harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{E_{0}\left( \ell , \mu , d \right)}{L^{d - 1}}\approx - C \left( d \right) \frac{\Gamma \left( \frac{d}{2}+ 1 \right)}{2^{d + 3}\pi^{d / 2}\Gamma \left( \frac{d}{2}\right) \ell^{d}}\left[ \left( \frac{d}{2}+ 1 \right) \mu^{\frac{d}{2}- 2}+ 2 \mu^{\frac{d}{2}- 1}\right] e^{- 2 \mu}.
Q_{BRST}=\oint dz \left\{c( -{1\over 2}\partial X\partial X +{1\over 2}\psi\partial \psi -{1\over 2}\partial \beta \gamma -{3\over 2}\beta \partial\gamma)+c\partial c b+{1\over 2}\gamma \psi\partial X -{1\over 4}\gamma^2b\right\},
\epsilon({\bf{q}}, \omega) = \frac{U_{ext}({\bf{q}},t)}{U_{eff}({\bf{q}},t)}= 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{n_{F}({\bf{k+q/2}}) - n_{F}({\bf{k-q/2}})}{\omega - \frac{{\bf{k.q}}}{m}}
\mu(u,qu)=\frac{R(u, qu, u, qu)}{g^{2}(u, u)-g^{2}(u, qu)}\ ,\quad\mu(u,q^{2}u)=\frac{R(u, q^{2}u, u, q^{2}u)}{g^{2}(u, u)-g^{2}(u, q^{2}u)}\ .
X_{(i;j)}:= \frac12(X_{i;j}+X_{j;i})=\frac12\left(X^a_{,j}\,g_{ai}+X^ag_{ai,j}+ X^a_{,i}\,g_{aj}+X^ag_{aj,i}-2X^ag_{ak}\Gamma^k_{ji}\right) \mathrm{div}X := X^i_{;i}=X^i_{,i}+ X^j\Gamma^i_{ij}=\frac{1}{\sqrt{\det g}}(\sqrt{\det g}\,X^i)_{,i}
\psi_{\sigma}= \frac{1}{(\omega_{\sigma}-E)}\chi_{\sigma}+ \frac{1}{(\omega_{\sigma}-E)}f^*_{\sigma}(a) \bigg[\frac1{\alpha}- \sum_{\tau}\frac{|f_{\tau}(a)|^2}{\omega_{\tau}(\omega_{\tau}-E)}\bigg]^{-1}\sum_{\nu}\frac{f_{\nu}(a)}{\omega_{\nu}}\frac{1}{(\omega_{\nu}-E)}\chi_{\nu}\;.
(\psi^\epsilon_{i,-\epsilon b^\epsilon_i})^{\pm 1}= (\phi^\epsilon_i)^{\pm 2}\cdot \prod_{j - i}(\phi^\epsilon_j)^{\pm c_{ji}},\ (\phi^\epsilon_i)^{\pm 1}\cdot (\phi^\epsilon_i)^{\mp 1}=1,\ [\phi^\epsilon_i,\phi^{\epsilon'}_j]=0,
\Delta^2_x|x-y|=\begin{cases}-\frac{(n-1)(n-3)}{|x-y|^3}\ \ \ \ \ \ \ \ \d\geq4, -4\pi\delta_{x=y}\ \ \ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \ \ \d=3.\end{cases}
\left( \begin{matrix}{{a^{\prime}}}&{{b^{\prime}}}\\{{c^{\prime}}}&{{d^{\prime}}}\\ \end{matrix}\right) y = y \left( \begin{matrix}{{a^{\prime}}}&{{q b^{\prime}}}\\{{q^{- 1}c^{\prime}}}&{{d^{\prime}}}\\ \end{matrix}\right)
\hat{\lambda}( z ) \sigma^{\mu}- \tilde{\hat{w}}( z ) \sigma^{\mu}+ \sigma^{\mu}\hat{w}( z ) = \hat{\lambda}( z ) \sigma^{\mu}- \sigma^{\nu}\hat{w}_{\nu}^{\mu}( z ) = \sigma^{\nu}\partial_{\nu}h^{\mu}( z ) ,
\hat{h}_{\mu \nu}^{+}= a^{2}\xi_{\mu \nu}+ 2 a^{2}\left( \int \frac{d z}{a^{4}}\right) \partial_{\mu}\partial_{\nu}\xi - 2 \frac{a^{2}}{a_{+}^{2}}\left( \int \frac{d z}{a^{2}}\right) \partial_{\mu}\partial_{\nu}\xi - \frac{U_{B}}{2}\frac{a^{2}}{a_{+}^{2}}\xi \eta_{\mu \nu}
= \frac{1}{4}[ L_{0}, L_{1}] + \frac{1}{4}( L_{0}^{2}g - L_{0}g L_{0}) + \, \frac{1}{4}( L_{1}g L_{1}- L_{1}^{2}g ) + \frac{1}{4}( L_{1}g L_{0}g - L_{0}g L_{1}g ) =
\begin{array}{ccl}p_{d+i}= h_i(p_1,\ldots,p_d)+ a_{i1}p_1^{b_{i1}}+\cdots+a_{id}p_d^{b_{id}}+a_{id+1}, p_{d+i}' = h_i(p_1',\ldots,p_d')+ a_{i1}(p_1')^{b_{i1}}+\cdots+a_{id}(p_d')^{b_{id}}+a_{id+1},\end{array}
\sum_{m=1}^M\psi_\mu(m,\chi)=E(\chi) \frac{M^{\mu+1}}{\mu(\mu+1)}-\sum_\rho\frac{M^{\rho+\mu}}{\rho+\mu}\cdot\frac{\Gamma(\rho)\Gamma(\mu)}{\Gamma(\rho+\mu)}+\frac{M^\mu}{\mu}\cdot\frac{L'}{L}(1,\overline{\chi^\ast}) + O\left(\frac{M^{\mu+1}}{T}(\log qM)^2+\frac{M^{\mu+1}}{T^\mu}(\log qT)+M^\mu(\log2q)(\log M)\right).
S \ll \widehat{Y}^{(n+3)/2+\varepsilon}\sum_{d=d_1d_2}|d_1|\sum_{d_2=ef}|e||f|\sum_{|t_2|\leq \widehat{Y}}\sum_{|t_1|\leq\frac{\widehat{Y}}{|t_2|}}|t_1|^{-1/2}\left(\widehat{V}^n +\widehat{Y}^{n/3}\right) \ll \widehat{Y}^{(n+3)/2+\varepsilon}|d|\left(\widehat{V}^n +\widehat{Y}^{n/3}\right),
\tau_1(p, p, 1) \geq \begin{cases}\frac{\pi(p-1)}{2p-1}-\frac{(\pi-2)(p-1)}{3p-2}1<p<p_0^* \frac{\pi(p-1)}{2p-1}-\frac{\pi^3(p-1)}{24(4p-3)}p_0^*<p<\infty.\end{cases}
g = \lambda_{1}\omega^{2}_{1}+\lambda_{2}\left(\omega_{2}+ \frac{\alpha_{3}}{2}\omega_{1}\right)^{2}+\lambda_{3}\left(\omega_{3}- \frac{\alpha_{6}}{2}\omega_{2}-\left(\frac{\alpha_{5}}{2}+ \frac{\alpha_{3}\alpha_{6}}{4}\right)\omega_{1}\right)^{2}
\lbrack (K+1)d\boldsymbol{x}^{k}]\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{i}^{\pm}=2d\boldsymbol{x}^{k}\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{k}^{\pm}-\frac{1}{2}d\boldsymbol{x}^{ijk},
e_{2}\psi = \left( \begin{array}{c}{-{\psi_{2}}}\\{{\psi_{1}}}\\{-{\psi_{4}^{*}}}\\{{\psi_{3}^{*}}}\\ \end{array}\right) \quad o r \quad e_{3}\mathbf{)}e_{1}\psi = \left( \begin{array}{c}{{\psi_{2}}}\\{{\psi_{1}}}\\{{\psi_{4}^{*}}}\\{-{\psi_{3}^{*}}}\\ \end{array}\right)
\omega= \omega_{\underline{i_1\dots i_\ell}}\, V^{\underline{i_1}}\wedge \dots \wedge V^{\underline{i_\ell}}\quad \Leftrightarrow \quad \star \omega = \frac{1}{(D-\ell)!}\, \epsilon_{\underline{a_1 \dots a_{D-\ell}b_1 \dots b_\ell}}\omega^{\underline{b_1\dots b_\ell}}\, V^{\underline{a_1}}\wedge \dots \wedge V^{\underline{a_{D-\ell}}}
H_\tau = \left( \frac{f(t_\tau)}{2m}(p_{1,\tau}^2 + p_{2,\tau}^2) + p_\tau + \frac{m\omega^2(t_\tau)f^{-1}(t_\tau)}{2}(x_{1,\tau}^2 + x_{2,\tau}^2) \right) \dot{t}_\tau.
r_n^{(k+1)}=\frac{\iint_{\, \mathbb{C}_n^{(k)}}r\cos{(\phi-\phi_n)}f(r) \, \mathrm{d}r \, \mathrm{d}\phi}{\iint_{\, \mathbb{C}_n^{(k)}}f(r) \, \mathrm{d}r \, \mathrm{d}\phi}.
\!\sum_{i=0}^p\! (-1)^i\nabla_{a_i}\eta(a_0,\dots ,\widehat{a_i},\dots ,a_p) + \!\sum_{i<j}\! (-1)^{i+j}\eta([a_i,a_j],a_0,\dots ,\widehat{a_i},\dots ,\widehat{a_j},\dots ,a_p);\!
\frac{\partial}{\partial q_{tA}}=\frac{\partial r}{\partial q_{tA}}\frac{\partial}{\partial r}+\frac{\partial e_B}{\partial q_{tA}}\frac{\partial}{\partial e_B}+\frac{\partial E_s}{\partial q_{tA}}\frac{\partial}{\partial E_s}+\frac{\partial y_{sB}}{\partial q_{tA}}\frac{\partial}{\partial y_{sB}}\;.
c_l = \int_0^{1/b}C_{b,\gamma}^2 \, e^{-2 \gamma t}e^{-2 \pi i \frac{l}{a}t}\, dt = C_{b,\gamma}^2 \, \frac{a \left(1-e^{-\frac{2 \gamma}{b}}\right)}{2 a \gamma+ 2 \pi i l}.
\begin{array}{c}max\Big\{min\Big[rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x);\qquad\qquad\qquad\qquad V^{ij}(x)-M^{ij}[V](x)\Big];V^{ij}(x)-N^{ij}[V](x)\Big\}=0,\end{array}
S_{{\rm YM}}[A] = \pm \frac{8\pi^{2}}{g^{2}}K[A]+\frac{1}{8\pi^{2}}\int d^{4}x \left(F_{\mu\nu}^{a}\mp \tilde{F}_{\mu\nu}^{a}\right) \left(F^{\mu\nu}_{a}\mp \tilde{F}^{\mu\nu}_{a}\right) \geq \frac{8\pi^{2}}{g^{2}}\left| K[A] \right| \, .
R_v := \sum_{k = 1}^K \min\big\{S^{\mbox{\tiny \upshape (\itshape k\upshape)}}+ \sum_{a \in \mathcal{A}}\max\{0, x_a - x^{\mbox{\tiny \upshape (\itshape k\upshape)}}_a\}; h \big \}= n_v \overline{S}_v + (K - n_v)h;
L_m\chi_y(z) = \sum\chi\left( \sum_{s \in S_\mu}\sum_{t \in T_\nu}Y_{s,t}(Z_{s,t}- Y'_{s,m}Z_{m,t}) + \sum_{s \in S_\mu}Y_{s,m}Y'_{s,m}\right) &= \sum\chi\left( \sum_{s \in S_\mu}\sum_{t \in T_\nu}Y_{s,t}Z_{s,t}\right)\chi\left( \sum_{s \in S_\mu}\left( Y_{s,m}- \sum_{t \in T_\nu}Y_{s,t}Z_{m,t}\right)Y'_{s,m}\right),
\|\nabla u\|_2^2 = \int_{\mathbb{R}^n}|\xi|^2|\hat{u}(\xi)|^2d\xi \leq \||\xi|^2\hat{u}(\xi)\|_2 \|\hat{u}\|_2 & = \|\Delta u\|_2 \|u\|_2 \leq \frac{\varepsilon^2}{2}\|\Delta u\|_2^2 + \frac{1}{2\varepsilon^2}\|u\|_2^2.
\frac{1-R_{ij}}{1+R_{ij}-\beta T_i}= 1+ \sum_{m> 0}\beta^m\,T_i^m + \sum_{\ell>0}(-1)^\ell \sum_{m\geq 0}\left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right)(\beta T_i)^m R_{ij}^\ell.
\lvert m_{n}\rvert_S= k+\sum_{j=1}^{k-1}\lvert m_j+n(\#a(s_j)-\#a^{-1}(s_j))\rvert = k+\sum_{j\in A^+}\lvert m_j+n\rvert+\sum_{j\in A^-}\lvert m_j-n\rvert+\sum_{j\in A^0}\lvert m_j\rvert
{\hat{\cal C}}^a | Q'_i,{\cal C}^{\prime a},\bar{\cal C}^{\prime}_{a},{\pi}'_a \rangle =\hat{\bar{\cal C}}_a | Q'_i,{\cal C}^{\prime a},\bar{\cal C}^{\prime}_{a},{\pi}'_a \rangle ={\hat{\pi}}_a | Q'_i,{\cal C}^{\prime a},\bar{\cal C}^{\prime}_{a},{\pi}'_a \rangle =0,
\int\frac{1}{\sqrt[4]{(ax^{2}+bx+c)^{3}}}dx=\frac{2\sqrt{2}}{\sqrt[4]{4a^{2}c-ab^{2}}}arcsl\frac{2ax+b}{\sqrt{4a(ax^{2}+bx+c)}+\sqrt{4ac-b^{2}}}
f(\rho)^{\frac{1}{4}}g(\rho)^{\frac{1}{4}}A^* g(\rho)^{\frac{1}{2}}f(\rho)^{\frac{1}{2}}A g(\rho)^{\frac{1}{4}}f(\rho)^{\frac{1}{4}}= f(\rho)^{\frac{1}{4}}g(\rho)^{\frac{1}{4}}A^* g(\rho)^{\frac{3}{4}}f(\rho)^{\frac{3}{4}}(f(\rho)g(\rho))^{-1}g(\rho)^{\frac{3}{4}}f(\rho)^{\frac{3}{4}}A f(\rho)^{\frac{1}{4}}g(\rho)^{\frac{1}{4}}.
f_{\omega_{\pi_\nu},s}\left(\tilde{\varepsilon}\underline{t}^\Delta\right)= \left|\det \underline{t}\right|^{ms+\frac{m}{2}}f_{\omega_{\pi_\nu},s}\left(\tilde{w}\underline{t}^\Delta\tilde{w}^{-1}\tilde{w}\right).
a\triangleleft b :=\mathcal{R}^{-1}(a_{1}\otimes b_{1})a_{2}\mathcal{R}(a_{3}\otimes b_{2}), b\triangleright a :=\mathcal{R}^{-1}(b_{1}\otimes a_{1})a_{2}\mathcal{R}(b_{2}\otimes a_{3}).
p_{X'Y'}(0,0) =\sum_{x,y,z}\frac{a(x)}{\bar{a}}\frac{b(y)}{\bar{b}}p_{XYZ}(x,y,z) = \frac{1}{\bar{a}\bar{b}}\sum_{x,y,z}q_{XYZ}(x,y,z)=\frac{1}{\bar{a}\bar{b}}>0
C_P\leq \frac{8R^2}{1}+\frac{D}{e}\left(32R^2+\frac{4R^2}{1}\right)\exp\left(n+\frac{R^2}{8t}+\frac{\sqrt{2(16nt+2R^2+R^2)}^2}{2t}\right) = 8R^2+\frac{36D}{e}R^2\exp\left(17n+\frac{25R^2}{8t}\right) \leq \left(8+\frac{36D}{e}\right)R^2\exp\left(17n+\frac{25R^2}{8t}\right).
\gamma^{-1/2}e^{\phi}\left[ \frac{\delta \mathcal{S}}{\delta \gamma_{ij}}\frac{\delta \mathcal{S}}{\delta \gamma_{kl}}\gamma_{ik}\gamma_{jl}+\frac{1}{2}(\frac{\delta \mathcal{S}}{\delta \phi})^{2}+\gamma_{ij}\frac{\delta \mathcal{S}}{\delta \gamma_{ij}}\frac{\delta \mathcal{S}}{\delta \phi}\right] =0~~~.
0 = \left( \frac{\partial U}{\partial u_{\nu}}\right)_{u_{\mu}= e_{\mu}}\rightarrow \frac{1}{g}+ \sum_{\mu = 1 ( \neq \nu )}^{N}\frac{2}{e_{\mu}- e_{\nu}}= \sum_{j = 1}^{\Omega}\frac{1}{z_{j}- e_{\nu}},
E = - 2 \pi^{4}L^{2}( N \alpha^{\prime})^{5 / 2}\lim_{r \to \infty}\left( \nu \left( R^{3}e^{- 2 \Phi}\right)^{\prime}- \sqrt{\nu_{B P S}}\, \left( R_{B P S}^{3}e^{- 2 \Phi_{B P S}}\right)^{\prime}\right) \, .
\sum_{d= O(B^{1-\frac{1}{(2+\lambda)r}})}O_{\varepsilon_i,\sigma}\left(\frac{B^{1-\frac{1}{2r}+\sigma(\frac{3}{4}+\frac{1}{8r})}}{d^{1+\frac{\sigma}{2}}}\right)=O_{\varepsilon_i,\sigma}(B^{1-\frac{1}{2r}+\sigma(\frac{3}{4}+\frac{1}{8r})}).
\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(1)_{k}(c)_{k}}x^kH_k(a-1) begin{align*}1mm] \:\:=-(1-x)^{c-a-b}\{\log(1-x)\}\sum_{k=0}^{\infty}\frac{(c-a)_k(c-b)_k}{(1)_{k}(c)_{k}}x^k begin{align*}1mm] \:\:\quad-(1-x)^{c-a-b}\sum_{k=0}^{\infty}\frac{(c-a)_k(c-b)_k}{(1)_{k}(c)_{k}}x^kH_k(c-a-1).
{\pmb{X^1}}E - E{\pmb{X^1}}= \displaystyle\frac{1}{2}({\pmb{X^1}}D + D{\pmb{X^1}}) - \displaystyle\frac{1}{4}({\pmb{X^1}}D - D{\pmb{X^1}}) - \big(c_3 + \displaystyle\frac{1}{16}-\displaystyle\frac{c_2^2}{4}\big)D,
K_{nn_1}^{lm}= \int_{0}^{\infty}x^{l+|m|+\delta_1+\delta_2}{\rm e}^{-x}\>{_1F}_1(-n_1 , |m|+\delta_1+1 ; x)\>{_1F}_1(-n+l+1, 2l+\delta_1+\delta_2 + 2 ; x) \, dx.
lim_{x^{0}\rightarrow-\infty}\int d^{3}x\langle\beta|e^{ip\cdot x}\overline{\Psi}(x)\gamma^{0}u_{p}^{s}|\alpha\rangle=\sqrt{Z}\int d^{3}x\langle\beta|e^{ip\cdot x}\overline{\Psi}_{in}(x)\gamma^{0}u_{p}^{s}|\alpha\rangle,
\{\, x, \, y, \, z \}\, \in \,{\cal M}_{\Gamma}\, \left( \, t_1, ..., t_{r}\, \right){}~~\longrightarrow~{\tilde W}_{\Gamma}\, \left ( \, x, \, y, \, z, \,; \, t_1, \, ....\, , t_r \, \right )~=~0,
T=\frac{1}{2}|det(\begin{matrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\ 1&1&1\end{matrix})|=\frac{1}{2}|x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{B}y_{A}+x_{C}y_{A}-x_{C}y_{B}|,
\begin{gathered}\sum_{k}u_{a}^{k*}u_{b}^{k}=\delta_{b}^{a}1=\sum_{k}u_{k}^{a}u_{k}^{b*}, \sum_{k}q^{(2\rho,\lambda_{k}-\lambda_{b})}u_{b}^{k}u_{a}^{k*}=\delta_{b}^{a}1=\sum_{k}q^{(2\rho,\lambda_{a}-\lambda_{k})}u_{k}^{b*}u_{k}^{a}.\end{gathered}
\theta'(x)= -\frac{m}{2}+(\frac{p}{2}-1)\cos 2\theta(x)+\frac{1}{2}(m-p)\big(|\sin 2\theta(x)|-\cos 2\theta(x)\big)+\delta_p = -\frac{m}{2}+(p-1-\frac{m}{2})\cos 2\theta(x)+\frac{1}{2}(m-p)|\sin 2\theta(x)|+\delta_p,
\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_r(x_0)}\vert x\vert^{2\alpha_{k,1}}U_k(x)e^{u_k}dx=\frac{\beta_0\left(4-a\beta_0\right)}{4\left(1-a\right)}.
L_{jk}(\lambda,\nu) =\begin{pmatrix}a(\lambda,\nu)\pi_k^{+}+b(\lambda,\nu) \pi_k^{-}c\sigma_k^{-}c\sigma_k^{+}b(\lambda,\nu)\pi_k^{+}+ a(\lambda,\nu)\pi_k^{-}\end{pmatrix}_{[\mathcal{V}_j]}.
\begin{array}{l}{{\hat{\lambda}^{\iota_{1}}={\frac{1}{2}}\left( \hat{\lambda}_{1_{w}4_{w}}- \hat{\lambda}_{2_{w}3_{w}}\right)}}\\{{\hat{\lambda}^{\iota_{2}}={\frac{1}{2}}\left( \hat{\lambda}_{1_{w}3_{w}}+ \hat{\lambda}_{2_{w}4_{w}}\right)}}\\{{\hat{\lambda}^{\iota_{3}}={\frac{1}{2}}\left( \hat{\lambda}_{1_{w}2_{w}}- \hat{\lambda}_{3_{w}4_{w}}\right)}}\\ \end{array}\hspace{5 . 9 c m}
\frac{vol(I_{n,m}^{\tilde{n}}\cap\hat{\mathfrak{D}}_{\tilde{n}})}{vol(\hat{\mathfrak{D}}_{\tilde{n}})}=\frac{1}{\sqrt{{\tilde{n}}}}vol(\{(x_1,\dots,x_{\tilde{n}})\in(0,1)^{\tilde{n}}\mid x_i\in I_{n,m},\sum x_i\in\mathbb{Z}\}).
\mathrm{P}_{1,i}^0 = \mathrm{P}\left(x_i>z_i, z_i<\frac{\epsilon_{1,1}}{\rho}\right)+\mathrm{P}\left(x_i<z_i<\frac{\epsilon_{1,i}}{\rho}\right) = \mathrm{P}\left( z_i<\frac{\epsilon_{1,1}}{\rho}\right)=1-e^{-\frac{\epsilon_{1,i}}{\rho}},
\int_{\partial B(0,r)}x_{1}^{4}dx=\frac{3r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)},\int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx=\frac{r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)}.
J = \operatorname*{d e t}\left[ \begin{array}{c c}{\partial w_{i}/ \partial z_{j}}{\partial w_{i}/ \partial \overline{{z}}_{j}}{\partial \overline{{w}}_{i}/ \partial z_{j}}{\partial \overline{{w}}_{i}/ \partial \overline{{z}}_{j}}\end{array}\right] ,
n(R_\Sigma-\epsilon_n) \leq \sum_{t=1}^nh(Y_{1}[t]|S_1^{n},Y_1^{t-1})+\sum_{t=1}^nh(Y_{2}[t]|S_2^{n},X_{3}^{n},Y_2^{t-1})-\sum_{t=1}^nh(Z_{1}[t])-\sum_{t=1}^nh(Z_{2}[t]).
{\cal T}_{11}(q) = -\frac{1}{2\pi}\int_{-\infty}^{\infty}d q_0^\prime\left( \frac{M^+(q_0^\prime, \vec{q})}{q_0-q_0^\prime +i \epsilon}- \frac{M^-(q_0^\prime, \vec{q})}{q_0-q_0^\prime -i \epsilon}\right)~,
-\cos\psi\csc\eta\cdot\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{2}[q_{0}](\frac{i}{\mu_{0}})w_{3}[q_{0}](\frac{i}{\mu_{0}})}{w_{1}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{1}\gamma^{0}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\max\limits_{k}\left|\sum\limits_{i=1}^kR_i-\frac{k}{n}\sum\limits_{i=1}^nR_i\right|=\max\limits_{k}\left|\sum\limits_{i=1}^k\sum\limits_{j=k+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right|,
\int_{-\infty}^{\infty}q^{\alpha^{2}-2\nu\alpha}\left(iq^{n/2}\right)^{-\alpha}S_{n}\left(-q^{2\nu-\alpha};q\right)J_{\alpha}^{(2)}\left(2iq^{n/2};q\right)d\alpha=\sqrt{\frac{\pi}{\log q^{-1}}}\frac{q^{-\nu^{2}}}{\left(q;q\right)_{\infty}\left(q;q\right)_{n}}.
H_{\gamma}(Y|X)=\textstyle \sum_{i=1}^s\sum_{j=1}^t -p_{i,j}\log_{\gamma}\frac{p_{i,j}}{p_i}; I_{\gamma}(X;Y)= \textstyle \sum_{i=1}^s\sum_{j=1}^t -p_{i,j}\log_{\gamma}\frac{p_ip'_j}{p_{i,j}}.
\sum_{m,n\geq 0}c_o(m,n;s)x^{m+n}=\sum_{l\geq 0}x^{l}\sum_{m=0}^l c_o(m,l-m;s) &=\sum_{l\geq 0}{(-d/c,qs/ab,s,qs^2/a^2c^2;q)_{l}\over (q,q^{2}s^2/abcd,q^{1/2}s/ac,-q^{1/2}s/ac;q)_{l}}(qx/d)^l{}_4\phi_3\left[{q^{-l},-q^{-l}ac/s,-b/a,qs/cd \atop -q^{-l+1}c/d,q^{-l}ab/s,-qs/ac};q,q\right].
\int x^3 \bar\rho v_t \left( \frac{r}{x}-\frac{x^2}{r^2}\right)dx =\frac{d}{dt}\int x^3 \bar\rho v \left( \frac{r}{x}-\frac{x^2}{r^2}\right)dx - \int x^2 \bar\rho v^2 \left( 1+2 \frac{x^3}{r^3}\right)dx
\mathbb{T}\Big[ \mathrm{H}\Big](\mathrm{x}) = \int_{\Omega}\nabla \nabla\cdot \Big(\mathbb{G}^{(\mathrm{k})}-\mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y})\mathrm{H}(\mathrm{y})d\mathrm{y}.
\dfrac{(\Tilde{\mathcal{N}}(f) \circ \rho_{\eta}(X))'}{\Tilde{\mathcal{N}}(f) \circ \rho_{\eta}(X)}= \sum_{w \in W_{\rho}^{m_0}}\dfrac{(f(X+w))'}{f(X+w)}= \sum_{w \in W_{\rho}^{m_0}}\dfrac{f'(X+w)}{f(X+w)}.
\begin{cases}A = (a_2^2-3a_4)({a_2'}^2-3a_4'), B=\frac{32}{27}\left( 2a_2^3-9a_2a_4+27a_6\right)\left(2{a'_2}^3-9a'_2a'_4+27a'_6\right), \Delta_{E_1}=16\left(a_2^2a_4^2-4a_2^3a_6+18a_2a_4a_6-4a_3-27a_6^2\right), \Delta_{E_2}=16\left({a'_2}^2{a'_4}^2-4{a'_2}^3{a'_6}+18{a'_2}{a'_4}{a'_6}-4{a'}_3-27{a'_6}^2\right). \end{cases}
F(X)=X^{2^{2m}+2^m}+X^{2^{2m}+1}+X^{2^m+1}+\delta^{2^{2m}}X^{2^m}+ \delta X^{2^{2m}}+X^2+(\delta^{2^{2m}}+\delta+1)X + \delta^{2^{2m}+1}.
\|{\cal R}\|_2^2 = \left(\beta^*_{S^* \backslash \widehat{S}_s}\right)^T \left(X_{S^* \backslash \widehat{S}_s}^T \Pi^{\perp}[\widehat{S}_s] X_{S^* \backslash \widehat{S}_s}\right)^2 \beta^*_{S^* \backslash \widehat{S}_s}\,.
\left\|U^{m,n}_{\psi,\overrightarrow{s}}\right\|_{L^{p_1}_{\omega_1}\left(\mathbb R^d\right)\times\cdots\times L^{p_m}_{\omega_m}\left(\mathbb R^d\right)\to L^{p}_{\omega}\left(\mathbb R^d\right)}={\mathcal A}.
f_{\Theta_\rho}(\theta ) = \begin{cases}1/\zeta, 0 < \theta \le P_{\rm{max}}/d^{\alpha}_{\rm j,w}, \rho=0 1/\zeta, \sigma_{\rm a}^2 < \theta \le \sigma^2_{\rm a}+ P_{\rm{max}}/d^{\alpha}_{\rm j,w},\rho=1, 0, ,\end{cases}
\frac{f(m+1,\ell)}{f(m,\ell)}\geq \left(\frac{\ell^{\frac{\ell}{6+\ell}}m}{m+1}\right)^{6+\ell}\geq \left(\frac{2^{\frac{2}{6+2}}21}{21+1}\right)^{6+\ell}> 1^{6+\ell}=1,
\P \{N_t \geq m + i\}\leq C_t e^{-t}\frac{t^{m + i}}{(m + i) !}< C_t e^{-t}\frac{t^{m + i}e^{m + i}}{(m + i)^{m + i}}= C_t e^{-t}\left( \frac{te}{m + i}\right)^{m + i}.
\exp\left(\sum^\infty_{n=1}\frac{a_n}{n}x^n_{n+1}\right)|\lambda^{sp}\rangle= (\sum^\infty_{i_1=0}Y_{-\lambda_1+i_1}x^{i_1}_{n+1})\cdots (\sum^\infty_{i_{l+n}=0}Y_{-\lambda_{l+n}+i_{l+n}}x^{i_{l+n}}_{n+1})|0\rangle = \sum_{\nu\prec \lambda}s_{\lambda/\nu}(x_{n+1})|\nu^{sp}\rangle = \sum_{\nu\prec \lambda}x^{|\lambda|-|\nu|}_{n+1}|\nu^{sp}\rangle,
(u^{2}+y)^{2}-(\sqrt{2y-a}u-\frac{b}{2\sqrt{2y-a}})^{2}=(u^{2}+y+\sqrt{2y-a}u-\frac{b}{2\sqrt{2y-a}})(u^{2}+y-\sqrt{2y-a}u+\frac{b}{2\sqrt{2y-a}})=0
0 > (\alpha_1 - \alpha_2 ) \int_{r_1}^{r_2}w f_1 f_2 = (PW)(r_2) -(PW)(r_1) & = P(r_2) [f_1(r_2) f_2'(r_2) - f_1'(r_2) f_2(r_2) ] - P(r_1) [f_1(r_1) f_2'(r_1) - f_1'(r_1) f_2(r_1) ] & = -P(r_2) f_1'(r_2) f_2(r_2) + P(r_1) f_1'(r_1) f_2(r_1).
q_{\beta}(n,t)=\begin{cases*}E_{\beta, 1}\left(-\alpha (e^{\theta}-1) t^{\beta}\right),\n=0, \displaystyle\sum_{\Omega_{n}}\prod_{j=1}^{n}\frac{\left(\alpha \theta^{j}/j!\right)^{x_{j}}}{x_{j}!}z_{n}!t^{z_{n}\beta}E_{\beta, z_{n}\beta +1}^{z_{n}+1}\left(-\alpha (e^{\theta}-1)t^{\beta}\right),\n\ge1,\end{cases*}
\gamma(t)=\left[\begin{array}{rrrrrrrr}\cos 2x(t) 0 -\sin 2x(t)\cos y(t) -\sin 2x(t)\sin y(t) 0 \cos 2x(t) \sin 2x(t)\sin y(t) -\sin 2x(t)\cos y(t) \sin 2x(t)\cos y(t) -\sin 2x(t)\sin y(t) \cos 2x(t) 0 \sin 2x(t)\sin y(t) \sin 2x(t)\cos y(t) 0 \cos 2x(t) \end{array}\right]
\gamma \alpha_{0}= \partial_{\mu}\left( \varepsilon^{\mu \nu \lambda \rho}f_{b c}^{a}\left( C_{\rho a}A_{\nu}^{b}A_{\lambda}^{c}+ C_{a}\partial_{\rho}\left( A_{\nu}^{b}A_{\lambda}^{c}\right) \right) \right) .
[c,u]=0 \ \ \u\in U(A_1^{(1)}), & [h(k), h(l)]=2k \delta_{k+l,0}c, & [h,d]=0, \ \ [h, h(k)]=0, & [d,h(l)]=l h(l), \ \ [d,e(k)]=ke(k), \ \ [d,f(k)]=kf(k), & [h,e(k)]=2e(k), \ \ [h,f(k)]=-2f(k), & [h(k), e(l)]=2e(k+l), \ \ [h(k), f(l)]=-2f(k+l), & [e(k), f(l)]=h(k+l)+k \delta_{k+l,0}c.
f_1(x) = \frac{x^{\nu -1}}{\Gamma(\nu)}E \left[e^{-x/S}S^{-\nu}\right]; f'_1(x) = \frac{\nu-1}{x}f_1(x) - \frac{x^{\nu -1}}{\Gamma(\nu)}E \left[e^{-x/S}S^{-\nu -1}\right].
\left( \begin{array}{cc}M_h -K_{h,\nu}-K_{h,\nu}- \lambda^{-1}M_h \end{array}\right) \left( \begin{array}{c}\underline{y}_0^c \underline{p}_0^c \end{array}\right) = \left( \begin{array}{c}{\underline{y}_d^c}_0 0 \end{array}\right).
\alpha(\ell,\pm \epsilon_{n,d}) = \mp \sum_{b=1}^{\ell -1}\zeta_{\ell}^{\tau^b}\epsilon_{n,d}^{N_b}&\equiv \mp \sum_{b=1}^{\ell-1}\zeta_{\ell}^{\tau^b}\bmod{d}&\equiv \mp 1 \bmod{d}.
\left[ \widetilde R_{1 N}^{( 2 )}( x ) \right]_{\mathrm{( I I I )}}\simeq \beta^{N}\simeq \operatorname{e x p}\left( -{\frac{N^{1 / 3}}{\alpha_{c}^{2}g^{2 / 3}}}\right) \rightarrow 0 .
\vartheta^{q+1}\mu_{0}=\sum\limits_{k=0}^{q}\sigma_{k}(c) \vartheta^{k}\mu_{0},q>p-1, (1-c) \vartheta^{q+1}\mu_{0}=\sum\limits_{k=0}^{q}\sigma_{k}(c) \vartheta^{k}\mu_{0},q=p-1, c\vartheta^{p}\mu_{0}=\sum\limits_{k=0}^{p-1}\sigma_{k}(c) \vartheta^{k}\mu_{0},q<p-1,
\frac{\partial^{2}\ln I ( \phi , \chi )}{\partial \phi_{a}\partial \chi_{b}}= \sum_{k = 0}^{N - 1}\sum_{l = 0}^{\infty}\left\langle \frac{P_{N - k}( x_{1}) \tilde{Q}_{N + l + 1}( y_{1})}{x_{1}- \phi_{a}}\right\rangle \left\langle \frac{P_{N + l + 1}( x_{2}) \tilde{Q}_{N - k}( y_{2})}{y_{2}- \chi_{b}}\right\rangle
\bigg(\int\limits_X\widetilde{\alpha}_{m,\,\varepsilon}\wedge\gamma_{m,\,\varepsilon}^{n-1}\bigg)\cdot\bigg(\int\limits_X\widetilde{\alpha}_{m,\,\varepsilon}^{n-1}\wedge\beta_{\varepsilon}\bigg) \geq \frac{1}{n}\,\int\limits_X\widetilde{\alpha}_{m,\,\varepsilon}^n = \frac{c_{\varepsilon}}{n}.
\mathcal{T}_t( \kappa(t,\tau)\,h(u,\theta)+(1-\kappa(t,\tau))\,g(u,\theta)) =\, \mathcal{T}_t(\kappa(t,\tau)\,h(u,\theta)+(1-\kappa(t,\tau))\,g(u,\theta)) \mathcal{T}_t\kappa(t,\tau)\,h(u,\theta)+\mathcal{T}_t(1-\kappa(t,\tau))\,g(u,\theta),
\int_X\Phi(u(x).f(x))d\mu =\int_Xe^{u(x).f(x)}-u(x)f(x)-1d\mu &<\int_X(1+u(x).f(x))log(1+u(x).f(x))d\mu &=\int_A(1+f(x))log(1+f(x))d\mu+\int_B(1+f(x))log(1+f(x))d\mu &<\infty.
f(x,y,n,z) :=\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi x}{n}\Big)+\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi y}{n}\Big) + z^2, g(x,y,n,z) :=\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi x}{n}\Big)+\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi y}{n}\Big) + z^2 -\frac{2 n^2}{3\pi^2}\sin^2\Big(\frac{\pi x}{n}\Big)\sin^2\Big(\frac{\pi y}{n}\Big).
W\left\{\operatorname{Re}\varphi\left( \mathrm{i}\alpha-0\right),\psi\left( \mathrm{i}\alpha\right) \right\}& =\operatorname{Re}W\left\{\varphi\left( \mathrm{i}\alpha-0\right),\psi\left( \mathrm{i}\alpha\right) \right\}=-2\alpha.
\Gamma \Bigl ({\frac{D}{2}}- 1 \Bigr ){{\frac{\dot{x}_{a}\cdot \dot{x}_{b}}{{\Bigl [{( x_{a}- x_{b})}^{2}\Bigr ]}^{{\frac{D}{2}}- 1}}}\quad}\nonumber \, -{\frac{1 - \alpha}{4}}\Gamma \Bigl ({{\frac{D}{2}}- 2}\Bigr ){\frac{\partial}{\partial \tau_{a}}}{\frac{\partial}{\partial \tau_{b}}}{\Bigl [{( x_{a}- x_{b})}^{2}\Bigr ]}^{2 -{\frac{D}{2}}}
\eta_+^A=\epsilon_\pi(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}\hat{\eta}_R -\hat{\eta}_L\end{array}\right)^A\ ,\qquad \eta_-^A=\epsilon_0(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}\hat{\eta}_L \hat{\eta}_R\end{array}\right)^A\ ,
u[k](x)=\sum_{j=0}^{+\infty}v_{q}^-[\omega,\hat{\mu}_j](x)k^j+\frac{1}{k^{l}}\sum_{j=0}^{+\infty}\hat{c}_jk^j+Bq^{-1}x \forall x \in \overline{\mathbb{S}[\Omega_Q]^{-}}\, , \forall k \in ]0,k_\#[\, ,
{\left( \frac{\rho_{\phi}}{\rho_{B I}}\right)_{e n d}}={\frac{3 \pi}{6 4}}\left({\frac{1}{h_{G W}^{2}\left( F_{1}( \alpha ) / F_{2}( \alpha ) \right)^{2}}}\right) \left({\frac{\rho_{g}}{\rho_{B I}}}\right)_{e q}
(T+S)^{\dagger}(I+ST^{\dagger}) =\big(ST^{\dagger}T+T\big)^{\dagger}(I+ST^{\dagger}) =\big((I+ST^{\dagger})T\big)^{\dagger}(I+ST^{\dagger}) =T^{\dagger}(I+ST^{\dagger})^{-1}(I+ST^{\dagger}) =T^{\dagger}.
f^{n}(x)=a+(x-a)f^{\prime}(a)^{n}+\frac{(x-a)^{2}}{2}(f^{\prime\prime}(a)f^{\prime}(a)^{n-1})\frac{f^{\prime}(a)^{n}-1}{f^{\prime}(a)-1}+\cdot\cdot\cdot
\Theta_1^+(x) = \dfrac{\delta}{\left(2\sqrt{x^2+\delta^2}(\sqrt{x^2+\delta^2}-x)\right)^{1/2}}= -\Theta_2^-(x) ; \Theta_2^+(x) = \dfrac{\sqrt{x^2+\delta^2}-x}{\left(2\sqrt{x^2+\delta^2}(\sqrt{x^2+\delta^2}-x)\right)^{1/2}}=\Theta_1^-(x),
\frac{\min(j,k)}{\min(j,k)+|j^2-k^2|}\frac{\min(k,\ell)}{\min(k,\ell)+|k^2-\ell^2|}\leq \frac{j}{j+|j^2-k^2|+|k^2-\ell^2|}\leq \frac{j}{j+|j^2-\ell^2|}=\frac{\min(j,\ell)}{\min(j,\ell)+|j^2-\ell^2|}.
0 = \int_M (-\Delta_g^F u^\psi(x) + u^\psi(x)) u^\psi(x) e^{\phi(x)}d\mu_g(x) = \|\nabla_gu^\psi\|_{L^2(M,e^\phi d\mu_g)}^2 + \|u^\psi\|_{L^2(M,e^\phi d\mu_g)}^2 - \langle e^\phi \mathcal{N}\psi,\psi \rangle,
D\Bigl(\frac{g(\lambda)}{H_\lambda}\Bigr) =\sum_{\lambda^{+}}\frac{g(\lambda^{+})-g(\lambda)}{H_{\lambda^+}}, \noalign{\noindent and}D^-\Bigl(\frac{g(\lambda)}{H_\lambda}\Bigr) =\sum_{\lambda^{-}}\frac{g(\lambda)-g(\lambda^-)}{H_{\lambda^-}}.
R_{\mu}(x)=\Phi_{\mu}(\xi)\sim\frac{\sqrt{\sinh\pi\mu}}{2\pi}\,\left\{\Gamma(i\mu)\left(\frac{mx_+}2\right)^{-i\mu}+\Gamma(-i\mu)\left(-\frac{mx_-}2\right)^{i\mu}\right\},\quad x\in R,\quad x_{\pm}\to 0.
\Pi^{(q)}_{k,> k^{-N_0}}(s^ke^{k\phi}\mathcal{S}_ku)|_D\in\Omega^{0,q}(D,L^k), &\hat\Pi^{(q)}_{k,> k^{-N_0},s}(\mathcal{S}_ku)|_D\in\Omega^{0,q}(D).
\Phi_{HO}(x,p)=\prod_{i=1}^{n}\frac{1}{4}(2 mn x + (i-k_{1})2m)(2 mn x +(i-k_{1})2m +m+(1-\epsilon_{1})\mu_{x}s_{x}) &\times \prod_{j=1}^{m}\frac{1}{4}( 2 mn (p+1) -2 mn x + 2 n(j-k_{2}))( 2 mn (p+1) -2 mn x + 2 n(j-k_{2})+n- 2\epsilon_{2}n\mu_{y}s_{y}),
\overline{\partial}u =\overline{\partial}\sum_ku_k\otimes e_k=\sum_k\overline{\partial}u_k\otimes e_k=\sum_k f_k\otimes e_k=f, ||u||^2_{I_F^\psi\otimes h_E,\omega}=\int_U|u|^2_{I_F^\psi\otimes h_E,\omega}dV_\omega=\sum_k\int_U|u_k|^2_{h_E,\omega}e^{-\psi}dV_\omega<+\infty.