t-\tau,u_{\tau}(t))(y_{\tau}(t)-H(t-\tau,u_{\tau}(t))z(t)) &{\rm with}\:{\rm initial}\: z(\bar{t}_{0})=0
\left|\Pi_{x}\left[\int_{s}^{t}F_{r}(\xi_{t-r}){\rm d}r\right]\right|\le \int_{s}^{t}\left|\Pi_{x}\left[F_{r}(\xi_{t-r})\right]\right|{\rm d}r=\int_{s}^{t}\left|\mathfrak{S}_{t-r}F_{r}(x)\right|{\rm d}r.
\tan(2\alpha_2') =\tan(\frac{\pi}{2}+2\phi_2)=|\cot(2\phi_2)|< 5.361\sqrt\epsilon, \alpha_2' <\frac{1}{2}\arctan(5.361\sqrt{\epsilon})< 2.681\sqrt{\epsilon}, \tan(2\beta_2') =\tan(\frac{\pi}{2}-2\psi_2)=\cot(2\psi_2)< 2.671\sqrt\epsilon, \beta_2' <\frac{1}{2}\arctan(2.671\sqrt{\epsilon})< 1.336\sqrt{\epsilon}.
p_{n+1}(\underline x_{n+1}) = \sum_{I \subseteq [2,n]}\left( h_{n-|I|+1}(\underline x_{n+1}^I) + h_{n-|I|}(\underline x_{n+1}^{I \cup \{n+1\}}) \right)= \sum_{I \subseteq [2,n+1]}h_{n+1-|I|}(\underline x_{n+1}^I).
\omega^{\rm rel}_w(t^{I_0}z^{J_0},\dots, t^{I_k}z^{J_k})=t^{I_0+\cdots+I_k}\omega^{\rm rel}_w(z^{J_0},\dots, z^{J_k})=t^{I_0+\cdots+I_k}(p^k_{2,w})^*\omega_{p_2^k(w)}(z^{J_0},\dots, z^{J_k}).
I_{\omega}=(-)^{r_5}(\omega^{1/2})^{r^{~2}_3+r_3r_4-r_1r_3-r_2r_3-r_3r_5-r_4r_5}\Bigg[\frac{\Phi(r_1)\Phi(r_2)}{\Phi(r_3)\Phi(r_4)}\Bigg]^{1/2}
S\circ\beta = \alpha, \big(S^{-1}h_{(2)}\big)_{(1)}\otimes_A\big(S^{-1}h_{(2)}\big)_{(2)}h_{(1)}= S^{-1}h\otimes_A 1_H, \big(S h_{(1)}\big)_{(1)}h_{(2)}\otimes_A\big(S h_{(1)}\big)_{(2)}= 1_H\otimes_A S h.
\mathrm{div}_T\Big( \tfrac{1}{2}\nabla_\xi H^2(\nabla w)\big)\Big)\,\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\cdot \nu - \nabla_T\big[\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\big] \,\big[\tfrac{1}{2}\nabla_\xi H^2(\nabla w)\big]_T\cdot \nu=H^2(\nabla w)\,H(\nu)\,\mathrm{tr}\,\mathcal{B}^H\,.
S_{0}= \frac{1}{4 e^{2}}\mathrm{R e}\int d^{4}x d^{2}\theta W_{a}C^{a b}W_{b}+ \frac{1}{4}\int d^{4}x d^{4}\theta \Big ( \phi^{*}e^{2 V}\phi + \tilde{\phi}^{*}e^{- 2 V}\tilde{\phi}\Big ) .
\rho_{\pm}= \frac{( 1 + \alpha^{2}) ( M \pm \sqrt{M^{2}- \left( 1 - \alpha^{2}\right) Q^{2}})}{\left( 1 \pm \alpha^{2}\right)}, a = \frac{2 ( 1 + \alpha^{2}) J}{( 1 + \alpha^{2}) \rho_{+}+ ( 1 - \alpha^{2}/ 3 ) \rho_{-}}.
\tilde{a}_0-\tilde{a}_0(T_1) =\frac{1}{\sqrt{2\pi K}}\int_{T_1}^\infty dt\; e^{itH^{(\Lambda)}}a_0e^{-itH^{(\Lambda)}}e^{-i(\Delta\tilde{E}+\tilde{\kappa})t}e^{-t^2/(2K)}+\frac{1}{\sqrt{2\pi K}}\int_{-\infty}^{-T_1}dt\; e^{itH^{(\Lambda)}}a_0e^{-itH^{(\Lambda)}}e^{-i(\Delta\tilde{E}+\tilde{\kappa})t}e^{-t^2/(2K)}.
u(t,x) =\frac{u_0(\overline{x})}{JX(t,\overline{x})}\exp\left(\int_{0}^{t}G(s,X(s,\overline{x}))ds\right) &+\frac{1}{JX(t,\overline{x})}\int_{0}^{t}f(\tau,X(\tau,\overline{x})) \exp\left(\int_{\tau}^{t}G(s,X(s,\overline{x}))ds\right) JX(\tau,\overline{x})d\tau,
\sum_{n_{1}, n_{2}}( C ( n_{1}, n_{2}) \bar{\chi}\partial_{\mu}\chi A_{\mu}( n_{1}, n_{2}) + c . c . ) + \sum_{( n_{1}, n_{2}) \neq ( 0 , 0 )}( m C ( n_{1}, n_{2}) \chi \bar{\chi}\phi ( n_{1}, n_{2}) + c . c . )
I_{2}=\sum_{k=0}^{m-1}\frac{k!^{2}\mathbb{A}_{k,k}^{2}}{(n-m+k)!^2}+2\sum_{j=1}^{m-1}\sum_{i=0}^{j-1}\frac{i!j!\mathbb{A}_{i,j}^{2}}{(n-m+i)!(n-m+j)!},
n(\lambda) = \langle (\lambda_1, \ldots, \lambda_n), (0,1,\ldots, n-1)\rangle= \langle \lambda, -\rho^\vee+\hbox{$\frac{(n-1)}{2}$}(1,1,\ldots, 1)\ \rangle = \hbox{$\frac{(n-1)}{2}$}\vert\lambda\vert - \langle \lambda, \rho^\vee\rangle
P_{\widehat{\mathcal{H}}_\chi^{\rm stiff}}\left(\mathcal{A}_{\chi,\varepsilon}^{\rm app}- zI \right)^{-1}|_{\widehat{\mathcal{H}}_\chi^{\rm stiff}}= \left(\frac{1}{\varepsilon^2}\mathcal{A}_\chi - \mathcal{B}_\chi(z) \right)^{-1}.
x_1^{p^n}=x_2^{p^4}=x_3^{p^4}=x_4^{p^2}=1 & [x_1,x_2]=[x_1,x_3]=x_2^{p^2}, \; [x_1,x_4]=x_3^{p^2}, & [x_2,x_3]=x_1^{p^{n-2}}, \; [x_2,x_4]=x_3^{p^2},\; [x_3,x_4]=x_2^{p^2}.
((e_{i}')^{k}(D_{u,u'}),x)_{L}=(1-q_{i}^{2})^{k}(D_{u,u'},f_{i}^{k}x)_{L}=(1-q_{i}^{2})^{k}(u,f_{i}^{k}x.u')_{\lambda}^{\varphi}=(1-q_{i}^{2})^{k}(e_{i}^{k}.u,x.u')_{\lambda}^{\varphi}=(1-q_{i}^{2})^{k}(D_{e_{i}^{k}.u,u'},x)_{L}.
\frac{e^{2 u_{k}}- q^{- 2 s}}{e^{2 u_{k}}q^{- 2 s}- 1}= \prod_{j = 1}^{2 s}\frac{e^{u_{k}}q - e^{u_{j}}}{e^{u_{k}}- e^{u_{j}}q}, k = 1 , \ldots , 2 s .
\int_{\mathbb{R}^d}\phi (x)dx=1,\int_{\mathbb{R}^d}y^{\beta_{1}}\phi (y)dy=0|\beta_{1}|\geq 1,\int_{\mathbb{R}^d}|y|^{m}|\partial_{\beta_{2}}\phi (y)|dy<\infty m\in \mathbb{N}.
\langle \mu_i( x \cdot \phi_0(T) - \phi_0(T) \cdot x ) \hat{a}, \hat{b}\rangle= \mu_i( [x, \varphi_0]) (b^* T a )\to \langle \mu( x \cdot \phi_0(T) - \phi_0(T) \cdot x ) \hat{a}, \hat{b}\rangle.
\begin{pmatrix}P Q\end{pmatrix}\partial_x u = \left[\begin{pmatrix}P Q\end{pmatrix}\partial_x\begin{pmatrix}P^* Q^*\end{pmatrix}\right]\begin{pmatrix}P Q\end{pmatrix}u = \begin{pmatrix}K_P 0 0 K_Q\end{pmatrix}\begin{pmatrix}u_P u_Q\end{pmatrix}.
\frac{1}{\epsilon}\int \frac{1}{\epsilon^N}Q\left(\frac{x-y}{\epsilon}\right)(\mathbf{v}(y) - \mathbf{v}(x))\, dy= \int Q(z)\frac{\mathbf{v}(x+\epsilon z) - \mathbf{v}(x)}{\epsilon}\, dz = \int Q(z) (z\cdot\nabla_x ) \mathbf{v}(\zeta)\, dz,
\frac{d^2 \gamma_\ell}{ds^2}(s)(s) = \sum_{i,j}^{n}\mathfrak\Gamma_{ij}^\ell \frac{d\gamma_i}{ds}(s)\frac{d\gamma_j}{ds}(s)\ ,\ \gamma(0)=e\ ,\ \frac{d\gamma}{ds}(0)=v\ ,
\gamma_{A,\ell}= \sum_{i\in I}\frac{\dim W_i}{\dim G_{A,\ell}- \dim (G_{A,\ell})_W}\leq \sum_{i\in I}\frac{\dim W_i}{\dim G_{A_i,\ell}- \dim (G_{A_i,\ell})_W}\leq \sum_{i \in I}\gamma_{A_i,\ell}.
\begin{aligned}E[K(x) K(y) ] = F(x \wedge y ) -F(x) F(y) + \ \sum_{d=1}^{\infty}(P(X_0 \leq x, X_d \leq y) -F(x) F(y)) + \ \sum_{d=1}^{\infty}(P(X_0 \leq y, X_d \leq x) -F(x) F(y)). \end{aligned}
\nabla F^{-1}(F(x)) \frac{\nabla\gamma(F(x))}{\gamma(F(x))}= \nabla F^{-1}(P) \frac{\nabla\gamma(P)}{\gamma(P)}+ \big( \nabla F^{-1}(F(x)) - \nabla F^{-1}(P) \big) \frac{\nabla\gamma(P)}{\gamma(P)}& + \nabla F^{-1}(F(x)) \Big( \frac{\nabla\gamma(F(x))}{\gamma(F(x))}- \frac{\nabla\gamma(P)}{\gamma(P)}\Big).
\frac{d^i x}{dt^i}= \sum_{j=i}^{p-1}j(j-1)\cdots(j-i+1) s_j(x)^p t^{j-i}(i=0,\dots,p-1) \frac{d^{p-1}(t^{p-1-i}x)}{dt^{p-1}}= -s_i(x)^p (i=0,\dots,p-1).
{h}(\theta) = \left(\sum_{j = 0}^{\infty}a_j e^{ij\theta}\right) - \left(\sum\limits_{j=0}^{\infty}(\bold{B}^{-1}\bold{a})_je^{ij\theta}\right)\left(b_{-1}e^{-i\theta}+ b_0+b_1 e^{i\theta}\right),
\mathcal{E}(t) \leq \tfrac{1}{\beta_{1}}\mathcal{L}(t) \leq\tfrac{1}{\beta_{1}}\mathcal{E}(0) e^{-\tfrac{C}{\beta_{2}}t}=: C \mathcal{E}(0) e^{-2 \alpha t}t \geq 0
{\hat Q}\frac{\partial^2{\hat H}^{(n)}}{\partial{\hat Q}^2}=\frac{1}{2}\frac{\partial^2{\hat H}^{(n)}}{\partial{\hat S}\partial{\hat P}}-\frac{1}{2}\frac{\partial^2({\hat P}{\hat H}^{(n)})}{\partial{\hat P}\partial{\hat Q}}\,.
\left[\partial_y^2 +\frac{4}{l}\coth(y/l)\partial_y -\frac{1}{(lH)^2\sinh^2(y/l)}(\partial_t^2 +3H\partial_t)+\frac{1}{l^2\sinh^2(y/l)e^{2Ht}}\partial_{\mbox{\boldmath $x$}}^2 \right]\delta\phi=0 .
\iint_{Q_T}|\mathbf{b}_x||\mathbf{b}_{xx}|dxdt =\frac{1}{\nu}\iint_{Q_T}|\mathbf{b}_x||\mathbf{b}_{t}+(u\mathbf{b})_x-\mathbf{w}_x|dxdt \leq C+C\int_0^T\|\mathbf{b}_x\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{w}_x|dxdt \leq C,
C\mathcal{H}^{\frac{\log 2}{\log 3}}_{\infty}(\Omega_{k})\leq\mathcal{H}^{\frac{\log 2}{\log 3}}_{\infty}(K_{1/3})\leq \mathcal{H}^{\frac{\log 2}{\log 3}}_{\infty}\Big (\bigcup_{I\in\mathcal{K}_k}I\Big )=\mathcal{H}^{\frac{\log 2}{\log 3}}_{\infty}(\Omega_{k})\le 1,
\int_\Omega |Dg_3-Dg|^p\,d\mu =\int_{\bigcup_{j=0}^{k-1}D(x_j,r'')}|Dg_3-Dg|^p\,d\mu &\leq K[g_3-g]_{\mathrm{Lip}}^p\sum_{j=0}^{k-1}\mu(D(x_j,r'')) &\leq K(1+cn)^p\pi k(r'')^2 \ .
\begin{gathered}(A\oplus B)_{i/o}=A_{i/o}\sqcup B_{i/o}\;, (A\oplus B)_M=A_M\oplus B_M\;, (A\oplus B)_I=A_I\oplus B_I\;, (A\oplus B)_O=A_O\oplus B_O\;.\end{gathered}
_{\tau}(\xi_{n}^{(0)})\left( \begin{array}{c}\Psi_{\tau}(h_{1},...,h_{n}=0,...,h_{\mathsf{N}}) \Psi_{\tau}(h_{1},...,h_{n}=1,...,h_{\mathsf{N}}) \vdots \Psi_{\tau}(h_{1},...,h_{n}=p-1,...,h_{\mathsf{N}})\end{array}\right) =\left( \begin{array}{c}0 0 \vdots 0\end{array}\right) ,
\lambda_{j,j}(t)=\lim_{x \to y_j}\,\psi_{f,j}(x,t)=-\frac{m(\zeta_{0j}+t\dot\zeta_{0j})-\dot\zeta_0}{4\pi}+\frac{m}{4\pi}\int_0^t \frac{J_1(m(t-s))}{(t-s)}(\zeta_{0j}+s\dot\zeta_{0j})ds.
(2\pi ik)(-2\pi ik)\int_{-\infty}^{t\wedge s}e^{-2\pi^2k^2\{(t-u)+(s-u)\}}du &=e^{-2\pi^2k^2(t+s-2u)}\big|_{u=-\infty}^{t\wedge s}=e^{-2\pi^2k^2|t-s|}=H_{|t-s|}(k).
\kappa_k^2 \quad=w_k'(I_d-P_{N_{(k-1)}})B\left[ B'(I_d -P_{N_{(k-1)}})B\right]^{-1}B'(I_d - P_{N_{(k-1)}})w_k &\quad=w_k'(I_d- \tilde{C}_{(k-1)}\tilde{C}_{(k-1)}')B\left[ I_p -xx'\eta_{(k-1)}\right]^{-1}B'(I_d - \tilde{C}_{(k-1)}\tilde{C}_{(k-1)}')w_k,
\widehat{\mathbf{C}}= \Big\{(\hat{c}^{\mbox{\tiny (\itshape k\upshape)}}_1, \ldots, \hat{c}^{\mbox{\tiny (\itshape k\upshape)}}_n)^\top, \; k \in \mathcal{K}:= \{1, \ldots, K\}\Big\},
A{}=A_{j-1}[X_{j+1};\sigma^*_{j+1}]\ldots [X_N;\sigma^*_N][X_j;\sigma_j',\delta_j'], \hat{A}{}=A_{j-1}[X_{j+1}^{\pm 1};\sigma^*_{j+1}]\ldots [X_N^{\pm 1};\sigma^*_N][X_j;\sigma_j',\delta_j'],
{\cal M}\displaystyle\int\limits^{\infty}_{-\infty}\displaystyle\frac{dx}{2\pi}\displaystyle\frac{1}{2}e^{-x}\,{\cal Q}(x)\doteq \displaystyle\frac{1}{L}\displaystyle\int\limits^{\infty}_{-\infty}\displaystyle\frac{dx}{2\pi}e^{-x}\,{\cal Q}_{-}(x)\, .
|\{(cl(i),cl(j)) \mid -j \in C_i, 1\le i,j\le l\}| &= \begin{cases}0 \mbox{if $1 \le l \le 2^{m/2}-2$}, 1 \mbox{if $2^{m/2}-1 \le l \le 2^{(m/2)+1}-4$}, 3 \mbox{if $2^{(m/2)+1}-3 \le l \le 2^{(m/2)+1}-2$}, 5 \mbox{if $2^{(m/2)+1}-1 \le l \le 2^{(m/2)+1}$}.\end{cases}
(-1)^{u-1}\tbinom p{u-1}\left(\tfrac mn\right)^{p-(u-1)}f_0(T_{p-(u-1),0}) + (-1)^u\tbinom pu \left(\tfrac mn\right)^{p-u}f_0(T_{0,u}) =(-1)^{u-1}\left(\tfrac mn\right)^{p-u}\left[\tbinom p{u-1}\left(\tfrac mn\right)(p-u+1)n - \tbinom pu um\right] =0,
A_{1}C_{1}^{\eta^{\ast}}=C_{1}A_{1}^{\eta^{\ast}}, B_{1}^{\eta^{\ast}}D_{1}=D_{1}^{\eta^{\ast}}B_{1}, &R_{A_{1}}C_{1}=0, D_{21}L_{B_{1}}=0, R_{M}R_{B_4}D_4=0, & R_{B_4}D_4\left(R_{C_4}\right)^{\eta^{\ast}}=0 .
u^{\infty}(\hat{x}; d, c_p, c_s) = \left(c_p u^{\infty}_{pp}(\hat{x}) + c_s u^{\infty}_{ps}(\hat{x})\right) \hat{x}+ \left(c_p u^{\infty}_{sp}(\hat{x}) + c_s u^{\infty}_{ss}(\hat{x})\right)\hat{x}^{\perp},
\frac{d}{dt}E^3_q(g)- \sum_{i}(\Delta_q b_i,\partial_i \Delta_q \partial_t a) +\Vert \nabla_x \Delta_qa \Vert^2_{L^2_x}= -5\sum_{i}(\partial_i \Delta_q c, \partial_i \Delta_q a) - \sum_{i,j}(\partial_j(v_i v_j \mu^{1/2},\Delta_q g_2), \partial_i \Delta_q a).
\aligned(I P_{> \frac{N}{8k}}u)(I P_{\leq \frac{N}{8k}}u)^{2k}- I((P_{> \frac{N}{8k}}u)(P_{\leq \frac{N}{8k}}u)^{2k}) = (I P_{> \frac{N}{2}}u)(P_{\leq \frac{N}{8k}}u)^{2k}- I((P_{> \frac{N}{2}}u)(P_{\leq \frac{N}{8k}}u)^{2k}).\endaligned
0 = (n)_{k+1}\langle \phi_j^i, \phi \rangle &= \sum_{\mathrlap{(v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_{k+1}) \in V_{k}}}\phi_j(v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_{k+1}) \psi_i (v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_{k+1}) &= (n)_k \langle \phi_j, \psi_i \rangle.
R_{n}(x) =I_{0}^{\left ( n+1 \right )\alpha}D_{0}^{\left ( n+1 \right )\alpha}g\left ( x \right ) &=\frac{1}{\Gamma \left ( \left ( n+1 \right )\alpha \right )}\int_{0}^{x}\left ( x-t \right )^{\left ( n+1 \right )\alpha -1}D_{0}^{\left ( n+1 \right )\alpha}g\left ( t \right )dt.
{\bar{{\cal G}}}_{( \mathrm{I})}^{\left( \rho \right) a}\Gamma^{\left( \rho \right)}= \Delta_{\mathrm{c l}( \mathrm{I})}^{\left( \rho \right) a}+ O \left( \rho^{2}\right) ,{\bar{{\cal G}}}_{( \mathrm{I I})}^{\left( \rho \right) a}\Gamma^{\left( \rho \right)}= \Delta_{\mathrm{c l}( \mathrm{I I})}^{\left( \rho \right) a}+ O \left( \rho^{2}\right) ,
\mathcal L = \frac{-1}{\tau_k}\int_{0}^{T_k}X_T(\gamma_k(t))\cdot \nabla \ln C(\gamma_k(t)) &+\frac{-1}{\tau_k}\int_{0}^{T_k}X_T(\gamma_k(t))\cdot \nabla \ln \Big(B\prod_{j\neq k}f_j^{\tau_j}\Big)(\gamma_k(t))\,.
G[s](w) =\frac{1}{1+u_1w+u_2w^2\cdots +u_sw^s}&=\sum_{n=0}^\infty (-1)^{n}(u_1w+u_2w^2\cdots +u_sw^s)^n &=\sum_{n=0}^\infty (-1)^{n}\sum_{\nu_1+\cdots\nu_s=n}\frac{n!}{\prod_{i=1}^{s}\nu_{i}!}\prod_{j=1}^s (u_jw^j)^{\nu_j}&=\sum_{\eta}(-1)^{\ell( \eta )}\frac{\ell(\eta)!}{\prod_{i=1}^sm_i(\eta)!}w^{|\eta|}\prod_{i=1}^s u_i^{m_i(\eta)}
W=W(x_0,t_0,u_0) - \frac{1}{2}\epsilon^\beta{u_0}^2{\tau}+\epsilon^\gamma u_0{{y}}+\epsilon^{\alpha+\gamma}{{y}}{\upsilon}+ \frac{1}{2}\epsilon^{\beta+2\alpha}{\tau}{\upsilon}^2 + \epsilon^{\alpha(k+2)}A_{k+2}{\upsilon}^{k+2}+\dots\, ,
lim_{x\rightarrow\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=lim_{x\rightarrow\infty}\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=lim_{x\rightarrow\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\cdot\cdot\cdot.
I = \left[\min I_1,\min I_1 + \sum_{j\in N}\lambda(I_j)\right], \nu = \nu_1+\sum_{j\in N,\,j\geq 2}\nu_j\Big(\cdot-\Big(\min I_1 + \sum_{k=1}^{j-1}\lambda(I_k) - \min I_j\Big)\Big).
f(s,t)f_{m_{st}}(s,t) =(ts)_{m_{st}-1}f_{m_{st}}(s,t) &=(ts)_{m_{ts}-1}f_{m_{ts}-1}(t,s) &=tf_1(t,s)f_2(t,s)\cdots f_{m_{ts}-1}(t,s) &=tf(t,s) &=t(st)_{m_{ts}-1}&=(ts)_{m_{ts}}.
\begin{array}{r c l}{{T_{c}}}&{{=}}&{{\displaystyle + [ 3 D_{m}( k_{1}) D_{m}( k_{2}) + D ( k_{1}) D ( k_{2}) ] D_{m}( k_{1}+ k_{2}) ( 1 + K_{1 2})}}\\{{}}&{{}}&{{\displaystyle + 4 ( N - 1 ) D ( k_{1}) D ( k_{2}) D_{m}( k_{1}+ k_{2}) .}}\\ \end{array}
n_j^\phi = -\phi(\omega(\tau_j^{p-1}, \tau_j)) = -\phi(a_j), n_{j,k}^\phi = \phi(\tilde \omega ( f_2(1_{j,k}))) = \phi(\tilde \omega ( [\tau_k \otimes \tau_j ] - [\tau_j\otimes \tau_k]) )= \phi(c_{j,k}).
R_{m,n}=\frac{{}_{1}F_{1}(-m;-m-n;z)}{{}_{1}F_{1}(-n;-m-n;-z)}=\frac{n!2^{m}\theta_{m}(\frac{z}{2};n-m+2,2)}{m!2^{n}\theta_{n}(-\frac{z}{2};m-n+2,2)}
\partial_t \Phi_{\beta,\varepsilon}(t,x;t_0) = - (t_0+t)^{-\beta-1}\left[ \beta \varphi_{\beta,\varepsilon}(z) + z \varphi_{\beta,\varepsilon}'(z) \right] = - (t_0+t)^{-\beta-1}\beta \varphi_{\beta+1,\varepsilon}(z) = - \beta \Phi_{\beta+1,\varepsilon}(t,x;t_0),
\log\|\langle s_0,\dots,s_d|Z\rangle\|_v=\log\|\langle s_0,\dots,s_{d-1}|Z\cap \mathrm{div}(s_d)\rangle\|_v+\int_{Z_{\overline{K}_v}^{\mathrm{an}}}\log\|s_d\|_v \prod_{i=0}^{d-1}c_1(\overline{L}_{i,v}),
y^{i}= 3 D ( x^{n})_{-}\ldots ( x^{i + 2})_{-}( x^{i + 1})_{-}x^{i}( x^{i + 1})_{-}^{- 1}( x^{i + 2})_{-}^{- 1}\ldots ( x^{n})_{-}^{- 1}
\Im \bigl( e^{(1-a)x}(e^x-\overline{z}) \bigr) = - e^{(1-a)x}\Im (\overline{z}) \,\,\begin{matrix}>0 \mbox{if}\Im (\overline{z}) <0 <0 \mbox{if}\Im (\overline{z}) >0\end{matrix}
\| \left((S_\varepsilon -I) b({\mathbf D}) \widetilde{\mathbf u}_0\right) (\cdot,t) \|_{H^l({\mathbb R}^d)}\leqslant 2 \| b({\mathbf D}) \widetilde{\mathbf u}_0 (\cdot,t) \|_{H^l({\mathbb R}^d)}&\leqslant 2 \alpha_1^{1/2}\| \widetilde{\mathbf u}_0 (\cdot,t) \|_{H^{l+1}({\mathbb R}^d)}.
\| P_{B\setminus A}(x-\mathcal{T}(x-z,\Delta_\alpha))\| \ \le\ 2\mathbf{A_p}\alpha \sup_{|\eta|=1}\| 1_{\eta (B\setminus A)}\| &\ \le\ 2\mathbf{A_p}C_{sd}\min_{n\in \Lambda}| e_n^*(x-z)| \| 1_{\varepsilon \Lambda}\| &\ \le\ 2\mathbf{A_p}C_{sd}C_{q}^2\eta_p(C_{q})\| x-z\|.
\|\Psi^{n}(dd^{c}\varphi)\|_U =\int_U\left(\Psi^{n}(dd^{c}\varphi)\right)\wedge \omega^{k-1}=\int_U\frac{1}{N^{n}}\left((g^{n})_*(dd^{c}\varphi)\right)\wedge \omega^{k-1}& = \frac{1}{N^{n}}\int_Udd^{c}\varphi\wedge (g^{n})^{*}\omega^{k-1}&=\frac{1}{N^{n}}\int_Udd^{c}\varphi\wedge (d^{mn}\omega+dd^{c}u_{nm})^{k-1}
p_{\epsilon}(x_i|U_\epsilon=0) =p(x_i)\mathbb{E}[f|X_i=x_i] p_{\epsilon}(x_i|U_\epsilon=1) =p(x_i)\left(\frac{1}{1-\epsilon}- \frac{\epsilon}{1-\epsilon}\mathbb{E}[f|X_i=x_i]\right)
\lim_{n\to\infty}R^{(k)}_{\Lambda^{(n)},N^{(n)}}\big(A(\Lambda^{(n)},N^{(n)};f_1),\dots,A(\Lambda^{(n)},N^{(n)};f_k)\big)=R^{(k)}(A(f_1),\dots,A(f_k))
\Delta_g^2 u_\delta = u_\delta^{(4)}+ 2Lu_\delta^{(3)}+L^2u_\delta''+L''u_\delta'+2L'u_\delta''+LL'u_\delta' \sim u_\delta^{(4)}+ 2Lu_\delta^{(3)}+L^2u_\delta''=s^2(s-L)^2e^{-s\rho}\sim s^4(1-p)^2e^{s\rho}.
(1-y)^{n-1}{\rm Tutte}_{n}^{(\beta)}(\{p_{i,j}\};x,y){}= \prod_{1\le i < j \le n}p_{ij}+ \sum_{\lambda \vdash n}{\rm Tutte}(K_{\ell(\lambda)-1}; x+y+\beta x y,0){\cal{L}}_{\lambda}^{(\beta)}(\{p_{ij}\}) .
v^{r e n}( q ) \, \simeq \, - \frac{\pi}{2 4}\, + \, \frac{\ln 2}{\pi}\, q \, + \, \frac{1}{2 \pi q}\,{,}\quad q \, = \, \frac{M_{0}^{2}\, R}{m}\,{,}\quad q \to \infty \,{.}
{\bf \Psi}_{n , k_{3}}^{T}( x , y ) = \left( c_{1}\Psi_{n - 1}( x , y ) , \; i c_{2}\Psi_{n}( x , y ) , \; c_{3}\Psi_{n - 1}( x , y ) , \; i c_{4}\Psi_{n}( x , y ) \right) \; .
\varphi([y_{i_{2m+1}},z_{j_1}]\dots[y_{i_{2m+n}},z_{j_n}])=\pm2^n \left(\prod_{k=2m+1}^{2m+n}\varphi(y_{i_k})_1\right)\cdot \left(\prod_{l=1}^{n+2p}\varphi(z_{j_l})_1\right).
k!\sum_{\substack{\Omega(n)=k n}}a(n)b(n)g(n)^2 =\frac{1}{(\alpha_k)^2}\sum_{\Omega(n)=k}a(n)b(n)\frac{k!}{\alpha_1!\cdots\alpha_{k-2}!}&=O_k\left(\sum_{p_1,\ldots,p_{k-2}}a(p_1\cdots p_{k-2})b(p_1\cdots p_{k-2})\right) &=O_k\left(\sum_{p\leq Y}a(p)b(p)\right)^{k-2}.
\begin{aligned}A_n =\frac{8n\pi \left[2 \cos^2\left(\frac{n\pi}{2}\right)-1\right]}{\sinh(2n\pi M)}\int_{0}^{\infty}\frac{(x-1)\sin\left(M\log x\right)}{x\left[4\pi^2n^2+\log^2(x)\right]\left(e^{x}+1\right)}dx \end{aligned}
\Delta_k(p)= \det \left( \begin{array}{ccccc}a_1 a_3 a_5 \dots a_{2k-1}a_0 a_2 a_4 \dots a_{2k-2}0 a_1 a_3 \dots a_{2k-3}0 a_0 a_2 \dots a_{2k-r}\dots \dots \dots \dots \dots 0 0 0 \dots a_{k}\end{array}\right)_{k\times k}.
(d^2 u_{\xi})_z(w,w) & = (p < \nabla_w \nabla B^{f(\xi)}, w > + p^2 < \nabla B^{f(\xi)}(z), w >^2) \exp(pB(z, \pi(\phi(\overrightarrow{x\xi})), f(\xi))) \geq (p ||w^{\bot f(\xi)}||^2 + p^2 < \overrightarrow{z f(\xi)}, w >^2) \exp(pB(z, \pi(\phi(\overrightarrow{x\xi})), f(\xi))) > 0
{\cal L}_{\mathrm{m a t t e r}}= \frac{1}{4}\sum_{i}\int \! \mathrm{d}^{2}\theta \mathrm{d}^{2}\bar{\theta}\bar{Q}_{i}e^{V}Q_{i}+ \frac{1}{2}\left\{\int \! \mathrm{d}^{2}\theta{\cal W}( Q_{i}) + \mathrm{H . c .}\right\}.
\overline{{{A}}}^{\mu}( \overline{{{q}}}) = A^{\nu}( q ) \, \overline{{{a}}}_{\nu}^{\mu}( q ) = \overline{{{a}}}_{\nu}^{\mu}( q ) A^{\nu}( q ) , \quad \overline{{{A}}}_{\mu}( \overline{{{q}}}) = A_{\nu}( q ) \,{a}_{\mu}^{\nu}( q ) ={a}_{\mu}^{\nu}( q ) A_{\nu}( q )
\begin{aligned}y= \sqrt{P_s}\zeta\sum_{l=1}^L\hat h_le^{j\phi_{l,n_t}}g_{l,n_t}+ & \sqrt{P_s(1-\zeta^2)}\sum_{l=1}^L\Delta{h}e^{j\phi_{l,n_t}}g_{l,n_t}+ n_0,\end{aligned}
\partial_\pm T = A_\pm T,{}~~~~A_\pm = \pm \left[ \frac{1}{2}\partial_\pm \Phi + \exp \left(\mp \frac{1}{2}{\rm ad}\Phi \right) \bar{\Psi}_\pm + \exp \left(\mp \frac{1}{2}{\rm ad}\Phi \right) \mu_\pm \right],
\rho_V (1-\pi_S(\rho_V, \rho_R)) = \frac{1}{Z(\rho_V, \rho_R)}\rho_V\sum\limits_{k+l < K}\frac{\rho_R^l}{l!}\rho_V^k &= \frac{1}{Z(\rho_V, \rho_R)}\sum\limits_{k>0, k+l \leq K}\frac{\rho_R^l}{l!}\rho_V^k &= 1-\pi_{0,.}(\rho_V, \rho_R).
\begin{cases}\mathcal{Y}^{n}\widetilde{\chi}^{m}+\mathcal{Y}^{m}\widetilde{\chi'}^{n}-(\widetilde{\chi}^{m})^{n}\,_{k}\mathcal{Y}^{k}=0,\cr(\widetilde{\chi}^{i})^{j}\,_{p}\widetilde{\chi}^{p}+(\widetilde{\chi}^{i})\widetilde{\chi}^{j}-\widetilde{\chi}^{j}\widetilde{\chi}^{i}=0.\end{cases}
\omega\mid U_{\pm}=\int_N\frac{-2 i S_{0}}{\left(1+z_{\pm}(\vec{x})\overline{z}_{\pm}(\vec{x})\right)^2}\delta \overline{z}_{\pm}(\vec{x})\wedge \delta z_{\pm}(\vec{x})d\mu(\vec{x})\,.
\sum_{j=1}^\infty \left( \alpha_{mj}\, \overline{\alpha_{nj}}- \beta_{mj}\,\overline{\beta_{nj}}\right) = \delta_{nm}\qquad\mbox{and}\sum_{j=1}^\infty \left( \alpha_{mj}\, \beta_{nj}- \beta_{mj}\,\alpha_{nj}\right) = 0,
\int_{\mathcal{H}_1}\dots \int_{\mathcal{H}_{r+1}}D^{\ell_1+\dots+\ell_i}f(x)(\underbrace{e_{j_1},\dots,e_{j_1}}_{\ell_1\},\dots,\underbrace{e_{j_i},\dots,e_{j_i}}_{\ell_i\}) &\times\prod_{t=i+1}^{r}:(\mathfrak{Re}\langle e_{j_{t}},\cdot\rangle)^{\ell_{t}}:(x_{t})\,dm_1(x_1)\dots dm_{r+1}(x_{r+1}),
\sum_{b\geq0}N (b) q^{b}=\prod_{m\geq1}(1-q^{m})^{-1}\cdot \bigg(1+\frac{p}{(1-p)^{2}}+ \sum_{d\geq1}\sum_{k|d}k (p^{k}+p^{-k})q^{d}\bigg) \,.
\mathfrak{A}=\sum_{\alpha\in\mathbb{N}^{n-1},j,k,l,m\in\mathbb{N}}(h^{2/3}T)^j(h^{2/3}\langle\lambda\rangle^{-1/2})^k(h^{1/3}\langle\lambda\rangle^{-1})^lh^m\mathcal{A}_{\alpha,j,k,l,m}(x',\xi',\lambda,z)D_{x'}^\alpha,
(x+y)^{n}=\sum_{k=0}^{n}a_{k}x^{n-k}y^{k}=a_{0}x^{n}+a_{1}x^{n-1}y+a_{2}x^{n-2}y^{2}+...+a_{n-1}xy^{n-1}+a_{n}y^{n}
B_{n}(f_{1},f_{2}) : = \frac{f_{1}(x)f_{2}(y)-f_{1}(y)f_{2}(x)}{x-y}= \sum_{i,j=1}^{n}\alpha_{ij}x^{n-i}y^{n-j}\in F[x,y], M_{n}(f_{1},f_{2}) : = (\alpha_{ij})_{1 \leq i,j \leq n}.
\pi_{2}(\phi_{\theta}) S_{2}e_{n}^{1}= e^{-i \left(n + 1 + a + \frac{\lambda}{2}\right) \theta}S_{2}e_{n}^{1}, n \geq 0;\,\,\,\pi_{2}(\phi_{\theta}) S_{2}e_{-n}^{1}= e^{i \left(n - 1 + b - \frac{\lambda}{2}\right) \theta}S_{2}e_{-n}^{1}, n \geq 1.
\frac{\vert a \vert}{\vert + \vert a \vert}+ \frac{\vert b \vert}{\vert + \vert b \vert}\geq \frac{\vert a \vert + \vert b \vert}{1 + \vert a \vert + b \vert}\geq \frac{\vert a + b \vert}{1 + \vert a \vert + b \vert}