harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[ \tilde{S}_{+}, \tilde{S}_{-}]_{P . B .}= - 2 i H ( S_{3}) \tilde{S}_{3}+ i S_{+}\displaystyle \frac{\partial A_{4}}{\partial S_{3}}\bar{C}_{3}C^{+}- i S_{-}\displaystyle \frac{\partial A_{4}}{\partial S_{3}}\bar{C}_{3}C^{-}~ , [ \tilde{S}_{3}, \tilde{S}_{\pm}]_{P . B .}= \mp i \tilde{S}_{\pm}~ ,
\sum_{k=0}^{m}\left( -1\right)^{k}\genfrac{[}{]}{0pt}{}{m}{k}\mathcal{E}_{k}^{\left( \alpha\right)}\left( \lambda\right)=\frac{2^{\alpha}\lambda^{m}m!}{\left( \lambda+1\right)^{\alpha+m}}\binom{\alpha+m-1}{m}.
\Lambda(r,Z)=\frac{\Gamma^2(1/2+ir)}{\Gamma(1+2ir)}\int_{0}^{\infty}\frac{g\left( a/y\right)}{y^{3/2-ir}}F\left( \frac{1}{2}+ir,\frac{1}{2}+ir,1+2ir;-y\right)dy,
a_n = \sum_{j=0}^n h_1(j) \prod_{i=j+1}^n f(i) = a_{n-d}\prod_{i=n-d+1}^n f(i) + \sum_{j=n-d+1}^n h_1(j) \prod_{i=j+1}^n f(i) & = a_{n-d}\prod_{i=n-d+1}^n f(i) + \sum_{j=0}^{d-1}h_1(n-j) \prod_{i=0}^{j-1}f(n-i) = a_{n-d}\prod_{i=n-d+1}^n f(i) + f_d(n)
A_{i,k}:= \left[ i a - 6 \left\lfloor \frac{i}{3}\right\rfloor, i a \right]_2, B_{i,k}:= \left[ i a + \frac{a}{2}- 3 - 6 \left\lfloor \frac{i-1}{3}\right\rfloor, i a + \frac{a}{2}- 1 \right]_2,
\widetilde{\mathsf{V}}\cdot \big( \mathsf{D}_{t}^{q}\cdot P \big) \big\vert_{U_{x,p}}= H_{0}^{q}\,P + H_{1}^{q}\, \big(\mathsf{D}_{t}^{1}\cdot P\big) + \dotsb + H_{p}^{q}\, \big(\mathsf{D}_{t}^{p}\cdot P\big){\scriptstyle(q=0,1,\dotsc,p)}.
\delta_{\lambda',\lambda}:Y_{\lambda'}=Y_{\lambda'}\times_{Y_\lambda}Y_\lambda\hookrightarrow Y_{\lambda'}\times_{Q_n}Y_{\lambda}= Z_{\lambda',\lambda}, \delta_{\lambda,\lambda'}:Y_{\lambda'}=Y_\lambda\times_{Y_\lambda}Y_{\lambda'}\hookrightarrow Y_{\lambda}\times_{Q_n}Y_{\lambda'}= Z_{\lambda,\lambda'}.
cr(\psi_\alpha(Q')) =\frac{((\sqrt 2+1)^\alpha+(\sqrt 2+1)^\alpha)((\sqrt 2-1)^\alpha+(\sqrt 2-1)^\alpha)}{-((\sqrt 2+1)^\alpha-(\sqrt 2-1)^\alpha)^2}&=-\frac{4(\sqrt 2+1)^\alpha(\sqrt 2-1)^\alpha}{((\sqrt 2+1)^\alpha-(\sqrt 2-1)^\alpha)^2}=-\frac{4}{((\sqrt 2+1)^\alpha-(\sqrt 2-1)^\alpha)^2}~.
\lim_{\varepsilon\to0}\frac{\int_{(-\varepsilon, \varepsilon)}u^2 m(\d u)}{\varepsilon^2|\log \varepsilon|}= \lim_{\varepsilon\to0}\frac{c\int_{(-\varepsilon, \varepsilon)}u^2 \int_{0}^\infty |r{\rm e}^{{\rm i}u}-1|^{-(2+\alpha)}r\d r \d u}{\varepsilon^2|\log \varepsilon|}=+\infty
d\tilde\theta_t=\begin{cases}\gamma_t F(\mathcal{\tilde X}_t,\tilde\theta_t)dt+\gamma_tH(\mathcal{\tilde X}_t,\tilde\theta_t)^{\top}dV_t, \tilde\theta_t\in\Theta 0, \tilde\theta_t\notin\Theta\end{cases},
\left\langle H \right\rangle_\Omega = \left\langle \Omega\right|\int{dx_1}\left({- \frac{1}{2}\Pi_1 \left({1 -\frac{{e^2}}{\pi}\frac{1}{{\partial_1^2}}}\right)^{-1}\Pi^1}\right)\left| \Omega \right\rangle .
J^w(\textbf{X}_{minRSSU}^{(n)}) = -\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}w(F^{-1}(u)) (1-u)^{2i-2}f(F^{-1}(u))du\right) & \geq -\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}kM (1-u)^{2i-2}du\right) &= -\frac{(n!)^2 k^n M^n}{2(2n-1)!!}.
\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}x=\frac{\Gamma(1+1)}{\Gamma(1-\frac{1}{2}+1)}x^{1-\frac{1}{2}}=\frac{\Gamma(2)}{\Gamma(\frac{3}{2})}x^{\frac{1}{2}}=\frac{1}{\frac{\sqrt{\pi}}{2}}x^{\frac{1}{2}}.
[f]_{\kappa}^{\bar{n}}=\prod_{j=0}^{n-1}f(z+j\kappa)=\prod_{j=0}^{n-1}\exp \left(\pi(z) \omega^{z/\kappa}\omega^{j}\right)=\exp \left(\pi(z) \omega^{z/\kappa}\sum_{j=0}^{n-1}\omega^j\right)\equiv e^0=1.
\int_{2^{k+2}I_Q}\|R_j\partial_jg_k(\cdot,t)\|_{L^1(2^{k+3}Q_1)}dt \lesssim\int_{2^{k+2}I}(2^{k+3}\ell(Q))^{N/p}\|R_j\partial_jg_k(\cdot,t)\|_{L^q(2^{k+3}Q_1)}dt &\lesssim\int_{2^{k+2}I_Q}(2^{k+3}\ell(Q))^{N/p}\|\partial_jg_k(\cdot,t)\|_{L^q(2^{k+3}Q_1)}dt &\lesssim (2^k\ell(Q))^{N/p}2^k\ell(Q)(2^k\ell(Q))^{N/q}\frac1{2^k\ell(Q)}&=(2^k\ell(Q))^N,
\left\lfloor \frac{e_{\mathcal{H}'_1 \cup \mathcal{H}'_2 \cup \mathcal{H}'_3 \cup \mathcal{H}'_4}(A',B')}2 \right\rfloor = \left\lfloor \frac{e_{G}(A',B')}2 \right\rfloor\ge \left\lfloor \frac{F}2 \right\rfloor
v^{\prime}* v^{\prime \prime}= ( b^{\prime}e^{- \lambda l_{-}^{\prime \prime}}+ e^{\lambda l_{-}^{\prime}}b^{\prime \prime}, l_{+}^{\prime}e^{- \lambda l_{-}^{\prime \prime}}+ e^{\lambda l_{-}^{\prime}}l_{+}^{\prime \prime}, l_{-}^{\prime}+ l_{-}^{\prime \prime})
\langle x_1 \boxtimes y_1 , x_2\boxtimes y_2 \rangle_{K_\lambda \boxtimes_N K_{\lambda^{-1}}}=\langle \langle x_2|x_1\rangle_N y_1, y_2\rangle_{K_{\lambda^{-1}}}=\langle (x_2^*x_1) y_1, y_2\rangle_{K_{\lambda^{-1}}}&=\phi(y_2^* x_2^*x_1y_1)=\tau(y_2^*x_2^*x_1y_1)=\langle x_1 y_1 , x_2y_2\rangle_{L^2(N)}.
p_D(t,x,y)\ge \int_{B_1}\ldots\int_{B_{k-1}}p_D(t/k,x,y_1)\ldots p_D(t/k, y_{k-1},y) dy_{k-1}\ldots dy_1 &\ge (c_2t(Tk)^{-1})^k\Pi^{k-1}_{i=1}|B_i| \ge c_3 (c_4t(Tk)^{-1})^{c_5k}\ge c_6 (c_7t(TR_1)^{-1})^{c_8R_1}\ge c_9 (t(TR_1)^{-1})^{c_{10}R_1}.
\Delta \Gamma_{\mu i}^{j}= \frac{1}{3}( \breve{D}_{\mu}\epsilon_{i l}) \epsilon^{l j}+ \frac{1}{3}\varepsilon_{\mu}^{k}\varepsilon_{i}^{\nu}( \breve{D}_{\nu}\epsilon_{k l}) \epsilon^{l j}+ \varepsilon_{\mu}^{k}\tilde{\zeta}_{( i l k )}\epsilon^{l j}
\psi_{x,y,z}(a,b)\psi_{x,y-1,z-1}(a,b)=\psi_{x,y-1,z}(a,b)\psi_{x,y,z-1}(a,b)+\psi_{x+1,y-1,z-1}(a,b)\psi_{x-1,y,z}(a,b),
\Sigma ( q ) = \int \frac{d^{4}r}{\left( 2 \pi \right)^{4}}\phi_{0}( r ) e^{- i q r}, \qquad \tilde{\pi}_{0}( q ) = \int \frac{d^{4}r}{\left( 2 \pi \right)^{4}}\pi_{0}( r ) e^{- i q r}.
\sum_{0\leq k\leq s/3}C_{0,s-3k,2k}(n_{1})b_{4}^{s-3k}(E_{1})b_{6}^{2k}(E_{1})=\sum_{0\leq k\leq s/3}C_{0,s-3k,2k}(n_{2})b_{4}^{s-3k}(E_{2})b_{6}^{2k}(E_{2})
\begin{pmatrix}\mathcal A \mathcal B \mathcal B^t \mathcal D \end{pmatrix}^{-1}= \begin{pmatrix}\mathbf I_\ell + \mathcal{BD}^{-1}\mathcal B^t -\mathcal{BD}^{-1}-\mathcal D^{-1}\mathcal B^t \mathcal D^{-1}\end{pmatrix}^{-1}.
=\frac{\epsilon^{0ijk5}}{(4!)^2}\left\{\partial_0\left[\frac{\sigma^\prime}{((\sigma^{\prime}/b)^2-(\dot \sigma/n)^2)^2}\right]-\partial_5\left[\frac{\dot\sigma}{((\sigma^\prime/b)^2-(\dot \sigma/n)^2)^2}\right]\right\}=0.
\Pi_{4}= \frac{g^{4}}{4}\int \frac{d^{3}p}{( 2 \pi )^{3}}\, \frac{N ( \omega )}{2 \omega}\left[ \frac{1}{\hbar^{2}k^{2}+ 2 \hbar k \cdot p}+ \frac{1}{\hbar^{2}k^{2}- 2 \hbar k \cdot p}\right] \, .
A_0^\prime(y)(4\eta(y,t)+\sigma(y,t))+\frac{\kappa^2}{3}\left(\xi_0^{\prime \prime}(y)\delta \xi(y,t)-\xi_0^\prime(y) \delta \xi^\prime(y,t)+(\xi_0^\prime (y))^2\frac{\delta b(y,t)}{b_0}\right)=0+ O(m_B^4/k^4).
V_{n,k}^{BB,\alpha,\theta}(N)=\frac{\binom{N}{k}\big(\frac{-\alpha \Gamma(\theta+\alpha)}{\Gamma(\theta+\alpha+n)}\big)^k}{\big(\frac{\Gamma(\theta+\alpha)\Gamma(\theta+n)}{\Gamma(\theta+\alpha+n)\Gamma(\theta)}\big)^{N}}.
\frac{1}{2}\left( \frac{4c^2(c-1)^2}{(c+1)^3}+ \sqrt{-3}\cdot \frac{2c(c-1)^2\sqrt{-3c}}{3(c+1)^2}\right) = \frac{(-\sqrt{c})^3 (c-1)^3}{(c+1)^3}\cdot \frac{\sqrt{c}-1}{\sqrt{c}+1}
\sum_{\gamma = \beta - \alpha}^{\beta}c_{\beta - \gamma \ \gamma}\left({\frac{1}{2 \alpha^{\prime}}}\right)^{\gamma + \alpha - \beta}\left( \begin{matrix}{{\gamma}}\\{{\beta - \alpha}}\\ \end{matrix}\right) B ( 2 \beta - \alpha - \gamma + \frac{1}{2}, \gamma + \alpha - \beta + \frac{1}{2}) = 0 \ .
\sum_{1\leq i, j\leq n}\Psi(v_iv_i^*v'_j){v'_j}^*= \sum_{j=1}^n \Psi\left(\sum_{i=1}^n v_iv_i^*v'_j\right){v'_j}^*= \sum_{j=1}^n \Psi\left(pv'_j\right){v'_j}^*= \sum_{j=1}^n \Psi(v'_j){v'_j}^*.
(h_{r})_{ab}= \frac{1}{2r}\partial_{s}|_{s=r}(s^{2}\sigma_{ab}) = \tilde{\sigma}_{ab}- \frac{2}{3}R_{ikjl}X^{k}X^{l}\partial_{a}X^{i}\partial_{b}X^{j}r^{2}+ \mathcal{O}(r^{4})
\Bigl(P_{{\bf b},d}~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d-1}}\Bigr)^{(-1)}&=\prod_{\substack i=1}^{s+1}(\theta-t)^{q^{d-1}}\mathbb{D}_{a_i-2}^{q^{d-a_i}}\alpha^{q^{d-2}}P_{{\bf b},d}+\prod_{j=1}^{s}(\theta-t)^{q^{d-1}}\mathbb{D}_{b_j-2}^{q^{d-b_j}}P_{{\bf b},d}~_{s+1}\mathcal{F}_s({\bf a};{\bf b})(\alpha)^{q^{d-1}}.
\int_0^tf(s)dD_s = \sum_{k=1}^{D_t}f(\tau_k) = \sum_{k=1}^{P_{\Lambda(t)}}f(\Lambda^{-1}(\sigma_k)) \sim \int_0^{\Lambda(t)}f(\Lambda^{-1}(s))ds = \int_0^tf(s)\kappa_sds.
\mathcal{E}_\pi(T_1, \ldots, T_n) = E\left( T'_{q_1}\cdots T'_{q_{jk_{m-p+2}-1}}L_{b'}T_\theta L_{\mathcal{E}_{\pi|_{V}}((T_1, \ldots, T_n)|_{V})}\right) &= \mathcal{E}_{\pi|_{W}}\left(\left.\left(T_1, \ldots, T_{\theta-1}, T_\theta L_{\mathcal{E}_{\pi|_{V}}\left((T_1,\ldots, T_n)|_{V}\right)}, T_{\theta+1}, \ldots, T_n\right)\right|_{W}\right)
T_{F,G,N}^{\Xi,L,\widetilde{\mathbf{r}}}(f_1,\dots,f_d) = \frac{1}{C_{\Xi,\chi}\eta^{h}}\widetilde{T}_{F,G,N}^{\Xi,L,\widetilde{\mathbf{r}}}(f_1\ast \chi,\dots,f_d\ast \chi) + O_{C,c,\varepsilon}(\eta/\sigma_G) + O_{C,c,\varepsilon}(\eta/\sigma_F N).
R(x) =\left[\left({n \choose g}-\alpha\right)-\left({n \choose g}-\alpha\right)\frac{n-g}{n}-\frac{n-g+1}{n}{n \choose g-1}\right] x^{n-g}+ +\sum_{j=g+1}^d\left[a_j-a_j\frac{n-j}{n}-a_{j-1}\frac{n-k+1}{n}\right] x^{n-k}+a_d\frac{n-d}{n}x^{n-d-1}
\operatorname{ad}^*_{\operatorname{Ad}_h\xi}\operatorname{Ad}^*_{h^{-1}}\alpha=\operatorname{Ad}^*_{h^{-1}}\operatorname{ad}^*_\xi\alpha,\frac{d}{dt}(\operatorname{Ad}^*_{h^{-1}}\alpha) =-\operatorname{ad}^*_{\delta^rh}\operatorname{Ad}^*_{h^{-1}}\alpha,
P(\partial_{\xi_1},\ldots,\partial_{\xi_n})\mathcal{F}(f)(\xi) =\mathcal{F}(P(i x_1,\ldots,i x_n)f)(\xi), \mathcal{F}(P(\partial_{x_1},\ldots,\partial_{x_n})f)(\xi) =P(-i \xi_1,\ldots,-i \xi_n)\mathcal{F}(f)(\xi)
D_i^{(r)}:= T_{i,i;i-1}^{(r)}1 \le i \le n, r > 0 E_i^{(r)}:= T_{i,i+1;i}^{(r)}1 \leq i < j \leq n, r > s_{i,j}F_i^{(r)}:= T_{i+1, i; i}^{(r)}1 \leq i < j \leq n, r > s_{j,i}.
\Pi_{h}^{i j}= - \left( \begin{matrix}{{0}}&{{2 h^{2}}}&{{2 h^{1}}}\\{{- 2 h^{2}}}&{{0}}&{{( ( h^{1})^{2}+ ( h^{2})^{2}+ 1 ) - e^{2 h^{0}}}}\\{{- 2 h^{1}}}&{{- ( ( h^{1})^{2}+ ( h^{2})^{2}+ 1 ) + e^{2 h^{0}}}}&{{0}}\\ \end{matrix}\right)
\Xi_n^{0}(z_1,z_2, s)=4\sum_{b> 0}\frac{(\bar{z_2}-\bar{z_1}-b)^n~(z_2-\bar{z_1}-b)^n}{|z_2-z_1-b|^{2s}~|\bar{z_2}-z_1-b|^{2s}}.
\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,q^{-1};q^2)_k}{(q^{2}/a,aq^2,bq^2,q^2;q^2)_k}(bq^4)^k begin{align*}5pt]\: \:\equiv[n](bq)^{(n+1)/2}\frac{(q^{-2}/b;q^2)_{(n+1)/2}}{(bq^2;q^2)_{(n+1)/2}}\pmod{[n](1-aq^n)(a-q^n)},
G^i(p_1,p_2) = \; G(p_1,p_2) - L(p_1,p_2) + \L (p_1,p_2) - L_{R_{i}}+\L_{R_{i}}(p_1,p_2) - L_{R_{j}}+\L_{R_{j}}(p_1,p_2) + L_{R_{ij}}-\L_{R_{ij}}(p_1,p_2)
{n(t-1)^2\over t+n}=1+nt-(1+n){t(1+n)\over t+n},{(t-1)^2\over t+(1/n)}=n+t-(1+n){t\bigl(1+{1\over n}\bigr)\over t+{1\over n}},{(t-1)^2\over t+s}={1\over s}+t-{1+s\over s}\cdot{t(1+s)\over t+s}.
E_{0,l}= \lambda[8ab_1 + 2a(2l+d) - 2b_1 - l(l+d-1)], \psi_{0,l}(r) \propto r^l (1+\lambda r^2)^{-a}\exp[-\lambda (b_1+2b_2) r^2 - \lambda^2 b_2 r^4],
S ={\frac{1}{1 6 \pi \alpha^{\prime 4}}}\int d^{1 0}x \sqrt{| \hat{G}|}\, e^{- \phi}\left[ \hat{R}_{1 0}+ ( \hat{\nabla}\phi )^{2}-{\frac{1}{1 2}}\hat{H}^{2}\right] \, ,
\dot{x}(t,s) = \lambda x(t,s) + \partial_s^2 x(t,s), s\in [0,1], &\dot{\hat x}(t,s) = \lambda \hat x(t,s) + \partial_s^2 \hat x(t,s) &\qquad+ \int_0^1 l(s) \left(\partial_s^2 x(t,\theta) - \partial_s^2 \hat x(t,\theta)\right)d\theta, &x(t,0) = 0, x(t,1) = 0, &\hat x(t,0) = 0, \hat x(t,1) = 0.
M^{( \pm )}( \nu ; k ) \equiv 1 + ( \nu^{2}- \frac{1}{4}) \int_{1}^{\infty}\frac{\psi ( \alpha ) \alpha^{\frac{1}{4}}[ \alpha^{\frac{\nu}{2}}+ \alpha^{\frac{- \nu}{2}}]}{( \alpha \pm i k )}d \alpha
\frac{1}{2}\| \partial_t \varphi_{\varepsilon}(t) \|^2_{V^*}+ \nu \int_{Q_t}|\nabla \partial_t \varphi_{\varepsilon}|^2 + \ell \gamma \int_{Q_t}|\partial_t \varphi_{\varepsilon}|^2 = \frac{1}{2}\| \partial_t \varphi_{\varepsilon}(0) \|^2_{V^*}
\omega_{\{3,3\}}:=\left\{(\cos\frac{2k\pi}{3},\sin\frac{2k\pi}{3},0,0)\right\}_{k=0}^2 \cup \left\{(0,0,\cos\frac{2k\pi}{3},\sin\frac{2k\pi}{3})\right\}_{k=0}^2.
z = e^{- 8 \pi^{2}/ e^{2}}M^{3 N_{c}- N_{f}}\Bigg ( \frac{m_{\lambda}^{2 / 3}}{B^{1 1 / 1 2}}\Bigg )^{N_{c}}\Big ( m_{q}^{2 / 3}m_{s q}^{1 / 3}\Big )^{N_{f}}
I_G^{\vee}= \bigcap_{1 \leq i_1 < i_2 < \dots < i_c \leq s}\ \bigcap_{k = 1,\ldots,c \atop 1 \le j_k \le e_k}(x_{i_1,j_1}, x_{i_2, j_2}, \dots , x_{i_c, j_c}),
|f_k(y_n^*) - \sum_{j=1}^{m_1}y_n^*(x^k_j)| \leq |f_k(y_n^*) - \sum_{j=1}^{r_n}y_n^*(x^k_j)| + |\sum_{j=m_1+1}^{r_n}y_n^*(x^k_j)| \leq \epsilon/8 + \sum_{j=m_1 + 1}^\infty |y_n^*(x^k_j)| < \epsilon/8 + \epsilon/8 < \epsilon/4.
\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)}&= \left< Op_h \left(\{\sigma(-h^2\Delta+V(x)-E(h)), \sigma(\chi_\alpha(x_n) hD_n)\}\right)u_h, u_h \right>_{L^2(\Omega_\Gamma)}+ O(h),
L_{k,p,q}\leq C_{H,l_{0}}|I_{k}|\cdot\left(\sum_{l=1}^{k-1}\frac{|I_{l}|}{(|I_{l}|+\cdots+|I_{k-1}|)^{H+\frac{1}{2}}}\cdot\left(\frac{|I_{1}|+\cdots+|I_{k}|}{|I_{1}|+\cdots+|I_{l-1}|}\right)^{H-\frac{1}{2}}\right)^{2}.
\{{\cal P}_{\alpha},{\cal J}_{\beta}\}= - 4 i P^{2}( J \gamma )_{\alpha \beta}+ 6 i ( P J ) ( P \gamma )_{\alpha \beta}- 4 \Gamma_{\alpha \beta}( P J ) + \frac{i}{4}\left( ( 1 - \nu^{2}) P^{2}- 1 6 ( P J )^{2}\right) \epsilon_{\alpha \beta},
\frac{\overline{W}_{\xi,\eta}(i)}{\overline{W}_{\xi,\eta}(i-1)}= M_{\xi}M_{\eta}\frac{X_{\xi}\omega - X_{\eta}\omega^{i}}{Y_{\eta}- Y_{\xi}\omega^{i}}\qquad \frac{\overline{W}_{\xi',\eta'}(i)}{\overline{W}_{\xi',\eta'}(i-1)}= M_{\xi'}M_{\eta'}\frac{X_{\xi'}\omega - X_{\eta'}\omega^{i}}{Y_{\eta'}- Y_{\xi'}\omega^{i}}\qquad
\frac{\partial^{n}}{\partial a^{n}}I=\int\frac{x^{2n}}{2^{n}}e^{-ax^{2}/2}dx=\frac{1\cdot3\cdot5...\cdot(2n-1)}{2\cdot2\cdot2...\cdot2}\sqrt{2\pi}a^{-\frac{2n+1}{2}}
\begin{pmatrix}1 0 a_{1,3}0 1 a_{2,3}0 0 0\end{pmatrix},\begin{pmatrix}1 a_{1,2}0 0 0 0 0 0 1\end{pmatrix},\textrm{or}\begin{pmatrix}0 0 0 0 1 0 0 0 1\end{pmatrix}.
-\left\{\int_{0}^{T}\int_{\Omega}\left(H(u,k)\,\frac{\partial\phi}{\partial t}+ \displaystyle\sum_{j=1}^{d}Q_{i}(u,k)\,\frac{\partial\phi}{\partial x_{j}}\right)\right\}\,dx\,dt\leq M\,d\,\int_{\Gamma}H(u,k)\,\phi\,dr.
\sum_{n}\frac{1}{\omega_{n}}\partial_{i}\phi_{n}\partial_{j}\phi_{n}^{*}=\lim_{\vec{y}\rightarrow \vec{x}}\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial y^{i}}\zeta(s=\frac{1}{2}\mid \vec{x},\vec{y}).
\begin{aligned}\Gamma:=\left\{g\in PGL(4,\mathbb{Z}[i])\mid g^{\dagger}Jg=J\right\},\;g^{\dagger}:=\,^{t}\bar{g},\,\,J:=\left(\begin{smallmatrix}0 E_{2}-E_{2}0\end{smallmatrix}\right), \Gamma_{T}:=\Gamma\rtimes\langle T\rangle,T:W\mapsto\,^{t}W\,\,(W\in\mathbb{H}_{2}), \Gamma_{M}:=\left\{gT^{a}\in\Gamma_{T}\mid(-1)^{a}\det(g)=1,a=0,1\right\}.\end{aligned}
Q(x)=\frac{x}{1-e^{-x}}=1+\frac{x}{2}+\sum_{i=1}^{\infty}\frac{B_{2i}}{(2i)!}x^{2i}=1+\frac{x}{2}+\frac{x^{2}}{12}-\frac{x^{4}}{720}+\cdot\cdot\cdot
\widehat{A}_1\xi =-\frac{106}{85}\xi^{(1)}(-1;1)-\frac{4}{85}\xi^{(1)}(3;1)-3\xi(-1) &=\frac{106}{85}\xi(-2)+\frac{149}{85}\xi(-1)+\frac{4}{85}\xi(2)-\frac{4}{85}\xi(3).
{\mathcal{L}}= i{\overline{{{\Psi}}}}\gamma^{\mu}\partial_{\mu}\Psi - m{\overline{{{\Psi}}}}\Psi - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}+ i e \overline{{{\Psi}}}{\gamma^{\mu}}{A_{\mu}}{\Psi}
E_{switch}=V^{2}\int_{T_{off}}^{T_{on}}\frac{dt}{M(q(t))}=V^{2}\int_{Q_{off}}^{Q_{on}}\frac{dq}{I(q)M(q)}=V^{2}\int_{Q_{off}}^{Q_{on}}\frac{dq}{V(q)}=V\Delta Q
c_{\mu}L_{r}(k)\xi(\eta, H_{v}', k, 0) = (-1)^{r'}2^{r}D_{\boldsymbol{K}}^{r'(k - r)}\left( H'_{\infty}\right)^{k-r}\times \prod_{i=1}^{r}L(i-k, \chi_{\boldsymbol{K}}^{i-1})^{-1}\textbf{e}(i HY).
\frak{h}_k=\frak{h}_k^++\frak{h}_k^-+\frak{h}_k^0\ \ \mbox{where}\ \ \frak{h}_k^+=\Re\sum_{n>0}h_{n,k}z^n\ \ ,\ \ \frak{h}_k^-=\Re\sum_{n<0}h_{n,k}z^n\ \ \mbox{and}\ \ \frak{h}_k^0=\frak{h}_{0,k}+C_{\eta,k}^0\,\log|z|\ .
\frac{\partial u}{\partial t}(t,x) =\big(Lu(t,\,.\,)\big)(x)+f(t,x)+bu(t,x)\frac{\d B^H}{\d t}, u(0,x) =x_0(x),\x\in \mathcal{O}, u(t,x) =0,\ (t,x)\in [0,T]\times \partial\mathcal{O},
\Psi_j\left({\bf r},t\right)=\frac{{\rm e}^{i{\bf k}_j\cdot{\bf r}-iE_{{\bf k}_j}t}}{\sqrt{2\vert E_{{\bf k}_j}\vert\Omega}}\,,\quad \Phi\left({\bf r},t\right)=\frac{g}{\mu_0^2}\frac{N}{2\vert E_{{\bf k}_j}\vert\Omega}\,,\quad\Omega\rightarrow\infty\,,
\operatorname{Cov}( \sqrt{n_1}\overline{X}_1, \sqrt{n_2}\overline{X}_2 ) = \frac{r_1b^2 \sigma_B^2 + r_1b \sigma_\epsilon^2}{\sqrt{r_1 r_2}b}= \sqrt{\frac{r_1}{r_2}}b \sigma_B^2 + \sqrt{\frac{r_1}{r_2}}\sigma_\epsilon^2.
(l_{(p,g)})^\ast[\gamma_p^\ast(\tau^\ast(\pi^\ast_2\omega))] =(l_{(p,g)})^\ast[(\pi_2\circ \iota_p)^\ast\omega] &=(\pi_2\circ \iota_p)^\ast\omega &=\gamma_p^\ast(\tau^\ast(\pi^\ast_2\omega))
\sum_{c \bmod{\varpi^{l-1}}}\widetilde{e}\left(\frac{c}{\varpi^{l-h-1}}\right) = N(\varpi^h) \sum_{c_2 \bmod{\varpi^{l-h-1}}}\widetilde{e}\left( \frac{c_2}{\varpi^{l-h-1}}\right).
a_2= [a_0, a_1]+ b_0 b_1^t -b_1 b_0^t =2 g_0 g_1 J^3 +2 \epsilon^{x y z}\bar a_0^x \bar a_1^y \bar J^z+\frac{1}{2}\vec \beta_0 \cdot \vec \beta_1 (J^3 + \bar J^3) = 0
\Sigma^{\left( 1\right)}\left({z_{1}},{z_{2}}\right) =-\mathcal{V}\left(z_{1}-z_{2}\right){{\mathcal{G}}^{\left( 0\right)}}\left({z_{1}},{z_{2}}\right) +\delta \left( z_{1}-z_{2}\right) \int dz\mathcal{V}\left( z_{1}-z\right){{\mathcal{G}}^{\left( 0\right)}}\left({z},{z}\right) \;.
\frac{d}{dt}\left[\begin{array}{c}u(t) u^{[1]}(t) \end{array}\right]=\left[\begin{array}{cc}Q(t) 1 \lambda-Q^2(t)-p(t) -Q(t) \end{array}\right]\left[\begin{array}{c}u(t) u^{[1]}(t)\end{array}\right]\ ,
{\rm ev}_{0}^{\mathfrak{m}}(\psi) =\sum_{n,m=0}^{\infty}\sum_{\substack{a_{1},\dots,a_{n}\in\{0,z\}b_{1},\dots,b_{m}\in\{1,1-z\}}}\big(1\otimes1\otimes{\rm ev}_{0}^{\mathfrak{m}}(e_{a_{1}}\cdots e_{a_{n}}){\rm ev}_{0}^{\mathfrak{m}}(e_{b_{1}}\cdots e_{b_{m}})\big)\cdot\tilde{L}\big({\rm Cut}_{(a_{1},\dots,a_{n};b_{1},\dots,b_{m})}(\psi)\mid_{z\to0}\big)
K_{T}( \zeta_{1}, \zeta_{2}) : = \left\langle \mathrm{T}\exp \zeta_{1}\mid \exp \zeta_{2}\right\rangle = \left\langle \exp \zeta_{1}\mid \mathrm{T}^{\prime}\exp \zeta_{2}\right\rangle = \left( \exp \zeta_{1}\mid \mathrm{T}^{\prime}\exp \zeta_{2}\right) .
\theta_{12}=(\frac{2C^1_{2,1}}{C^1_1}-\frac{C^1_1}{\mu})\omega_1,~~~~d\omega_1=(\frac{C^1_1}{\mu}-\frac{C^1_{2,1}}{C^1_1})\omega_1\wedge\omega_2,~~~~d\omega_2=0.
\begin{array}{l}\lbrack q^{-},\alpha_0^{+}]=i\,, \left[ \alpha_n^{-}\,,\alpha_m^{+}\right] =n\,\delta_{n+m,0}\,\,\,, \left[ s_n\,,s_m\right] =\left( \frac k2-1\right) \,n\,\delta_{n+m,0}\,\,\,, \left[ L_n^{\prime}\,,L_m^{\prime}\right] =(n-m)L_{n+m}^{\prime}\end{array}
P_{\cal T}^{1/2}\left(k\right) = \sqrt{k^3 \over 2 \pi^2}\left|{v_k \over m_{\rm Pl}a}\right| ={2^{\nu - 3/2}\over 2 \pi}{\Gamma\left(\nu\right) \over \Gamma\left(3 /2\right)}\left({H \over m_{\rm Pl}}\right) \left|C_{\rm+}\left(k\right) + C_{-}\left(k\right)\right| \left({y \over 1 - \epsilon}\right)^{3/2 - \nu},
\nabla_{\mu}\phi_{\mu_1\dots\mu_k}= (\partial_{\mu}+ \omega_{\mu})\phi_{\mu_1\dots\mu_k}- \sum_{i=1}^{k}\Gamma_{\mu_{i}\mu}^{\lambda}\phi_{\mu_1\dots\mu_{i-1}\lambda\mu_{i+1}\dots\mu_k}
\mathcal R_n := \left\{\frac{p(z)}{q(z)}=\frac{p_{n-1}z^{n-1}+p_{n-2}z^{n-2}+\cdots+p_0}{z^n+q_{n-1}z^{n-1}+\cdots+q_0}:~ p\in\mathcal P_{n-1},~q\in\mathcal Q_n\right\}
\Delta W^0_4(p,p_1,p_2,p_3) =1\otimes W^0_4(p,p_1,p_2,p_3)+W^0_4(p,p_1,p_2,p_3)\otimes 1+ W^0_3(p,p_1,\bar{q})\otimes W^0_3(\bar{q},p_2,p_3)+W^0_3(q,p_1,p_2)\otimes W^0_3(p,q,p_3).
\left\Vert e^{k\left\vert x\right\vert^{p}}u\left( x,0\right) \right\Vert_{X}=a_{k}\leq a_{2}e^{2a_{1}k^{\frac{q}{q-p}}}=a_{2}e^{2a_{1}k^{\frac{1}{2-p}}}.
B_j^-\Omega_\lambda = \sum_{i=j}^{n}\sum_{s=1}^{i-j+1}\sum_{I\in\mathcal{I}_{ji}(s)}d_{i}^-(\lambda)\frac{1}{\prod_{\ell\in I,\ell\neq i}(\lambda_i-\lambda_\ell-i+\ell)}E^{e_I}\Omega_{\lambda-\epsilon_i},
|u(r)| = \Big|-\int_r^1 u'(s) ds\Big| &\leq \Big( \int_r^1 |u'(s)|^2 s^{\beta -1}ds \Big)^{1/2}\Big( \int_r^1 s^{1-\beta}ds \Big)^{1/2}&\leq \Big( a \frac{r^{2-\beta}-1}{\beta -2}\Big)^{1/2}.
\frac{1}{\sqrt{N}}\sqrt{\frac{\pi}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}= \frac{1}{2 \sqrt{\pi}}\left[ \frac{\theta}{N}- \arcsin \left( \frac{e^{2}}{2 m c}\sqrt{\frac{N}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}\right) \right] .
T_{1 2}= \frac{N_{1}N_{2}}{\det \left( 2 \left( M_{1}+ M_{2}\right) \sigma \right)^{d / 2}}\exp \left( \frac{1}{4}\left( \lambda_{1}+ \lambda_{2}\right)^{T}\left( M_{1}+ M_{2}\right)^{- 1}\left( \lambda_{1}+ \lambda_{2}\right) \right)
\overline{M}= \begin{pmatrix}1 0 \dots 0 \overline{m}_{21}1 \dots 0 \vdots \vdots \ddots \vdots \overline{m}_{n1}\overline{m}_{n2}\dots 1 \end{pmatrix}, \begin{cases}\overline{m}_{ij}= (-1)^{i+j}e_{i-j}(\overline{S}_i) \overline{S}_i = \{x_2,\dots,x_i\}.\end{cases}
R_{(1,16)}(n) = 12\sigma(n) - 36\sigma(\frac{n}{3}) + 12\sigma(\frac{n}{16}) -36\sigma(\frac{n}{48}) + 144\, W_{(1,16)}(n) + 1296\, W_{(1,16)}(\frac{n}{3}) - 432\, \biggl( W_{(3,16)}(n) + W_{(1,48)}(n) \biggr).
\left[\frac{d^2}{dr^2}-\frac{M^2\omega^2r^2}{\hbar^2}-\frac{\ell(\ell+1)}{r^2}+\frac{E^2-\left(Mc^2/2\right)}{\hbar^2c^2}-\frac{M\omega}{\hbar}\left[j(j+1)-\ell(\ell+1)+1\right]\right]R_{n\ell}(r)=0.
\alpha =\alpha_{M}\left( \left[ F_{ij}^{a}\right] ,\left[ \pi_{a}^{\mu}\right] ,\left[ y^{\alpha_{0}}\right] ,\left[ \mathcal{P}_{1a}\right],\left[ \mathcal{P}_{2a}\right] \right) e^{M}\left( \eta_{2}^{a}\right) ,
Y^{b \beta}{}_{a \alpha}= ( ( \textstyle{\frac{4}{{\cal N}}}- 1 ) \kappa - \eta ) \delta^{b}{}_{a}\delta^{\beta}{}_{\alpha}+ \delta^{b}{}_{a}s_{\alpha}{}^{\beta}+ 2 s^{b}{}_{a}\delta^{\beta}{}_{\alpha}.
\begin{cases}\overline u_t\geq \overline u(t)( a_0(t)-a_1(t)\overline u(t) -a_2(t)\underline v(t)) \underline v_t\leq \underline v(t)( b_0(t)-b_1(t)\overline u(t) -b_2(t)\underline v(t)).\end{cases}
\langle x | \frac{1}{E - \hat{H}_{0}\pm i \epsilon}\hat{V}| \psi^{( + )}\rangle = \left( \frac{- i m e^{- i \pi / 4}}{\sqrt{2 \pi}\hbar^{2}}\right) \frac{e^{i k r}}{\sqrt{k r}}\int d^{2}x^{\prime}\, e^{- i \vec{k}^{\prime}. \vec{x}^{\prime}}\langle x^{\prime}| \hat{V}| \psi^{( + )}\rangle \; ,
\int_0^1 \int_M \langle \nabla f, \nabla \eta \rangle \: d\mu_t \: dt \: = \:\int_M \langle \nabla f(1), \nabla \eta_1 \rangle \: d\mu_1 \: = \:\int_M \langle \nabla \eta_1, \nabla \eta_1 \rangle \: d\mu_1.
e^{2 \beta z_{+}^{\prime \prime}}= e^{- 2 \beta z_{-}^{\prime \prime}}= - 1 - \frac{2 \beta^{2}}{\chi^{2}}+ \frac{2 \beta}{\chi^{2}}\left( \chi^{2}+ \beta^{2}\right)^{1 / 2}> 1 .
\Sigma_{\mathcal{T}}^{\mathsf{M}}=\left\{\tau (\lambda ):\tau (\lambda)\equiv \frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}\text{\, if}Q_{\mathsf{M}}(\lambda )\in \Sigma_{\mathcal{Q}}^{\mathsf{M}}\right\}.
F(\rho,\sigma)=(tr\sqrt{|\psi_{\rho}\rangle\langle\psi_{\rho}|\sigma|\psi_{\rho}\rangle\langle\psi_{\rho}|})^{2}=\langle\psi_{\rho}|\sigma|\psi_{\rho}\rangle(tr\sqrt{|\psi_{\rho}\rangle\langle\psi_{\rho}|})^{2}=\langle\psi_{\rho}|\sigma|\psi_{\rho}\rangle.
\mu(s\widetilde{\alpha},\sigma)=\begin{cases}\gamma(G/P)^2q_F^{n(\sigma)+n(\sigma\otimes\omega)}\frac{(1-\omega(\varpi)q_F^{-2s})(1-\omega(\varpi)^{-1}q_F^{2s})}{(1-\omega(\varpi)q^{-1-2s})(1-\omega(\varpi)^{-1}q_F^{-1+2s})}\gamma(G/P)^2q_F^{n(\sigma)+n(\omega)+n(\sigma\otimes\omega)}\end{cases}
\sum_{n=1}^NKp_ny^n_0+\sum_{s=1}^{K-1}\left[\frac{K-s}{s+1}\sum_{g\in\mathcal{G}_{s+1}}\pi_{s+1}^g \max_{n\in g}y^n_s+\frac{s}{K}\gamma \sum_{n=1}^Ny_s^n\right]+\gamma\, \sum_{n=1}^Ny_K^n.
r(\theta\,|\,\phi)=\frac{(rN)^{\frac{1}{2}}\,\sin(2\theta)^{\frac{1}{2}}}{\left(2\,\sin(\phi\pm\theta)\right)^{\frac{rN-1}{2rN}}\,P(\frac{\pi}{2}-\theta\,|\,\phi)}\,\prod_{k=1}^{rN}\sin\left(\frac{2\theta-\pi k}{rN}\right)^{\frac{rN-2k}{2rN}}\,,
\prod_{j=1}^k\prod_{l=0}^m a_{j,l}=\prod_{j=1}^ka_{j,0}\prod_{j=0}^{k-1}a_{k-j,1}\prod_{j=1}^ka_{j,2}\cdots \prod_{i=1}^{\lfloor \frac{m+1}2\rfloor(k-1)}(u_i,v_i).
f- \chi_{[0,\infty)}M_1 u_0 - \chi_{[0,\infty)}Au_0 =(\partial_{0,\nu}M_0 + M_1 +A )(u-\chi_{[0,\infty)}u_0) = \partial_{0,\nu}M_0(u-\chi_{[0,\infty)}u_0) + (M_1 +A )(u-\chi_{[0,\infty)}u_0),
\lambda(a^3) \oplus \lambda(a^4) \oplus \lambda(a^5) \oplus \lambda(a^6) = = (\lambda(a^1) \oplus \lambda(a^2) \oplus \lambda(a^3) \oplus \lambda(a^4)) \oplus (\lambda(a^1) \oplus \lambda(a^2) \oplus \lambda(a^5) \oplus \lambda(a^6)) = = 0 \oplus 1 = 1.
\tilde\alpha_{i,n}= 0, \mbox{for}i<0,\n<1\mbox{or}i\geq n, \tilde\alpha_{0,1}= 1, \tilde\alpha_{i,n+1}= \frac{2i}{n}\tilde\alpha_{i,n}+\frac1n\tilde\alpha_{i-1,n}+4\tilde\alpha_{i,n},{}\mbox{in the other cases}.
{\rm ii})\;\; m_{\alpha}\,{\bf j^{\alpha}}\ne 0\;\; ,\qquad\qquad\frac{\partial}{\partial u}{\bf{\hat C}}=(\log r)\frac{1}{2}m_{\alpha}\,{\bf j^{\alpha}}(u,\phi)\quad .\qquad\qquad\qquad\qquad
\lambda ( \beta ) = \lambda_{0}- \frac{3}{2}\frac{\lambda_{0}^{2}}{( 2 \pi )^{D / 2}}\sum_{n = 1}^{\infty}\biggl ( \frac{m_{0}}{\beta n}\biggr )^{\frac{D}{2}- 2}K_{\frac{D}{2}- 2}( m_{0}n \beta ) ,