harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
( ( \frac{d}{d x})^{N - 1}- k^{k}e^{x}( \frac{d}{d x}+ \frac{1}{k}) ( \frac{d}{d x}+ \frac{2}{k}) \cdots ( \frac{d}{d x}+ \frac{k - 1}{k}) ) w ( x ) = 0 ,
A\coloneqq\ A_1\bigtimes_{i=1}^{n-1}(d_{i,-1}d_{i,0}d_{i,2},d_{i,0}d_{i,1}d_{i,2},d_{i,1}d_{i,2}d_{i,4},d_{i,2}d_{i,3}d_{i,4},d_{i,3}d_{i,4}d_{i,6},d_{i,4}d_{i,5}d_{i,6})A_{i+1}=\ A_1\bigtimes_{i=1}^{n-1}(b_nm_{4n-2}d_{i,2},m_{4n-2}d_{i,1}d_{i,2},d_{i,1}d_{i,2}m_0,d_{i,2}d_{i,3}m_0,d_{i,3}m_0t_{i+1},m_0b_1t_{i+1})A_{i+1}.
\int_0^{\left(\frac{r}{2k}\right)^{1/4}}\log (- 6\log \gamma ){\rm d}\gamma = \log(-6\log \gamma) \gamma \bigg\vert_0^{r^{1/4}/(2k)^{1/4}}- \int_0^{r^{1/4}/(2k)^{1/4}}\frac{1}{\log \gamma}{\rm d}\gamma \hphantom{\int_0^{\left(\frac{r}{2k}\right)^{1/4}}\log (- 6\log \gamma ){\rm d}\gamma}{}= O\big(r^{1/4}\log (-\log r) \big),
\langle\lambda,\mu;l-\lambda,m-\mu|lm\rangle=(\begin{matrix}l+m\\ \lambda+\mu\end{matrix})^{1/2}(\begin{matrix}l-m\\ \lambda-\mu\end{matrix})^{1/2}(\begin{matrix}2l\\ 2\lambda\end{matrix})^{-1/2}.
\int d^{1 0}x s q r t{- \hat{g}}\ \left[{\textstyle \frac{3}{4}}\hat{G}^{2}\right] = \int d^{9}x \sqrt{g}\ \left[{\textstyle \frac{3}{4}}k G^{2}-{\textstyle \frac{3}{4}}k^{- 1}\left( H^{( 2 )}- \ell H^{( 1 )}\right)^{2}\right] .
q_{2t}= \frac{\mu_t}{\chi}- \frac{\mu}{\chi^2}\chi_t =\frac{2}{3}\frac{u_t}{u}- \frac{\mu}{\chi}\left(\frac{2}{3}\frac{u_t}{u}\frac{\mu}{\chi}- \frac{1}{3}\frac{\nu}{\chi}\right)=\frac{2}{3}\frac{u_t}{u}- q_2\left(\frac{2}{3}\frac{u_t}{u}q_2 - \frac{1}{3}q_1\right),
x_{\Delta}= \sqrt{\frac{A}{4 \pi}}\ \ \mathrm{w i t h}\ \A = 2 \pi \int_{0}^{\pi}d \theta \sin \theta x_{h}^{2}\frac{\sqrt{l_{2}( x = x_{h}, \theta ) m_{2}( x = x_{h}, \theta )}}{f_{2}( x = x_{h}, \theta )}\ .
\rho_r^*\mathcal{U}^{C_r}=\Big(\bigoplus_{\alpha \geq 0}\bigoplus_{n \geq 0, r\mid n}\ \mathbb{C}\left(\frac{n}{r}\right)\Big) \bigoplus \Big( \bigoplus_{\alpha \geq 0}\bigoplus_{n\geq 0, r \mid n}\ \mathbb{C}\left(\frac{n}{r}\right) \oplus \mathbb{C}\left(-\frac{n}{r}\right)\Big).
A_{p,1}= \{v_i\in V : f(x_i)=p\textup{and}f(w_i)=f(u)\textup{or}f(u)+1\textup{(mod}b-1)\}, A_{p,q}= \{v_i\in V : f(x_i)=p\textup{and}f(w_i)=f(u)+q\textup{(mod}b-1)\}\textup{, for $q\ge 2$.}
\log \Upsilon ( x ) = \int_{0}^{\infty}\frac{d t}{t}\; \biggl [ \biggl ( \frac{Q}{2}- x \biggr )^{2}e^{- t}- \frac{\sinh^{2}\bigl ( \frac{Q}{2}- x \bigr ) \frac{t}{2}}{\sinh \frac{b t}{2}\sinh \frac{t}{2 b}}\biggr ] .
\frac{1}{|\gamma+r\sin\theta|^\tau}=\frac{|\theta-\theta_1|^\tau}{r^\tau|\sin\theta-\sin\theta_1|^\tau}\frac{1}{|\theta-\theta_1|^\tau}\leq\frac{A(\tau)}{r^\tau|\cos\theta_1|^\tau}\frac{1}{|\theta-\theta_1|^\tau},
\langle \mu^{sp}|\Gamma_+(\{\mathbf{x}^{\pm},\mathbf{z}\})|\lambda^{sp}\rangle=sp_{\lambda/\mu}(\mathbf{x}^{\pm};\mathbf{z})=\det(M_{ij})^{l+n+1}_{i,j=1}, &\langle \mu^{o}|\Gamma_+(\{\mathbf{x}^{\pm},\mathbf{z}\})|\lambda^{o}\rangle=o_{\lambda/\mu}(\mathbf{x}^{\pm};\mathbf{z})=\det(N_{ij})^{l+n+1}_{i,j=1},
\nu^{\pi^*}_{n-1}(1|1)\equiv{{c}_1(n-1)=\frac{2^{\mu_1(n-1)}}{2^{\mu_1(n-1)+\Delta{C}_n}}},~\pi^*_{n-1}(1|1)\equiv{{d_1(n-1)}=\frac{\beta_{n-1}(1+2^{\mu_1(n-1)+\Delta{C}_n})-1}{(\beta_{n-1}-\delta_{n-1})(1+2^{\mu_1(n-1)+\Delta{C}_n})}}
\Gamma_{6,k}(\alpha_2,\alpha_3,\alpha_5,\alpha_6) =\left\{Id,\tau_{6}(\alpha_2, \alpha_3, \alpha_5, \alpha_6),\cdots,\tau_{6}^{k-1}, \ \ \alpha_{2}\alpha_{3}\leq -1+ \displaystyle\cos^2\frac{2\pi}{k}\right\}.
+\left[|x| -{4 \over 945}|y|^3 +{2 \over 45}|x||y|^2+{1 \over 3}|x|^2|y| +{8 \over 15}|x|^{5/2}|y|^{1/2}+{1 \over 3}|x|^3 \left(-{17 \over 6}-\psi\left( 1 \over 2 \right)+\ln{|x|^2 \over |y|}\right)\right]\cos{\bar\theta \over 2}
\int\limits_0^\pi \int\limits_0^\infty f(r\sin \psi, r\cos \psi)\sin\psi dr d\psi =\int\limits_0^\pi \int\limits_{-\infty}^0 f(-r\sin \psi, -r\cos \psi)\sin\psi dr d\psi &=\int\limits_0^\pi \int\limits_{-\infty}^0 f(-r\sin \phi, r\cos \phi)\sin\phi dr d\phi.
D|_{N_R}(s\mathfrak{c}^b_0-s\mathfrak{h}^b_0)=s\chi_0\left( \begin{array}{c}-i\frac{\partial_t a^+\eta_0}{\sqrt{z}}i\frac{\partial_t a^-\bar{\eta}_0}{\sqrt{\bar{z}}}\end{array}\right)+\Theta^0_{s}(\psi)-D_{pert}(s\mathfrak{k}^b_0).
\left. \begin{array}{r}{h ( a , c ) A + l ( a , b , c ) \bar{A}\bar{B}}\\{h ( b , c ) \bar{B}+ l ( a , b , c ) B A}\\{e^{c}+ j^{2}. f ( c , a ) A \bar{C}+ f ( c , b ) C \bar{B}+ j^{2}. g ( c , a , b ) ( \bar{A}\bar{B}\bar{C}+ j . C B A )}\\ \end{array}\right)
\Gamma_{PV}[A]=\left. \frac{d}{ds}\zeta_{PV}(s)\right |_{s=0}=\lim_{s\to 0}\pm\frac{i}{\Gamma\left(\frac{s+1}{2}\right)}\int^\infty_0\!\!\!\!dt~t^{\frac{s-1}{2}}~\frac{d G_{\mp}(t,s)}{d s}{\rm Tr}[{D\!\!\!\!/}\exp(-t{D\!\!\!\!/}^{~2})].
\frac{1}{1-q^r P^{-1}}= \frac{1}{(1-q^r) +q^r (1-P^{-1})}= \frac{1}{1-q^r}\sum_{m = 0}^{N}\left( \frac{q^r}{1-q^r}(1-P^{-1}) \right)^m \in K\left( \mathbb{P}^N \right) \otimes \mathbb{C}(\!(q)\!)
\mathring{A}(a,b,c) = \Gamma \big ( \tfrac{c}{2}\big ){_1 F}_{1}\big ( \tfrac{c}{2}\, ; \, \tfrac{1}{2}\, ; \, \tfrac{b^2}{4 a}\big ), \mathring{B}(a,b,c) = - \tfrac{b}{\sqrt{a}}\Gamma \big ( \tfrac{c+1}{2}\big ){_1 F}_{1}\big ( \tfrac{c+1}{2}\, ; \, \tfrac{3}{2}\, ; \, \tfrac{b^2}{4 a}\big ),
z_{hi}:= \binom{n}{h}\frac{(2i+\sigma)(k+l-n)_{i-k-l}(\alpha+2l+1)_{n-l-h}(\beta+2k+1)_{h-k}}{(\alpha+2l+1)_{i-k-l}(i+k+l+\sigma)_{n+1-k-l}}, begin{align*}1ex] w_{hi}:= Q_{i-k-l}(h-k;\, \beta+2k,\, \alpha+2l,\, n-k-l),
\langle \nu, u \rangle = \lim_{j \rightarrow \infty}\langle \nu, \varphi_{j}\rangle \leq \lim_{j \rightarrow \infty}\int_{\Omega}\nabla u \cdot \nabla \varphi_{j}\;dx=\int_{\Omega}| \nabla u |^{2}\;dx < +\infty.
\begin{cases}-y_k''(x)+a(x)y_k(x)=(k^2+\widetilde a+\widetilde\lambda(\widetilde \varphi_k,k))y_k(x),\quad[0,2\pi], y_k(0)=y_k(2\pi), y_k'(0)=y_k'(2\pi),\end{cases}
H=\sum_{n,n^{\prime}}\int d^{3}xa_{n^{\prime}}^{\dagger}\psi_{n^{\prime}}^{(+)^{*}}(x)E_{n}^{0}\psi_{n}^{(+)}(x)a_{n}=\sum_{n,n^{\prime}}E_{n}^{0}a_{n^{\prime}}^{\dagger}a_{n}\delta_{nn^{\prime}}=\sum_{n}E_{n}^{0}a_{n}^{\dagger}a_{n}=\sum_{n}E_{n}^{0}\hat{N}
\begin{aligned}g_1 =\max(f_1,f_2,...,f_k), g_2 =\min_{i\leq k}\max(f_1,..,\breve{f_i},..,f_k), g_3 =\min_{i<j\leq k}\max(f_1,...,\breve{f_i},..,\breve{f_j},..,f_k),\end{aligned}
A^{S_1}(z)=\frac{1}{z^{5/3}}\begin{pmatrix}0 1 0 0 0 0 -t 0 0 \end{pmatrix}+\frac{1}{z^{4/3}}\begin{pmatrix}0 0 0 0 0 0 0 q_1 0 \end{pmatrix}\hphantom{A^{S_1}(z)=}{}+\frac{1}{z}\begin{pmatrix}p_2q_2 0 0 0 p_1q_1-p_2q_2-\theta^0_2-1/3 0 0 -p_1q_1-\theta^0_1-2/3 \end{pmatrix}+\cdots.
H(t,\xi)=E\bigg(\varepsilon_0+\frac{\varepsilon_0\eta_0}{t^2}\bigg)+C_0^{1/3}\sum_{|k|> k_0/2}H(t,k,\xi),H(t,k,\xi)=\sum_{q\neq 0}e^{-2|k-q|}\frac{\varepsilon_0\eta_0}{(\xi-tq)^2+q^2},
\frac{1}{2\pi i}\int_{(4)}\bigg(\frac{1}{2}W(s; f, \overline{g}) + \frac{W(s-1; f,\overline{g})}{s + k - 2}\bigg) X^s \Gamma(s) ds \quad+ \frac{1}{(2\pi i)^2}\int_{(4)}\int_{(-1 + \epsilon)}W(s-z; f, \overline{g})\zeta(z) \frac{\Gamma(z) \Gamma(s - z + k - 1)}{\Gamma(s + k - 1)}dz X^s \Gamma(s) ds.
u^{(s-1)k+q}= (1 - \frac{q}{k}) u^{(s-1)k}+ \frac{q}{k}u^{sk}+ \frac{u''(\xi)}{2}(t_{(s-1)k+q}- t_{(s-1)k})(t_{(s-1)k+q}- t_{sk}),\xi\in(t_{(s-1)k},t_{sk}).
(1-q) \frac{\partial}{\partial x^\alpha}S(x, t, q)^{-1}= S(x, t, q)^{-1}\circ (\phi_\alpha \bullet - ), L_i^{-1}q^{Q_i \frac{\partial}{\partial Q_i}}\circ S(x, t, q)^{-1}= S(x, t, q)^{-1}\circ B_i q^{Q_i \frac{\partial}{\partial Q_i}},
\mathrm{ann}_{l}(A):=\{a\in\mathcal{A}\mid a\,\mathrm{k}\langle A\rangle=\{0\}\}\quad \quad\mathrm{ann}_{r}(A):=\{a\in\mathcal{A}\mid\mathrm{k}\langle A\rangle\,a=\{0\}\}
S=\epsilon\int_{t_{1}}^{t_{2}}\frac{1}{c}\sqrt{\eta_{\alpha\beta}\frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt}}dt\Rightarrow L=\frac{\epsilon}{c}\sqrt{\eta_{\alpha\beta}\frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt}}
u^{(k+1)}(t) = \cos(t\sqrt{-\Delta}) u_0 + \frac{\sin(t \sqrt{-\Delta})}{\sqrt{-\Delta}}u_1 + \int_0^t \frac{\sin((t-s)\sqrt{-\Delta})}{\sqrt{-\Delta}}2u^{(k)}_x \wedge u^{(k)}_y (s) ds.
\theta_{\alpha_{j}}^{nd_{j}}(\exp(t)) =e^{nd_{j}\alpha_{j}\left(\frac{1}{n}\left((m_{1}+1)\omega_{1}^{*}+\cdots +(m_{r}+1)\omega_{r}^{*}\right)\right)}=e^{d_{j}\alpha_{j}\left((m_{1}+1)\omega_{1}^{*}+\cdots +(m_{r}+1)\omega_{r}^{*}\right)}&=e^{d_{j}\alpha_{j}\left((m_{j}+1)\omega_{j}^{*}\right)}=e^{2\pi i(m_{j}+1)}=1.
\frac{\mathrm{d}}{\mathrm{d}r}\left\{r^{4/(p_i-1)}v_i(0)\overline v_i(r)\right\}\Big|_{r=1}=v_i(0)\frac{\mathrm{d}}{\mathrm{d}r}\left\{r^{4/(p_i-1)}\overline v_i(r)\right\}\Big|_{r=1}=0,
I_{2}= -\int (u\cdot\nabla) \Lambda^{2r_{1}}u \cdot \Delta u\leq \lVert u\rVert_{L^{\frac{1}{r_{1}- \frac{1}{2}- \epsilon}}}\lVert \Lambda^{2r_{1}}\nabla u\rVert_{L^{2}}\lVert \Delta u\rVert_{L^{\frac{1}{1-r_{1}+ \epsilon}}}
\int \Pi_{\mu\nu\lambda\sigma}A_\mu A_\nu A_\lambda A_\sigma =\frac{1}{2880m^4}\int d^4x\,[14 F_{\mu\nu}F_{\nu\alpha}F_{\alpha\beta}F_{\beta\mu}- 5(F_{\mu\nu}F_{\mu\nu})^2] + \cdots.
\frac{(n+1)p_{n+1}f(n+1,r_{n+1},p_{n+1})}{n p_{n}f(n,r_{n+1},p_{n})}= \left(1-\Theta\left(\frac{g_n}{np_n}\right)\right) \left(1+O \left( \frac 1n \right)\right) < 1,
M(t) =\sum_{i=1}^{d}\sum_{k;k\neq i}\int_{0}^{t}\frac{dW_i(s)}{X^{1/2,\mu}_i(s)-X^{1/2,\mu}_k(s)}, Z(t) =\exp\left(\nu M(t)-\frac{\nu^2}{2}\langle M \rangle(t)\right).
\frac{1}{(q;q)_\infty(x^\alpha;q)_\infty(qx^{-\alpha};q)_\infty}= \frac{1}{\big((1-q)(1-q^2)\dotsm\big)\big((1-x^\alpha)(1-qx^{\alpha})\dotsm\big)\big((1-qx^{-\alpha})(1-q^2x^{-\alpha})\dotsm\big)}&=(1+x^\alpha+x^{2\alpha}+x^{3\alpha}+\dotsm)+q(x^{-\alpha}+2+3x^\alpha+3x^{2\alpha}+\dotsm)+q^2\dotsm.
\frac{1}{\phi_{\bar X_n}(t_1)}\left. \frac{\phi_{(\bar X_n,s_n^2)}(t_1, t_2)}{\partial t_2}\right|_{t_2=0}=\left. \frac{\partial \phi_{s_n^2}(t_2)}{\partial t_2}\right|_{t_2=0}.
\theta[\int_I x^S q(\bar{y}^S)dE]-\theta[\int_I x^R q(\bar{y}^R)dE]=\sum_{s=1}^4 \frac{1}{s!}\theta^{(s)}(\int_I x^R q(\bar{y}^R)dE)\left[\int_I x^S q(\bar{y}^S)dE-\int_I x^R q(\bar{y}^R)dE\right]^s+o(N^{-2}).
\langle \, \rangle =-2i\int_{-\infty}^{0}dx\cos (q_{1}x)e^{-i\omega (q_{1})t}{\mathop{\partial}^{\leftrightarrow}}_{t}\,_{out}\langle p_{1},\dots ,p_{k}|\Phi_{in}(x,t)|q_{2},\dots ,q_{l}\rangle_{in}\; .
\mathbb{F(}z)=\mathbb{F}_{\alpha_{j},\beta_{j},\lambda_{j},\zeta}(z)=\left\{\zeta \int\limits_{0}^{z}t^{\zeta -1}\prod_{j=1}^{n}\left( \frac{\mathbb{E}_{\alpha_{j},\beta_{j}}(t)}{t}\right)^{1/\lambda_{j}}dt\right\}^{1/\zeta},
2 G_{A B}( q_{\phantom{A}\mu}^{A}q_{\phantom{B}\nu}^{B}+ n^{A}n^{B}q_{\mu \nu}) / 3 - G_{\phantom{A}A}^{A}q_{\mu \nu}/ 6 = - 2 \kappa_{5}^{2}[ \tau_{\alpha \beta}q_{\phantom{\alpha}\mu}^{\alpha}q_{\phantom{\beta}\nu}^{\beta}- \tau_{\phantom{\alpha}\alpha}^{\alpha}q_{\mu \nu}/ 4 ] \delta ( y ) / 3
\begin{array}{l}\xi=e_0, \ \ \ \varphi e_1=e_3, \ \ \ \varphi e_2=e_4,\ \ \ \varphi e_3=e_1,\ \ \ \varphi_4=e_2, g(e_i,e_i)=1, \ \ \g(e_i,e_j)=0, \ \ \i,j\in \{0,1,\ldots,4\}, \i\neq j.\end{array}
h \, \left( \, t ;{\bf x},{\bf x}\, | \, M \, \right) = \frac{1}{\left( 4 \, \pi \, t \right)^{d / m}}\: \left[ \, a_{0}\, ( \, x \, | \, M \, ) + a_{1}\, ( \, x \, | \, M \, ) \: t + \dots \, \right] \ .
\sum_{a , b}\eta_{a b}\vartheta \left[ \begin{array}{c}{{a}}\\{{b}}\\ \end{array}\right] \vartheta \left[ \begin{array}{c}{{a}}\\{{b + v_{5}}}\\ \end{array}\right] \prod_{r = 3}^{4}\vartheta \left[ \begin{array}{c}{{a + \frac{1}{2}}}\\{{b + v_{r}}}\\ \end{array}\right] = 0
I_1 ={\sum}^* (e^{g(z_a)}-e^{g(z_a')})e^{ik\cdot h_n(f^{\varphi(a)-n}z_a)}\hat v(z_a,\varphi(a)-n), I_2 ={\sum}^* e^{g(z_a')}(e^{ik\cdot h_n(f^{\varphi(a)-n}z_a)}-e^{ik\cdot h_n(f^{\varphi(a)-n}z_a')})\hat v(z_a,\varphi(a)-n), I_3 ={\sum}^* e^{g(z_a')}e^{ik\cdot h_n(f^{\varphi(a)-n}z_a')}(\hat v(z_a,\varphi(a)-n)- \hat v(z_a',\varphi(a)-n)).
[df|\omega_2|\ldots|\omega_n]=D_{x_0}(1\otimes f\otimes [\omega_2|\ldots|\omega_n])+[p^*(f)\omega_2|\ldots|\omega_n]-f(x_0)[\omega_2|\ldots|\omega_n] \in \mathrm{im}(D_{x_0})+F_{n-1}\mathrm{Sh}(\Omega(C))
0 \longrightarrow R \xrightarrow{\begin{pmatrix}x y z^2\end{pmatrix}}R^3 \xrightarrow{\begin{pmatrix}0 z^2 -y -z^2 0 x y -x 0 \end{pmatrix}}R^3 \xrightarrow{\begin{pmatrix}x y z^2\end{pmatrix}}R \longrightarrow R/I \longrightarrow 0
\mathcal{E}\left(\boldsymbol{\phi}\right) = \mathbf{r}^{H}\mathbf{r}- \frac{1}{N_c}\mathbf{r}^{H}\boldsymbol{\Phi}_{\boldsymbol{\phi}}\mathbf{B}\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{H}\mathbf{r}& = \frac{1}{N_c}\boldsymbol{\phi}^{H}\mathbf{J}_1\left(\mathbf{R}^{H}\mathbf{R}- \mathbf{R}^{H}\mathbf{B}\mathbf{R}\right)^{T}\mathbf{J}_1\boldsymbol{\phi},
\zeta^{[1]}_{n+1}(x)=x \zeta^{[1]}_{n}(x)-\alpha \frac{n(n+3)}{(n+1)(n+2)}\zeta^{[1]}_{n-1}(x)-\gamma \frac{(n-1)(n+4)}{(n+1)(n+2)}\zeta^{[1]}_{n-2}(x)\, , n \geq 2, & \zeta^{[1]}_{0}(x)=1,\zeta^{[1]}_{1}(x)=x, \zeta^{[1]}_{2}(x)=x^{2}-\frac{2}{3}\alpha.
\pi(X K_\lambda)_{j}^{i}c_{k}^{j}\pi(K_{\lambda^\prime}Y)_{\ell}^{k}= \pi(X K_{\lambda}K_{\lambda^{\prime}})_{j}^{i}c_{k}^{j}\pi(Y)_{\ell}^{k}= \pi(X)_{j}^{i}c_{k}^{j}\pi(K_{\lambda}K_{\lambda^{\prime}}Y)_{\ell}^{k}.
q_1(h)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=1}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+1,n_2]_<^t d'_1=n_1+1}}\det\left(-E_{2r+s+1}+M_h\right),
\left( \lambda^{L_{0}}\otimes \lambda^{L_{0}}\right) \; \Delta_{\zeta , z}( A ) \; \left( \lambda^{- L_{0}}\otimes \lambda^{- L_{0}}\right) = \Delta_{\lambda \zeta , \lambda z}\left( \lambda^{L_{0}}A \lambda^{- L_{0}}\right) ,
x \longleftrightarrow\frac{1}{\sqrt{6}}\frac{\mu_{0}}{\mu_{1}}(A+B),y=\frac{i}{\sqrt{6}}\frac{\mu_{0}}{\mu_{1}}(-A+B),z=\frac{1}{\sqrt{6}}\frac{\mu_{0}}{\mu_{1}}(AB-BA) xy \longleftrightarrow\frac{-i}{\sqrt{30}}\frac{\mu_{0}}{\mu_{2}}(A^{2}-B^{2}),x^{2}-y^{2}\longleftrightarrow\sqrt{\frac{2}{15}}\frac{\mu_{0}}{\mu_{2}}(A^{2}+B^{2})
S ( t_{j}^{i}) = \overline{{{t}}}{}_{j}^{i}\, , \ \ \S ( \overline{{{t}}}{}_{j}^{i}) = \Omega_{m}^{i}t_{n}^{m}\, \overline{{{\Omega}}}{}_{j}^{n}\, , \ \ \S ( E_{i}) = - E_{j}\, \overline{{{t}}}{}_{i}^{j}
(\begin{matrix}E_{x}(t)\\E_{y}(t)\\ 0\end{matrix})=(\begin{matrix}E_{0x}e^{i(kz-\omega t+\phi_{x})}\\E_{0y}e^{i(kz-\omega t+\phi_{y})}\\ 0\end{matrix})=(\begin{matrix}E_{0x}e^{i\phi_{x}}\\E_{0y}e^{i\phi_{y}}\\ 0\end{matrix})e^{i(kz-\omega t)}.
M^{(3)}=\begin{pmatrix}1 8 0 0 0 0 0 1 9 0 0 0 0 0 1 9 9 9 0 0 0 1 9 -9 9 0 0 0 1 9 0 0 9 9 0 1 9 0 0 -9 9 0 1 9 9 9 9 9 9\end{pmatrix}
(D_A)_{ii}=\frac{\widehat{\Theta}_{ii}^{-1}-\Theta_{ii}^{-1}}{1 + \delta^2 ({\Theta}_{ii}^{-1}+\rho)(\widehat{\Theta}_{ii}^{-1}+\rho)+\delta ({\Theta}_{ii}^{-1}+\rho)+\delta(\widehat{\Theta}_{ii}^{-1}+\rho)}.
\frac{d}{dx}\bigg({}_2F_1\bigg[\begin{matrix}-a+x \beta &\gamma\end{matrix}\bigg|\,1\bigg]_c\bigg)\bigg|_{x=0}\equiv \frac{d}{dx}\bigg({}_2F_1\bigg[\begin{matrix}p-a+x -b &p-c\end{matrix}\bigg|\,1\bigg]_c\bigg)\bigg|_{x=0}= \frac{\Gamma(p-c)}{\Gamma(p-c+b)}\cdot \frac{d}{dx}\bigg(\frac{\Gamma(a+b-c-x)}{\Gamma(a-c-x)}\bigg)\bigg|_{x=0}\pmod{p^2}.
\left\lbrace\begin{aligned}q^\epsilon(t) =q_0^\epsilon+\frac{1}{\epsilon}\int_{0}^{t}p^\epsilon(s)ds, p^\epsilon(t) =e^{-\frac{t}{\epsilon^2}}p_0^\epsilon+\frac{1}{\epsilon}\int_{0}^{t}e^{-\frac{t-s}{\epsilon^2}}f(q^\epsilon(s))ds+\frac{1}{\epsilon}\int_{0}^{t}e^{-\frac{t-s}{\epsilon^2}}d\beta(s).\end{aligned}\right.
I(\tau) = \exp \left(\int \frac{d^2p}{(2\pi)^2}\; e^{-2\frac{|p|}{n}}\; \hat{K}(p) \;2 \sum_{i,j=1}^{M}\Big( e^{-ipw_i}- e^{-ipz_i}\Big)\Big( e^{+ipw_j}- e^{+ipz_j}\Big) \right)
\|U_h^{n}\|^2+\Delta t\sum_{k=2}^n\|\nabla U_h^{k-\theta}\|^2+\gamma\Delta t\sum_{k=2}^n\|\Sigma_h^{k-\theta}\|^2\leq C(\|U_h^{0}\|^2+\|u_H^{0}\|^2+\Delta t\sum_{k=1}^n\|g^{k}\|^2).
\partial_tU=\Delta U-\nabla P-hu_\infty\cdot\nabla U+\bar g,\quad\mbox{div $U$}=0 (x\in\mathbb R^3,\,t>0), &U\to 0 \quad\mbox{as $|x|\to\infty$}, &U(x,0)=\bar v_0(x).
\nabla\times F\cdot\hat{n}=[(\frac{\partial0}{\partial y}-\frac{\partial M}{\partial z})i+(\frac{\partial L}{\partial z}-\frac{\partial0}{\partial x})j+(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y})k]\cdot k=(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}).
\gamma_{\mathfrak n}(x,y,v) =B\left([x,y]_{\theta,\gamma},v \right)=B_{\mathfrak a}(\theta(x,y),v); \gamma_{\mathfrak n}(x,v,y) =B\left([x,v]_{\theta,\gamma},y \right)=B_{\mathfrak a}(\theta(x,y),v).
a(-A)\nabla_{\eta}P_{s,t}^0f(x) =\sum_{i=1}^\infty a(\lambda_i)\nabla_{\eta}P_{s,t}^0\langle f,e_i\rangle(x)e_i &=\sum_{i=1}^\infty\nabla_{\eta}P_{s,t}^0\langle a(-A)f,e_i\rangle(x)e_i &=\nabla_{\eta}P_{s,t}^0(a(-A)f)(x).
\left( \begin{array}{c c}{{\Pi \partial_{i}\Pi}}&{{0}}\\{{0}}&{{\partial_{i}}}\\ \end{array}\right) + \left( \begin{array}{c c}{{B_{i}}}&{{\langle I_{i}|}}\\{{| J_{i}\rangle}}&{{\hat{R}_{i}}}\\ \end{array}\right)
\begin{gathered}\frac{1}{t-s}\left ( X' \left (X^{-1}(x,t),t \right ) - X' \left (X^{-1}(x,s),s \right ) \right ) = \int_0^1 \left ( (\partial_t X' ) \left (X^{-1}(x,\beta_\tau),\beta_\tau \right ) + \left (\partial_t X^{-1}\right )(x, \beta_\tau) (\nabla_a X')\left (X^{-1}(x, \beta_\tau),\beta_\tau \right ) \right ) d\tau, \end{gathered}
~~~~~~~\Lambda (\alpha,\beta,w,\theta) =-\sum_{n=0}^{\infty}log(1-e^{2\pi i \theta}e^{-2\pi((n+\alpha)w-i\beta)})+log(1-e^{-2\pi i \theta}e^{-2\pi((n+1-\alpha)w+i\beta)}),
\displaystyle \nu = |w|^{2^{*}_{\alpha}}+ \sum_{j\in J}\nu_{j}\delta_{x_{j}},\;\;\;\displaystyle \mu\geq y^{1-2\alpha}|\nabla w_0|^{2}+ \sum_{j\in J}\mu_{j}\delta_{x_{j}},\;\;\; \displaystyle\mu_{j}\geq S(\alpha,n)\nu_{j}^{\frac{2}{2^*_{\alpha}}}.
|(\psi \circ g)'(x)-(\psi \circ g)'(y)| = |\psi'(g(x))g'(x)-\psi'(g(y))g'(x)+\psi'(g(y))g'(x)-\psi'(g(y))g'(y)| \leq |\psi'(g(x))-\psi'(g(y))| \cdot |g'(x)|+|\psi'(g(y))| \cdot |g'(x)-g'(y)|,
\sum^{\lfloor n/e\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{r^{2}}\equiv -J_{e}(n)n^{\phi(n)-2}\phi_{J_{e}}^{(2-\phi(n))}(n)\frac{B_{\phi(p^{l})-1}(\frac{1}{e})}{\phi(p^{l})-1}\pmod{p^{l}},
\sum_{\ell = 1}^{5}x_{\ell}^{\frac{D}{w_{\ell}}}+ a_{0}\prod_{\ell = 1}^{5}\left( x_{\ell}\right) + \sum_{i = 7}^{r + 3}a_{i}\prod_{\ell = 1}^{5}x_{\ell}^{n_{i}^{\ell}}= 0 ,
\begin{aligned}\int_X -u dd^c v \wedge \Omega^{n-1}\leq \int_X -u \Omega_u \wedge \Omega^{n-1}+ &+ B_\Omega \|u\|_\infty^2 \|v\|_\infty C \int_X (\omega^2 \wedge \Omega^{n-2}+ \omega^3 \wedge \Omega^{n-3}+ \omega^4 \wedge \Omega^{n-4}).\end{aligned}
\sum_{1 \leq h_2,\dots,h_k \leq H}\left|\sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j)\right| \ll \frac{kA^2}{\log H}H^{k-1}X.
g^{}([z])(A,B)=\sum_{h,i=1}^n\left[\frac{\left(1+\sum_{k=1}^n\vert t^k_j\vert^2\right)\delta_{h,i}-t^i_j\bar{t}^h_j}{\left(1+\sum_{k=1}^n\vert t^k_j\vert^2\right)^2}\right](\alpha^h\overline{\beta}^i+\overline{\alpha}^i\beta^h),
[ \delta_{1}, \delta_{2}] A_{\mu}^{a}= D_{\mu}( f_{b c}^{a}\epsilon_{1}^{( 0 ) b}\epsilon_{2}^{( 0 ) c}) + f_{b c}^{a}B_{\mu}^{( n ) b}f_{d e}^{c}\epsilon_{1}^{( 0 ) d}\epsilon_{2}^{( 0 ) e},
\rho_{e}^{\textrm{opt}}= \frac{\theta \xi \| \mathbf{h}\|^{2}(N_{T}- 1) \{[1 - (1-\varepsilon)^{\frac{1}{K}}]^{\frac{1}{1-N_{T}}}- 1 \}\delta_{e}^{2}}{(1-\theta)\gamma_{e}^{2}}.
II_{4}=-2\sum_{m=1}^{6}{6\choose m}\int_{0}^{2\pi}(\partial_{x}^{6}\mathcal{J}_{\varepsilon}u^{\varepsilon})(\partial_{x}^{m}\mathcal{J}_{\varepsilon}u^{\varepsilon})(\partial_{x}^{7-m}\mathcal{J}_{\varepsilon}u^{\varepsilon})\dx.
\varphi (g^{n}_1 g_3) = \varphi \big( g^{j}_1 \big) \varphi \big( g^{i}_1 g_3 \big) \mbox{or}\varphi (g^{n}_1 g_3) = \varphi \big( g^{i}_1 g_3 \big) \varphi \big( g^{j}_1 \big) \,.
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}K_{3}(t)g\right\|_{2}\le C(1+t)^{-\frac{n}{4 \sigma}- \frac{\ell+ k-1}{2\sigma}}\| g \|_{1}+C e^{-ct}\| \nabla^{(k-2 \sigma(1-\ell))_{+}}_{x}g\|_{2},
\widehat{D}_0 \pi_a + i \varsigma{{\partial E_a}\over{\partial\phi_i}}\tau_i-i \overline{p}'{{\partial \overline{J}^a}\over{\partial\overline{\phi}_i}}\overline{\tau}_i+iE_a \beta -i \overline{J}^a \overline{\gamma}=0
\frac{p_i}{p_{N+1}}= \frac{-\delta_{i,0}Q + (1-q)^i \sum_{0 \leq j_1 < \cdots < j_i \leq N}\Lambda_{j_1}\cdots \Lambda_{j_i}\prod_{k \in \{0,\dots,N\}-\{j_1, \dots, j_i\}}(1-\Lambda_k)}{(1-q)^{N+1}\Lambda_0 \cdots \Lambda_N}
\frac{l_{i}^{x}\mu_{1}^{x}}{l_{1}^{x}M_{i}^{x}}\le \varphi_{i}\le \frac{l_{i}^{x}M_{1}^{x}}{l_{1}^{x}\mu_{i}^{x}},\thinspace i=1,\thinspace \mathellipsis ,\thinspace m,
2x_{i}\sum_{j\ne i}\sum_{k\ne i,j}x_{j}^{3}x_{k}^{4}+3x_{i}^{2}\sum_{j\ne i}\sum_{k\ne i,j}x_{j}^{2}x_{k}^{4}+4x_{i}^{3}\sum_{j\ne i}\sum_{k\ne i,j}x_{j}^{2}x_{k}^{3}
\begin{aligned}-h_r^{-2}\frac{dh_r}{dr}\alpha+h_r^{-1}\frac{d\alpha}{dr}=h_r^{-1}\mathcal{Q}_rh_ra+\frac{1}{2}*_{g_r}([\mathfrak{A},h_r^{-1}\alpha]+[h_r^{-1}\alpha,\hat{\mathfrak{A}}]), \frac{d\alpha}{dr}=\mathcal{Q}_r\alpha+\frac{1}{2}*_{g_r}([\mathfrak{A},\alpha]+[\alpha,\hat{\mathfrak{A}}])+h_r^{-1}\frac{dh_r}{dr}\alpha\end{aligned}
c \sqrt{\frac{m}{M}}\cdot\frac{n+1-i}{n}\leq \sqrt{1-\xi_i^w(0)}\leq C\sqrt{\frac{M}{m}}\cdot\frac{n+1-i}{n}\;\quad c\sqrt{\frac{m}{M}}\cdot\frac{i}{n}\leq \sqrt{1+\xi_i^w(0)}\leq C\sqrt{\frac{M}{m}}\cdot\frac{i}{n}.
a_{ij}^{(s)}=s\Bigg({a_{ii}^{(s-1)}a_{ij}^{(1)}-\frac{a_{ij}^{(s-1)}}{x_i-x_j}}\Bigg),\ \i\ne j,\ 0\leq i\leq M, &a_{ii}^{(s)}=-\sum\limits_{j=0,j\ne i}^M{a_{ij}^{(s)}},\ \i = j,
\widehat{{m}}_{13,2}=\widehat{{m}}_{12,3}, 2\, \widehat{{m}}_{11,3}-2\,\widehat{{m}}_{13,1}=\widehat{{m}}_{22,3}+\widehat{{m}}_{33,3}2\, \widehat{{m}}_{11,2}-2\,\widehat{{m}}_{12,1}=\widehat{{m}}_{33,2}+\widehat{{m}}_{22,2},
( (U^{(1)}= V^{(k_1)}\mapsto \ldots \mapsto V^{(k_2)}), &((a_h^{(k_1)}, a_j^{(k_1)}), \ldots, (a_h^{(k_2 - 1)}, a_j^{(k_2 - 1)})), &(b^{(1)}= c^{(1)}, \ldots, c^{(k_2)}))
\limsup_{m\to\infty}\frac{1}{m^\beta \theta(m)}\sum_{j=1}^m z_j \ < \infty\,, \qquad \limsup_{m\to\infty}\frac{1}{m^\beta \theta(m)}\sum_{j=1}^m (\log z_j)_- \ < \infty\,, \qquad \liminf_{m \to \infty}\frac{1}{m^{\beta'}\theta(m)}\sum_{j=1}^m z_j = \infty \,, \beta':=\beta^2/(1+\beta),
\left\langle E_{\mathrm{e x t}}^{\mathrm{( s u r f )}}\right\rangle_{S U B}= \frac{\left( 4 \xi - 1 \right) L^{N}a^{2 - D}}{2^{D - 2}\pi^{D / 2 - 1}\Gamma ( D / 2 - 1 )}
\mathcal{A}_{s h G}= \int_{- \infty}^{\infty}d t \left( \int_{\infty}^{0}d x \left[ \frac{1}{2}\partial_{\mu}\Phi \partial^{\mu}\Phi - \frac{m_{0}^{2}}{b^{2}}\cosh b \Phi \right] - M_{0}\cosh \frac{b}{2}\left( \Phi - \phi_{0}\right) \right) \, ,
\overset{\,\,1}{A}_{m}=T^{tr}(\alpha_{+,m},\beta_{+,m})-T^{tr}(\alpha_{+,m},\beta_{-,m}),\overset{\,\,2}{A}_{m}=T^{tr}(\alpha_{+,m},\beta_{-,m})-T^{tr}(\alpha_{-,m},\beta_{-,m}).
\int_{t_0}^0 \int_{\partial\Omega_k}|\partial_tu_k|^2ds\;dt =\int_{t_0}^0\int_{\partial\Omega_k}|d\pi_N(u_k)\partial_{\nu_k}u_k|^2ds\;dt =\int_{t_k+r_kt_0}^{t_k}\int_{\partial B}|u_t|^2d\phi\;dt \le\int_{t_k+r_kt_0}^{T_0}\int_{\partial B}|u_t|^2d\phi\;dt\to 0
e_{j}=(e_{j1},e_{j2},...,e_{jn})=\left( m_{1j}(x_{1}),...,m_{1j}(x_{n})\right) m_{1j}(x_{r}) =\frac{1}{\delta_{j}(x_{1},x_{j})}\left( 1-\frac{k_{j1}k_{1r}}{k_{11}k_{jr}}\right) =\left( 1-k_{jj}^{-1}\right)^{-1/2}\left( 1-\frac{1}{k_{jr}}\right)
\Delta(q, t, p) := \frac{(p^2 q^2; p^2, q^{2})}{(p^2 t^2; p^2, q^{2})}\frac{(p^2 t^{2n}; p^2, q^{2n})}{(p^2 q^{2n}; p^2, q^{2n})},
- \bar{R}_{\mu \nu}F^{\rho \mu}F_{\rho}^{\nu}+ \frac 1 4 F^{\lambda \nu}F^{\mu \rho}F_{\lambda \mu}F_{\nu \rho}-{\frac{1}{1 6}}( F^{\mu \nu}F_{\mu \nu})^{2}+ \frac 1 2 F_{\mu \rho}^{; \rho}F^{\mu \lambda}_{; \lambda}.
q_{a}^2=\frac{\displaystyle \prod_{b=1}^N(\bar{\sigma}_a^2-\lambda_b)}{\displaystyle \prod_{b=1,b\neq a}^N(\bar{\sigma}_a^2-\bar{\sigma}_b^2)}=\frac{\Lambda(\bar{\sigma}_a^2)}{A'(\bar{\sigma}_a^2)}
t_0 = \frac{1}{\sqrt{2 \pi}}\left(\sqrt{\frac{t_3}{v_2}}\right)^{\rho}\sqrt{\frac{t_4}{v_1}}t_5 = \left( 1 - \frac{\rho \lambda^2 v_2}{1 + \lambda v_2}-\frac{(1- \lambda \rho)^2 v_1}{1 + (1 - \lambda \rho) v_1}\right)^{-1}.
\begin{array}{c}\varphi (n) = q^{n-1}(S_{0}(n,q^2) + 2\nu S_{0}(n,-q^2) begin{align*}4mm]= q^{1-n}\{[n]^{(a)}_{q^2}+ 2\nu (-)^{n-1}[n]^{(a)}_{-q^{2}}\}begin{align*}4mm]= [n]^{(b)}_{q}+ 2\nu (i)^{1-n}[n]^{(b)}_{iq}. \end{array}
P(|Y_x^- (t)| \leq \varepsilon) \leq P\left(\bigcap_{i=1}^d \{|Y_{x,i}^-(t)|\leq \varepsilon\}\right) &= P\left(\bigcap_{i=1}^d \{Z_{x,i}^-(t)\leq \varepsilon^2\}\right) &\leq \prod_{i=1}^d P\left( V_i(t)\leq \varepsilon^2\right)
a_n\!=\!\Vert (T_{1n}\!-\!c_{1n}){\bf 1}\Vert^2=1\!-\!c_{1n}^2, \,\,t_n\!=\!\frac{b_n}{a_n}\Big(\sum_{n=3}^m\frac{b_n^2}{a_n}\Big)^{-1}\!=\!-\frac{1}{1+c_{1n}}\Big(\sum_{n=3}^m\frac{1-c_{1n}}{1+c_{1n}}\Big)^{-1}.
\Gamma_{f_a}^{(f_a)_b}(1-t/b)\Gamma^{f_a}_f(1-t/a)= \Gamma_{f_a}^{f_b}(\tfrac{1-t/b}{1-t/a})= \Gamma_{f_b}^{(f_b)_a}(1-t/a)\Gamma^{f_b}_f(1-t/b).