harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi(2,\phi)= \frac{\phi}{\ln 2}\int^{\infty}_{0}\frac{1}{1+x\phi}\left(-K \sum^{K-1}_{p=1}{K-1 \choose p}(-1)^p e^{-px}\right. \left. + (K-1) \sum^{K}_{l=1}{K \choose l}(-1)^l e^{-lx}\right) dx.
\int_0^t s^\frac{k-1}{k}V^{\delta_2+\delta_3}(V')^{1-\delta_2}ds \leq t^\frac{k-1}{k}\int_0^t V^{\delta_2+\delta_3}(V')^{1-\delta_2}ds &\leq t^{\frac{k-1}{k}}t^{\delta_2}\left(\int_0^t V^{(\delta_2+\delta_3)/(1-\delta_2)}(V')ds\right)^{1-\delta_2}&= t^\frac{n-k-2}{n-k-1}\left(\frac{1-\delta_2}{1+\delta_3}\right)^{1-\delta_2}V^{1+\delta_3}.
\left\langle L_{F'}e_{k}',e_{j}' \right\rangle = \begin{cases}\mu^{|\alpha(j) |- |\alpha(k) |}\sum_{l=1}^n \alpha_l(k) \, a_{l,(\alpha(j)-\alpha(k))_l}|\alpha(j) |> |\alpha(k)| \sum_{l=1}^n \alpha_l(k) \, a_{l,l}j= k 0 .\end{cases}
a_1 =A-\sum_{i=2}^{(k-5)/2-t}ia_i= \sum_{i=2}^{(k-5)/2-t}\left(2t+i(t+\frac{k+1}{2})\right)\frac{\binom{n}{k-2-i}}{n}-\sum_{i=2}^{(k-5)/2-t}ia_i \ge \sum_{i=2}^{(k-5)/2-t}\left[\left(2t+i(t+\frac{k+1}{2})\right)\frac{\binom{n}{k-2-i}}{n}-i\right]\ge \sum_{i=2}^{(k-5)/2-t}\left(\frac{k+1}{2}-1\right)i\ge 0,
I_{N}^{}\left( f\right) :=\frac{1}{2}{\displaystyle\sum_{k=0}^{\infty}}{\displaystyle\sum_{\ell=1}^{a_{k}}}{\displaystyle\sum_{j=1}^{N}}\lambda_{j,\left( k,\ell\right)}\cdot t_{j,\left( k,\ell\right)}^{-\frac{k}{2}}\cdot f_{\left( k,\ell\right)}\left( \sqrt{t_{j,\left(k,\ell\right)}}\right) .
( \ln a_{1}^{2})^{\prime}( \ln a_{2}^{2})^{\prime}+ ( \ln a_{2}^{2})^{\prime}( \ln a_{3}^{2})^{\prime}+ ( \ln a_{3}^{2})^{\prime}( \ln a_{1}^{2})^{\prime}- ( a_{1}^{2}- a_{2}^{2})^{2}= A^{2},
\mathcal L f(x) = \int_{\alpha_-(x)}^{\alpha_+(x)}\frac{f(y)}{(b-a)y(1-y)}\ \d y - \int_{\beta_-(x\wedge a/4)}^{\beta_+(x\wedge a/4)}\frac{f(y)}{(b-a)y(1-y)}\ \d y ,
1 > \frac{\Phi(-y_2)}{\Phi(-y)}> 1 - \frac{y^2 + 1}{y}e^{\frac{y^2}{2}}\frac{C_1}{\sigma \sqrt{n}}e^{-\frac{1}{2}(y + \frac{C_1}{\sigma \sqrt{n}})^2}= 1 - (y + \frac{1}{y}) \frac{C_1}{\sigma \sqrt{n}}e^{-\frac{C_1 y}{\sigma \sqrt{n}}- \frac{C_1^2}{2 \sigma^2 n}}= 1 + O( \frac{y}{\sqrt{n}}).
B=\frac{1}{(m|g|)^2}\sum_{\substack{\ell=1 (\ell\neq j,k)}}^n\sum_{i,h,h'=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{k,j}[\tilde{v}_{i-h'-1}\tilde{w}_{i-h'-1}]_{\ell,k}[\tilde{v}_{i+h+1}\tilde{w}_{i+h+1}]_{j,\ell}\frac{\lambda_\ell^{2m-h-h'-2}\lambda_j^h\lambda_k^{h'}}{(\lambda_\ell^m-\lambda_j^m)(\lambda_\ell^m-\lambda_k^m)},
\left(\frac{k'_{\perp}}{k^{'+}}e^{-i\lambda\varphi'}+\frac{k_{\perp}}{k^+}e^{+i\lambda\varphi}\right)\delta^{\lambda}_{-\lambda'}-m\lambda\left(\frac{1}{k^{'+}}+\frac{1}{k^+}\right)\delta^{\lambda}_{\lambda'}
(E_1=M\oplus K_X^{n-1}\oplus K_X^{n-2}\oplus \cdots \oplus K_X^{2-n}\oplus K_X^{1-n}\oplus M^{-1}, \quad\theta_1=\begin{pmatrix}0 0 0 &1 0 & \ddots \ddots & 1 0 & 0 0\end{pmatrix}).
\frac{\partial z_A}{\partial z_3}= \frac{\partial H}{\partial b_A}=\frac{z_A(z_A-1)(z_A-z_3)}{z_3(z_3-1)}\left[2b_A -\frac{\mu_1}{z_A}-\frac{\mu_2}{z_A-1}-\frac{\mu_3-1}{z_A-z_3}\right]
\mathbb E\left[e^{-s \overline{X}_{e(q)}}\right] =\prod_{k=1}^{m^+}\left(\frac{s+\eta_k}{\eta_k}\right)^{m_k}\prod_{k=1}^{M}\left(\frac{\beta_{k,q}}{s+\beta_{k,q}}\right)=\sum_{k=1}^{M}\frac{C^q_k}{s+\beta_{k,q}}:=\psi_q^+(s),
\frac{n(n+1)}{3}\left(\rho^n +\frac{\rho^{n-1}}{2}\right) \leq n\rho^n + \frac{n(n-1)}{2}\rho^{n-1}\leq \sum_{i=0}^{n}i\rho^i =\frac{n}{3}\sum_{i=0}^{n}\rho^i \leq \frac{n(n+1)}{3},
\sum_{n=0}^\infty \frac{(-1)^nq^{n(n+1)/2}}{(q;q)_n}=\frac{(-q;q)_\infty}{(q;q)_{\infty}}\sum_{n=0}^{\infty}\sum_{|m|\leq n}(-1)^{m}(1-q^{2n+1})q^{(3n^2+n)/2-m^2}&= \frac{(-q;q)_\infty}{(q;q)_{\infty}}(f_{1,5,1}(q,q;q)+q^2f_{1,5,1}(q^4,q^4;q)) &=\frac{(-q;q)_\infty}{(q;q)_{\infty}}J_1J_{1,2}&=J_1.
\sum_{i=1}^{3}\sum_{j=1(j\neq i)}^{3}[(2_{1}\alpha^{i},2_{2}\alpha^{j})+(2_{2}\alpha^{i}, 2_{1}\alpha^{j})+(2_{3}\alpha^{i},2_{4}\alpha^{j})+(2_{4}\alpha^{i},2_{3}\alpha^{j})+(2_{2}\alpha^{i},2_{1}\alpha^{i})+(2_{4}\alpha^{i},2_{3}\alpha^{i})]
\mathcal{L}^{\mathrm{( p r o j )}}= \frac{1}{\sqrt{1 - v^{2}}}\, \left( \begin{array}{c | c}{{1}}&{{\mathbf{v}^{T}}}\\ \hline{{\mathbf{v}}}&{{\sqrt{1 - v^{2}}\,{\bf 1}\, + \, \frac{\mathbf{v}\mathbf{v}^{T}}{v^{2}}\, \left( 1 - \sqrt{1 - v^{2}}\, \right) \,}}\\ \end{array}\right) ~ ,
(0,0)_{p^n+1}= p^n-2, (0,i)_{p^n+1}= (i,0)_{p^n+1}= (i,i)_{p^n+1}= p^n(p^n-1) \textrm{for}i \ne 0, (i,j)_{p^n+1}= (p^n-1)^2 \textrm{for}i \ne j \textrm{and}i,j \ne 0,
Y_{\delta_{t}\delta_{s}}^{a_{s}a_{t}}+1=c_{1}^{(s,t)}\frac{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I})\cdot\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\delta_{s}\delta_{t}}^{a_{s}a_{t}})}{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\delta_{s}}^{a_{s}})\cdot\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\delta_{t}}^{a_{t}})}.
(T_{i}-1)(T_{i}+q)=0 (1 \le i<k), T_{i}T_{i+1}T_{i}=T_{i}T_{i+1}T_{i}(1 \le i \le k-2), T_{i}T_{j}=T_{j}T_{i}(|i-j|>1), X_{i}X_{j}=X_{j}X_{i}(i, j=1, \ldots , k), X_{i+1}T_{i}-T_{i}X_{i}=T_{i}X_{i+1}-X_{i}T_{i}=(1-q)X_{i+1}+\alpha (1\le i<k), X_{i}T_{j}=T_{j}X_{i}(i\not=j, j+1).
\varphi(K_1^+(w) \phi^+(w), K_1^-(z)) = \varphi(\phi^+(w),K_1^-(z)) \varphi(K_1^+(w),K_1^-(z)) = g(\frac{w}{z}), & \varphi(K_1^+(w) E^+(w), F^-(z)) = \varphi(E^+(w),F^-(z))\varphi(K_1^+(w),1) = h(\frac{w}{z}).
D_{P_4}= \frac{1}{4}\sum_{\left\{u,v\right\}\in E(G)}\left(d(u,-v)+d(v,-u)\right)^2+o(n^4) &\geq \frac{1}{4}\cdot e(G)\left[\frac{1}{e(G)}\sum_{\left\{u,v\right\}\in E(G)}\left(d(u,-v)+d(v,-u)\right)\right]^2 +o(n^4) &= \frac{1}{4}\cdot \frac{n^2}{4}\cdot \left(\frac{n}{2}\right)^2+o(n^4) = \frac{n^4}{64}+o(n^4) \,.
\begin{array}{ll}\partial_y (\frac{u_0^1-u_0^2}{\partial_y(u_0^1 + u_0^2)}) = (u_0^1-u_0^2)[\frac{1}{(u_0^1 + u_0^2)_y}]_y + \partial_y(u_0^1-u_0^2)\frac{1}{(u_0^1 + u_0^2)_y}\equiv 0.\end{array}
{\cal L}_{\Lambda}={\cal L}_{\Lambda}^{S}+{\cal L}_{\Lambda}^{F}= \sum_{\kappa \in{( I , I I , T_{I}, T_{I I})}}\left\{N_{\kappa}^{F}{\cal L}_{\Lambda}^{F}( \kappa ) + N_{\kappa}^{S}{\cal L}_{\Lambda}^{S}( \kappa ) \right\}.
0 = \int_\Omega \sum_{i=1}^{m}\sum_{\ell=1}^Q \left( \langle D_iu_\ell , \nabla \hat{f}(u_\ell) \rangle D_i \varphi + \langle D_i u_\ell, D \nabla \hat{f}\cdot D_i u_\ell \rangle \varphi \right) &= \int_\Omega \sum_{i=1}^{m}\left( D_i (f \circ u) D_i \varphi + \sum_{\ell=1}^Q \nabla^2f(u_\ell)(D_iu_\ell, D_iu_\ell) \varphi \right).
\sum_{M^g \in M^G}\frac{|x^G \cap M^g|}{|x^G|}\frac{|y^G \cap M^g|}{|y^G|}= |G:M| \frac{|x^G \cap M|}{|x^G|}\frac{|y^G \cap M|}{|y^G|}&= \operatorname{fpr}(x,M^G) \operatorname{fix}(y,M^G).
\mathbb{E}\left[\int (\mu(s,x) - \mu(s,y))^2d[B]s \right] = \mathbb{E}\left[(M(x)-M(y))^2 \right] \leq \mathbb{E}[(M(x) - M^n(x))^2]+ \mathbb{E}[(M^n(x)- M^n(y))^2] + \mathbb{E}[(M^n(y) - M(y))^2] .
\sum_{i \in M_C}\sum_{j=1}^{r_i - 1}a_{i r_i}x_{ij}+ \sum_{i \in M_C}\sum_{j=r_i}^{n_i}\max \left\{a_{ij}, b - \sum_{k \in M_C - i}a_{k r_k}\right\}x_{ij}\leq b
\Delta^{( 2 )}( x , x^{\prime}) = \frac{1}{2 \sqrt{\pi}}\int_{\Gamma_{l}+ \Gamma_{r}}f ( x , x^{\prime}, s ) e^{- \alpha}\alpha^{1 / 2}\left[ 2 + ( 4 / 3 ) \alpha \right] d s \quad , \, [ 0 . 3 c m ]
F(i,j,m,n,{\bf q}, r, c,k,{\bf \epsilon}) := \Pr\left( \sum_{\ell=1}^{j-1}2\xi''_{i,\ell}(q_\ell^2, c_j, \epsilon) + 2\eta''_{i,j,i}+ \sum_{\ell=j+1}^{n}\xi'_{i,\ell}(q_\ell) = r_i - k\right) \ \ \ \times \Pr\left( \sum_{\ell=1}^i 2\xi_{\ell,j}(q_j^2) + \sum_{\ell=i+1}^{m}\xi_{\ell,j}(q_j) = c_j - k\right), k \in \{0,1\}.
\begin{aligned}\bar{\nabla}_{aB+U+xZ}(bB+V+yZ)={}\Bigl(\frac{1}{2}\langle U,V\rangle+xy\Bigr)B-\frac{1}{2}\bigl(bU+yJU+xJV\bigr) &{}+\Bigl(\frac{1}{2}\langle JU,V\rangle-bx\Bigr)Z.\end{aligned}
E_N^{\mathrm{HC}}\le\; 4\pi \mathfrak{a}(N-1)+e_\Lambda \mathfrak{a}^2 &-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+8\pi\mathfrak{a}-\sqrt{|p|^4+ 16\pi \mathfrak{a}p^2}-\frac{(8\pi \mathfrak{a})^2}{2p^2}\right]+ CN^{-\varepsilon},
\|f\|_1^2 + \frac{1}{\gamma_i}\|f\|_2^2 =\|f\|_{1+ k_{\gamma_i}}^2 =\langle f,1 \rangle_1^2 + \| f - \langle f,1 \rangle_1\|_{k_{\gamma_i}}^2= \frac{1}{\gamma_i}\|f\|_k^2.
a(v_k,v_k) \preceq \left(1+ \max_{\mathcal{F}_{k l}\subset \partial\Omega_k}\frac{1}{\lambda_{\mathcal{F}_{k l}}^{m_{k l}+1}}\right) \sum_{\partial\Omega_l\cap \partial \Omega_k=\mathcal{F}_{k l}}a_{|\Omega_l}(u,u).
^{\left( \alpha \right)}\bar{T}\!=\! \frac{\left( 1\!-\!4\alpha \right)|\delta|^{2\alpha-2}}{2w^{4\alpha+1}}\left[ \left( 2\kappa v^{3}\!+\!\delta \,\acute{}\,v^{2}\!+\!2\delta^{2}\kappa v+\delta^{2}\delta \,\acute{}\, \right) \bar{e}\!+\!2\delta v^{2}\bar{n}\!+\!2\delta^{2}v\bar{z}\right] .
\begin{cases}x_j^{k+1}=\mathrm{prox}_{\frac{\alpha_j}{\beta}f_j}(x_j^k-\alpha_jA_j^\top(\sum_{i=1}^j A_ix_i^{k+1}+\sum_{i=j+1}^sA_i(2x_i^k-x_i^{k-1})-b)- \frac{\alpha_j}{\beta}A_j^\top y^k), j\in\mathbb{N}_s, y^{k+1}=y^k+\beta(\sum_{i=1}^sA_ix_i^{k+1}-b). \end{cases}
\psi(v^{(\gamma)})=vp_{\gamma}, \quad\psi(e_i^{(\gamma)})=e_ip_{W(e_1)^{-1}\gamma},\quad\psi((e_i^{(\gamma)})^*)=e_i^*p_{W(e_i^{s(e)})W(e_1^{s(e)})^{-1}\gamma}
\mathcal{E}\left( x\right) =\int \mathrm{d}^3k\sqrt{\frac{2\omega}{\left(2\pi \right)^3}}\left\{\begin{array}{c}\left[ c^1e_1+c^3\left( e_3+e_4\right) \right] \mathrm{e}^{-ikx}+ \left[ c^{*2}e_1+c^{*4}\left( e_3+e_4\right) \right] \mathrm{e}^{ikx}\end{array}\right\},\quad \omega \equiv \sqrt{\overrightarrow{k}^2},
t_\lambda P(\sigma_\lambda^{-1}(Z(\eta)) ) = \sum_{(\rho,\xi) \in \Lambda^{min}(\lambda, \eta)}P(\sigma_\lambda(Z(\rho))) t_\lambda = \sum_{(\rho, \xi)\in \Lambda^{\operatorname{min}}(\lambda, \eta)}t_\lambda t_\rho t_\rho^* .
\psi_{_{WKB}}= C{\left( \pi \chi_{0}^{3}a_{0}^{2}(\eta - \frac{1}{2}sin{2\eta}) \right)}^{-1/2}exp \left( \frac{i}{\hbar}\pi \chi_{0}^{3}a_{0}^{2}(\eta - \frac{1}{2}sin{2\eta}) \right) ~.
T_{m}(x_{1},\ldots ,x_{m})= (x_{m}^{1}+x_{m}^{2})T_{m-1}(x_{1},\ldots,x_{m-1}) &+(x_{m}^{1}-x_{m}^{2})T_{m-1}(B^{2^{m-2}}(x_{1}),B^{2^{m-2}}(x_{2}),B^{2^{m-3}}(x_{3})\ldots ,B^{2}(x_{m-1})),
E_\infty^{(K)}(t,[s]_K) = \sum_{i=K+1}^\infty \int_{s_{K+1}=0}^\infty \! ... \! \int_{s_{i}=0}^\infty \varphi_i([s]_i) n_\infty^{(i)}(t, [s]_i) \,d[s]_{K+1, i},
C \Big [ \begin{array}{c}{\alpha + \beta}\\{\gamma}\\ \end{array}\Big ] = ( - 1 )^{\gamma ( \psi^{\mu}, \tilde{\psi}^{\mu})}\exp \Big \{{\frac{i \pi}{2}}n ( \alpha \beta \gamma ) \Big \}C \Big [ \begin{array}{c}{\alpha}\\{\gamma}\\ \end{array}\Big ] C \Big [ \begin{array}{c}{\beta}\\{\gamma}\\ \end{array}\Big ] .
\chi(\beta^{j}+\ldots + \beta^{i+j}) =\frac{(\beta^{j}+\ldots + \beta^{i+j})\cdot(\beta^{j}+\ldots + \beta^{i+j}-K_S)}{2}+ \chi(\O_S) &=\sum_{j\leq k< l\leq i+j}\beta^k\beta^l + \chi(\O_S)\,.
\widehat{K_1}(y) =\int_{|x|<\frac{1}{|y|}}e^{2\pi ix\cdot y}K_1(x)\,dx+\int_{\frac{1}{|y|}\leq|x|\leq\frac{2}{\beta}}e^{2\pi ix\cdot y}K_1(x)\,dx &=\int_{|x|<\frac{1}{|y|}}(e^{2\pi ix\cdot y}-1)K_1(x)\,dx+\int_{\frac{1}{|y|}\leq|x|\leq\frac{2}{\beta}}e^{2\pi ix\cdot y}K_1(x)\,dx.
(\|\tilde{u}\|_{\nu}^{\sharp})^2 =\sum_{j+k\leq\nu}\int_{-2T}^{2T}\|(-\partial_t^2)^{j}(-\triangle)^{k}\tilde{u}(t,\cdot)\|^2dt &\simeq\sum_{a,b}\sum_{j+k=\nu}|c_{ab}|^2\lambda_a^{2j}\mu_b^{2k}&\simeq \sum_{a,b}\sum_{j+k=\nu}|c_{ab}|^2(a^jb^k)^4.
\theta_{v}^{n,k}(x)=\frac{1}{n}\left(1-F(a;x)\right)\left(1-F_{n-1,k}(a;x)\right)+\left(1-\frac{1}{n}\right)\left(1-F(a;x)\right)^2\left(1-F_{n-2,k}(a;x)\right).
\frac{1}{n}\int_{-m^*}^{m^*}\frac{1}{|\phi_X(u)|^{2(K-1)}}\d u\leq \frac{2u_n^{(a)}}{n}|\phi_X(u_n^{(a)})|^{-2(K-1)}\leq 2 C_2 n^{\frac{1}{2\beta K}-1}a n^{\frac{2(K-1)}{2K}}= 2a C_2 n^{-\frac{2\beta -1}{2\beta K}}\leq 2a C_2 n^{-\frac{2\beta -1}{2b K}}.
\frac{{\rm d}}{{\rm d}t}\int\frac{\big|\widetilde{U}\big|^2}{2h}+ \int\frac{W^{\rm T}Q(w^*)W}{2h}+\int\widetilde{W}\cdot\mathrm{L}(w^*) = \int\Big[\phi\cdot \big(b-b^*\big) +\psi\cdot \big(d-d^*\big) + \varphi\cdot \big(v-v^*\big) \Big]
(\nabla^2\Psi(P_j))^2 = 2\begin{pmatrix}M_1^2 0 0 -M_1^2 0 0 0 M_2^2 0 0 -M_2^2 0 0 0 j^2 0 0 -j^2 -M_1^2 0 0 M_1^2 0 0 0 -M_2^2 0 0 M_2^2 0 0 0 -j^2 0 0 j^2\end{pmatrix}.
B_e=\begin{pmatrix}1 0 \cdots \cdots 0 0 -1 1 \cdots \cdots 0 0 0 -1 \ddots 0 0 \vdots \vdots \ddots \ddots \vdots \vdots 0 0 \ddots 1 0 0 0 \cdots \cdots -1 1\end{pmatrix},
\nabla F_{i_0}= (-2n_1 q_1^2Y_1, -2n_2q_2^2Y_2, \cdots ,-2n_{i_0-1}\,q_{i_0-1}^2Y_{i_0-1}, & 2p_{i_0}-2(n_{i_0}+1)\,q_{i_0}^2Y_{i_0}, -2n_{i_0+1}\,q_{i_0+1}^2Y_{i_0+1}, \cdots, -2n_mq_m^2Y_m ).
K_{1}(w,\lambda(t))-\frac{\lambda(t)^{2}}{4 w}= -\int_{0}^{\infty}\frac{R dR}{(1+R^{2})^{3}}\frac{4 R^{2}\lambda(t)^{2}}{2w\left(\sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}+1+w^{2}-R^{2}\lambda(t)^{2}\right)}
\Pi_{2}'\left(\widetilde{\psi}(d_{1}\overline{N}(^{\ast}b)(d_{2}n))\right) =-d_{1}\hat{\psi}_{i}\gamma' \xi_{be_{k}}(d_{2}n) &=-d_{1}\gamma' \hat{\psi}_{o}\xi_{be_{k}}(d_{2}n) &=-d_{1}\gamma' \xi_{be_{k}}(d_{2}\widetilde{\psi}(n)) &=\Pi_{2}' \left(d_{1}(^{\ast}b)d_{2}\cdot \psi(n)\right)
\int_{\mathbb{R}^{N}}\left( 1+ 2w^2 \right) \nabla w\cdot \nabla h\, dx+ \int_{\mathbb{R}^{N}}2w |\nabla w|^2 \, h \, dx + \int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) wh\,dx=\int_{\mathbb{R}^{N}}K(|x|) g(w) h\,dx ,
\sum_{k=1}^{H-1}(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj}){+}c_i = x_i -\sum_{k=1}^{H-1}(a_{ik}b_{k1}- a^0_{ik}b^0_{k1}) +\sum_{k=1}^{H-1}(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj}) &=x_i +\sum_{k=1}^{H-1}\{(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj})- (a_{ik}b_{k1}- a^0_{ik}b^0_{k1})\}
P(dx_1d[x_2,x_3]) + P([dx_2,x_3]dx_1) + P([x_2,dx_3]dx_1) = P(d[x_2,x_3]dx_1) + P([dx_1,d[x_2,x_3]]) + P([dx_2,x_3]dx_1) + P([x_2,dx_3]dx_1) = P((d[x_2,x_3] + [dx_2,x_3] + [x_2,dx_3])dx_1) + P([dx_1,d[x_2,x_3]]) = P((d[x_2,x_3] + d[x_2,x_3])dx_1) + P([dx_1,d[x_2,x_3]]) = P([dx_1,d[x_2,x_3]]).
\begin{aligned}m_1 =(q^{1/2}+q^{-1/2})\left(\frac{(b-b^{-1})(c-c^{-1})-(a-a^{-1})\Upsilon}{(q-q^{-1})^2}\right), m_2 =(q^{1/2}+q^{-1/2})\left(\frac{(c-c^{-1})(a-a^{-1})-(b-b^{-1})\Upsilon}{(q-q^{-1})^2}\right), m_3 =(q^{1/2}+q^{-1/2})\left(\frac{(a-a^{-1})(b-b^{-1})-(c-c^{-1})\Upsilon}{(q-q^{-1})^2}\right). \end{aligned}
D^{s,p}(\mathbb{R}^N)=\left\{u\in L^{{N\,p/(N-s\,p)}}(\mathbb{R}^N):\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy<\infty\right\}.
W^{\prime}( r )^{2}= \frac{1}{V^{2}( r )}\left( \Delta_{L}E^{2}- V \left( m^{2}+ \frac{4 j^{2}}{r^{2}}\right) - \frac{4 \Delta}{r^{4}}a j_{R}E - \frac{4 g^{2}a^{2}}{r^{6}}j_{R}^{2}\right) ,
e(\widetilde{S_i},\widetilde{T_i}) \geq |\widetilde{S_i}|\cdot (\frac{n}{2(2l+1)}-1)-(4k-2)kn &= |\widetilde{S_i}|\cdot \frac{2n}{4(2l+1)+1}+|\widetilde{S_i}|\cdot (\frac{n}{32l^2+36l+10}-1)-(4k-2)kn &>|\widetilde{S_i}|\cdot \frac{2n}{4(2l+1)+1}
S_{\mathrm{e f f}}[ \chi ] = \int \! d^{2}x \Bigl ({\frac{1}{2}}\partial_{\alpha}\chi \partial_{\alpha}\chi -{\frac{\delta v}{2}}\chi^{2}+{\frac{\pi V}{N}}\chi^{4}+{\cal O}\bigl ( \chi^{6}/ N^{2}\bigr ) \Bigr ) ,
j(\tau)=1728\frac{g_{2}(\tau)^{3}}{\Delta(\tau)}=1728\frac{g_{2}(\tau)^{3}}{g_{2}(\tau)^{3}-27g_{3}(\tau)^{2}}=1728\frac{g_{2}(\tau)^{3}}{(2\pi)^{12}\eta^{24}(\tau)}
( \begin{array}{c c}{{H \, ,}}&{{\Theta}}\\ \end{array}) \rightarrow ( \begin{array}{c c}{{H \, A^{- 1}\, ,}}&{{- H \, A^{- 1}C^{\prime}B^{- 1}+ \Theta \, B^{- 1}}}\\ \end{array}) \quad .
\phi_{K_i}\cdots \phi_{K_1}= \phi_{K_i}[p^{k-1}]\phi_{L_s}\cdots\phi_{L_1}& = [p^{k-1}]\phi_{L_s}^2\phi_{L_{s-1}}\cdots \phi_{L_1}& = [p^{k}]\phi_{L_{s-1}}\cdots \phi_{L_1}.
| \Pr (f_{0}\in \mathcal{C}(\hat{f},\hat{R}_{\alpha}) ) - (1-\alpha) | &\leq \tilde{\phi}_{\Pi_{\beta}}\left( c_{1}\sqrt{p\log n}\right) + \delta_{2}+ \delta_{3}+ c_{1}(n^{-2\delta}+ \delta_{1}p\log n + \zeta_{n}+ \gamma_{n}),
H(z'_2,..,z'_{\cal N}, P'_2,..,P_{\cal N})\!=\!\!-\!\!\int_{\{z_0\}}^{\{z'_B\}}\!\!\sum_B\frac{\partial \beta_B}{\partial \mu}(z'_2,..,z'_{\cal N},z''_A,..,\mu) dz''_B +f(z'_2,..,z'_{\cal N}).
{}_2 F_1\bigg(\frac{d}{2},\frac{d+1}{2};\frac{d}{2}+1;\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) = \sin^{-(d-1)}\bigg(\frac{\zeta}{2}\bigg) \,{}_2 F_1\bigg(1,\frac{1}{2};\frac{d}{2}+1;\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg).
{\cal L}_{\mathrm{k i n}}^{( 0 )}= e^{- \phi}k^{-{\frac{1}{2}}}\sqrt{\dot{X}^{\mu}\dot{X}^{\nu}\bigl ( g_{\mu \nu}- k^{2}A_{\mu}^{( 2 )}A_{\nu}^{( 2 )}\bigr ) - k^{2}{\dot{S}}^{2}- 2 k^{2}\dot{S}{\dot{X}}^{\mu}A_{\mu}^{( 2 )}}.
f = x_1^{a_1}, f = x_1^{a_1}+ x_1 x_2^{a_2}+ \dots + x_{N-1}x_N^{a_N}, f = x_1^{a_1}x_2 + x_2^{a_2}x_3 + \dots + x_{N-1}^{a_{N-1}}x_N + x_N^{a_N}x_1 .
\begin{matrix}{{e^{\tilde{h}^{0}}= e^{- h^{0}}\bigl [ \sin^{2}t \bigl ( 1 + ( h^{1})^{2}+ ( h^{2})^{2}\bigr ) \bigr ] + e^{h^{0}}\cos^{2}t + 2 ( \cos t \sin t ) h^{2}}}\\{{\tilde{h}^{1}= h^{1}}}\\{{\tilde{h}^{2}= \cos t \sin t \bigl [ e^{- h^{0}}\bigl ( 1 + ( h^{1})^{2}+ ( h^{2})^{2}\bigr ) - e^{h^{0}}\bigr ] + ( \cos^{2}t - \sin^{2}t ) h^{2}}}\\ \end{matrix}
\lim_{n\to\infty}\,\sup_{0<\tau<\frac{t_n}{16}}\frac{1}{\tau}\int_{|t_n-t|<\tau}\,\int_{|x|<Ct}\,\left(\partial_tu+\frac{x}{t}\cdot\nabla u+\left(\frac{d}{2}-1\right)\frac{u}{t}\right)^2\,dx\,dt = 0.
(\sum_g g^{-1}(\alpha)g)(v_i,\sigma v_i) =\, (v_i,\sigma v_i)\left(\begin{array}{cc}\alpha_1-\alpha_2-\alpha_5+\alpha_6 \alpha_2-\alpha_3+\alpha_4-\alpha_6 -\alpha_2+\alpha_3+\alpha_4-\alpha_5 \alpha_1-\alpha_3+\alpha_5-\alpha_6\end{array}\right) = \,(v_i,\sigma v_i){\bf{A}}, ,
(g_{1-\alpha}*W)(t_{*}) \leq g_{1-\alpha}((t_{2}-t_{1})/4)\int_{0}^{t_{2}-t_{0}}\int_{\rho B_{1}}w^{2}dxds = \frac{4^{\alpha}}{\Gamma(1-\alpha)(\sigma\eta)^{\alpha}(\rho-\rho')^{\alpha}}\int_{0}^{t_{2}-t_{0}}\int_{\rho B_{1}}w^{2}dxds.
s_{(2k_1,2k_1)}s_{(2k_2,2k_2)}\cdots s_{(2k_r,2k_r)}=s_{(2k_1,2k_1,\ldots,2k_r,2k_r)}+\sum_{\mu>(2k_1,2k_1,\ldots,2k_r,2k_r)}a_\mu s_\mu
M =\begin{pmatrix}\rho_1 1 0 \cdots 0 - \rho_2 0 1 \cdots 0 \vdots \vdots \vdots \cdots \vdots (-1)^{N-2}~ \rho_{N-1}0 0 \cdots 1 (-1)^{N-1}~ \rho_N 0 0 \cdots 0\end{pmatrix}.
R_{T}(\tau,\mathbf{v):=}T^{-1/2}\sum_{t=k+1}^{T}\big \{1[\mathbf{y}_{t,k}\leq\mathbf{q}_{t,k}(\tau)+T^{-1/2}\mathbf{v}_{t,k}]-E\big[F_{\mathbf{y}|\mathbf{x}}^{(k)}(\mathbf{q}_{t,k}(\tau)|\mathbf{x}_{t,k})\big]\big\}& -\left( \mathbb{V}_{T,k}(\tau)+\nabla G^{(k)}(\tau)^{\top}\mathbf{v}\right) .
I\left(\frac{u}{v}\right)=\frac{L^2(1,\chi)}{\sqrt{uv}}\int_0^\infty k\left(\frac{T}{x}\right)h\left(\frac{x}{u}\right)h\left(\frac{x}{v}\right)\frac{dx}{x}+O(T^{-\frac{1}{9}})
c_{j}^-(\lambda)=\bigg(\prod_{\ell=1}^{j-1}\frac{\lambda_j-\lambda_\ell-j+\ell}{\lambda_j-\lambda_\ell-j+\ell-1}\bigg)^{\frac{1}{2}}F_j(\lambda_1,\dots,\lambda_{j-1},\lambda_j-1,\lambda_{j+1}\dots,\lambda_n),
k-\Lambda=-\frac{k\sigma}{1-\sigma}+\left( \frac{k^2(k+1)(k-1)^3\sigma +\theta^tk^3(k^3-3k^2+O(k))}{(k-1)^3(k^2+k-k\theta^{t-3})(1-\sigma)}\right) \phi^u.
\alpha: (x_1, ..., x_7) \longmapsto (x_1, x_2, x_3, -x_4, -x_5, -x_6, -x_7) \beta: (x_1, ..., x_7) \longmapsto (x_1, -x_2, -x_3, x_4, x_5, \frac{1}{2}-x_6, -x_7) \gamma: (x_1, ..., x_7) \longmapsto (-x_1, x_2, -x_3, x_4, \frac{1}{2}-x_5, x_6, \frac{1}{2}-x_7)
(\nabla_{e_i}\eta)(e_i) =e_i[\eta(e_i)]-\eta(\nabla_{e_i}e_i) =g(\nabla_{e_i}\xi,e_i)+g(\xi,\nabla_{e_i}e_i)-g(\xi,\nabla_{e_i}e_i) =g(\nabla_{e_i}\xi,e_i),
-\cos\psi\cdot\left(\frac{F[-q,p](i\mu_{0})w_{1}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{2}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
G(s)= p^{5s}\pi^{-8s}\Gamma\left(\frac{s}{2}\right)^3 \Gamma\left(\frac{s+1}{2}\right)^3 \Gamma\left(\frac{s+k-1}{2}\right)^4 &\times \Gamma\left(\frac{s+k}{2}\right)^4 \Gamma\left(\frac{s+2k-2}{2}\right) \Gamma\left(\frac{s+2k-1}{2}\right).
\gamma (s,0)=\tau_{V\alpha}(\Phi_0^{\Gamma_{\tilde{L}}}(\tilde{R}_h^{e-}(g(s),g_1)))=\tau_{V\alpha}(\tilde{R}_h^{e-}(g(s),g_1))=g(s) & \gamma (s,h)=\tau_{V\alpha}(\Phi_h^{\Gamma_{\tilde{L}}}(\tilde{R}_h^{e-}(g(s),g_1)))= \tau_{V\alpha}(\tilde{R}_h^{e+}(g(s),g_1))=g_1,
\frac{6}{A_{3,r}^2}\cdot{}_2F_{1}\left(\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{27}{A_{3,r}^{12}}\right)=3\sqrt[3]{2k_{4r}}\cdot{}_2F_1\left(\frac{1}{3},\frac{1}{6};\frac{7}{6};k_{4r}^2\right)=\Pi(r)
h_{}(\lambda,\mu) =-\intop f_{D}(x)\log f_{D}(x)dx =-\intop f_{D}(x)\log\left[\mu e^{-\mu x}\frac{\rho}{1-\rho}\left(e^{-\left(\lambda-\mu\right)x}-1\right)\right]dx =-\log\mu+\mu E\left[D\right]\log e-\log\left(\frac{\rho}{1-\rho}\right) \qquad-\intop f_{D}(x)\log\left(e^{-\left(\lambda-\mu\right)x}-1\right)dx.
D_{\alpha}^+(x,y) := \frac{1}{4\pi^2}\chi_{(-\alpha,\alpha)}(x)\chi_{(-\alpha,\alpha)}(y)\int\limits_{\alpha+1}^\infty \frac{1}{(z-x)(z-y)}\, dz, D_{\alpha}^-(x,y) := \frac{1}{4\pi^2}\chi_{(-\alpha,\alpha)}(x)\chi_{(-\alpha,\alpha)}(y)\int\limits_{-\infty}^{-\alpha-1}\frac{1}{(z-x)(z-y)}\, dz.
\xi_1^2+\frac{\xi_2^2}{a(t)^2}+ \frac{\xi_3^2 \csc^2(\eta )}{a(t)^2}+\frac{\xi_4^2 \sec^2(\eta )}{a(t)^2}= u_1^2 + \frac{1}{a(t)^2}( u_2^2 + u_3^2+u_4^2 ),
\sigma_{AB}(x,\xi) =u_{\xi}^{-1}(x)(A(Bu_{\xi}))(x) &=u_{\xi}^{-1}(x)\int_{\Omega}K_{A}(x,y)\Big[\int_{\Omega}K_{B}(y,z)u_{\xi}(z)dz\Big]dy &=u_{\xi}^{-1}(x)\int_{\Omega}K_{A}(x,y)u_{\xi}(y)\sigma_{B}(y,\xi)dy.
\int_{u}^{\infty}( x^{2}- u^{2})^{\nu - 1}e^{- \mu x}\, d x = \frac{1}{\sqrt{\pi}}( \frac{2 u}{\mu})^{\nu - \frac{1}{2}}\Gamma ( \nu ) K_{\nu - \frac{1}{2}}( u \mu ) ,
\#\mathcal{Z}_{M,0}= \sum_{k=1}^n \tfrac{(\#S-1)!}{(\#S-1-k)!}p_k(M) \leq C\sum_{k=1}^n \tfrac{(\#S-1)!}{(\#S-1-k)!}M^{k-1}\left(\tfrac{e}{k}\right)^{2k}\leq C M^{n-1}
\langle E_{eff}(x_0,x_1) \rangle \;=\; \frac{1}{6}\left[\frac{\partial^2}{\partial x_0^2}\frac{\partial^2}{\partial y_1^2}-\frac{\partial^2}{\partial x_0 \partial x_1}\frac{\partial^2}{\partial y_0 \partial y_1}\right] \Delta_\varphi(x_0,x_1;y_0, y_1)|_{y_\mu \to x_\mu}\;,
S~=~\oint~d\tau \int~d^2r_{\perp}~ \left({1 \over 2}{\Omega}^{-}~{\vec \nabla}^2{\dot{\Omega}}^{+}-{1 \over2}{\Omega}^{+}~{\vec \nabla}^2{\dot{\Omega}}^{-}+{\dot k}^{+}{\Omega}^{-}-{\dot k}^{-}{\Omega}^{+}\right) .
\psi = \left( \begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\ \end{array}\right) \propto \left( \begin{array}{c}{\exp - 2 \pi \beta^{- 1}\int_{- \infty}^{x}d y \left( \dot{\phi}+ \frac{\beta}{4 \pi}\phi^{\prime}\right)}\\{\exp - 2 \pi \beta^{- 1}\int_{- \infty}^{x}d y \left( \dot{\phi}- \frac{\beta}{4 \pi}\phi^{\prime}\right)}\\ \end{array}\right) ,
\widehat{\mathbb{I}}^q(\ell') = \widehat{\mathbb{I}}^q(\ell'')\widehat{\mathbb{I}}^q(\ell) &= q^{-\sum_{i<j}\varepsilon_{ij}a_ia_j}\prod_iX_i^{a_i}\cdot\widehat{\mathbb{I}}^q(\ell)
\int_A\!R_u(y)\,\Pi(dy) = \Pi_u(A) = \int_0^1\!P_{u-t}(x,A)\,\Pi_t(dx) = \int_0^1\!R_t(x)\,P_{u-t}(x,A)\,\Pi(dx) &\geq a(t) \int_0^1\!P_{u-t}(x,A)\,\Pi(dx) = a(t)\,\Pi(A)=\int_A\!a(t)\,\Pi(dy).
\sum_{i=0}^6 |g_i|{\bf 1}_{|\xi|<1}+ \sum_{i=0}^{10}|g_i'|{\bf 1}_{|\xi|<1}\leq{\sf c}_0 (T-t)^{{\sf c}_1}\lambda^{-\frac{n+2}{2}}\sigma (1+|y|^2)^{-(1+{\sf c}_2)}{\bf 1}_{|\xi|<1}.
C_{4}= 4 L^{4}\frac{q}{3 !}\left( \psi \omega^{2}- \frac{1}{3}A \wedge \eta \wedge \omega \right) - \frac{r^{4}}{L^{4}}d^{4}x + \frac{q}{9}L^{3}( \star_{5}d A ) \wedge \eta \ .
|J(y,t)-Q_0|\leq n_0\epsilon \mbox{for}|y|\leq \min I_1(t), &|J(y,t)-Q_k|\leq n_0\epsilon \mbox{for}|y|\in [\max I_k(t), \min I_{k+1}(t)],\; k=1,..., n_0-1, &0<J(y,t)<n_0\epsilon \mbox{for}|y|\geq\max I_{n_0}(t),
X(i,j) = \begin{cases}\; 1 \; j - i = 1 \mod m \; 0 \end{cases}, Y(i,j) = \begin{cases}\; \delta(m\cdot 1 - n \cdot 1) \; i=j=1 \; (-1)^{m+n}\; i=n,j=1 \; 1 \; j - i = 1 \; 0 \end{cases},
\frac{d^{n-1}K}{dv^{n-1}}(v,v)=-\int_0^v\frac{\partial^{n-1}\nu}{\partial u^{n-1}}(v,v')dv'-\sum_{l=0}^{n-2}\binom{n-1}{l+1}\frac{\partial^{n-2}\nu}{\partial u^{n-2-l}\partial v^l}(v,v).
\{x_5, x_1, [x_2, x_3, x_4]_C\}= \{\{x_5, x_1, x_2\}, x_3, x_4\}- \{\{x_5, x_1, x_3\}, x_2, x_4\}+ \{x_2, x_3, \{x_5, x_1, x_4\}\}^{*}, \{x_1, x_2, \{x_5, x_3, x_4\}\}^{*}\;= \{\{x_1, x_2, x_5\}^{*}, x_3, x_4\}+ \{x_5, [x_1, x_2, x_3]_C, x_4\}+ \{x_5, x_3, [x_1, x_2, x_4]_C\},
\|x-P_A(x)\|_\omega \ =\ \left\|x-P_A(x)-P_B(x)+\sum_{n\in B}e_n^*(x)e_n\right\|_\omega \ \leqslant\ \left\|P_{(A\cup B)^c}(x)+t1_{B}\right\|_\omega \ \leqslant\ \left\|P_{(A\cup B)^c}(x) + t1_F\right\|_\omega \ \leqslant\ \left\|P_{(A\cup B)^c}(x) + P_F(x)\right\|_\omega\ =\ \left\|x-S_k(x)\right\|_\omega,
G^{(I)}_{MN}dx^Mdx^N={{r^2}\over{\sqrt{Q_eQ_p}}}(-dt^2+dx^2_1)+{{\sqrt{Q_eQ_p}}\over{r^2}}dr^2+\sqrt{{Q_e}\over{Q_p}}\sum_{i=2}^5 dx^2_i+\sqrt{Q_eQ_p}d\Omega^2_3.
+ \frac{2}{A B - 1}( A X_{n}^{0}\tilde{X}_{n}^{1}- B X_{n}^{1}\tilde{X}_{n}^{0}) e^{- n l_{0}}e^{- n l_{1}}] e x p \sum_{n \geq 1}[ X_{- n}^{0}\tilde{X}_{- n}^{1}- X_{- n}^{1}\tilde{X}_{- n}^{0}] | 0 >
H_n=\int d{\bf x}\int dk dk' e^{ikx}e^{ik'x}\phi (k)\phi (k')(k_0'^2-k_0 k_0')\sum_{s+t=n-1}(i^2)^{s+t+1}(k^2)^s(k'^2)^t-\int d{\bf x}{\cal L}_n
\overline{X}_0 := X_0, &\overline{X}_3 := \left\lbrace (\eta, w) \in [H^3(0,L)\cap H^1_0(0,L)]^2: \eta_x(0)=w_x(L)=0\right\rbrace, &\overline{X}_{3\theta}:= [\overline{X}_0,\overline{X}_3]_{[\theta]},