harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left. \begin{array}{l}{{( \gamma ) ={}^{g_{4}}[ S_{4}] \hspace{2 m m}{}^{g_{3}}[ S_{3}] \hspace{2 m m}{}^{g_{2}}[ S_{2}] \hspace{2 m m}{}^{g_{1}}[ S_{1}] \hspace{2 m m}[ \xi ]}}\\{{\hspace{6 m m}= [ \xi ] \hspace{2 m m}{}^{h_{4}}[ S_{4}] \hspace{2 m m}{}^{h_{3}}[ S_{3}] \hspace{2 m m}{}^{h_{2}}[ S_{2}] \hspace{2 m m}{}^{h_{1}}[ S_{1}] ,}}\\ \end{array}\right.
\Omega(\underline s)\circ h(\underline s) =\Omega'(\underline s)\circ h_{w_M}(\underline s)\circ N(\underline s, \left( X(1)\otimes I(\underline\varepsilon',\lambda') \right)) &=\Omega'(\underline s)\circ N(\underline s,X_M)\circ h(\underline s)
\left\|\sum_{j\in \mathbb{S}}F_j^*A_jh\right\|^2=\left\langle\sum_{j\in \mathbb{S}}F_j^*A_jh, \sum_{k\in \mathbb{S}}F_k^*A_kh\right\rangle= \sum_{j\in \mathbb{S}}\left\langle A_jh, F_j\left(\sum_{k\in \mathbb{S}}F_k^*A_kh\right) \right\rangle=\sum_{j\in \mathbb{S}}\|A_jh\|^2.
\pi(XK_{\lambda})_{j}^{i}c_{k}^{j}\pi(K_{\lambda^{\prime}}Y)_{\ell}^{k}=\pi(X)_{j}^{i}\pi(K_{\lambda})_{j}^{j}c_{k}^{j}\pi(K_{\lambda^{\prime}})_{k}^{k}\pi(Y)_{\ell}^{k}.
\int_{\mathcal{O}_K^{\times}}{\chi(ac (x^m))|dx|}=\int_{\mathcal{O}_K^{\times}}{\chi^m(ac (x))|dx|}&=\int_{\mathcal{O}_K^{\times}}{|dx|}&=\int_{\mathcal{O}_K}{|dx|}-\int_{\pi\mathcal{O}_K}{|dx|}&=\int_{\mathcal{O}_K}{|dx|}-q^{-1}\int_{\mathcal{O}_K}{|dx|}&=1-q^{-1}
\lim_{\varepsilon\to{0}^{+}}\frac{f(\mathbf{x}_{0}+\varepsilon \mathbf{x})}{\varepsilon^{\alpha}L(\varepsilon)}=g(\mathbf{x}) \ \ \ \mbox{weakly$^{\ast}$ in}\mathcal{K}_{M,r}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'_{r}(\mathbb{R}^{n})).
\delta^{+}( k ) \stackrel{k \rightarrow \infty}{\cong}\frac{1}{k}\int_{0}^{\infty}d x V ( x ) \cos^{2}( k x ) \hspace{1 c m}\delta^{-}( k ) \stackrel{k \rightarrow \infty}{\cong}\frac{1}{k}\int_{0}^{\infty}d x V ( x ) \sin^{2}( k x )
c_{0}= \alpha_{+}^{2}\; , \quad c_{1}= \frac{\epsilon \left( \alpha_{+}+ \epsilon \alpha_{-}\right)^{2}}{\epsilon - \sin \frac{n \gamma}{1 6}}+ 2 \epsilon \; , \quad c_{2}= \frac{\left( \alpha_{+}+ \epsilon \alpha_{-}\right)^{2}}{\left( \epsilon - \sin \frac{n \gamma}{1 6}\right)^{2}}\; ,
\mid{P}_{0}\rangle \: = \: f ( p ) \; \exp \left[ 2 \, \,{({\cal W}^{- 1})}_{\nu \rho}^{\mu \sigma}{\mathbf a}^{\dagger \nu}{\mathbf b}_{\mu}^{\dagger}{\mathbf a}^{\dagger \rho}{\mathbf b}_{\sigma}^{\dagger}\right] \, \mid 0 \rangle .
\langle W(C) \rangle_{YM}= e^{[\cdots]}\int d\mu_C(\xi) \Biggr\langle \exp \left[ 2i \rho^{-1}K^{1/2}Jg \int_{S_C}dS^{\mu\nu}(x){\cal D}[\partial_x] h_{\mu\nu}^\xi(x) \right] \Biggr\rangle_{APEGT},
[J_\omega\psi](n) = \begin{cases}\overline{p(T^{n-1}\omega)}\psi(n-1)+q(T^n\omega)\psi(n) + p(T^n\omega)\psi(n+1) n \ge 1 q(\omega)\psi(0) + p(\omega)\psi(1) n = 0. \end{cases}
\rho_0=\frac{|\phi|^{q-2}\,\phi_+}{\displaystyle\int_\Omega |\phi|^{q-2}\,\phi_+\,dx}\cdot \mathcal{L}^N\mbox{and}\rho_1=\frac{|\phi|^{q-2}\,\phi_-}{\displaystyle\int_\Omega |\phi|^{q-2}\,\phi_-\,dx}\cdot \mathcal{L}^N.
dX^{(j)}_t =\bigg(f^{(j)}(X^{(j)}_t)+\nu\big(X^{(j-1)}_t-2X^{(j)}_t+X^{(j+1)}_t\big)\bigg)dt &+\sum\limits_{i=1}^mc_{i}X^{(j)}_t\circ dW_t^{(i)},j=1,\dots,N
\left(V^{[n]}_{z_1z_1}- V^{[n]}\right)_{z_1}- U_{z_2}=0, V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, \left(\Omega^{[1]}\,U\right)_{z_1}-\frac{U}{2n}- U_{z_2}=0, \Omega^{[j]}_{z_1}=U\,V^{[j]}_{z_1}, j=1\dots n.
m=\alpha^{-1}\left( \sum_{j=0}^{m-2}\omega \left( t_{j},t_{j+1}\right)\right) +1\leq 2\alpha^{-1}\left( \sum_{j=0}^{m-1}\omega \left(t_{j},t_{j+1}\right) \right) \leq 2\alpha^{-1}\omega^{\alpha}\left(s,t\right) .
\mathcal{Q}_i^{-1}= \mathcal{P}^{-1}- \mathcal{P}^{-1}([\sigma \otimes K] \mathcal{P}^{-1}) + \mathcal{P}^{-1}([\sigma \otimes K]\mathcal{P}^{-1})^2 + ... (-1)^{i-1}\mathcal{P}^{-1}([\sigma \otimes K] \mathcal{P}^{-1})^{i-1},
X= \begin{pmatrix}J^{K\textnormal{th},\textnormal{eq}}(q,Q) \delta_q J^{K\textnormal{th},\textnormal{eq}}(q,Q) \vdots \left( \delta_q \right)^N J^{K\textnormal{th},\textnormal{eq}}(q,Q) \end{pmatrix}
\left( \beta_{\mu}^{( 1 )}\partial_{\mu}+ m \right) \Psi^{( 1 )}( x ) = 0 , \Psi^{( 1 )}( x ) = \left( \begin{array}{c}{\psi_{\mu}( x )}\\{\psi_{[ \mu \nu ]}( x )}\\ \end{array}\right) ,
\vert x\rangle=\phi^{-}(x)\vert0\rangle = \int d^n\mu(\mathbf{p}) \,e^{ipx}a_{c}^{*}(\mathbf{p})\vert0\rangle = \int d^n\mu(\mathbf{p}) \,e^{ipx}\vert \mathbf{p}\rangle,
\nu(x_{2m,i}) =\nu{\left(\sum_{k=1}^\infty x_{2m-1,k}\alpha_{k,i}\right)}=\nu{\left(x_{2m-1,1}\alpha_{1,i}+\sum_{k=2}^\infty x_{2m-1,k}\alpha_{k,i}\right)}\geq\min_{k\geq2}\{\nu(x_{2m-1,k})+\nu(\alpha_{k,i})\}.
\int_{-\infty}^{+\infty}z^{\ell}p^{(k)}(z)\,\mathrm{d}z =\left[z^{\ell}p^{(k-1)}(z)-\ell\,z^{\ell-1}p^{(k-2)}(z)+\ell(\ell-1)\,z^{\ell-2}p^{(k-3)}(z)\right. &\hphantom{=\;}\left.+\dots+(-1)^{\ell-1}\ell !\,z\,p^{(k-\ell)}(z)\right]_{-\infty}^{+\infty}+(-1)^{\ell}\ell!\int_{-\infty}^{+\infty}p^{(k-\ell)}(z)\,\mathrm{d}z.
\gamma_{\psi'}(\pm e_{i_1}e_{i_1+k}\cdots e_{i_r}e_{i_r+k}) = \pm(-1)^{r+\frac{r(r-1)}{2}}\gamma_{\psi'}(e_{i_r+k}\cdots e_{i_1+k}e_{i_r}\cdots e_{i_1}) = \pm(-1)^{r+\frac{r(r-1)}{2}}\gamma_{\psi'}(\mathfrak{s}(\varepsilon_{I\cup(I+k)}) = \pm(-1)^{r+\frac{r(r-1)}{2}}\gamma_{\psi'}((-1)^r) = \pm(-\sqrt{-1})^r.
\frac{1}{2}\sum_{\substack{pq\in E_\Gamma \alpha(pq)=\lambda\alpha_i}}\left(\frac{f(p)c_q\prod_{\substack{e\in E^q e\neq qp}}\alpha(e)-f(q)c_p\prod_{\substack{e\in E^p e\neq pq}}\alpha(e)}{c_pc_q\cdot\alpha(pq)\cdot\prod_{\substack{e\in E^p e\neq pq}}\alpha(e)\cdot\prod_{\substack{e\in E^q e\neq qp}}\alpha(e)}\right).
C_+:=\begin{cases}\overbrace{\sigma_1\otimes\sigma_2\otimes\sigma_1\otimes\sigma_2\otimes\cdots\otimes\sigma_1\otimes\sigma_2}^n, ; \underbrace{\sigma_1\otimes\sigma_2\otimes\sigma_1\otimes\sigma_2\otimes\cdots\otimes\sigma_1\otimes\sigma_2\otimes\sigma_1}_n, ,\end{cases}
H = \frac{z_{A}( z_{A}- 1 ) ( z_{A}- z_{3})}{z_{3}( z_{3}- 1 )}\left\{b_{A}^{2}- ( \frac{\mu_{1}}{z_{A}}+ \frac{\mu_{2}}{z_{A}- 1}+ \frac{\mu_{3}- 1}{z_{A}- z_{3}}) b_{A}+ \frac{\kappa}{z_{A}( z_{A}- 1 )}\right\}.
\{X,Y,Z\}=\left.{\partial_r X\over \partial\Phi^A}\right|_K\left.{\partial_l^3 W\over \partial J_A\partial J_B \partial J_C}\right|_K\left.{\partial_l Y\over \partial\Phi^C}\right|_K\left.{\partial_r Z\over \partial\Phi^B}\right|_K(-1)^{\varepsilon_B(\varepsilon(Y)+\varepsilon(Z)+1)}.
\left\lbrace\begin{array}{lll}u''_re_r- \Delta u_re_r + u_rAe_r=0 \hbox{in}(T,+\infty) \times \Omega, \noalign{\medskip}\partial_\nu u_re_r + u_r Be_r=0 \hbox{on}(T,+\infty) \times \Gamma.\end{array}\right.
f(x,t) =\sum_{(\alpha,\beta)\in\mathbb{Z}_+^{m+k}}\frac{1}{\alpha!\beta!}\partial^{(\alpha,\beta)}f(0,0)x^\alpha t^\beta= &=\sum_{\alpha\in\mathbb{Z}_+^m}\frac{1}{\alpha!}\left(\sum_{\beta\in\mathbb{Z}_+^k}\frac{1}{\beta!}\partial^{(\alpha,\beta)}f(0,0)t^\beta\right)x^\alpha= &=\sum_{\alpha\in\mathbb{Z}_+^m}\frac{1}{\alpha!}\partial^{(\alpha,0)}f(0,t)x^\alpha.
\#\left(P_{f}^{-1}\alpha\right)=\sum\limits_{b\in\alpha}\#\left(P_{f_b}^{-1}\gamma_{|b|}\right)-\left(\#\alpha-1 -\sum\limits_{b\in\alpha}\textbf{1}_{f_b\equiv1}+\textbf{1}_{f\equiv 1}\right).
\sup\left\{\mathcal{E}_f(V)\ :\ \int_\Omega |V|^p\,dx=1\right\}=-\frac{1}{2}\,\sup_{u\in W^{1,2}_0(\Omega)\setminus\{0\}}\frac{\langle f,u\rangle^2}{\displaystyle\int_\Omega|\nabla u|^2\,dx+\left(\int_\Omega|u|^{2p/(p-1)}\right)^{(p-1)/p}},
\frac{{\partial^2_{y}{\partial^n_x u}}}{u^s_y +{u}_y}=\partial^2_y \left(\frac{{{\partial^n_x u}}}{u^s_y +{u}_y}\right) + 2\left(\frac{{\partial_y{u}}}{u^s_y +{u}_y}\right)\frac{u^s_{yy}+{u}_{yy}}{{u^s_y +{u}_y}}-{\partial^n_x u}\, \partial^2_y\left(\frac{1}{u^s_y +{u}_y}\right),
{\cal Z}_{\mathrm{G - L}}^{\mathrm{3 D}}= \int{\cal D}A_{\mu}{\cal D}\theta^{\mathrm{s i n g .}}{\cal D}\theta^{\mathrm{r e g .}}\exp \left\{- \int d^{3}x \left[ \frac{1}{4 q^{2}}F_{\mu \nu}^{2}+ \frac{\eta^{2}}{2}\left( \partial_{\mu}\theta - A_{\mu}\right)^{2}\right] \right\},
\left\|\left(I-c_0^{-1}C\right)a\right\|_X=\left(\sum_{k\in\mathbb{Z}}\left|\left(a-c_0^{-1}c*a\right)_k\right|^2\right)^{1/2}=\left(\sum_{k\in\mathbb{Z}}\left|c_0^{-1}\sum_{m\neq k}c_{k-m}a_m\right|^2\right)^{1/2}\le\frac{\sum_{m\not=0}|c_m|}{|c_0|}\|a\|_X<\|a\|_X.
S^{\star}= \sum_{C}\left\{- \frac{1}{\oint_{C}d x_{t}}\oint_{C}d x_{t}\biggl | \left( \frac{\delta}{\delta \sigma^{\mu t}}- i \frac{2 \pi}{e}W_{\mu t}\right) \Phi [ C ] \biggr |^{2}- M_{0}^{2}\biggl | \Phi [ C ] \biggr |^{2}\right\},
\mathbb{P}_{\mu}(\mathbb{D}_{T}(i,j) > 2t \,| (X_{T})^{*}_{i}= u, (X_{T})^{*}_{j}= v) = \overline{\mathbb{P}}_{0, \overline{\xi^{*}}}^{h^{T}}(\inf\{r \in [0,T]: (\overline{X}_{r})^{\mbox{\tiny $J$}}_{i}= (\overline{X}_{r})^{\mbox{\tiny $J$}}_{j}\}> t)
| D^{i}W | = | h | \Big ( A_{i}^{2}e^{- 2 J \beta x}+ B_{i}^{2}e^{- 2 J \beta ( 1 - x )}+ 2 A_{i}B_{i}e^{- \beta J}\cos ( 2 \alpha - \beta \chi ) \Big )^{\frac{1}{2}}\; .
F(z+\omega_2\,|\,\overline\omega_1,\omega_2)\cdot F(z\,|\,\overline\omega_1,-\omega_2)= \prod_{k\geq 0}\big(1-x_2 (q_2 \widetilde{q}_2)^{k/2}\big)\cdot \prod_{k\geq 1}\big(1-x_2^{-1}(q_2 \widetilde{q}_2)^{k/2}\big)^{-1},
\hat{H}( t ) = \sum_{\alpha}\Omega_{\alpha}^{( D )}( t ) \Bigl ( \hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}+ \frac{1}{2}\Bigr ) + \Omega_{\alpha}( t ) \Bigl ( \hat{a}_{\alpha}^{\dagger 2}+ \hat{a}_{\alpha}^{2}\Bigr ) \equiv \sum_{\alpha}\hat{H}_{\alpha}( t ) .
\left| \int_{\mathbb{R}^2}(F(y) - F(x))\partial_1G(y) \partial_1 \psi_T(x-y) \, \textrm{d}y \right| \lesssim C(G) \left[ F\right]_{\alpha}\int_{\mathbb{R}^2}|x_2 |^{\frac{\alpha -1}{2}}d^{\alpha}(x,y) |\partial_1\psi_T(x - y) | \, \textrm{d}y \lesssim C(G) \left[ F\right]_{\alpha}(T^{\frac{1}{4}})^{2 \alpha -2}.
\int D{\cal B}_{+}d e t \left( \begin{array}{c c}{{0}}&{{i \partial_{t}+{\cal B}_{+}}}\\{{i \partial_{-}}}&{{0}}\\ \end{array}\right) = d e t \left( \begin{array}{c c}{{0}}&{{i \partial_{t}}}\\{{i \partial_{-}}}&{{0}}\\ \end{array}\right) ~ \int D U ~ e^{- ( 1 + 2 N ) \dot{W}[ U ]},
\rho(x)=\begin{cases}1 \mathrm{on}B_{R}(x_{0}) \frac{\left( 1+\delta\right) R-r(x)}{\delta R}\mathrm{on}B_{\left(1+\delta\right) R}(x_{0})\backslash B_{R}(x_{0}) 0 \mathrm{elsewhere},\end{cases}
\lambda_{(m-i,k)}= \, \prod_{j=1}^{m-i}(1-q^{k-1+j}r) \frac{(k+m-i)_q^!}{(k)_q^!},{k+m+1\choose i+1}_{\!\!q}= \, \frac{(k+m+1)_q^!}{(i+1)_q^!(k+m-i)_q^!},
H B^{-1}H^{-1}= a^{-1}A - a^{-2}B, \;\;H K^{-1}H^{-1}= a A - a^{2}K, &H (B^\downarrow)^{-1}H^{-1}= a^{-1}A - a^{-2}B^\downarrow, H (K^\downarrow)^{-1}H^{-1}= a A - a^{2}K^\downarrow.
( J_{2}- K_{1}- 1 ) D_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}; 0}+ ( J_{2}- K_{2}- 1 ) D_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}; J_{1}}.
\begin{aligned}D_1(U)= \{\hbox{\#$T(\mathbf n)$: $\mathbf n=\left\lceil A\right\rceil$, where $A$ is a sentence of $\mathcal L$ and \#$A$ is in $U$}\}, D_2(U)= \{\hbox{\#[$\neg T(\mathbf n)$]: $\mathbf n=\left\lceil A\right\rceil$, where $A$ is a sentence of $\mathcal L$ and \#[$\neg A$] is in $U$}\}.\end{aligned}
T_{nk}(x|g)=\left ( \begin{array}{cc}\Lambda_{\mu \nu}(x|g) 0 0 -\varepsilon \end{array}\right )\;,\;\;\Lambda_{\mu \nu}(x|g)=\eta_{\mu \nu}\varepsilon-\frac{e_{0}}{2}\varepsilon gF_{\mu \nu}(x)\varepsilon\;.
\langle W^R_K \rangle = \hbox{\rm dim}\, R \sum_{i=0}^\infty\sum_{j=1}^{d_i}\alpha_{ij}(K) r_{ij}(R) x^i,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x={2\pi i\over k+h}=ig^2/2,
a'(n;i+1)-a'(n;i) =a(n-1;i)+a(n;i+1,i-1)-a(n;i,i-1)+\sum_{j=1}^{i-2}a(n;i+1,j)-a(n;i,j) &=a(n-1;i)-\sum_{i'=i-2}^{n-5}w_{i',i-2}+\sum_{j=0}^{i-2}w_{i-3,j},
\begin{aligned}{\boldsymbol r}_{SD}[i] ={\boldsymbol H}_{SD}[i]{\boldsymbol A}_S[i]{\boldsymbol s}[i] +{\boldsymbol n}_{SD}[i],{\boldsymbol r}_{{SR}_{k}}[i] ={\boldsymbol F}_{SR_{k}}[i]{\boldsymbol A}_S[i]{\boldsymbol s}[i] +{\boldsymbol n}_{SR_k}[i], k = 1,2,~...~ n_{r}, ~~i = 1,2,~...~,\end{aligned}
(1\otimes_\nabla X_j)(\delta_k \otimes c \otimes \delta_n \otimes b) = \delta_k \otimes (c + [X_j,c]) \otimes \delta_n \otimes b \mapsto \alpha_d^k(c + [X_j,c])\cdot \delta_{(n,k)}\otimes b = \alpha_d^k(X_j c)\cdot \delta_{(n,k)}\otimes b = X_j\left( \alpha_d^k(c)\cdot \delta_{(n,k)}\otimes b \right)
G_1^1 + F_{(3,0),1}+ F_{(1,2),1}= 0 ; G_1^2 + F_{(2,1),1}+ F_{(0,3),1}= 0 \, , G_2^1 + F_{(3,0),2}+ F_{(1,2),2}= 0 ; G_1^2 + F_{(2,1),2}+ F_{(0,3),2}= 0 \, .
P_{k+1}^1 \circ (Q_B)^{k+1}_{k+1}= L_{\infty,k+1}\circ H_{k+1}\circ (Q_B)_{k+1}^{k+1}= L_{\infty,k+1}- L_{\infty,k+1}\circ (Q_B)_{k+1}^{k+1}\circ H_{k+1}-L_{\infty,k+1}\circ (i\circ p)^{\vee (k+1)}= L_{\infty,k+1}+ (Q_A)_1^1 \circ P_{k+1}^1
\begin{cases}T_s(tst\cdots)_k = (sts\cdots)_{k+1}, 0\leq k\leq m-3, T_t(sts\cdots)_k=(tst\cdots)_{k+1}, 0\leq k\leq m-3, T_s(sts\cdots)_k=(sts\cdots)_k, 1\leq k\leq m-2, T_t(tst\cdots)_k = q_t(sts\cdots)_{k-1}+ (1-q_t)(tst\cdots)_k, 1\leq k\leq m-2, T_s(tst\cdots)_{m-2}=T_t(sts\cdots)_{m-2}= (sts\cdots)_{m-2}.\end{cases}
\frac{d}{dt}\mathcal{H}^{d-1}(M_t) =- \int_{M_t}h\cdot v \, d\mathcal{H}^{d-1}= -\int_{M_t}(v+\langle h\cdot \nu \rangle \nu) \cdot v \, d\mathcal{H}^{d-1}= -\int_{M_t}|v|^2 \, d\mathcal{H}^{d-1}-\langle h \cdot \nu \rangle \int_{M_t}\nu \cdot v \, d\mathcal{H}^{d-1}=-\int_{M_t}|v|^2 \, d\mathcal{H}^{d-1}.
C_{m_{1},m_{2},m_{3}}^{j_{1},j_{2},j_{3}}\ =\ \delta_{m_3,m_1+m_2}\sqrt{2j+1}\ \Delta(j_{1},j_{2},j_{3})\S_{m_{1},m_{2},m_{3}}^{\j_{1},\ \j_{2},\ \j_{3}}& \cdot{\sum_{z}}\frac{(-1)^{z}}{z!(j_{1}+j_{2}-j_{3}-z)!(j_{1}-m_{1}-z)!(j_{2}+m_{2}-z)!(j_{3}-j_{2}+m_{1}+z)!(j_{3}-j_{1}-m_{2}+z)!}
\begin{aligned}\phi(f_1,1) +\phi(2,f_2) - \phi(2,1) - L_1(f) = (2^{0.6}-f_1^{0.6}) - 2^{0.6}f_2^{0.6}& \leq \min\bigl\{-f_1^{0.6}- f_2^{0.6}+ 1, 2^{0.6}(f_1^{0.6}- 2^{0.6}) - 2^{0.6}f_2^{0.6}\bigr\}\leq \min\bigl\{R_1(f), R_2(f)\bigr\}.\end{aligned}
\sum\limits_{\substack{j=1 (j,q_1)=1}}^{q_1}\left(\left|Z_G(j)\right|+\left|Z_\Omega(j)\right|\right)\ll X^{\varepsilon}\cdot \frac{|m^{\ast}|}{(d,|m^{\ast}|)^{1/2}}\cdot \sum\limits_{i=1}^{4}u^{\alpha_i}\left|m^{\ast}\tilde{m}\right|^{\beta_i}(hd)^{\gamma_i}X^{\delta_i},
y_N^{1+s}\int_0^{\eta/y_N}\int_0^{1/y_N}\frac{t\,\rho^{N-2}}{\left(\rho^2+|t-1|^2\right)^{N/2-s}}\cdot\frac{d\rho}{\left(\rho^2+|t-1|^2+t\right)^s}\;dt\ \leq\ \eta\,y_N^s=\eta\,\delta(y)^s
f(x)=\frac{1}{2\pi}\int\nolimits_{0}^{2\pi}f_{t}(x)dt,\frac{1}{2\pi}\int\nolimits_{0}^{2\pi}\int_{0}^{1}\left\vert f_{t}(x)\right\vert^{2}dxdt=\int_{\mathbb{R}}\left\vert f(x)\right\vert^{2}dx,
{\mathcal H}(\overline{\mathcal S},\overline{\mathcal U})_k={\rm End}_{k[\overline{\mathcal S}]}({\rm ind}_{\overline{\mathcal U}}^{\overline{\mathcal S}}{\bf 1}_k)^{\rm op}\cong{\rm End}_{k[I_0^{j}]}({\rm ind}_{I_0}^{I_0^{j}}{\bf 1}_k)^{\rm op}{}\hookrightarrow{\mathcal H}(G,I_0)_{{\rm aff},{k}}\quad\subset{\mathcal H}(G,I_0)_{k}.
A(x_x \mid x_{[k-1],}p_{[k+1..\ell]}) + 8 \gamma_{k+1}+ O(\log n) \stackrel{(a)}{\leq}C(x_k \mid x_{[k-1]}, p_{[k+1..\ell]}) + 16 \gamma_{k+1}+ O(\log n) & \stackrel{(b)}\leq n_k + 16 \gamma_{k+1}+ O(\ell-k) \log n & \stackrel{(c)}< n_k + \eta_k + 1 = \hat{n}_k.
a_1\beta_{Y^2}+a_2\beta_{XY^2}+a_3\beta_{Y^3}+a_4\beta_{X^2Y2}+2a_5\beta_{XY^3}= a_1(\beta_1-\beta_{X^2})+a_2(\beta_X-\beta_{X^3})+a_3(\beta_Y-\beta_{X^2Y})+a_4(\beta_{X^2}-\beta_{X^4}) +2a_5(\beta_{XY}-\beta_{X^3Y}) = \beta_{X^3}-p.
{\{\pi_{i}, \pi_{j}\}}_{D ( \Phi )}= g G_{i j}^{a}( x ) I^{a}+ i g \left( \nabla_{k}^{a b}G_{i j}^{b}( x ) \right) I^{a}\xi_{k}({\cal P}_{m}\xi_{m}) \frac{( b \kappa + a )}{\tilde{\beta}( \omega + \tilde{m})}= g G_{i j}^{a}( q ) I_{\varphi}^{a},
|(j_{12}j_{3})jm\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{3}}C_{m_{1},m_{2},m_{12}}^{j_{1},j_{2},j_{12}}C_{m_{12},m_{3},m}^{j_{12},j_{3},j}|j_{1}m_{1}j_{2}m_{2}j_{3}m_{3}\rangle
\begin{array}{l}\displaystyle \int_{I \cap (-\infty;c_{i,j}']}s_{i,j}[\det(s_{k,l})]^{(n-N-2)/2}ds_{i,j}+ +\displaystyle \int_{I \cap [c_{i,j}'';+\infty)}s_{i,j}[\det(s_{k,l})]^{(n-N-2)/2}ds_{i,j}= =\alpha \int_I s_{i,j}[\det(s_{k,l})]^{(n-N-2)/2}ds_{i,j}\end{array}
\Gamma(2-\alpha)\tau^{\alpha}\sum_{j=1}^{n}p_{n-j}g^j\leq \Gamma(2-\alpha)\tau^{\alpha}\max_{1\leq j\leq n}g^j\sum_{j=1}^{n}p_{n-j}\leq \frac{t_n^{\alpha}}{\Gamma(1+\alpha)}\max_{1\leq j\leq n}g^j,n\geq1.
A=\frac{1}{(\beta-1)(\alpha+\beta-1)}\sum_{i}p^{\alpha+\beta-1}_{i}+\frac{1}{\alpha(\alpha+\beta-1)}\sum_{i}q^{\alpha+\beta-1}_{i}&B=\frac{1}{\alpha(\beta-1)}\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}
\Pi_{k \boldsymbol{\nu}}\big (x_{1}, x_{2}\big)\sim \Pi_{k \boldsymbol{\nu}}\big (x_{1}, x_{2}\big)_1=\Pi_{k \boldsymbol{\nu}}\big (x_{1}, x_{2}\big)_{11}+\Pi_{k \boldsymbol{\nu}}\big (x_{1}, x_{2}\big)_{12},
\begin{aligned}\widetilde{\theta}_1:=\theta+\frac{(1-\theta)}{l_1+l_2\left(\frac{1}{N}\sum\limits_{j=1}^Ne^{-\beta\tau_j^{*}}x_j(\tau_j^{*})\right)}&=\theta +\frac{(1-\theta)}{l_1+\frac{l_2}{N}e^{-\beta\tau_i^{*}}x_i(\tau_i^{*})+\frac{l_2}{N}\sum\limits_{j=1,j\neq i}^Ne^{-\beta\tau_j^{*}}x_j(\tau_j^{*})}\end{aligned}
P ( l ) = \frac{2 \pi \lambda}{l !^{2}( l + 1 ) !^{2}}\left( \frac{\omega r_{0}}{2}\right)^{2 l + 2}\left| \frac{\Gamma^{2}\left( \frac{l + 2}{2}- \frac{i \lambda}{2}\right)}{\Gamma \left( 1 - i \lambda \right)}\right|^{2}.
\lambda_{0,1}(t) = \int_{t}^{\infty}\int_{t_{1}}^{\infty}\frac{-b^{2}\log(\log(t_{2}))}{t_{2}^{2}\log^{b+2}(t_{2})}dt_{2}dt_{1}=\frac{-b^{2}\log(\log(t))}{(b+1)\log^{b+1}(t)}+ O\left(\frac{1}{\log^{b+1}(t)}\right)
\frac{(1 - \theta)a_N}{\ell}= \frac{1 - \theta}{s\ell}\log \bigg( \frac{s}{\mu}\bigg) \leq \tau_{\ell+1}- \tau_{\ell}\leq \frac{1 + \theta}{s\ell}\log \bigg( \frac{s}{\mu}\bigg) = \frac{(1 + \theta) a_N}{\ell}.
\tilde{H}= \frac{\sqrt{\alpha^\prime}}{2}g^{\mu \nu}(P_\mu - \tilde{Q}_{_{E}}{\cal A}_\mu - \tilde{Q}_{_{M}}\tilde{\cal A}_\mu)(P_\nu - \tilde{Q}_{_{E}}{\cal A}_\nu - \tilde{Q}_{_{M}}\tilde{\cal A}_\nu) + \frac{\sqrt{\alpha^\prime}}{2k^2}({\tilde{Q}_{_{E}}}^2 +{\tilde{Q}_{_{M}}}^2),
\sum_{{k=1,k\neq j}}^n\dfrac{1}{z_{\nu,k}^2-z_{\nu,j}^2}=\lim_{z\to z_{\nu,j}}\left(\dfrac{K_{\nu+1}(z)}{2zK_\nu(z)}-\dfrac{2\nu}{z^2}-\dfrac{1}{z_{\nu,j}^2-z^2}\right)-\dfrac{K_{\nu+1}(-z_{\nu,j})}{2z_{\nu,j}K_\nu(-z_{\nu,j})}=:\mathbf{K}_1-\mathbf{K_2}\,.
\Psi_{\mathrm{{0}}}( r^{\prime}, \theta^{\prime}; 0 ) ={\frac{1}{\sqrt{2 \pi}\xi}}\exp \left\{i k r^{\prime}\cos \theta^{\prime}-{\frac{1}{4 \xi^{2}}}( r^{2}+ r_{0}^{2}+ 2 r r^{\prime}\cos \theta^{\prime}) \right\}.
\frac{2}{r}( f^{'}\cos 2 \theta \frac{\partial I^{2}}{\partial \theta}+ 2 f \sin 2 \theta \frac{\partial I^{2}}{\partial r}- f^{'}\sin 2 \theta \frac{\partial I^{1}}{\partial \theta}+ 2 f \cos 2 \theta \frac{\partial I^{1}}{\partial r}) \nonumber
\sum_{k =1}^K \delta_{k,\ell}P_{k,\ell}\alpha_{k,\ell}N_k = \sum_{k =1}^K P_{k,\ell}\alpha_{k,\ell}N_k - (2^{\rho_{\ell}R_{sum}}-1)(\sigma^2 + \mathcal{I}_{\ell}),~\forall \ell \in \mathcal{L}\Rightarrow ~~ - \sum_{k=1}^K\sum_{\ell =1}^L (N_k -1)\log_2(\delta_{k,\ell}) = B.
\int_{pr_{g+1}}\gamma^* (\delta_{\Theta_{\alpha}}) \gamma^*\omega_0^{g}= \sum_{j=1}^g\int_{pr_{g+1}}\delta_{\lbrace P_j=P_{g+1}\rbrace}\gamma^*\omega_0^g &+\int_{pr_{g+1}}\delta_{\lbrace P_g\in\sigma(P_1,\dots,P_{g-1})\rbrace}\gamma^*\omega_0^{g},
\sum_{\lambda^{+}}\frac{\prod_{\square\in\lambda^{+}}\rho(h_\square,z)}{H_{\lambda^{+}}}-\frac{\prod_{\square\in\lambda}\rho(h_\square,z)}{H_\lambda}= z\sum_{\lambda^-}\frac{\prod_{\square\in\lambda^-}\rho(h_\square,z)}{H_{\lambda^-}}.
\mathcal{J}(y(v) ,v) - \mathcal{J}(y(u),u) = \,\frac{1}{2}\|y(u) - y(v)\|^2 + \int_\Omega p(u)(f+v-f-u) \, d\boldsymbol{x}+ \frac{\lambda}{2}\|u-v\|^2 + \lambda \int_\Omega (u-u_d)(v-u) \, d\boldsymbol{x}= \,|||u-v|||^2 + \int_\Omega (p(u)+\lambda(u-u_d))(v-u) \, d\boldsymbol{x}.
\langle 0 \vert h_{i j}( \eta ,{\bf x}) h^{i j}( \eta ,{\bf x}+{\bf r}) \vert 0 \rangle \equiv \int_{0}^{+ \infty}\frac{\mathrm{d}n}{n}\frac{\sin n r}{n r}n^{3}P_{\mathrm{h}}( \eta , n ) ,
\int_{D_0}|\tilde F-(1-b(k\Psi))fF^{2k}|^2e^{-k\varphi+v(k\Psi)-k\Psi}\le (2-e^{-t_0-1})\int_{D_0}\mathbb{I}_{\{-t_0-1<k\Psi<-t_0\}}|f|^2e^{-k\Psi},
\mathcal{SSO}=\{v\in \mathbb{C}^{\mathbb{N}}: \langle v,S(s_1,s_2)v \rangle=\delta_{0s_1}\delta_{0s_2}\hbox{for all $s_1$ and $s_2$}\}&=\{v\in \mathbb{C}^{\mathbb{N}}: \langle S(s'_1,s'_2) v,S(s_1,s_2)v \rangle=\delta_{s'_1s_1}\delta_{s'_2s_2}\hbox{for all $s_1$, $s'_1$, $s_2$ and $s'_2$}\}.
U_k=\left(\begin{array}{ll}\frac{1}{p^T Hp}Hp \frac{1}{p^T Bp}Bp\end{array}\right) \left(\begin{array}{cc}\hat{m}_{1,1}\hat{m}_{1,2}\hat{m}_{1,2}\hat{m}_{2,2}\end{array}\right)\left(\begin{array}{c}\frac{1}{p^T Hp}p^TH \frac{1}{p^T Bp}p^TB \end{array}\right).
\Vert g \Vert_{X_r}= \displaystyle\sup_{n\in\mathbb{N^*}}\Big(n^3\Vert g_n \Vert_{L^r(S_2)}\Big) + \displaystyle\sup_{n\in\mathbb{N^*}}\Big(n\Big\Vert\mathcal{G}\cdot\dfrac{\partial g_n}{\partial u}\Big\Vert_{L^r(S_2)}\Big)
(\sigma\circ\alpha^r)_0 =r; (\sigma\circ\alpha^r)_{i-r}=i; (\sigma\circ\alpha^r)_{i+1-r}=j+1; (\sigma\circ\alpha^r)_{k-j+i-r}=k; (\sigma\circ\alpha^r)_{k-j+i+1-r}=i+1; (\sigma\circ\alpha^r)_{k-r}=j; (\sigma\circ\alpha^r)_{k+1-r}=k+1; (\sigma\circ\alpha^r)_{n-r}=n; (\sigma\circ\alpha^r)_{n+1-r}=0; (\sigma\circ\alpha^r)_n =r-1,
\begin{gathered}\mathbb{E}\big( d_p(\pi_n(X) , \pi_n(Y_0))\big)^p = \mathbb{E}\big( d_p(\pi_n(X) , \pi_n(S \circ Z) ) \big)^p \leq \leq \mathbb{E}\big(d_p(\pi_n(X) , Z) + d_p(Z ,\pi_n(S \circ Z))\big)^p \leq 2^p\mathbb{E}\big( d_p(\pi_n(X), Z)\big)^p = 2^p\mathbb{E}\big( d_p(\tilde{X}, \tilde{Y})\big)^p.\end{gathered}
\{a_{n}\}=1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\frac{1}{7},-\frac{1}{8},\frac{1}{9},-\frac{1}{10},\frac{1}{11},-\frac{1}{12},\frac{1}{13},-\frac{1}{14},\frac{1}{15},\cdot\cdot\cdot
E(t)=\sum_{0\leq k\leq s}\Big\{\int \frac{1}{A}|D_t\theta_k|^2+\int \frac{1}{A}|D_t\sigma_k|^2 +i\int \theta_k\overline{\partial_{\alpha}\theta_k}+i\int \sigma_k\overline{\partial_{\alpha}\sigma_k}\Big\}.
\sum_{j\geq 0}\begin{pmatrix}m j \end{pmatrix}(a_{(n+j)}b)_{(m+k-j)}&=\sum_{j\geq 0}(-1)^j \begin{pmatrix}n j \end{pmatrix}\left(a_{(m+n-j)}b_{(k+j)}-(-1)^n b_{(n+k-j)}a_{(m+j)}\right).
(\partial-{\bar\partial})(\hat{{\cal\phi}^0},\hat{{\cal A}_z^0})\sim{\sqrt{3}(-1+a_i)a_i^3\biggl[(b_ie_i)^2c_if_i(c_i-f_i){\bar z}+(-b_ic_i^2e_i^2f_i(f_i-b_i)+ e_ib_i^2c_if_i^2(c_i-e_i))z\biggr](z-{\bar z})\over|z|^4},
S_{1}=1=(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}})c^{2}\dot{t}^{2}-(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}})^{-1}\dot{r}^{2}-r^{2}\dot{\theta}^{2}.
\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x,\,\varphi(T)=y}}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t))\, \Bigl \vert \,\varphi(t) \in D_{-\delta}\cup \partial D_{-\delta}, \,\, \forall t \in [0, T] \biggr\}< T\bigl(r({\mathcal{K}}) + 2\varkappa\bigr),
s^{-k}\int_{B_s(x)}D_\mu(y,t)\,d\mu(y) \leq Cs^{-k}\int_{B_s(x)}t^{-k}\int_{B_t(y)}W_t(z)\,d\mu(z)\,d\mu(y) &= Cs^{-k}t^{-k}\int_{B_s(x)}\mu(B_t(y)) W_t(y)\,d\mu(y) &\leq Cs^{-k}\int_{B_s(x)}W_t(y)\,d\mu(y)\, ,
R_1(t) := \{(x_1, x_2, x_3) \mid (x+y+z \ge 0) \wedge (2x+y-z\ge t) \wedge (x-y+2z \ge t) \wedge (-x+2y+z\ge t)\}, R_2(t) := \{(x_2,x_3) \mid 0 \le x_2, x_3 \le t\}, R_3(t) := \{(x_2, x_3) \in R_2(t) \mid \{x_1 \in \mathbb{R}\mid (x_1, x_2, x_3) \in R_1(t)\}= \varnothing\}.
\omega = \omega_{\underline{{{i_{1}\dots i_{\ell}}}}}\, V^{\underline{{{i_{1}}}}}\wedge \dots \wedge V^{\underline{{{i_{\ell}}}}}\quad \Leftrightarrow \quad \star \omega = \frac{1}{( D - \ell ) !}\, \epsilon_{\underline{{{a_{1}\dots a_{D - \ell}b_{1}\dots b_{\ell}}}}}\omega^{\underline{{{b_{1}\dots b_{\ell}}}}}\, V^{\underline{{{a_{1}}}}}\wedge \dots \wedge V^{\underline{{{a_{D - \ell}}}}}
\{{\psi}_{\alpha}( x ) ,{\psi}_{\beta}( y ) \}= 0 , \quad \{{\psi}_{\alpha}( x ) ,{\psi}_{\beta}^{\ast}( y ) \}= ( ({\gamma}^{0}{\gamma}^{-})^{- 1})_{{\alpha}{\beta}}{\delta}( x^{+}- y^{+}) .
F ({\cal P}, \kappa :{\cal P}_{1},{\kappa}_{1};{\cal P}_{2},{\kappa}_{2}) = ( - 1 )^{\gamma + \sigma + m a x ( \gamma , \sigma ) + I_{1}+ I_{2}- I}\, F ( \tilde{\cal P}, \tilde{\kappa}: \tilde{\cal P}_{1}, \tilde{\kappa}_{1}; \tilde{\cal P}_{2}, \tilde{\kappa}_{2})
\zeta(s) = \begin{cases}\sum_{n=1}^\infty \frac{1}{n^s}= \prod_{p}\frac{p^s}{p^s-1}\; , \Re(s) >1 (1-2^{1-s}) \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}\;, 0<\Re(s)<1 2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma(1-s)\zeta(1-s)\;, \Re(s)<0\end{cases}
|\Psi\rangle_{\nu}=|\Psi_{0}\rangle_{\nu}\oplus|\Psi_{1}\rangle_{\nu}\oplus|\Psi_{2}\rangle_{\nu}\oplus\cdot\cdot\cdot=a|0\rangle\oplus\sum_{i}a_{i}|\psi_{i}\rangle\oplus\sum_{ij}a_{ij}|\psi_{i},\psi_{j}\rangle_{\nu}\oplus\cdot\cdot\cdot
\mathcal{A}_{\varepsilon}(u) = \left\{(t,l) \,\,\Big\vert\,\,|t-u|\leq\frac{l}{2}\,\,\,\,\varepsilon\leq l\leq 1\,\,\,\,|t-u|\leq\frac{1}{2}\,\,\,\,l\geq1\right\}.