harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(h_1,\widetilde e) \circ (g_{1},g_{2}) \cdot (v,k) = (h_1 g_{1}\cdot v, g_{2}(g_1\cdot v) \cdot k), (g_{1},g_{2})\circ (h_1,\widetilde e) \cdot (v,k) = (g_{1}h_{1}\cdot v, g_{2}( g_1h_1\cdot v) \cdot k ),
\widetilde{x}^{\rm max}_{2T}(t) < 0 \mbox{for $1 \leq t \leq T-1$}, & \widetilde{x}^{\rm max}_{2T}(t) =0 \mbox{at $t=T$}, & \widetilde{x}^{\rm max}_{2T}(t) > 0 \mbox{for $T+1 \leq t \leq 2T-1$}.
\frac{f(\omega h_j,1)}{g(\omega h_j)}= e^{\omega h_j}+ \frac{(-1)^{p+1}(\omega h_j)^{p+1}}{2g(\omega h_j)}\left[\sum_{k=p+1}^\infty \left(\frac{\omega h_j}{2}\right)^k R^{-,(-k)}_{p+1}(1) \right],
\zeta ( s | A ) = \, \mathrm{T r}\, A^{- s}= \int_{{\cal M}}A^{- s}( x , x ) d^{N}x = \frac{1}{\Gamma ( s )}\int_{0}^{\infty}t^{s - 1}\, \mathrm{T r}\, e^{- t A}\, d t \: ,
\boldsymbol{D}_{\!n}= \left( \left( X_0, X_1 \right), \ldots, \left( X_{n-1}, X_n \right) \right) & = \left( \left( S_0 + \varepsilon_0, T(S_0) + \varepsilon_1 \right), \ldots, \left( T^{n-1}(S_0) + \varepsilon_{n-1}, T^n(S_0) + \varepsilon_n \right) \right)
C_0^{S^{\star}}=\begin{cases}\,\,\int_{-\beta^{\star}\eta^{\star}}^{\eta^{\star}-\beta^{\star}\eta^{\star}}{p(\gamma)d\gamma}, \eta^{\star}> 0, \,\,\int_{\eta^{\star}-\beta^{\star}\eta^{\star}}^{-\beta^{\star}\eta^{\star}}{p(\gamma)d\gamma}, \eta^{\star}< 0;\end{cases}
\begin{array}{l}\mbox{H}_{2j}^{\left( k\right)}=\varepsilon^j\left( 1-\varepsilon \right)^j \mbox{H}_{2j+1}^{\left( k\right)}=\varepsilon^j\left( 1-\varepsilon\right)^{j+1}\end{array}\;\forall \;0\leq j\leq \frac{k-1}2
\int_{-\delta}^{\delta}\frac{\vert \varphi(t)\vert dt}{(s-t)^2}=\int_{-\delta}^{0}\frac{\vert \varphi(t)\vert dt}{(s-t)^2}+\int_{0}^{2s}\frac{\varphi(t)dt}{(s-t)^2}+\int_{2s}^{\delta}\frac{\varphi(t)dt}{(s-t)^2},
K_{n}(z)= \frac{1}{2}\left(\frac{z}{2}\right)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\frac{-z}{4}\right)^{k}+(-1)^{n+1}\ln\left(\frac{z}{2}\right)I_{n}(z) &+\frac{1}{2}\left(\frac{-z}{2}\right)^{n}\sum_{k=0}^{\infty}\left(\psi(k+1)+\psi(n+k+1)\right)\frac{\left(\frac{z^{2}}{4}\right)^{k}}{k!(n+k)!},
[\overline{\nabla}_{\phi_k^{(c)}},\overline{\mathcal R}(\phi^{(c)}_i,\phi^{(c)}_j)]=\left(\begin{array}{ccc}0 \ \delta_i^{(c)}\pi \cdot \overline{R}_{ij}^{p^c}\cdot \textup{Fr}^{3c}\ \ \ 0 \ 0 \end{array}\right).
n(2x+y)+n(2x-y) &~=\frac{4n(x)n(y)}{\left(4n(y)^{\frac{2}{9}}-n(x)^{\frac{2}{9}}\right)^9}\Big[256n(y)+2304n(x)^{\frac{2}{9}}n(y)^{\frac{7}{9}}+2016n(x)^{\frac{4}{9}}n(y)^{\frac{5}{9}}&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+336n(x)^{\frac{6}{9}}n(y)^{\frac{3}{9}}+n(x)^{\frac{8}{9}}n(y)^{\frac{1}{9}}\Big]
A_{7}=(\begin{matrix}1&0&0&0&0&0&0\\ 1&2&0&0&0&0&0\\ 1&3&3&0&0&0&0\\ 1&4&6&4&0&0&0\\ 1&5&10&10&5&0&0\\ 1&6&15&20&15&6&0\\ 1&7&21&35&35&21&7\\ \end{matrix})
|E_1(1)-E_2(1)|\le \left|\int_0^1\Big[(n_1-b_1)-(n_2-b_2)\Big]dy\right| \le \int_0^1|n_1-n_2|(y)dy+\|b_1-b_2\|_{C[0,1]}= |n_1-n_2|(\xi)+\|b_1-b_2\|_{C[0,1]}, \quad\exists\xi\in[0,1],
\mathrm{Tr}_{\perp,r}(A):= \sum_{\alpha \ge1}\langle u_{r,\alpha},A\, u_{r,\alpha}\rangle,\mathrm{Tr}_{\perp,\ell}(A):= \sum_{\alpha \ge1}\langle u_{\ell,\alpha},A\, u_{\ell,\alpha}\rangle
d s^{2}= ( 1 - \frac{2 m ( r )}{r})^{- 1}d r^{2}+ r^{2}( d \theta^{2}+ s i n^{2}\theta d \varphi^{2}) - ( 1 - \frac{2 m ( r )}{r}) e^{- 2 \delta ( r )}d t^{2}.
\left\vert \int(\psi\circ R_{k})\,\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}(u-u^{\ast})\,d\mu\right\vert \leq\sup\left\vert\psi\right\vert \,\left\Vert \frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}(u-u^{\ast})\right\Vert_{1}& <\frac{\varepsilon}{3}u,u^{\ast}\in\mathcal{U}
E_n^{-1}(z) A_n^{(3)}(s) E_n(z) = - n^6 \left(E_n^{-1}(z) B_n^{(1)}(s) E_n(z) \right)\left( E_n^{-1}(z) A_n^{(1)}(s) E_n(z) \right)\left(E_n^{-1}(z) B_n^{(1)}(s) E_n(z) \right) &= n^6 \cdot \mathcal{O}\left(n^{-\frac{1}{2}}\right) \cdot \mathcal{O}(1) \cdot \mathcal{O}\left(n^{-\frac{1}{2}}\right)= \mathcal{O}\left(n^{5}\right)
L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}(k_{AF})^{\kappa}\epsilon_{\kappa\lambda\mu\nu}A^{\lambda}F^{\mu\nu}-\frac{1}{4}(k_{F})_{\kappa\lambda\mu\nu}F^{\kappa\lambda}F^{\mu\nu}.
H(x,y,z^I)=\left[\begin{array}{c}\displaystyle A^Ty^E+{\cal J}c(x)^Ty^I begin{align*}4pt]Ax-b+y^E begin{align*}4pt]c(x)+y^I-z^I begin{align*}4pt]\min\{y^I,z^I\}\end{array}\right],
\tilde{Z}_\uparrow(w, z) =\sum_{n=0}^\infty \frac{1}{z^{n+1}}\prod_{i=0}^{n-1}(\lambda+w -\lambda p^i) &= \frac{1}{z}+ w \sum_{n=1}^\infty \frac{1}{z^{n+1}}\prod_{i=1}^{n-1}(\lambda+w -\lambda p^i)
G_n^3-G_{n-1}^3-\frac{3\alpha^{n+2}}{(\alpha-\beta)\alpha(\alpha^5-1)}+\lambda_2<\left(\sum_{k=n}^\infty\frac{1}{G_k^3}\right)^{-1}<G_n^3-G_{n-1}^3- \frac{3\alpha^{n+2}}{(\alpha-\beta)\alpha(\alpha^5-1)}+\lambda_1,
(dK - Kd)(\omega) = -K\bigg(\pi^*(d\sigma) \cdot f + (-1)^k\pi^*(\sigma) \bigg[ dx^i \cdot \frac{\partial f}{\partial x^i}+ d\psi^j \cdot \frac{\partial f}{\partial \psi^j}+ d\eta \cdot \frac{\partial f}{\partial \eta}\bigg] \bigg) &=(-1)^{k-1}\pi^*(\sigma)[f - f(x, \psi, 0)],
\big((\nabla u^h)(A^h)^{-1}\big)(x',hx_3) = Q_0\bar A^{-1}\Big(Id_3 + h\bar A^{-1}J_1^h \bar A^{-1}+ h^2 \bar A^{-1}J_2^h \bar A^{-1}+ J_h^3\Big),
\left({\tau_-}_{\gamma\cdot (x,\xi)}\right)^{-1}\left({U_{\gamma *}}_{| B_-(x,\xi)}(\zeta)\right)=\frac{1}{T_\gamma ( B_-(x,\xi) )}\,\gamma \cdot \left( \left({\tau_-}_{(x,\xi)}\right)^{-1}(\zeta) \right)
\Psi(x_1, x_2, x_3) =\frac{(2x_1, x_1^2 + x_2^2 + x_3^2 - 2x_3, x_1^2 + x_2^2 + x_3^2 - 2x_2)}{x_1^2 + x_2^2 + x_3^2 - 2x_2 - 2x_3 + 2}.
l=1: h_{1}^{a}{}''(0) = \frac{2\sqrt{2}}{3}\frac{\cos(a+\pi/8)}{\sin a}\rho h_{1}^{a}{}'(0) , h_{1}^{a}(0) = h_{1}^{a}{}'''(0) = 0 , l=2: h_{2}^{a}{}'''(0) = \frac{2\sqrt{2}}{5}\frac{\cos(a-\pi/8)}{\sin a}\rho^{3}h_{1}^{a}(0) , h_{1}^{a}{}'(0) = h_{1}^{a}{}''(0) = 0 .
\sup_{y\in \bar{B}(x,r)}d(f(x),f(y)) =\sup_{y\in \bar{B}(x,r)}\big(h_{f(x)}(f_*V,f_*V)^{1/2}+ o(d_g(x,y))\big) &=\sup_{|V|_{g(x)}\leq r}|Df V|_{h(f(x))}+ o(r)
L_{1}=\dfrac{\left\Vert \mathbf{x}_{\ast}(S)\right\Vert_{2}^{2}-\overset{n}{\underset{i=1}{\sum}}\dfrac{1}{d_{i}}}{2}\leq R_{-1}(G)\leq \dfrac{\left\Vert \mathbf{x}^{\ast}(S)\right\Vert_{2}^{2}-\overset{n}{\underset{i=1}{\sum}}\dfrac{1}{d_{i}}}{2}=U_{1}.
F(\mathbf{m}):=\begin{pmatrix}\displaystyle -q m_2 +\frac{1}{2}\int_0^L(m_1\varphi_1+m_2\varphi_2 + g(m_1\varphi_1+m_2\varphi_2))^2\varphi'_1dx \displaystyle q m_1 +\frac{1}{2}\int_0^L(m_1\varphi_1+m_2\varphi_2 + g(m_1\varphi_1+m_2\varphi_2))^2\varphi'_2dx\end{pmatrix},
e^{t/\kappa}g(x+vt,v) - g(x,v) = \int_0^t \frac{e^{s/\kappa}}{\kappa}\rho_g(x+vs) \left( \alpha\mathcal{M}_{T(x+vs)}(v) + (1-\alpha) \mathcal{M}_{\tau(x+vs)}(v) \right) \mathrm{d}s.
\lim_{n\to\infty}\frac{1}{n^{1/4}}\sum_{k=0}^{n^{1/4}}\sigma_\ast^k \widehat{\nu}(E_{n,t_1, \ldots, t_d}(x)) = \lim_{n\to\infty}\mu\left( y \in \Sigma_A: S_ny(t_1,\ldots,t_d) \in \prod_{i=1}^{d}(-\infty, x_i] \right) &= F_{t_1,\ldots,t_d}(x).
{\displaystyle\sum\limits_{n=1}^{N}}d_{4}(n) \sim\frac{1}{6}N\ln(N)^{3}+\frac{4\gamma_{0}-1}{2}N\ln(N)^{2}+(-4\gamma_{1}+6\gamma_{0}^{2}-4\gamma_{0}+1)N\ln(N) & \ \ +(2\gamma_{2}-12\gamma_{1}\gamma_{0}+4\gamma_{1}+4\gamma_{0}^{3}-6\gamma_{0}^{2}+4\gamma_{0}-1)N
\begin{array}{llllllllll}(A_1 \oplus J)^1 = \langle x_1, y_1, h_1 \rangle \oplus \langle h_2 \rangle, (A_1 \oplus J)^2 = \langle x_1, y_1, h_1 \rangle \oplus \langle x_2 \rangle, (A_1 \oplus J)^3 = \langle x_2, y_2, h_2 \rangle \oplus \langle h_1 \rangle, (A_1 \oplus J)^4 = \langle x_2, y_2, h_2 \rangle \oplus \langle x_1 \rangle.\end{array}
\underset{\{\mathbf{p}_{i}\}_{i=1}^N}{\min}\sum_{i=1}^{M}C_i(\mathbf{p}_i,k) - \sum_{i=M+1}^{N}U_i(\mathbf{p}_i,k) & \underline{\mathbf{p}}_i^{[k]}\leq \mathbf{p}_i \leq \overline{\mathbf{p}}_i^{[k]}, \, i =1,\hdots,N & \sum_{i=1}^N \mathbf{p}_i = \mathbf{P}^{[k]},
B_2(t)=\exp\bigg(\mathbf{1}_{t\leq T_0}\cdot\int_{t}^{T_0}D\bigg(\varepsilon_0\cdot\varepsilon_0^{1/100}+\mathbf{1}_{t'\leq T_2}\cdot\varepsilon_0^{1/2}+\frac{1}{t'}+\mathbf{1}_{t'\leq T_3}\cdot\varepsilon_0(\log k_0)^4+T_0^{-1/4}\bigg)\,\mathrm{d}t'\bigg),
C_{p}= [ A \Gamma ( 1 + p ) \zeta_{R}( 1 + p ) ]^{\frac{1 - 2{\cal D}( 0 )}{2 ( 1 + p )}}\frac{\exp \left( \frac{d}{d s}{\cal D}( s ) |_{s = 0}\right)}{[ 2 \pi ( 1 + p ) ]^{\frac 1 2}}\mathrm{,}
R_{p}= q^{\ell_{p}}\prod_{n=1}^{\infty}(1 - q^{n})^{\chi(n)}, S_{p}= q^{a_{p}}\prod_{n=1}^{\infty}\frac{(1 - q^{np})^{b_{p}}}{(1 - q^{n})^{b_{p}}}, \ell_{p}= \sum_{n=1}^{\frac{p-1}{2}}\frac{n(n-p)}{2 p}\chi(n), \frac{p-1}{24}= \frac{a_{p}}{b_{p}}, \gcd(a_{p}, b_{p}) = 1.
\sum_{\substack{1\le i < r 1 \le j \le r}}\bigl([j-i-1] - (j-i-1)\bigr)h_{ij}+ \sum_{1 \le j \le r}([j-r-1] -(j-1))h_{rj}&= r\sum_{i=1}^{r-1}h_{i, \le i}.
\eta(u) = (y^{-1}- y)(\hat{q}^{-1}x^{-1}y^{-1}- \hat{q}yx)^{-1}&=-\hat{q}(y^{-1}- y)(xy -x^{-1}y^{-1})^{-1}&=-\hat{q}v^{-1}\eta(v) = (\hat{q}y^{-1}yx - \hat{q}^{-1}yx^{-1}y^{-1})(\hat{q}^{-1}x^{-1}y^{-1}- \hat{q}yx)^{-1}&= -q(x-x^{-1})(xy-x^{-1}y^{-1})^{-1}&= -v^{-1}u
\varphi_{n}\circ f-\varphi_{n}-\beta = \varphi_{n}( f_{n}+ g_{n})-\varphi_{n}-\beta & = \varphi_{n}\circ f_{n}-\varphi_{n}-\beta+ \sum_{i=1}^{+\infty}\frac{\varphi_{n}^{(i)}( f_{n})}{i!}g_{n}^{i}& = o(\mathrm{e}^{-(\beta_{n}+\nu_{n})\zeta}),
\mu\tau =\mu(\sigma\alpha)(0, d(\lambda)\vee d(\mu)-d(\mu))=(\mu\sigma\alpha)(0, d(\lambda)\vee d(\mu)) &=(\lambda\beta)(0, d(\lambda)\vee d(\mu))=\lambda\beta(0, d(\lambda)\vee d(\mu)-d(\lambda))\in \Lambda^{d(\lambda)\vee d(\mu)},
\kappa_{2p}( \nu_t ) = \nu_t (X^{2p}) - \sum_{\begin{array}{c}{\scriptstyle \sigma \in NCE(2p)}{\scriptstyle \sigma \neq 1_{2p}}\end{array}}\ \Bigl( \, \prod_{W \in \sigma}\kappa_{{}_{|W|}}( \nu_t ) \, \Bigr).
P_L(\tilde{r};0) = -P_{\rm affine}\left( -\tilde{r}; \lambda_L, \omega_L, \frac{\beta}{a_2}\right), T_L(\tilde{r};0) = T_{\rm affine}\left( -\tilde{r}; \lambda_L, \omega_L, \frac{\beta}{a_2}\right),
(p_{1,5}+p_{4,5})(9) =4,9=1+1+1+1+1+1+1+1+1=4+1+1+1+1+1=4+4+1, (p_{1,5}+p_{4,5})(8) =3, 1+1+1+1+1+1+1+1=4+1+1+1+1=4+4, (p_{1,5}+p_{4,5})(5) =2, 1+1+1+1+1=4+1, (p_{1,5}+p_{4,5})(2) =1, 1+1.
S_{e f f}^{2 + 1}= - 2 \pi \int d \tau \sqrt{g_{\tau \tau}}, g_{\tau \tau}= ( \frac{d \xi^{0}}{d \tau})^{2}- h_{\alpha \beta}( \xi ) \frac{d \xi^{\alpha}}{d \tau}\frac{d \xi^{\beta}}{d \tau}\; \; ,
r-\sum_{j\ne i}z_j^*(|v_j|-|v_i|) \ge r-\sum_{j=1}^{i-1}z_j^*(|v_j|-|v_i|) & \ge r-\sum_{j=1}^{i-1}(|v_j|-|v_i|) = r-A(i-1)>0.
\sum_{\substack{a_{i,j}\in \mathbb F_q, i=1,\dots,n, j=1,\dots m \prod_{j=1}^m (1+ \sum_{i=1}^n a_{i,j}x^i) \equiv 0 \mod x^{n+1}}}\psi \left( \sum_{j=1}^m a_{n,j}\right) = q^{m \frac{n+1}{2}-1}\mu
\varphi_{\alpha_{1}, \alpha_{2}+ m , \alpha_{3}- m}\, , \; \; \varphi_{\alpha_{1}- n , \alpha_{2}, \alpha_{3}+ n}\, , \; \; \varphi_{\alpha_{1}+ p , \alpha_{2}- p , \alpha_{3}}\, , \; \; m , n , p \in \mathrm{\bf ~ N}\, ,
A_1 \,=\, \begin{pmatrix}\widehat u_1^2 + p \widehat u_1 \widehat u_2 - \langle \widehat u_1\widehat u_2\rangle \end{pmatrix}~, A_2 \,=\, \begin{pmatrix}(m+\widehat u_2)\widehat u_1 (m+\widehat u_2)\widehat u_2 +p \end{pmatrix}~, B = \begin{pmatrix}0 \widehat u_1 \partial_1 m\end{pmatrix}~.
A \ = \ \{1, 6, 10, 22, 24, 30 \}, B \ = \ \{7, 12, 13, 15, 19, 27 \}, C \ = \ \{3, 4, 17, 18, 23, 28 \}, D \ = \ \{2, 9, 11, 16, 26, 29 \}, E \ = \ \{5, 8, 14, 20, 21, 25 \},
\sum_{j=1}^\infty \frac{A_j^2}{Z_j^p}\leq \frac{\cos^2(Z_1)}{Z_1^{p-1}\left[Z_1 + \sin(Z_1)\cos(Z_1) \right]}+ \frac{1}{\pi^p}\sum_{j=2}^\infty\frac{1}{(j-1)^p}.
B_{p}^{(\alpha ,\beta ;m)}(x,y) =\int_{0}^{1}t^{x-1}(1-t)^{y-1}{}_{1}F_{1}(\alpha ;\beta ;\frac{-p}{t^{m}(1-t)^{m}})dt. (Re(p) >0,Re(x)>0,Re(y)>0,Re(\alpha )>0,Re(\beta )>0)
A(z)=A_{\infty,0}(x) + A_{0,1}(x)z^{-1}+ A_{1,1}(x)(z+1)^{-1}+ A_{a,1}(x)(z+a)^{-1}, begin{align*}0.3cm] B(z)=\begin{pmatrix}0 1 z+u(x) 0\end{pmatrix}, \mbox{where}u(x) = \frac{-2 v^{\prime}(x^{2})x^{2}+v(x^{2})^{2}-2q(x^{2})+v(x^{2})}{x^{2}},
\{f, g\}_{PB}=\frac{\partial f}{\partial x_1}\frac{\partial g}{\partial p_1}+\frac{\partial f}{\partial x_2}\frac{\partial g}{\partial p_2}- \frac{\partial f}{\partial p_1}\frac{\partial g}{\partial x_1}-\frac{\partial f}{\partial p_2}\frac{\partial g}{\partial x_2}\;,
\lim_{k\rightarrow\infty}\frac{f(n_k)-1}{\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)^{\delta}}=\lim_{k\rightarrow\infty}\frac{f\left(\epsilon_k\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)\right)-1}{\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)^{\delta}}=q.
{\mathcal L}_{D3}={\cal O}(\Theta^0) + Z^{-1}\left(-\frac{i}{2}\overline{\Theta}\Gamma^i D_i \Theta - \frac{i}{48}e^{\phi}\overline{\Theta}\Gamma^{mnp}\Theta Re \left[(*_6 G)_{mnp}- i G_{mnp}\right]\right)
\big\|m_r-m_{n,r}\big\|_1 = \int^1_0 |W_r(t) - W_{n,r}(t)|\,d\mu(t) = \sum^{2^n-1}_{j=0}\int^{(j+1)/2^n}_{j/2^n}|W_r(t) - W_{n,r}(t)|\,d\mu(t) \le \sum^{2^n-1}_{j=0}\int^{(j+1)/2^n}_{j/2^n}\epsilon \,d\mu(t) = \epsilon,
\int_{|u-\alpha_1|>\frac{d}{M}}\frac{\,\mathrm{d}u}{\big((u-\alpha_1)^2+(v-\alpha_2)^2\big)^{\frac{p}{2}}}\leqslant{}\int_{|u-\alpha_1| >\frac{d}{M}}\frac{\,\mathrm{d}u}{|u-\alpha_1|^p}={}\frac2{p-1}d^{1-p}M^{p-1},
R_{+}( q ) = \left( \begin{matrix}{q^{1 / 2}}&{0}&{0}&{0}\\{0}&{q^{- 1 / 2}}&{q^{1 / 2}- q^{- 3 / 2}}&{0}\\{0}&{0}&{q^{- 1 / 2}}&{0}\\{0}&{0}&{0}&{q^{1 / 2}}\\ \end{matrix}\right) \ .
5 x_{1}+ 1 0 \dot{4}0 0 0 = 5 [ - 4 ( a - 2 5 )^{2}+ 2 5 0 0 ] + 1 0 4 0 0 0 = - 2 0 ( a - 2 5 )^{2}+ 1 1 6 5 W
\mathrm{SO}(n)_{\mathbb{C}}=\mathrm{SO}(n;\mathbb{C}),\quad\mathrm{SU}(n)_{\mathbb{C}}=\mathrm{SL}(n;\mathbb{C}),\quad\mathrm{U}(n)_{\mathbb{C}}=\mathrm{GL}(n;\mathbb{C}).
[ \delta_{{\tilde{\epsilon}}{'}}, \delta_{\tilde{\epsilon}}] \psi_{i}= - 2 i{\tilde{\epsilon}}{\tilde{\epsilon}}{'}f_{i j k}J_{j}\psi_{k}- 2 i{\tilde{\epsilon}}{\tilde{\epsilon}}{'}f_{j l m}f_{j i k}x_{m}x_{k}{\dot{\psi}}_{l}\; ,
\psi_{h}= \varphi_{b_1h}^B \circ \varphi_{a_1h}^A\circ\varphi_{b_2h}^B \circ \varphi_{a_2h}^A\circ\varphi_{b_2h}^B \circ \varphi_{a_1h}^A\circ \varphi_{b_1h}^B,
\zeta_{AV,2}(s_1,s_2,s_3) = \sum_{n=1}^\infty \sum_{m>n}\frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}&= \left( \sum_{n \leq N}\sum_{n<m \leq M}+ \sum_{n \leq N}\sum_{m> M}+ \sum_{n=N+1}^\infty \sum_{m>n}\right) \frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}&= A + B + C,
\hat{U}^\dagger(t_2,t_1)=\hat{U}^{-1}(t_2,t_1)=\hat{U}(t_1,t_2)\ \ ,\ \ \hat{U}(t_3,t_2)\,\hat{U}(t_2,t_1)=\hat{U}(t_3,t_1)\ ,
\beta = \alpha\wedge \left(\omega^{15}+\omega^{26}+\omega^{37}+\omega^{48}-s \, \omega^{58}\right)\ =\ \omega^{1567}- \omega^{2568}- \omega^{3578}+ \omega^{4678}-s \, \widetilde{\kappa},\quad \phi = \, \hat{\phi}+s\,\beta +\frac{s^2}{2}\, \widetilde{\kappa}
\sum_{k=0}^{M}[4k-1]\frac{(q^{-1};q^2)_k^4}{(q^2;q^2)_k^4}q^{4k}\equiv-(1+3q+q^2)[n]^4\pmod{[n]^4\Phi_n(q)}, begin{align*}5pt] \sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a;q^2)_k(q^{-1};q^2)_k^2}{(q^{2}/a,aq^2;q^2)_k(q^2;q^2)_k^2}q^{4k}\equiv0\pmod{[n]^2(1-aq^n)(a-q^n)},
N(t) =\int_{\mathbb{R}}g_+p(t,g)dg=\int_{\mathbb{R}}g_+dg\int_{\mathbb{R}}G_t(g-y)p_{t,0}(y)dy &=\int_{\mathbb{R}}p_{t,0}(y)dy\int_{\mathbb{R}}g_+G_t(g-y)dg &=\int_{\mathbb{R}}(N(B(t)+y,C(t)))p_{t,0}(y)dy.
\delta_\pi(f)= \ [\pi, f]_{\Omega}, d_T(\theta)= \ \lambda P[\pi, \theta]_{\Omega}+P\big[[\pi, T]_{\Omega}, \theta\big]_{\Omega}= \ \lambda[\mu_V, \theta]_{\Omega}+\big[[\pi, T]_{\Omega}, \theta\big]_{\Omega}, h_T(f)= \ \sum_{k=1}^n\frac{1}{k!}\lambda^{n-k}P\big[\ldots[f, \underbrace{T]_{\Omega},\ldots, T}_k\big]_{\Omega}.
\phi_{k,i}= \frac{2-n}{2}\frac{\lambda_{i}\gamma_{n}\nabla_{a_{i}}G^{\frac{2}{2-n}}_{a_{i}}}{1+\lambda_{i}^{2}\gamma_{n}G^{\frac{2}{2-n}}_{a_{i}}}\varphi_{i}+\frac{\nabla_{a_{i}}u_{a_{i}}}{\lambda_{{i}}u_{a_{i}}}\varphi_{i},
|u_i| \leq ||\vec{v}_1||_2\cdot ||\vec{v}_2||_2 \cdots ||\vec{v}_{n-1}||_2 \leq ||\vec{v}_1||_1\cdot ||\vec{v}_2||_1 \cdots ||\vec{v}_{n-1}||_1 \leq \left(\frac{||\vec{v}_1||_1 + ||\vec{v}_2||_1 + \cdots + ||\vec{v}_{n-1}||_1}{n-1}\right)^{n-1}= \left(\frac{||\vec{v}||_1}{n-1}\right)^{n-1}
S = \sum_{a_1,\dots,a_n=0}^{b-1}\left(\frac{\sigma_n(a_n)}{b}+\dots+\frac{\sigma_1(a_1)}{b^n}\right)\left(\frac{a_1}{b}+\dots+\frac{a_n}{b^n}\right) = \sum_{a_1,\dots,a_n=0}^{b-1}\sum_{k_1,k_2=1}^{n}\frac{\sigma_{k_1}(a_{k_1})a_{k_2}}{b^{n+1-k_1}b^{k_2}}.
{\rm Tutte}(\Gamma,x,y)= \{(0,11,25,20,7,1)_{x}, (11,46,39,8)_{x}, (32,52,12)_{x}, \hphantom{{\rm Tutte}(\Gamma,x,y)=}{}(40,24)_{x}, (29,6)_{x}, 15, 5, 1 \}_{y}.
(u \otimes v)^{\natural}_{st}:= - \int_s^t \big[ \dot{\mu}_{r}\otimes \delta v_{sr}+\delta u_{sr}\otimes \dot{\nu}_r\big] dr + u_{st}^{\natural}\otimes v_s + u_s \otimes v^{\natural}_{st}+ u_{st}^{\flat}\otimes v_{st}^{\flat}+ u_{st}^{\flat}\otimes B_{st}^1 v_s + A_{st}^1 u_s \otimes v_{st}^{\flat}.
\begin{array}{c}{{D_{T M}^{( - 1 )}( t ) = \displaystyle{\frac{- \left({n_{0}}^{2}\, R^{2}\, t \, \left( 8 \, a^{2}+{n_{0}}^{2}\, R^{2}\, t^{2}\right) \right)}{8 \,{\left( a^{2}+{n_{0}}^{2}\, R^{2}\, t^{2}\right)}^{2}\,{\sqrt{1 + \frac{{n_{0}}^{2}\, R^{2}\, t^{2}}{a^{2}}}}}- \frac{\left( 2 +{n_{0}}^{2}\, t^{2}\right) \, Z^{2}}{2 \, t^{3}\,{\sqrt{1 +{n_{0}}^{2}\, t^{2}}}}},}}\\ \end{array}
\eta^{2}\cdot \tau^{2}{\cal{G}}\; \; : = \; \;{\cal{Y}}_{[ p_{1}, . . . , p_{N}]}\circ \left({\frac{1}{4}}\sum_{i , j , k , l = 1}^{N}\left( \eta^{( i j )}\eta^{( k l )}+ 2 \eta^{( i k )}\eta^{( l j )}\right) \tau^{( i j )}\tau^{( k l )}\right){\cal{G}}
\frac{8}{\pi^6}\sum_{i=1}^\infty \frac{i}{(2i-1)^{3}}= \frac{1}{\pi^6}\Big(\zeta\Big(2,-\frac{1}{2}\Big)-4+\frac{1}{2}\Big(\zeta\Big(3,-\frac{1}{2}\Big)-8\Big)\Big).
\gamma \Delta \vec{k}_{l}= \gamma ( \vec{k}_{l + 1}- \vec{k}_{l}) = \frac{\Lambda_{l}}{4 M^{4}}\vec{n}_{l}, ( l = 1 , 2 , \dots , L ) , \mathrm{w i t h}\vec{k}_{L + 1}= \vec{k}_{1},
\sum_{i_1,i_2,\ldots,i_n \geq 0}\frac{m_{i_1 i_2 \cdots i_n}}{i_1 ! i_2 ! \cdots i_n !}t_1^{i_1}t_2^{i_2}\cdots t_n^{i_n}\,\, =\,\,\sum_{\ell = 1}^k \lambda_\ell \,\cdot\,{\rm exp}(t_1 \mu_{\ell 1}+ \cdots + t_n \mu_{\ell n}) \cdot{\rm exp}\biggl( \frac{1}{2}\sum_{i,j=1}^n \sigma_{\ell ij}t_i t_j \! \biggr).
\|(-B)^{-1}\|=(1-\lambda_1(B))^{-1}\leq\max_{1\leq i\leq k}(1-\lambda_1(B_i))^{-1}+1 \leq n(\ln n)^{-\frac{1}{2}}e^{3C_0-c_0-(\ln n)^{\frac{1}{2}}/2+1}+1=O\left(n(\ln n)^{-\frac{1}{2}}e^{-(\ln n)^{\frac{1}{2}}/2}\right).
\delta^{k_{1}^{j}-1}(\varphi^{j}) =\varphi_{[k_{1}^{j}-1]}^{j}(\zeta_{[-q_{1},-1]},x) \delta^{k_{1}^{j}}(\varphi^{j}) =\varphi_{[k_{1}^{j}]}^{j}(\zeta_{[-q_{1},-1]},x,u)
B_{p}(\chi)v =\lim_{\zeta\to\chi}D_{\zeta}(\chi/\zeta)(1+\zeta/\chi)^{2}A_{p}(\chi,\zeta)v &=\lim_{\zeta\to\chi}D_{\zeta}(\chi/\zeta)(1+\zeta/\chi)^2 \sum_{N\in\mathbb{Z}}\chi^{N}f_{N}(1,\zeta/\chi)v.
\sup_{\ell=1,\ldots,k}\P_{\bf U}(\tau_{0,\ell}\ge j)\leq \P_{\bf U}({\theta_k}\ge j)\le \P_{\bf U}\left(\cap_{i=1}^{j-1}\{U_i\le q^\star_{k+i-1}\}\right)=\prod_{i=k}^{k+j-2}q_i^\star
{D}^{(1/2)}(R_w):={D}^{(1/2)}({\bf p},{\bf n})=\frac{p_0-{\bf p}\cdot{\bf n}+m-i{\sigma}\cdot({\bf p}{\times}{\bf n})}{\sqrt{2(p_0+m)(p_0-{\bf p}\cdot{\bf n})}},\quad{D}^{(1/2)}({\bf p},{\bf n})\,{D}^{\dagger(1/2)}({\bf p},{\bf n})=1.
e^{(n-\nu-2)t}\int_X|Ric(\omega(t))+\omega(t)|^2_{\omega(t)}\omega(t)^n &=e^{(n-\nu-2)t}\partial_t\left(\int_XS(t)\omega(t)^n\right)+e^{(n-\nu-2)t}\int_X(S(t)+1)(S(t)+n)\omega(t)^n &=\partial_t\left(e^{(n-\nu-2)t}\int_XS(t)\omega(t)^n\right)+e^{(n-\nu-2)t}\int_X(S(t)^2+(\nu+3)S(t)+n)\omega(t)^n.
R_{\hat{g}}( z , \bar{z}) = - 1 \to R_{g}( z , \bar{z}) = - e^{4 \pi G ( z , w )}\left( 1 + 8 \pi e^{- \varphi ( z , \bar{z})}\delta^{( 2 )}( z - w ) +{\frac{1}{4 \chi ( \Sigma )}}\right) .
\frac{1}{2}\int_{0}^{2\pi}\int_{0}^{2\pi}\left[ \left( \frac{1}{w^{1/\alpha}\left( x\right)}-\frac{1}{w^{1/\alpha}\left( y\right)}\right) \left(w\left( x\right) -w\left( y\right) \right) \right] dxdy=0.
\partial_i \tilde{T}^{(r-1)}+ \sum_{l=0}^r (\sum_{j=1}^{m}(\partial_{i}{h^{j}}) (A_j^{(l)}\circ h) \tilde{T}^{(r-l)}-\tilde{T}^{(r-l)}\tilde{A}_i^{(l)})=0 & (r-1)\tilde{T}^{(r-1)}+\sum_{l=0}^r ( (B^{(l)}\circ h ) \tilde{T}^{(r-l)}-\tilde{T}^{(r-l)}\tilde{B}^{(l)}) =0,
\gamma_4 = \frac{\wp'(U_1)^2 - 4 \wp(U_1)^3 - \big(\wp'(\alpha)^2 - 4\wp(\alpha)^3\big)}{4\big(\wp(U_1)-\wp(\alpha)\big)}, \gamma_6 = -\frac{\wp(\alpha) \big(\wp'(U_1)^2 - 4 \wp(U_1)^3\big) - \wp(U_1) \big(\wp'(\alpha)^2 - 4\wp(\alpha)^3\big)}{4\big(\wp(U_1)-\wp(\alpha)\big)},
\Psi_{( 3 , 3 , 2 )}= \prod_{i < j}( z_{i}^{( 1 )}- z_{j}^{( 1 )})^{3}\prod_{i < j}( z_{i}^{( 2 )}- z_{j}^{( 2 )})^{3}\prod_{i , j}( z_{i}^{( 1 )}- z_{j}^{( 2 )})^{2}\; \Psi_{0}.
\mathbf{R}_n^2\left\|\frac{f}{|\cdot|^2}\right\|^2+\widetilde{\mathbf{R}}_n\left\|\frac{P^{\perp}f}{|\cdot|^2}\right\|^2 =\mathbf{R}_n^2\left\|\frac{Pf}{|\cdot|^2}\right\|^2+(\mathbf{R}_n + n-1)^2\left\|\frac{P^{\perp}f}{|\cdot|^2}\right\|^2 \le\|\Delta Pf\|^2+\|\Delta P^{\perp}f\|^2=\|\Delta f\|^2.
f_{ij}\,^{k}\widetilde{f}^{mn}\,_{k}=\widetilde{f}^{m'n}\,_{j}f_{im'}\,^{m}+\widetilde{f}^{m'n}\,_{i}f_{m'j}\,^{m}, &f_{ij}\,^{k}\widetilde{f}^{mn}\,_{k}=\widetilde{f}^{mn'}\,_{j}f_{in'}\,^{n}+\widetilde{f}^{m'n}\,_{j}f_{im'}\,^{m}, &f_{ij}\,^{k}\widetilde{f}^{mn}\,_{k}=\widetilde{f}^{mn'}\,_{j}f_{in'}\,^{n}+\widetilde{f}^{mn'}\,_{i}f_{n'j}\,^{n}.
\frac{\partial_r [w(\theta)]^\top \mathbb{E}_\theta [f(\textbf{X})]}{\mathbb{E}_\theta [h(\textbf{X}) + w(\theta)^\top f(\textbf{X}) ]}= \frac{\partial_r [w(\theta)]^\top \bar{f}}{\bar{h}+ w(\theta)^\top \bar{f}},\quad~r\in\{1,\ldots,k\}.
P(x_2x_1x_3x_4) = \P(x_2x_1x_4x_3) + P (x_2x_1dx_4dx_3) + P (x_2x_1[x_3,x_4]) P(dx_2dx_1x_3x_4) = \P(dx_2dx_1x_4x_3) + P (dx_2dx_1dx_4dx_3) + P (dx_2dx_1[x_3,x_4]) P([x_1,x_2]x_3x_4) = \P([x_1,x_2]x_4x_3) + P ([x_1,x_2]dx_4dx_3) + P ([x_1,x_2][x_3,x_4]).
K_{\bf a}^{( 1 )}( x , x^{\prime}, t ) \simeq -{\frac{e^{-{\frac{( | x | + | x^{\prime}| )^{2}}{4 t}}}}{4 \sqrt{\pi t}}}{\frac{1}{\sqrt{| x | | x^{\prime}|}}}{\frac{1}{\ln{\frac{2 t k_{0}}{| x | + | x^{\prime}|}}}}.
\frac{d}{dt}\sum_{|\alpha|=0}^{m}\langle A_{2}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle +G_{0}\left(\|v\|_{m}^{2}+\|\nabla v\|_{m}^{2}\right)\leq\frac{4C\epsilon}{2}\|\nabla v\|_{m}^{2}+C\left\lbrace\|f_{2}\|_{m-1}^{2}\right. &+C\left.(\mu_{0}(t)+\mu_{1}(t))\left(\|u\|_{m}^{2}+\|v\|_{m}^{2}+\|w\|_{m}^{2}\right)\right\rbrace,
\langle \Psi | \Psi^{\prime}\rangle = \int_{{\cal M}_{L}\times{\cal E}_{L}}{\cal D}^{e}e \;{\cal D}^{e}\widetilde{x}\left( \mathrm{V o l}_{e}{\cal D}_{L}\right)^{- 1}\overline{{\Psi}}\Psi^{\prime}\; \; \; .
\tilde{D}_n(V):=\begin{cases}-\mathrm{i}\boldsymbol{\sigma}\cdot \nabla + \sigma_3 + Vn=2 -\mathrm{i}\boldsymbol{\alpha}\cdot \nabla + \beta + V n=3 \end{cases}\mathsf{C}_{0}^{\infty}(\mathbb{R}^n\setminus\{0\};\mathbb{C}^{2(n-1)}),
h^{\underline{a}}_{\i}\, \dot{\xi}^{\underline{a}}_{\alpha}-\varepsilon_{ij}\, B^{\underline{a}}_{\alpha}\,\xi^{\underline{a}}_{\alpha}\, \dot{x}^{j}-{\cal E}^{\underline{a}}_{j;\alpha}\,\xi^{\underline{a}}_{\alpha}= 0.
[Free~ Channel]~~\quad \Omega_F(\psi_0)\equiv s\lim_{t\to \infty}U_0(0,t)J_c\psi(t) &\psi(t)=U(t,0)\psi_0, &J_c= U_0(t,0)J(|x|/t^{\alpha}\leq 1)U_0(0,t), &0\leq \alpha\leq a(n,p)<1,
\begin{bmatrix}D_{-\beta}\alpha C \beta B A_{-\alpha}\end{bmatrix}\begin{bmatrix}u_1 u_2\end{bmatrix}=\begin{bmatrix}D_\alpha -\beta C -\alpha B A_\beta \end{bmatrix}\begin{bmatrix}u_1 u_2\end{bmatrix}- \gamma \begin{bmatrix}v_1 v_2\end{bmatrix},
[v,w]+J[Jv,w] + J[v,Jw]-[Jv,Jw] = 0+J[Xu,w]+J[v,Au]-[Xu+Yv+Zw,Au+Bv+Cw] = XJv-XJw+AJv-X[u,Bv+Cw]+A[u,Yv+Zw] = XJv-XJw+AJv+XBv-XCv+XCw-AYv+AZv-AZw = (XB-XC-AY+AZ)v+(XC-AZ)w+(X+A)Jv-XJw = (XB-XC-AY+AZ+XY+AY-XB)v + &+(XC-AZ+XZ+AZ-XC)w+(X^2+AX-XA)u
f^{[2]}(z_0,1,z_0) =\frac{d}{dz}f^{[1]}(z,1)_{\vert z=z_0}=\frac{d}{dz}\Big(\frac{f(z)-f(1)}{z-1}\Big)_{\vert z=z_0}= \frac{d}{dz}g(z)_{\vert z=z_0}=g^{[1]}(z_0,z_0).
A_{0}^{\left( 1 \right)}= \frac{c \hbar}{\left| e \right|}f_{1}\left( r \right) \, , \, \; \, A_{1}^{\left( 1 \right)}\, \, = \frac{c \hbar}{\left| e \right|}\frac{A \left( r \right)}{r}\sin \varphi \, , \; \, A_{2}^{\left( 1 \right)}= - \frac{c \hbar}{\left| e \right|}\frac{A \left( r \right)}{r}\cos \varphi \, , \; \, A_{3}^{\left( 1 \right)}= \frac{c \hbar}{\left| e \right|}
\check{\Phi}_{1}= \check{x}_{-}- \check{\zeta}\tau = 0 , \check{\Phi}_{2}= \check{\pi}_{-}- \frac{\left[ \check{\pi}_{a}+ e A_{a}( \check{x}_{-}) \right]^{2}+ m^{2}}{2 \check{\pi}_{+}}.
\delta m = ( \dots ) \int_{- \infty}^{\infty}d k \sqrt{k^{2}+ 2 m^{2}}\left[ \int d x \frac{e^{i k x}}{\cosh^{2}\left( \frac{m x}{\sqrt{2}}\right)}\right] \left[ \int d y \frac{e^{- i k y}}{\cosh^{2}\left( \frac{m y}{\sqrt{2}}\right)}\right]
Q_0v^iv^j\geq ~ (\alpha+1)CF^{-\alpha-1}\left(\sum_k\dot{f}^k\kappa_k^2-F\kappa_1\right) = ~ (\alpha+1)CF^{-\alpha-1}\sum_k\dot{f}^k\kappa_k\left(\kappa_k-\kappa_1\right)~\geq~0