harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ \partial_{y}^{2}+ \frac{4}{l}\coth ( y / l ) \partial_{y}- \frac{1}{( l H )^{2}\sinh^{2}( y / l )}( \partial_{t}^{2}+ 3 H \partial_{t}) + \frac{1}{l^{2}\sinh^{2}( y / l ) e^{2 H t}}\partial_{\mathrm{\boldmath ~ x ~}}^{2}\right] \delta \phi = 0 .
\overline{u}_0(t) =e^{-tA}v_0, \overline{u}_\ell(t) =A^\ell\left(\sum_{j=0}^\ell\begin{pmatrix}2\ell-1 \ell+j-1\end{pmatrix}\frac{(-tA)^j}{j!}e^{-tA}v_0+\sum_{k=0}^{\ell-1}\begin{pmatrix}2\ell-1 \ell+k\end{pmatrix}\frac{(-tA)^k}{k!}e^{-tA}u_1\right).
\eta(\tau_n^{q-\ell}\tau_j, \tau_i) =\tau_n^q\tau_1 \eta(\tau_{j+1},\tau_{i+1})=(\tau_n^q \tau_1)^{n-j}\tau_n^{q-1+D(n-j+i)}\tau_{n-j+i+1}=(\tau_n^q \tau_1)^{n-j}\tau_n^{\ell-1+D(n-j+i)+D(i)}\tau_{n-j+i+1}.
a_h(\tilde u - v_h, \tilde g) = (\tilde u - v_h, -\Delta\tilde g + \tilde g)_{\Omega_h}+ (\tilde u - v_h, \partial_{n_h}\tilde g)_{\Gamma_h}= (u - v_h, \partial^m\eta)_{\Omega_h\cap\Omega}+ \operatorname{Res}_g(\tilde u - v_h) &= (\tilde u - v_h, \partial^m\eta)_{\Omega_h}+ \operatorname{Res}_g(\tilde u - v_h).
\Gamma ( p , P ) = - e^{2}\int \mathrm{\frac{d^3 q}{( 2 \pi )^3}}D ( p - q ) \gamma_{\mu}S ( \mathrm{\frac{1}{2}}P + q ) \Gamma ( q , P ) S ( - \mathrm{\frac{1}{2}}P + q ) \gamma_{\mu},
f_n(t) =\biggl(a-b+c+nd+\int_{[1/n,n]}s^{-1}\,d\mu(s)\biggr) &\qquad+\biggl(b+nc+d+\int_{[1/n,n]}d\mu(s)\biggr)t &\qquad-c(1+n){t(1+n)\over t+n}-d(1+n){t\bigl(1+{1\over n}\bigr)\over t+{1\over n}}&\qquad-\int_{[1/n,n]}{1+s\over s}\cdot{t(1+s)\over t+s}\,d\mu(s) &=f_n(0^+)+f_n'(+\infty)t-h_n(t)
V_{1}|_{\phi = 0}= \frac{\partial V_{1}}{\partial \phi}|_{\phi = 0}= \frac{\partial^{2}V_{1}}{\partial \phi^{2}}|_{\phi = 0}= \frac{\partial^{3}V_{1}}{\partial \phi^{3}}|_{\phi = 0}= \frac{\partial^{4}V_{1}}{\partial \phi^{4}}|_{\phi = 0}= 0
\frac{[Ext^1_U(j_{*}\hat{T},\mathbb{G}_m)]}{[Ext^1_U(j_{*}\hat{T}',\mathbb{G}_m)]}=\frac{\prod_{v\in S_{\infty}}[H^0(K_v,\hat{T}'')][\mathrm{cok}(\hat{\lambda}(H^1_c))]}{[H^0_{c}(U,\mathcal{Q})]q(\lambda(O_{K,S}))}.
\begin{cases}\left(\sigma_y(t), \partial D \right)\to 0\ \,\ \,u(\sigma_y(t))\to 0\ \,\ \,t\to t_y^-,\ \ |\sigma_y(t)|\to 0 \ \,\ \,u(\sigma_y(t))\to u(0)\ \,\ \, t\to t_y^+. \end{cases}
[ r_{1 2}( z_{1}- z_{2}) , r_{2 3}( z_{2}- z_{3}) ] + [ r_{1 2}( z_{1}- z_{2}) , r_{1 3}( z_{1}- z_{3}) ] + [ r_{1 3}( z_{1}- z_{3}) , r_{2 3}( z_{2}- z_{3}) ] = 0 .
\begin{array}{ccllllll}\theta (q_a,z_a) = - 1.6\ldots \times 10^{-15}+ i\, 2.8\ldots \times 10^{-16}=: \chi_0~, \theta_z(q_a,z_a) = - 8.0\ldots \times 10^{-16}+ i\, 0.0\ldots \times 10^{-15}=: \lambda^*~.\end{array}
-{\frac{2}{3}}\sum_{i}\mathrm{t r}F_{9 i}\mathrm{t r}F_{9 i}^{3}-{\frac{2}{3}}\sum_{i}\mathrm{t r}F_{5 i}\mathrm{t r}F_{5 i}^{3}-{\frac{1}{6}}\sum_{i j}w_{i j}\mathrm{t r}F_{5 i}^{3}\mathrm{t r}F_{9 j}-{\frac{1}{6}}\sum_{i j}w_{i j}\mathrm{t r}F_{5 i}\mathrm{t r}F_{9 j}^{3}.
\begin{aligned}0 \leq W_\nu = - \left(u_{\nu\nu}+ \sum_k u_{k}\rho_{k\nu}- \phi_\nu - \phi_{u}\phi - q_{\nu}\right) + (A + \frac{M}{2}) &\leq - u_{\nu\nu}+ C + \frac{M}{2},\end{aligned}
\int_{-\infty}^{\infty}d\rho\, \psi_{\rho}(x) \psi^{*}_{\rho}(y) = \int_{-\infty}^{\infty}d\rho\, x^{-\frac{1}{2}+ i\rho}\, y^{-\frac{1}{2}- i\rho}& = 2\pi \delta(x-y).
Mf(s)=\int_{0}^{\infty}x^{s}e^{-px}\frac{dx}{x}=\int_{0}^{\infty}(\frac{u}{p})^{s}e^{-u}\frac{du}{u}=\frac{1}{p^{s}}\int_{0}^{\infty}u^{s}e^{-u}\frac{du}{u}=\frac{1}{p^{s}}\Gamma(s).
\Lambda(P_A c_{\pi_1}, P_A c_{\pi_2}, P_A c_{\pi_3}) = \Lambda(c_{\pi_1}, c_{\pi_2}, P_A c_{\pi_3}) = \sum_{B \subset A}(-1)^{|A \setminus B|}\Lambda(c_{\pi_1}, c_{\pi_2}, Q_B c_{\pi_3}) = \sum_{B \subset A}(-1)^{|A\setminus B|}\Lambda(Q_B c_{\pi_1}, Q_B c_{\pi_2}, Q_B c_{\pi_3}),
exp(ad \: X_{k}) exp(-ad \: X_{k})exp(ad \: X_{k})(h_{k}) =exp(ad \: X_{k})exp(-ad \: X_{-k})(h_{k}-2X_{k}) = exp(ad \: X_{k})(h_{k}-2X_{k}+ [h_{k}-2X_{k},X_{-k}] - [[X_{k},X_{-k}],X_{-k}]) = exp(ad \: X_{k})(h_{k}-2X_{k}+ -2X_{-k}-2h_{k}+2 X_{-k}) = exp(ad \: X_{k})(-h_{k}-2X_{k}) = -h_{k}-2X_{k}+ [X_{k},-h_{k}-2X_{k}] = -h_{k},
\frac{H_{Q M}[ \rho , \Pi ]}{N^{2}}~ = ~ \frac{\gamma}{8}\int d \theta \left\{\rho ( \theta ) \left( \frac{\partial}{\partial \theta}\frac{\delta ~ S}{\delta \rho ( \theta )}\right)^{2}+ \frac{\pi^{2}}{3}\rho^{3}( \theta ) \right\}+ \frac{1}{N^{2}}V [ \rho ] ~ = ~ \frac{E}{N^{2}}\; .
\frac{d T}{d x}= - \frac{4 c \varepsilon^{2}H Z^{4}n}{\nu_{0}^{2}v^{2}}\int_{0}^{\infty}\xi d \xi \left\{1 - e^{- \alpha \xi \int_{\xi}^{\infty}\frac{d u}{u^{2}}B ( u )}\right\}\frac{\int_{\xi}^{\infty}\frac{d u}{u}B ( u )}{\alpha \xi \int_{\xi}^{\infty}\frac{d u}{u^{2}}B ( u )}~ ~ .
\left( A_p -C\right)^{-1}= A_p^{-1}+ A_p^{-1}C^{\frac{1}{2}}\left( Id - C^{\frac{1}{2}}A_p^{-1}C^{\frac{1}{2}}\right)^{-1}C^{\frac{1}{2}}A_p^{-1}.
{\cal R}\left( \lambda , \mu \right) = \left( \begin{array}{c c c c}{{\frac{\lambda q - \mu q^{- 1}}{\lambda - \mu}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{\frac \mu{\lambda - \mu}\left( q - q^{- 1}\right)}}&{{0}}\\{{0}}&{{\frac \lambda{\lambda - \mu}\left( q - q^{- 1}\right)}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\frac{\lambda q - \mu q^{- 1}}{\lambda - \mu}}}\\ \end{array}\right)
\begin{array}{r l}{{3 H^{2}\ =}}&{{{\displaystyle{\frac{2 \pi \lambda \phi^{4}}{M_{\mathrm{p}}^{2}- 8 \pi \xi \phi^{2}}}\ ,}}}\\{{( 1 - 6 \xi ) \ 3 H \dot{\phi}\phi \ =}}&{{{\displaystyle - \{\frac{\lambda \phi^{4}M_{\mathrm{p}}^{2}}{M_{\mathrm{p}}^{2}- 8 \pi \xi \phi^{2}}}\ .}}}\\ \end{array}
T^{(i)}_B (z) =\displaystyle\sum_{\mu= 0}^{n-1}G_{a+\bar{\varepsilon}_\mu}\phi^{*(a,a+\bar{\varepsilon}_\mu )}(z)K^{(i)}\left( \left. a+\bar{\varepsilon}_\mu \begin{array}{c}a a \end{array}\right| z \right) \phi^{(a+\bar{\varepsilon}_\mu ,a)}(z).
\left|\frac{b_{t,s}^{2-\epsilon}}{a_{t,s}}\right|=\left|\frac{b_{t,s}}{a_{t,s}^x}\right|^{1/x}\cdot |b_{t,s}|^{2-\frac{1}{x}-\epsilon}\leq K_2^{1/x}D^{-2^j\frac{x+1}{x^2}}D^{(t-s)(2-\epsilon-\frac{1}{x})}.
\frac 1 q =\frac 1{(m+2)(n-1)}\left( \frac 8{\kappa-1}- \frac m{\mu_\ast}\right) -\frac{n-1}{2(n+1)}\intertext{and}\frac 1 s = \frac{(m+2)(n-1)}4 \left( \frac 1 2 -\frac 1 q\right) +\frac m{4 \mu_\ast}.
x f(x) = \frac{1}{\pi}\lim_{\varepsilon \to 0^+}\mathcal{I}(T_\mu(x - i\varepsilon)) = \frac{1}{\pi}\lim_{\varepsilon \to 0^+}\mathcal{I}( \sqrt{a - x+i\varepsilon}(c_{1/2}+ c_{3/2}(a-w) + c_{5/2}(a-w)^2 + \ldots)) = \frac{1}{\pi}\lim_{\varepsilon \to 0^+}\sqrt{x-a}\cdot h(x),
\|x^{t+1}- x^t\| + (\|x^{t+1}- x^t\| - \|x^{t}- x^{t-1}\|) \le \frac{C}{D}[\varphi(L_\beta(x^t,y^t,z^t) - l^*) - \varphi(L_\beta(x^{t+1},y^{t+1},z^{t+1}) - l^*)].
\rho^{(S^2_+)}(r) = -\frac{1}{4\pi}\int_0^{2\pi}\int_0^{\pi/2}\partial_\theta \sin\alpha d\theta d\varphi = -\left. \frac{1}{2}\sin\alpha \right|_{\theta=0}^{\theta=\pi/2}= \Theta(r-z_0)\ .
\sum_{j,k = 1}^{n}\langle D^2 \Theta(\log u) \partial_j \log u, \partial_k \log u \rangle X_j X_k & = \bigg\langle D^2\Theta(\log u)\bigg(\sum_{j=1}^n X_j \partial_j \log u\bigg), \sum_{k=1}^n X_k \partial_k \log u \bigg\rangle
\sum_{r,s=1}^n \frac{\partial g^k_{rs}}{\partial t}\frac{\partial}{\partial z^{r}_k}\wedge\frac{\partial}{\partial z_k^{s}}=\sum_{r,s,a,b=1}^n \frac{\partial g_{rs}^k}{\partial t}\frac{\partial f_{jk}^a}{\partial z_k^r}\frac{\partial f_{jk}^b}{\partial z_k^s}\frac{\partial}{\partial z_j^a}\wedge \frac{\partial}{\partial z_j^b}
s_{i}(p)+(p-C_{i}'(s_{i}(p))-\lambda_{i}+\mu_{i})\cdot(D'(p)-{\textstyle \sum}_{j\ne i}s_{j}'(p))=0,{\textstyle \sum_{j}}s_{j}(p)=D(p)+\varepsilon, s_{i}(p)\le Cap_{i}, s_{i}(p)\ge0, \lambda_{i}(Cap_{i}-s_{i}(p))=0, \mu_{i}s_{i}(p)=0, \lambda_{i},\mu_{i}\ge0,
\bigg|\sum_{m=0}^{i-1}\alpha(i-m,m)x_m \bigg|^2 \leq C_1^2 \bigg(\sum_{m=0}^{i-1}\Big(\frac{i+1}{m+1}\Big)^{1/4}x_m\bigg)^2 &\leq C_1^2 (i+1)^{1/2}\sum_{m=0}^{i-1}\Big(\frac{1}{m+1}\Big)^{1/2}\sum_{m=0}^{i-1}|x_m|^2 &\leq C_2 (i+1)^{1/2}\sqrt{i}\|x\|_E^2 \leq C_2 (i+1)\|x\|_E^2 .
T = \left( \begin{array}{c c c c}{{0}}&{{- g_{z}}}&{{g_{y}}}&{{f_{x}}}\\{{g_{z}}}&{{0}}&{{- g_{x}}}&{{f_{y}}}\\{{- g_{y}}}&{{g_{x}}}&{{0}}&{{f_{z}}}\\{{- f_{x}}}&{{- f_{y}}}&{{- f_{z}}}&{{0}}\\ \end{array}\right) .
f(x,a,b) =- r \overline{\overline{W}}^{(q+r)}(x) + I_a^{(q,r)}(x) f(0,a,b) + \frac{W^{(q+r)}(x)}{W^{(q+r)}(b)}\Big[ r \overline{\overline{W}}^{(q+r)}(b) - I_a^{(q,r)}(b) f(0,a,b)\Big].
c_{k-1}- 2z c_{k}+ c_{k+1}= \lambda_{+}^{p-1}(1+\lambda_{+}\lambda_{-}- 2z\lambda_{+}) - \lambda_{-}^{p-1}(1+\lambda_{-}^{2}- 2z\lambda_{-}) & = \lambda_{+}^{p}(\lambda_{-}- \lambda_{+}) \neq 0.
\langle e^{X^{0}( t_{1})}\cdots e^{X^{0}( t_{n})}\rangle = \prod_{i < j}| e^{i t_{i}}- e^{i t_{j}}|^{2}= 4^{\frac{n ( n - 1 )}{2}}\prod_{i < j}\sin^{2}({\frac{t_{i}- t_{j}}{2}}) ~ .
\mathbb{E}|\widehat{g}(\psi)|^4 = \sum_{-Q\leq n_1,n_2,n_3,n_4\leq 2Q}\mathbb{E}g(n_1)g(n_2)g(n_3)g(n_4)\cdot e((n_1+n_2-n_3-n_4)\psi) &\leq \sum_{-Q\leq n_1,n_2,n_3,n_4\leq 2Q}\left|\mathbb{E}g(n_1)g(n_2)g(n_3)g(n_4)\right|
\begin{aligned}\frac{d}{dt}\int |\nabla^{N+1}n|^2 dx+\frac{N+3}{1+t}\int |\nabla^{N+1}n|^2 dx &\lesssim \left(\frac{N+3}{1+t}\right)^2 \int |\nabla^{N}n|^2 dx+(1+t)^{-(N+4)}&\lesssim (1+t)^{-(\frac{7}{2}+N)}+(1+t)^{-(N+4)}&\lesssim (1+t)^{-(\frac{7}{2}+N)}.\end{aligned}
\frac{1}{\gamma}(|{\sf a}|_2^2 + 2\mu_P({\sf n}|P_dDd) + \mu_P^2|P_dDd|_2^2) + \mu_L |P_d Dd|_2^2 +\mu_0 (Dd|d)^2 &= \frac{1}{\gamma}|{\sf a}-\mu_PP_dDd|_2^2 + \mu_L |P_d Dd|_2^2 +\mu_0 (Dd|d)^2,
\xi^{2}( x^{5}) = \frac{2 \alpha}{\alpha + \beta}\; e^{- \sqrt{\frac 3 2}\, ( \beta - \alpha ) \, x^{5}}\, , \qquad A ( x^{5}) = - \frac{\alpha \, x^{5}}{{\sqrt{6}}}+ \frac{1}{6}\log \frac{\alpha + \beta - 2 \alpha e^{- \sqrt{\frac{3}{2}}\, ( \beta - \alpha ) \, x^{5}}}{\beta - \alpha}\, .
\partial_{t}\mathcal{F}(t)\leq -2 \tfrac{\min\big\{1, \min\big\{\tfrac{1}{2}, C_{\mathcal{K}, 1}\big\}^{-1}\big\}}{\max\{1, \rho_{1}, \rho_{2}, \rho_{3}, \tau_{0}, K\}}\mathcal{E}(t) =: C \mathcal{E}(t).
\langle x \vert x \rangle = \sum_{i=1}^{N_v}\left\langle i \vert X^* X \vert i \right\rangle = \sum_{\mu=1}^{N_v}\sum_{\nu=1}^{N_v}\left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2 \ .
\mu^b(p(l)=k, \xi_1=x_1, \ldots, \xi_k=x_k, \xi_{k+1}=x_{k+1},\tau_1\in dt^1, \ldots, \tau_{k+1}\in dt^{k+1}) &=1_{\{x_1=x_{k+1}\}}Q_{x_2}^{x_1}\cdots Q_{x_1}^{x_k}\frac{1}{t^1+\cdots+t^{k+1}}(-L^{x_1}_{x_1})e^{L_{x_1}^{x_1}t^1}\cdots (-L^{x_k}_{x_k})e^{L_{x_k}^{x_k}t^k}e^{L_{x_{k+1}}^{x_{k+1}}t^{k+1}}\,dt^1\cdots dt^{k+1}
\sigma_{1}= \left[ \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right] , \; \; \ \ \ \sigma_{2}= \left[ \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right] , \sigma_{3}= \left[ \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right]
\bigg| \frac{\sum_{i = 1}^n a_{i,n}[b_{i,n}- a_{i,n}]}{\sum_{i = 1}^n a^2_{i,n}}\bigg|\leq \sqrt{\frac{\sum_{i = 1}^n (a_{i,n}- b_{i,n})^2}{\sum_{i = 1}^n a^2_{i,n}}}= o_{\mathbb{P}_n}(1).
E_{2n+1}(f) =\frac12\det [u_{j-i}-u_{j+i+2}]_{0}^{n-1}\times \det [u_{j-i}+u_{j+i}]_0^n, E_{2n}(f) =\,\,\,\,\, \det [u_{j-i}-u_{j+i+1}]_{0}^{n-1}\times \det [u_{j-i}+u_{j+i}]_0^{n-1}.
\begin{bmatrix}\partial^2 f_k / \partial z_i\partial z_i \partial^2 f_k / \partial z_j\partial z_i \partial^2 f_k / \partial z_i \partial z_j \partial^2 f_k / \partial z_j\partial z_j \end{bmatrix}= \begin{bmatrix}0 \partial^2 f_k / \partial z_i\partial z_j \partial^2 f_k / \partial z_i \partial z_j 0 \end{bmatrix},
\bigl[ s [y_i, x_1], y_{i+1}\dots y_{i' - 1}\bigr] [y_{i'}, x_2] + \bigl[ s [y_i, x_2], y_{i+1}\dots y_{i' - 1}\bigr] [y_{i'}, x_1] & \equiv \ \sum_{m=i+1}^{i'-1}y_{i+1}\dots y_{m-1}\bigl( [s, y_m] [y_i, x_1] y_{m+1}\dots y_{i'-1}[y_{i'}, x_2] & + \ [s, y_m] [y_i, x_2] y_{m+1}\dots y_{i'-1}[y_{i'}, x_1] \bigr) \pmod{I}.
\sigma_{i}= 2^{1 / 2}\left( \begin{array}{c c}{e_{i}^{3}}&{e_{i}^{2}}\\{e_{i}^{1}}&{- e_{i}^{4}}\\ \end{array}\right) ; \qquad \bar{\sigma}_{i}= 2^{1 / 2}\left( \begin{array}{c c}{- e_{i}^{4}}&{- e_{i}^{2}}\\{- e_{i}^{1}}&{e_{i}^{3}}\\ \end{array}\right) .
\protect Z = \sum_{T \in{\tilde{\cal{T}}}_n}(\prod_i q_i)^{d/2 + \alpha}\int \prod_{i,\mu}dX_i^{\mu}\exp \left(- \sum_{i,j}C_{ij}\sum_{\mu=1}^{d}(X_i^{\mu}- X_j^{\mu})^2 \right),
\frac{1}{h}\int_{x_{1}}^{x_{2}}(f(x+h)-f(x))dx=\frac{1}{h}(\int_{x_{2}}^{x_{2}+h}f(x)dx-\int_{x_{1}}^{x_{1}+h}f(x)dx)\rightarrow f(x_{2})-f(x_{1})
\ln(2) \phi_{B_r}^+\Big{(}\frac{t}{2}\Big{)}=\int_{t/2}^t \tau^{p-1}\frac{\phi_{B_r}^+(t/2)}{\tau^p}\,d\tau\le\psi_{B_r}(t)\le \int_0^t t^{p-1}L_p \frac{\phi_{B_r}^+(t)}{t^p}\,d\tau= L_p\phi_{B_r}^+(t).
\beta_{-2}:= \alpha_0 \cdot \beta_1 - \frac{1}{2^2}\beta_0 - \frac{1}{2^3}\beta_1 - \frac{8}{3}\beta_{-1}= \frac{4}{3}((a_2 + a_{-2}) \cdot v_{(1,3)}- (a_3, a_{-3}) \cdot v_{(1,2)}) \in M_{\frac{1}{2^2}}^{(a_1)}.
U_j(X,t) = \frac{U(r_jX, r_j^2 t)}{\bigg(\frac{1}{r_j^{n+3+a}}\int_{\mathbb Q_{r_j}^+}U^2 x_{n+1}^a dX dt\bigg)^{1/2}},\,\,\,\ \ \ \j\geq 1.
\theta_0 = x, \theta_1 = x, \theta_2 = x^3 + q_3, \theta_3 = x^6 + 5q_3x^3 + q_5x - 5q_3^2, \theta_4 =\textstyle x^{10}+ 15q_3x^7 + 7q_5x^5 - 35q_3q_5x^2 + 175q_3^3x - \frac{7}{3}q_5^2 + q_7x^3 + q_3q_7.
g^{(q)}(b;h) =Z^{(q+r)}(b)\dfrac{\Phi(q+r)}{r+q}\Bigg[-\dfrac{\rho^{(q)}_{b}(b;h)+g^{(q)}(b;h)}{Z^{(q)}(b)}r\int_b^{\infty}e^{-\Phi(q+r)y}Z^{(q)}(y)dy-\int_b^{\infty}e^{-\Phi(q+r)u}h(u)du &+r\int_{b}^{\infty}e^{-\Phi(q+r)y}\rho_b^{(q)}(y;h)dy\Bigg]+g^{(q)}(b;h)+\dfrac{r}{r+q}Z^{(q+r)}(b)e^{-\Phi(q+r)b}g^{(q)}(b;h).
\Big\vert\mathbf{P}(A\neq \emptyset) - \mathbf{P}(B\neq \emptyset) \Big\vert\le \Big\vert \mathbf{P}(\{A= \emptyset \}\cap \{B \neq \emptyset\})+\mathbf{P}(\{A\neq \emptyset \}\cap \{B =\emptyset\})\Big\vert
\Psi^{{\vec k}{\vec m}}_{l \zeta}= \exp \left[ i\frac{N_{A}X}{L_1}\left(\theta + \frac{2\pi N_A}{\kappa L_2}Y \right)\right] \prod_{\alpha < \beta}\left(\frac{\theta_{1}^{\ast}\left(z_{\alpha}-z_{\beta}\mid\tau \right)}{\theta_{1}\left(z_{\alpha}-z_{\beta}\mid\tau\right)}\right)^{\frac{1}{2\kappa}}\tilde{\Psi}^{{\vec k}}_{l \zeta 0}
\mathrm{DTM}_{(p,q)}(X^{n}) =\mathrm{E}[(X-\mathrm{DTE}_{(p,q)}(X))^{n}|x_{p}<X<x_{q}] &=\mathrm{E}\left[\sum_{k=0}^{n}\binom{n}{k}X^{k}(-\mathrm{DTE}_{(p,q)}(X))^{n-k}|x_{p}<X<x_{q}\right] &=\sum_{k=0}^{n}\binom{n}{k}(-\mathrm{DTE}_{(p,q)}(X))^{n-k}\mathrm{E}[X^{k}|x_{p}<X<x_{q}].
|\dot f_{y,r}(x_1)-\dot f_{y',r'}(x_1)| = \Big|\frac{\Phi_1(x,y)}{\Phi_2(x,y)}-\frac{\Phi_1(x',y')}{\Phi_2(x',y')}\Big| &\geq \Big|\frac{\Phi_1(x',y)}{\Phi_2(x',y)}-\frac{\Phi_1(x',y')}{\Phi_2(x',y')}\Big|-\Big|\frac{\Phi_1(x,y)}{\Phi_2(x,y)}-\frac{\Phi_1(x',y)}{\Phi_2(x',y)}\Big|.
\phi_1 := \phi_{11A}(\tau, z) = 2 \zeta^{\pm 1}- 2 + (-2\zeta^{\pm 2}+ 4 \zeta^{\pm 1}- 4)q + (2 \zeta^{\pm 3}- 4 \zeta^{\pm 2}+ 8 \zeta^{\pm 1}- 12) q^2 + O(q^3)
\#(\widehat{\mathcal{P}}\cap RD)=\#\left((\mathcal{P}_*\cap RD)\setminus\bigcup_{c\in C}(\mathcal{P}_*\cap c\mathcal{P}_*\cap RD)\right) &=\#(\mathcal{P}_*\cap RD)-\#\left(\bigcup_{i=1}^n(\mathcal{P}_*\cap c_i\mathcal{P}_*\cap RD)\right),
E[J_{vw}]=E[\sum_{i=1}^{k_{v}}I_{i}^{(v,w)}]=\sum_{i=1}^{k_{v}}E[I_{i}^{(v,w)}]=\sum_{i=1}^{k_{v}}\frac{k_{w}}{2m-1}=\frac{k_{v}k_{w}}{2m-1}
(\sigma_p^{s,0})^2 =(\sigma_p^{s-1,\hat{k}_s})^2, (\tau_p^{s,k})^2 =\sigma_W^2+\sum_{u\neq p}(\sigma_u^{s,0})^2 + (\sigma_p^{s,k})^2, (\sigma_p^{s,k+1})^2 =\delta_p^{-1}\mathbb{E}\left[\left(\eta_{s,k}(X+\tau_p^{s,k}Z)-X\right)^2 \right],
HPM(x, y, z, t) = HPL([x, y],\alpha (z), \alpha (t)) - HPL([y, x], \alpha (z), \alpha (t)) & \quad+ HPL(\alpha (x), \alpha (y), [z, t])+ HPL(\alpha (y), \alpha (z), [x, t]) & \quad- [\alpha^{2}(z), HPL(x, y, t)] + [\alpha^{2}(y), HPL(x, z, t)].
{\frac{\cosh^{2}\tau ( r_{+}^{2}- \alpha^{2})}{( 1 + \alpha^{2}l^{- 2})^{2}}}\left({\frac{( r_{+}^{2}- \alpha^{2})}{r_{+}^{2}}}\cosh^{2}\tau - ( 1 + \alpha^{2}l^{- 2}) \sinh^{2}\tau \right) ,
\frac{M_n(t)}{\sigma^2(t)\mathfrak{s}_n(t)}-F_n(t) &=\frac{M_n(t)-\sigma^2((t-\ell h)_+)M^0_n(t)}{\sigma^2(t)\mathfrak{s}_n(t)}+\frac{\{\sigma^2((t-\ell h)_+)-\sigma^2(t)\}M^0_n(t)}{\sigma^2(t)\mathfrak{s}_n(t)}&=:\mathbf{I}_n(t)+\mathbf{II}_n(t),
K^{1_{++}1_{++}}(x,b') = \sum_{n=-\infty}^{\infty}\left\{- \sqrt{\alpha-1}\cosh\left[{(r_+^2 - r_-^2) \over r_+}\Delta t - r_- \Delta \phi_n) \right] + \sqrt{\alpha}\cosh( r_+\Delta \phi_n) \right\}^{-2h_+}.
\le \sum_{n=1}^\infty \sum_{p\in S(A_n,A_{n+1})}\exp\bigg(-\frac{1}{n}d(x,px)-\delta_\P d(x,px)\bigg) &\le \sum_{n=1}^\infty \sum_{p\in S(A_n,\infty)}\exp\bigg(-\bigg(\frac{1}{n}+\delta_\P\bigg)d(x,px)\bigg) &\le \sum_{n=1}^\infty \frac{1}{n^2},
\int_\Omega v(\textbf{x},t_2)\varphi(\textbf{x},t_2)d\textbf{x}+\int_{t_1}^{t_2}\int_\Omega \nabla v:\nabla\varphi d\textbf{x}dt = \int_\Omega v(\textbf{x},t_1)\varphi(\textbf{x},t_1)d\textbf{x}+\int_{t_1}^{t_2}\int_\Omega[v\cdot\partial_t\varphi -(v\cdot\nabla_H)v\cdot\varphi-w\partial_zv\cdot \varphi]d\textbf{x}dt,
f(z)=z+\sum\limits_{k=2}^\infty\frac{1-\alpha}{\lambda_k[k]^m_{p,q}-\alpha u_k[k]^n_{p,q}}x_kz^k+\sum\limits_{k=1}^\infty\frac{1-\alpha}{\mu_k[k]^m_{p,q}-(-1)^{n+j-(m+i)}\alpha v_k[k]^n_{p,q}}y_k\overline{z}^k,
q_{y_0}(s,s) = \tilde{q}_s(y_0,s)= \tilde{q}_s(\phi_{0,s}(\tilde{y}_0)), q_{y_n}(s,s-t_n) = \tilde{q}_s(y_n,s-t_n) = \tilde{q}_s(\phi_{t_n,s}(\tilde{y}_n)),
\alpha \beta = 0, \beta \gamma = 0, \gamma \alpha = 0, \delta \xi = 0, \xi \eta = 0, \eta \delta = 0, \alpha^r = (\beta \delta \eta \gamma)^s, \ \ \quad (\gamma \beta \delta \eta)^s = (\delta \eta \gamma \beta)^s, \ \ \quad \xi^t = (\eta \gamma \beta \delta)^s.
\begin{dcases}\partial_t \rho + \partial_\theta (v[\rho] \rho) = 0, \theta \in \mathbb T,~~t > 0, v[\rho](\theta, t) = - K \int_{\mathbb T}\sin (\theta - \theta_*) \rho(\theta_*, t) ~ d\theta_*.\end{dcases}\mbox{or}\partial_t \rho - \partial_\theta \Big(K R \rho \sin(\theta- \phi) \Big) = 0.
\langle \zeta^{\prime}| p_{\mu}| \zeta^{\prime \prime}\rangle = - i \hbar \frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle + \frac{\partial F ( \zeta^{\prime \prime})}{\partial \eta^{\prime \prime}{}^{\mu}}\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle \, ,
g(\zeta) = - \frac{2^{5/2}\, \sqrt{\pi}\, \Gamma((d+3)/2)}{\Gamma(d/2+1)}\, \sin^{d-2}\bigg(\frac{\zeta}{2}\bigg)\, \cos^4 \bigg(\frac{\zeta}{2}\bigg), \alpha_0 \leq \zeta \leq \pi.
B(T) = - \frac{\pi^{3/2}\sigma^{3}}{\sqrt{3}}\sqrt{\frac{kT}{\varepsilon}}\exp \left( \frac{\varepsilon}{kT}\right) \left( 1 + \frac{35}{36}\frac{kT}{\varepsilon}+ \frac{5005}{2592}\left( \frac{kT}{\varepsilon}\right)^{2}+ \mathcal{O}(T^{3}) \right).
g=h_1\cdot h_2, &h_1:=f_1^{r_1}\cdots f_{j_0-1}^{r_{j_0-1}}f_{j_0}^{r_{j_0}+\alpha_{j_0}}f_{j_0+1}^{r_{j_0+1}}\cdots f_{s}^{r_{s}}, &h_2:=f_1^{\alpha_1q_1}\cdots f_{j_0-1}^{\alpha_{j_0-1}q_{j_0-1}}f_{j_0}^{\alpha_{j_0}(q_{j_0}-1)}f_{j_0+1}^{\alpha_{j_0+1}q_{j_0+1}}\cdots f_{s}^{\alpha_sq_s}.
p_{2\alpha}(x_1,x_2) =\left[\frac{(-1)^{\alpha}}{2^{2\alpha}\,\alpha!\,(\mu_3+1/2)_{\alpha}}\right](\sigma_1 T_1+\sigma_2 T_2)^{2\alpha}p(x_1,x_2), p_{2\alpha+1}(x_1,x_2) =\left[\frac{(-1)^{\alpha+1}}{2^{2\alpha+1}\,\alpha!\,(\mu_3+1/2)\,(\mu_3+3/2)_{\alpha}}\right](\sigma_1 T_1+\sigma_2 T_2)^{2\alpha+1}p(x_1,x_2),
\bar{\rho}_{{}_{\mathrm{R E G}}}( r ) = - \frac{m r^{- 1}}{( 2 \pi )^{\frac{3}{2}}}\int_{0}^{\infty}d s \exp ( - s^{2}- \frac{m^{2}r^{2}}{2{s}^{2}}) [ I_{1 - F}( s^{2}) - I_{F}( s^{2}) ] ,
\sum_{j=1}^N a_j e^{2\pi i n_j y/R}P(x\delta_j\tfrac R{\pi Y}) = \sum_{\ell=1}^n b_\ell\,y^\ell\sum_{j=1}^N a_j \delta_j^\ell e^{2\pi in_j y/R}&= \sum_{l=1}^n b_\ell \hat f_\ell(y) y^\ell
< \alpha_{p}( t ) > ={}( - i )^{p /{\bar{n}}}{\frac{\lambda ( p )}{\lambda ( p ) - 1}}({\frac{C_{\bar{n}}^{\frac{1}{2}}}{{C_{\bar{n}}^{\frac{1}{2}}}^{*}}})^{{\frac{p}{\bar{n}}}}J_{p /{\bar{n}}}( \Phi ({\bar{n}}, p ) )
\begin{aligned}v(x,\zeta_{1};h)=e^\frac{i x\cdot\zeta_{1}}{h}(1+h r_{1}(x,\zeta_{1};h)), &u_{2}(x,\zeta_{2};h)=e^\frac{i x\cdot\zeta_{2}}{h}(1+h r_{2}(x,\zeta_{2};h))\end{aligned}
\begin{array}{l l}{{U = - 2 \pi \int_{0}^{r_{0}}\tilde{T}_{_{( 0 )}t}^{t}r d r ;}}\\{{T = - 2 \pi \int_{0}^{r_{0}}\tilde{T}_{_{( 0 )}z}^{z}r d r .}}\\ \end{array}
\begin{array}{ccc}D : \Omega^p \longrightarrow \Omega^{p+1}\hskip 4cm D((DF_1)....(DF_p) F) \equiv (-1)^{-1}(DF_1)...(DF_p) (DF), ~~~\forall F, F_1,..., F_p \in \Omega^p({\cal A}),\end{array}
:\! |u_N |^{2m}\!: \, = \, :\! \big((\Re u_N)^2 + (\Im u_N)^2\big)^m\!: \, & = \sum_{\l = 0}^m \begin{pmatrix}m \l\end{pmatrix}\, :\! (\Re u_N)^{2\l}\!: \, :\! (\Im u_N)^{2(m - \l)}\!:.
\prod^{k-1}_{i=0}\parallel\psi^\ast_{t_{i+1}-t_i}|\mathcal{E}_{\varphi^X_{t_i}(x)}\parallel\leq e^{-\eta t_k},~\prod^{l-1}_{i=k}m(\psi^\ast_{t_{i+1}-t_i}|\mathcal{F}_{\varphi^X_{t_i}(x)})\geq{\rm e}^{\eta (T-t_k)}.
\Re\Big\{\frac{\gamma_L'}{\gamma_L}(\sigma+it) \Big\}= \frac{1}{2}\Big[ (r_1+r_2) \Delta(\sigma, t) + r_2 \Delta(\sigma+1,t) - (r_1+2r_2) \log \pi \Big] & = \frac{1}{2}\Big[ r_1 ( \Delta(\sigma,t) - \log \pi) + 2r_2 \cdot ( \tfrac{\Delta(\sigma,t) + \Delta(\sigma+1,t)}{2}- \log \pi )\Big] .
G_{\beta}(t,t')=\frac{1}{\beta}\Sigma_{m=-\infty}^{\infty}\exp[-i\omega_{m}(t-t')] \frac{1}{\omega_{m}^{2}+\Omega^2}=\frac{\cosh[\frac{\Omega}{2}(|t-t'|-\beta)]}{\Omega \sinh[\frac{\beta\Omega}{2}]},
\hat{\Gamma}_{b c}^{a}\equiv \{_{b c}^{\; a}\}_{h a t}- \frac{1}{2}( \psi_{, b}\; \delta_{c}^{a}+ \psi_{, c}\; \delta_{b}^{a}- \hat{g}_{b c}\hat{g}^{a n}\; \psi_{, n}) .
(\mathcal{A}u)(\xi):=-\sum_{i,j=1}^d a_{ij}(\xi)\frac{\partial^2 u}{\partial \xi_i\partial \xi_j}+\sum_{i=1}^d b_{i}(\xi)\frac{\partial u}{\partial \xi_i}+ c(\xi)u(\xi), \ \u\in\mathcal{C}^2(\mathcal{O}), \ \ \xi\in\mathcal{O},
\left(f_{n}\left(g_{s,i,j}\right)\right)\left(T_{b_{i}}\left(v\right)\right)=\varphi_{s,i,j}\left(T_{b_{i}}\left(v\right),0\right)=\varphi_{s,i,j}\left(T_{(b_{i},s_{i})}\left(v\right)\right)=T_{b_{j}}\left(v\right).
\begin{array}{r l}{}&{\hat{\Gamma}( k_{1}, k_{2}, k_{3}, k_{4})}\\{=}&{\displaystyle - g_{\pi}^{2}( 2 \pi )^{4}\delta^{4}( \sum_{i}k_{i}) \sum_{\{\lambda_{j}\}}\sum_{\lambda}\prod_{j = 1}^{4}\left[ \eta_{\lambda_{j}}( k_{j}) \right] \left( A_{s}+ A_{t}+ A_{u}\right) ,}\\ \end{array}
D_{pert, s\eta^i}(\psi+s\mathfrak{c}^g_{i+1}-s\mathfrak{h}^g_{i+1}) &\mbox{}\mbox{}\mbox{}\mbox{}\mbox{}\mbox{}+D_{pert, s\eta^i}|_{N_{\frac{\mathfrak{r}}{T^{i+1}}}}(s\mathfrak{c}^b_{i+1}+s\mathfrak{c}^s_{i+1}-s\mathfrak{h}^b_{i+1}-s\mathfrak{h}^s_{i+1}) &=s^2\mathcal{A}\mbox{}mod(\ker(D|_{L^2_1}))
r_{f_1,f_2,\frac{d}{c}}(\tau) := \int_{\frac{d}{c}}^{i\infty}\int_{w_1}^{\frac{d}{c}}\frac{f_1(w_1) f_2(w_2)}{(-i(w_1+\tau))^{2-\kappa_1}(-i(w_2+\tau))^{2-\kappa_2}}dw_2 dw_1, r_{f_1, \frac dc}(\tau) :=\int_{\frac dc}^{i\infty}\frac{f_1(w)}{\left(-i(w+\tau)\right)^{2-\kappa_1}}dw.
|\mathcal{T}_r|\leq (2d)^n \prod_{j=1}^n (2r_j+1)^{d-1}\leq (2d)^n \left(\frac{2\sum_{j=1}^n r_j+n}{n}\right)^{(d-1)n}=(2d)^n \left(\frac{2\ell+n}{n}\right)^{(d-1)n}.
V^{n+1}_{abc_n d_n}(p, T, M(T)) = \frac{1}{2}\rho^{n+1}\Bigl(\frac{\Sigma^1_{ren.}(p, M(T)) + \Sigma^1_T(p, M(T), T)}{2}\Bigr)^n v^{(n+1)}_{abc_n d_n}.
\left( \begin{matrix}{a^{\prime}}&{b^{\prime}}\\{c^{\prime}}&{d^{\prime}}\\ \end{matrix}\right) x = \left( \begin{matrix}{x a^{\prime}+ ( 1 - q^{2}) y c^{\prime}}&{q^{- 1}x b^{\prime}+ ( q - q^{- 1}) y ( a^{\prime}- d^{\prime})}\\{q x c^{\prime}}&{x d^{\prime}+ ( 1 - q^{2}) y c^{\prime}}\\ \end{matrix}\right)
\big(F''(V(\vec p))[V'(\vec p)\widehat{\vec p}_1,V'(\vec p)\,\cdot]^*\vec g\big)\widehat{\vec p}_2 = \int_0^T \langle V'(\vec p)\widehat{\vec p}_1({u}_2^\prime(t)+Q{u}_2(t)),w(t)\big\rangle_X\d t &\qquad\qquad+\int_0^T \langle V'(\vec p)\widehat{\vec p}_2({u}_1^\prime(t)+Q{u}_1(t)),w(t)\big\rangle_X\d t.
A_N:= \sqrt{\frac{2}{\tilde\mu}}\sum_{i=1}^N\sqrt{\frac{\delta_i}{a_i}}+ +\left(\|x^*-v_0\|^2+\frac{M^2N(r+\frac{\rho}{2}(N+3))}{\tilde\mu}+\frac{2}{\tilde\mu}\sum^N_{i=1}\frac{\delta_i}{a_i}+\frac{4}{\tilde\mu}\sum_{i=1}^N\frac{\varepsilon_i}{a_i^2}+\frac{2}{\mu}\left(\sum_{i=1}^N \sqrt{\frac{\delta_i}{a_i}}\right)^2\right)^{1/2}.
S_{\rm ct}=-\frac{1}{8\pi G_{p+2}}\int d^{p+1}x\sqrt{-h}\frac{c_0}{l_{\rm eff}},\ \ \c_0=\sqrt{p(p+1)},\ \ \ \frac{1}{l_{\rm eff}}=b e^{-a\phi/2}\sqrt{\frac{1}{2p(p+1)}},
[X(\zeta),R_{p}(\chi)]= 2(q+1)D\delta(-\zeta/\chi)R_{p-1}(\chi) &-4\delta(-\zeta/\chi)\left(B^{-}(\chi)R_{p-1}(\chi)+R_{p-1}(\chi)B^{+}(\chi)\right) &-2\delta(-\zeta/\chi)D_{\chi}R_{p-1}(\chi).
\nabla_{0}\nabla_{0}X_{\mu}= q \cos{( \mathrm{I m}\Phi / 2 )}\in^{i j}t^{j}n_{\mu}^{i}\ , \n a b l a_{0}\nabla_{1}X_{\mu}= q \sin{( \mathrm{I m}\Phi / 2 )}t^{i}n_{\mu}^{i}\ ,
Z^{(0)}_2(\xi)=\begin{cases}\xi_2 + \mathcal{O}(\exp(-2\pi \xi_2)), \xi_2 \to +\infty, begin{align*}3mm] \displaystyle{\frac{\xi_2}{h_0}}+C_2 + \mathcal{O}(\exp(\pi h^{-1}_0 \xi_2)), \xi_2 \to -\infty,\end{cases}