harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_d\left(x,R\right) = \left\{z\in\mathbb{Z}^2\mid d(x,z)\leq R\right\},\ \mbox{particularly:}D_d\left(x,R\right) =\left\{x\right\}\cup\left(\mathop{\bigcup}\limits_{r=1}^RC_d(x,r)\right), \ \mbox{where}\R\in\mathbb{Z}.
\nabla \zeta^{\bar{A}t}= d \zeta^{\bar{A}t}-{\frac{1}{4}}\gamma_{a b}\omega^{a b}\zeta^{\bar{A}t}- \Delta_{\bar{B}s}^{\bar{A}t}\zeta^{\bar{B}s}+{\frac{i}{2}}{\cal Q}\zeta^{\bar{A}t}
\Sigma_{\pm}( q_{3}, \left|{\bf q}\right| ) = - \frac{1}{4 \pi}\ln \left[ \frac{1}{2 m}\left( - \frac{1}{2}\delta \pm i q_{3}+ \frac{\left|{\bf q}\right|^{2}}{2 m}\right) \right] \, + \, O \left( \frac{1}{m^{2}}\right) .
U_{0}(x,y)=-\frac{j}{\lambda}\frac{e^{jkz}}{z}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{j\frac{k}{2z}[(x-x_{i})^{2}+(y-y_{i})^{2}]}U_{i}(x_{i},y_{i})dx_{i}dy_{i},
\mathbb{E}_{f}\!\left[S^{2}\right]=\lim_{q\to 1}\mathbb{E}_{f}\!\left[T^{2}\right]=\frac{\Gamma(mn)}{2}\left(-\frac{2}{\Gamma(mn+q)}\mathbb{E}_{g}\!\left[L\right]+\frac{1}{\Gamma(mn+2q)}\mathbb{E}_{g}\!\left[L^{2}\right]\right)^{\prime\prime}\bigg|_{q=1},
[\begin{matrix}u^{\prime}\\v^{\prime}\\w^{\prime}\end{matrix}]=[\begin{matrix}1&0&0\\ 0&cos\Omega t&sin\Omega t\\ 0&-sin\Omega t&cos\Omega t\end{matrix}][\begin{matrix}u^{\prime\prime}\\v^{\prime\prime}\\w^{\prime\prime}\end{matrix}],
\begin{cases}-\Delta\eta'- k^2\eta'=0\,\,\,\,\Omega begin{align*}0.3cm] \frac{\partial \eta'}{\partial n}=\left(-\frac{\partial^2 \eta}{\partial n^2}\right) V\cdot n+\nabla \eta\cdot\nabla_\Gamma(V\cdot n)\,\,\,\,\partial\Omega, \end{cases}
\beta_{j,j''}(a_{i,j}) = \bar{\beta}_{j,j''}(a_{i,j}) = \bar{\beta}_{j',j''}\circ \bar{\beta}_{j,j'}(a_{i,j}) & = \bar{\beta}_{j',j''}\circ\beta_{j,j'}(a_{i,j})= \beta_{j',j''}\circ\beta_{j,j'}(a_{i,j}).
Z^{\, dual}[\,h\,]=\left(\, det\; L_h\, \right)^{1/2}\,\exp \left(ik_A^{\, non-inv}S_A[\,h\,]\right)\, \int[\,{\cal D}\hat\phi\,]_h\,\exp\left[ -i\int d^2x \sqrt{-h}\,\hat\phi\, L_h\,\hat\phi\,\right]
\begin{array}{c}{\pi_{x}= p_{x}+ \frac{\lambda}{2}\left( x \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) x \right) ,}{\pi_{y}= p_{y}+ \frac{\lambda}{2}\left( y \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) y \right) .}\end{array}
( \Gamma_{1}^{\mathrm{s c a l a r s}})^{\wedge \wedge \vee}= - \frac{N_{s}g^{3}}{4 \pi^{2}}\frac{a}{m}\ln ( 1 / a ) \frac{p^{+}K^{\wedge}}{p_{1}^{+}p_{2}^{+}}\sum_{l = 1}^{M_{1}- 1}\frac{f_{s}( l / M )}{M}- \, ( \, 1 \leftrightarrow 2 \, ) .
\mathrm{length}(\mathcal{O}_Z)=\chi(\mathcal{O}_X(\Delta))+\chi(\omega_X^{-1}\otimes \mathcal{O}_X(-\Delta))-\chi(T_X)= 2\chi(\mathcal{O}_X)+K_X^2+K_X\cdot \Delta +\Delta^2-\chi(T_X)=K_X\cdot \Delta +\Delta^2 +c_2(X),
D_{s\chi\eta}=(\phi_s^{-1})_*\circ D_s= e_1\cdot(\phi_s^{-1})_*(\partial_{\tau})+e_2\cdot(\phi_s^{-1})_*(\partial_u)+e_3\cdot(\phi_s^{-1})_*(\partial_{\bar{u}}) &+\frac{1}{2}\sum_{i=1}^3e_i\sum_{k,l}(\phi_s^{-1})_*(\omega_{kl}(e_i))e_ke_l.
(X_L^{\prime i}(e^{2\pi i}z), X_R^{\prime i}(e^{-2\pi i}\bar{z}))= (X_L^{\prime i}(z)+2\pi \ell v_L^i, X_R^{\prime i}(\bar{z})-2\pi \ell v_R^i)+ ({\rm torus \shift}),
\min_{\mathbf{v}}\frac{\lambda_0 + \sum\limits_{l=1}^{2N-2}\lambda_l |v_l|^2}{2N-1-\frac{1}{K}-\frac{1}{K}\sum\limits_{l=1}^{2N-2}|v_l|^2}, ~~~\mathbf{w}=\frac{1}{2N-1}\mathbf{F}^{}_{2N-1}\mathbf{v}\geq 0.
\hat{H}^{+}_2\partial_2\varphi=\dot{H}^{+}_{N}+\varphi\partial_1\hat{H}^{+}_{N}\quad\mbox{and}\hat{H}^{-}_2\partial_2\varphi=\dot{H}^{-}_{N}-\varphi\partial_1\hat{H}^{-}_{N}\qquad\mbox{on}\,\,\,\Gamma_T\,.
{{\mathbf{V}}_1}= \left[{\begin{array}{*{20}{c}}{{\mathbf{V}}_1^1}{{\mathbf{V}}_1^2}\end{array}}\right],{{\mathbf{V}}_2}= \left[{\begin{array}{*{20}{c}}{{\mathbf{V}}_2^1}{{\mathbf{V}}_2^2}\end{array}}\right]
{\mathbf O}^{-1}\Delta(e_{0}){\mathbf O}=F_{1}\otimes 1 ,{\mathbf O}^{-1}\Delta(e_{1}){\mathbf O}=x E_{1}\otimes 1 + q^{-H_{1}}\otimes x E_{2},{\mathbf O}^{-1}\Delta(h_{0}){\mathbf O}= - H_{1}\otimes 1 - 1 \otimes H_{2},{\mathbf O}^{-1}\Delta(h_{1}){\mathbf O}=H_{1}\otimes 1 + 1 \otimes H_{2},
\sum_{j=0}^{n}\left(a_{j}\left(\frac{x}{1-2x}\right)-(1-2x)a_{j}(x)\right) -\frac{2x^{3}(3-6x+2x^{2})}{(1-x)^{2}(1-2x)}&= -\frac{2x}{1-x}\cdot\frac{1+(n+2)x}{1-(n+3)x}\cdot \left((n+3)(x-1)^2-(n+2)(2x-1)\right) a_{n+1}(x).
\tilde{T}_{\vec s^1,k_1}\star \tilde{T}_{\vec s^2,k_2}=\sum_{\vec s^3,k_3}\frac{\partial^3 \Phi_0}{\partial t_{\vec s^1,k_1}\partial t_{\vec s^2,k_2}\partial t_{\vec s^3,k_3}}\tilde{T}_{-\vec s^3}^{k_3}.
\frac{1}{n^{2-q}}\ln_{q}P\left( \frac{1}{n}\sum_{i=1}^{n}X_{i}<x\right) \geq\frac{1}{n^{2-q}}\ln_{q}b_{q}\left( \left\lfloor nx\right\rfloor;n,r\right) & \simeq-\frac{1}{2-q}D_{2-q}\left( x\left\Vert r\right. \right).
\widetilde{C}_{X^{\ast}}( s_1(-1)\otimes I_2, I_{1}\otimes g_2) =\widetilde{C}_{X^{\ast}}([-1,1], [1, I_{1}\otimes g_2]) &=\nu(1, 1)\widetilde{c}_{X^{\ast}}(1 , I_{1}\otimes g_2)=1;
\ell_{R}\left(R/{I_{q}}\right)= \ell_{R}\left(\frac{R \cap \mathbb{F}_{\geqslant \boldsymbol{u}}}{I_{q}\cap \mathbb{F}_{\geqslant \boldsymbol{u}}}\right) + \ell_{R}\left(\frac{R}{R \cap \mathbb{F}_{\geqslant \boldsymbol{u}}}\right) - \ell_{R}\left( \frac{I_{q}}{I_{q}\cap \mathbb{F}_{\geqslant \boldsymbol{u}}}\right)
\widehat{H}= \displaystyle \frac{2 \pi}{L}\left( \widehat{L_{0}}+ \widehat{\overline{{L}}_{0}}- \displaystyle \frac{c}{1 2}\widehat{I}+ \lambda \displaystyle \frac{L^{2 - h}}{\left( 2 \pi \right)^{1 - h}}\widehat{B}\right) \: ,
0 =\sum_{j=1}^{n-1}\bar h_{X_i X_j}\, \frac{\partial X_j}{\partial x_n}+ \bar h_{X_i t}\, \frac{\partial t}{\partial x_n}=\sum\limits_{j=1}^{n-1}\bar h_{X_i X_j}\, \frac{\partial X_j}{\partial x_n}- \frac{\bar h_{X_i t}}{\bar h_t}.
\int \frac{\mathrm{d}\Omega}{4 \pi}{\cal L}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left( k_{1}, k_{2}, k_{3}, \hat{Q}\right) = - i f^{a_{1}a_{2}a_{3}}{\cal A}_{\mu_{1}\mu_{2}\mu_{3}}\left( k_{1}, k_{2}, k_{3}\right) .
[\frac{1}{2}\left(\tilde{\varphi_t}^\star F_0\right)^2]_{\tilde y^{i}\tilde y^{j}}=\frac{1}{2}{\frac{\partial^2 (\tilde{\varphi_t}^{\star}F_0^2)}{\partial \tilde y^{i}\partial \tilde y^{j}}}=\tilde{\varphi_t}^\star(\frac{1}{2}\frac{\partial^2 F_0^2}{\partial y^i \partial y^j})=\tilde{\varphi_t}^\star([\frac{1}{2}F_0^2]_{y^i y^j}).
1 = \|x^*\| = \Bigl\|\sum_{i=1}^\infty x^*_i\Bigr\| \geqslant \Bigl\|\sum_{i=1}^\infty a_iy^*_i\Bigr\|-\|x_1^*\|-\Delta \geqslant\frac{1}{C}\Bigl\|\sum_{i=1}^\infty a_ie^*_{t_i}\Bigr\|_{T^*}-2-\Delta \geqslant \frac{1}{C}\Bigl\|\sum_{i=1}^\infty \|x^*_{i+1}\|e^*_{t_i}\Bigr\|_{T^*}-2-2\Delta.
\mathcal{E}_f(V)-\mathcal{E}_f(V_0) \le\frac{c_1^2}{2}\int_\Omega(V-V_0)\,V_0^{p-1}\,dx=\frac{c_1^2}{2}\,\left[\int_\Omega V\,V_0^{p-1}\,dx-1\right].
\phi_{m+1}\pi^s c =\sum_{1\leq i<j}\phi_iu_i + \phi_j ( \sum_{1\leq i<j}\phi_ie_i +\pi f)+\sum_{j+1\leq i\leq n}\pi \phi_i v_i\cr =\sum_{1\leq i<j}\phi_i(u_i+\phi_j e_i)+\pi \phi_j f+ \sum_{j+1\leq i\leq n}\pi \phi_i v_i
f_{TTT}= \frac{96 i}{\pi}\frac{(\Phi_T (T))^2}{\Phi (T)(\Phi(T) -\Phi(\frac{i}{\sqrt{2}}))}\frac{\Phi^{\frac{1}{2}}(U)}{\Phi_U (U)}\frac{{(\Phi(U)- \Phi(\frac{i}{\sqrt{2}}))}^{\frac{1}{2}}}{(\Phi(U) -\Phi(T))}.
\tilde{\psi}_{\mu}( x ) = a^{1 / 2}( t ) e^{- K / 1 2}\hat{\psi}_{\mu}( x ) , t i l d e \psi_{5}( x ) = \sqrt{\frac{3}{2}}a^{- 1 / 2}( t ) e^{K / 1 2}\hat{\psi}_{5}( x ) .
\gamma_i =\frac{\frac1r}{1-\frac1r}=\frac1{r-1}, \alpha_i =\frac{\frac1r-(\frac1r)^h}{1-\frac1r}=\frac{1-\frac1{r^{h-1}}}{r-1}, \frac{\gamma_i}{1+\gamma_i}=\frac 1r, \frac{\alpha_i}{1+\alpha_i}=\frac{r^{h-1}-1}{r^h-1}.
\prod_{1 \le i < j \le m}( x_{j}- x_{i}) = \left| \begin{array}{c c c c c}{{1}}&{{x_{1}}}&{{x_{1}^{2}}}&{{\dots}}&{{x_{1}^{m - 1}}}\\{{1}}&{{x_{2}}}&{{x_{2}^{2}}}&{{\dots}}&{{x_{2}^{m - 1}}}\\{{\vdots}}&{{\vdots}}&{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{1}}&{{x_{m}}}&{{x_{m}^{2}}}&{{\dots}}&{{x_{m}^{m - 1}}}\\ \end{array}\right|
t^{2}\left(1 + \frac{m}{2t}\right)^{2}\alpha''(t) + t\left(1 + \frac{m}{2t}\right)\left( - a + (a+2)\frac{m}{2t}\right) \alpha'(t) - 2(2 a + 1) \frac{m}{2t}\alpha(t) =0
\chi ( r ) = \frac{\sqrt{2}}{2}\, \left( \begin{array}{c}{{1}}\\{{e^{i \phi}}}\\ \end{array}\right) \, \delta_{r ( + )}\, + \, \frac{\sqrt{2}}{2}\, \left( \begin{array}{c}{{- e^{- i \phi}}}\\{{1}}\\ \end{array}\right) \, \delta_{r ( - )},
\Re \left\{\frac{- \nu^{- s}}{\pi}\, e^{\displaystyle{- i \frac{\pi}{2}( s + 1 )}}\int_{- i z}^{\infty - i z}t^{- s}\, \displaystyle{\frac{d \left( \ln{I_{\nu}\left( \nu \, t \right)}\right)}{d \, t}}\, d t \right\}.
H^*(X_{D}) =\mathbb{Z}[x_1,x_2,x_3]/\langle x_1^{b}, x_2(x_1+x_2)^{a-1}, x_3(x_1+x_2+x_3)^{c-1}\rangle, H^*(X_{\widetilde{D}}) =\mathbb{Z}[y_1,y_2,y_3]/\langle y_1^{b}, y_2(y_1+y_2)^{c-1}, y_3(y_1+y_2+y_3)^{a-1}\rangle.
|\widetilde{h}(x)-\widetilde{h}(y)|\le \limsup_{i\to\infty}\left|\sum_{l=i}^{N_i}a_{i,l}h_l(x)-\sum_{l=i}^{N_i}a_{i,l}h_l(y)\right|\le \limsup_{i\to\infty}\int_{\gamma}\sum_{l=i}^{N_i}a_{i,l}g_{h_l}\,ds= \int_{\gamma}g\,ds
{m\over2\pi}\delta_{j,0}\,\cosh\theta= \sigma_j(\theta)+\widetilde{\sigma}_j(\theta)+ s_j\,\ \sum_{k=0}^2 \int_{-\infty}^\infty \Phi_{j,k}(\theta-\theta') \,\,\sigma_k(\theta') \,\, d\theta', \qquad j=0,1,2.
\mathcal{M}\left\lbrace{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y)\right\rbrace =\int_{0}^{1}t^{x+s-1}(1-t)^{y+s-1}dt\int_{0}^{\infty}v^{s-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-v\right)dv.
E_{n_{1}}= \frac{\pi}{2 \lambda_{L}^{2}e^{2}}\frac{V}{L_{1}^{2}}\left( \frac{n_{1}}{N_{f}}\! + \! \frac{\Phi}{\Phi_{0}}\! + \! \frac{Q}{2}\right)^{\! 2}\! \! + \! 2 \pi N_{f}^{2}\lambda_{L}^{2}e^{2}\frac{Q^{2}}{V}.
\frac{|\mathcal{A}_m|}{\binom{4k+6}{2k+2}}= \frac{7\binom{4k}{2k}+\binom{7}{2}\binom{4k}{2k+1}+\binom{7}{1}\binom{4k}{2k+2}+\binom{4k}{2k+3}}{\binom{4k+6}{2k+2}}\rightarrow\frac{36}{64}=0.5625.
\sum_{j=1}^{\infty}\sum_{e_{1}\cdots e_j}(\alpha_1(e_{1}\cdots e_j))^{-(q-1)s}(\mu[e_{1}\cdots e_j])^q = \sum_{j=1}^{\infty}\sum_{e_{1}\cdots e_j}\phi^s(e_1 \cdots e_j)^{1-q}(\mu[e_{1}\cdots e_j])^q
\Psi_N(qz,qz,1/z,1/z) = \sum_{k = 0}^N q^kz^k{N\brack k}_q , \Phi_N(qz,qz,1/z,1/z) = \frac{\Psi_N(qz,qz,1/z,1/z)}{(q;q)_N}= \sum_{k = 0}^N \frac{q^k z^k}{(q;q)_k(q;q)_{N-k}}.
0=[\begin{matrix}x&y&w\end{matrix}][\begin{matrix}\frac{1}{a^{2}}&0&\frac{L}{a^{2}}\\ 0&\frac{1}{b^{2}}&-\frac{d}{b^{2}}\\ \frac{L}{a^{2}}&-\frac{d}{b^{2}}&\frac{L^{2}}{a^{2}}+\frac{d^{2}}{b^{2}}-1\end{matrix}][\begin{matrix}x\\y\\w\end{matrix}]
P=[\begin{matrix}|&&|\\v_{1}&\cdot\cdot\cdot&v_{n}\\ |&&|\end{matrix}]\equiv[\begin{matrix}(v_{1})_{1}&\cdot\cdot\cdot&(v_{n})_{1}\\ \vdots&&\vdots\\ (v_{1})_{n}&\cdot\cdot\cdot&(v_{n})_{n}\end{matrix}]
\psi(\alpha \star \beta) = \psi(\lfloor p_{i}; h_{i}\rceil \star \lfloor p_{j}; h_{j}\rceil ) = \psi( \lfloor \gamma_{1}p_{i}p_{j}; h_{i}+ h_{j}\rceil) = \lfloor \phi(\gamma_{1}p_{i}p_{j}); h_{i}+ h_{j}\rceil = \lfloor \phi(\gamma_{1})\phi(p_{i}p_{j}); h_{i}+ h_{j}\rceil.
\sum_{k=n-4}^nc_k\leq 9\cdot \frac{2}{5}n\prod_{j=1}^{\frac{n-7}{2}}\left(\frac{n-2j-1}{2}\right)^2 =9\cdot \frac{2}{5}n\cdot \frac{\left(\left(\frac{n-3}{2}\right)! \right)^2}{(2!)^2}=\frac{9}{10}n\left(\left(\frac{n-3}{2}\right)! \right)^2.
J ( M , T ) = \int \frac{d^{3}p}{( 2 \pi )^{3}}\frac{1}{p^{2}+ M^{2}}+ 2 \int_{- \infty + i \epsilon}^{\infty + i \epsilon}\frac{d z}{2 \pi}\int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{1}{z^{2}+{\bf p}^{2}+ M^{2}}\frac{1}{e^{- i \beta z}- 1}\; .
\varphi(x,y) \equiv - \sum_{n>0}\hbar \frac{e_{+}q}{\pi{\rm L}}(1+{\rm N}^2) \frac{1}{{\rm E}_n^2({\rm N})}\cos(\frac{2\pi n}{\rm L}(x-y)) =- \frac{q{\rm M}_{\rm N}}{2e_{+}}\frac{\cosh(\frac{\rm L{\rm M}_{\rm N}}{2}-{\rm M}_{\rm N}|x-y|)}{\sinh{\frac{\rm L{\rm M}_{\rm N}}{2}}}.
\begin{pmatrix}G^{1}G^{2}\end{pmatrix}=-\frac{1}{2}\begin{pmatrix}f_{1}y^{1}+f_{2}y^{2}g_{1}y^{1}+g_{2}y^{2}\end{pmatrix}=-\frac{1}{2}\begin{pmatrix}f_{1}f_{2}g_{1}g_{2}\end{pmatrix}\begin{pmatrix}y^{1}y^{2}\end{pmatrix}=-\frac{1}{2}J\cdot y,
|\nabla\nu_{0}\cdot\overline{\nu}|^{2}+\left|\nabla\left(1-\left(\overline{\nu}\cdot\nu_{0}\right)^{2}\right)^{1/2}\right|^{2}=\frac{|\nabla\nu_{0}\cdot\overline{\nu}|^{2}}{1-(\nu_{0}\cdot\overline{\nu})^{2}}
\delta_{f}L_{3}= - \frac{6 \kappa_{1}^{2}}{r}\left( ( 1 - f^{2}) f_{r r}- f \: f_{r}^{2}- \frac{1}{r}( 1 - f^{2}) f_{r}+ \frac{1}{r^{2}}( 1 - f^{2})^{2}f \right)
(PW)' = P[f_1f_2''- f_2 f_1''] + P'[f_1f_2' - f_2 f_1'] & = f_1[ (Pf_2')' + (\alpha_2 -Q) f_2] -f_2 [ (Pf_1')' + (\alpha_1w - Q) f_1] + (\alpha_1 - \alpha_2 )w f_1 f_2 & =(\alpha_1 - \alpha_2 )w f_1 f_2.
\begin{array}[pos]{rcl}x_1\!(k) \!\!=\!\! x_1(\!k\!-\!1\!)\!\left(r\!+\!1\!-\!rx_1(k\!-\!1)\!-\!bx_2(k\!-\!1)\!\right)\!+\!w_1(\!k\!-\!1\!), x_2\!(k) \!\!=\!\! cx_1(k\!-\!1)x_2(k\!-\!1)+(1-d)x_2(k\!-\!1)\!+\!w_2(\!k\!-\!1\!),\end{array}
x_t = 1 - t - \frac{t}{d'}, y_t = \frac{t}{d'-5}\left( \frac{1}{4}- \frac{1}{d'}\right), z_t = \frac{t}{d'-5}\left( \frac{1}{4}- \frac{4}{3 d'}\right), t \leq \frac{d'}{d'+1}
g = g_{R}g_{L}= \left( \begin{array}{c c}{{A}}&{{U}}\\{{- V}}&{{B}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a \tilde{a}- u \tilde{v}}}&{{a \tilde{u}+ u \tilde{b}}}\\{{- v \tilde{a}- b \tilde{v}}}&{{b \tilde{b}- v \tilde{u}}}\\ \end{array}\right)
\frac{d}{dt}(J(u(t))) = \frac{d}{dt}(\langle p(t), u(t) \rangle) = \langle \partial_t p(t), u(t) \rangle + \langle p(t), \partial_t u(t) \rangle = - \frac{1}t \langle \phi(t), u(t) \rangle - \Vert p(t) \Vert^2.
P_{| n_{1}, n_{2}, \dots , n_{k}\rangle}^{\pm}( \eta , \vartheta , u ) = P^{\pm}( \eta , \vartheta , u ) \prod_{i \textrm{o d d}}a_{n_{i}}( \eta , u ) \prod_{i \textrm{e v e n}}a_{n_{i}}( \bar{\eta}, u ) ,
\frac{F}{V}= (-1)^f\frac{1}{\beta}\int_0^\infty \frac{dp^-}{\sqrt{2}\pi}\int \frac{d^{D-2}p_T}{(2\pi)^{D-2}}\ln\left( 1-(-1)^f e^{-\frac{\beta}{\sqrt{2}}\left(\frac{{\vec p_T}^{~2}+M^2}{2p^-}+p^-\right)}\right)
D^C = \{(P_f)_{f\in G}\in\wp(G)^G\; |\; \mbox{$\exists g, h \in G$ with $g \neq h$ such that $gh^{-1}\notin P_h$ and $hg^{-1}\notin P_g$}\}&= \bigcup_{g,h\in G, g \neq h}(W_{(h, gh^{-1})}^C \cap W_{(g,hg^{-1})}^C),
p_{eff}(\phi)=\frac{1}{4}\left(1+\frac{b_{k}}{b_{g}}\right)\dot{\phi}^{2}-\frac{\Delta^{2}}{16b_{g}M^{4}}\dot{\phi}^{4}e^{2\alpha\phi/M_{p}}- \frac{M^{4}}{4b_{g}}e^{-2\alpha\phi/M_{p}}
\begin{array}{c}v_L(\lambda_n)=-b,v_L(\lambda_{n+1}-\lambda_n)\geq v_L(\lambda_n-\lambda_{n-1})+1, v_M((\sigma-1)\wp_\alpha(\lambda_{n+1}))\geq v_M((\sigma-1)\wp_\alpha(\lambda_n))+1,\end{array}
\sum_{i=2}^{\infty}[z^i]\frac{1}{2}\left(-\frac{z^3}{(1+z)^3}+\frac{z^2}{(1+z)^2}\eta\right)\left(-1-\frac{1}{z}+ \frac{(1+z)^2}{z^2}\eta\right)u^{i-2}\alpha_i = \sum_{i=2}^{\infty}[z^i]\left(-\frac{z}{1+z}\eta+\frac{1}{2}\eta^2\right)u^{i-2}\alpha_i+\sum_{i=2}^{\infty}[z^i]\frac{z^2}{(1+z)^3}u^{i-2}\frac{\partial}{\partial q_i}.
\underline{{h s ( 2 , 2 )}}: f ( a , \bar{a}) = \alpha + \alpha_{i \bar{j}}a^{i}\bar{a}^{\bar{j}}+ \alpha_{i_{1}i_{2}\bar{j}_{1}\bar{j}_{2}}a^{i_{1}}a^{i_{2}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}+ \ldots
f_0=\begin{pmatrix}\delta_2-\delta_1 \mu_2-\mu_1 \end{pmatrix}begin{align*}1em] f_1=\begin{pmatrix}(1-\alpha)\delta_1\mu_1 + \delta_2(\mu_1-\alpha\mu_2)-\gamma \mu_1(\mu_1-\alpha\mu_2) \delta_2(\delta_2-\alpha\delta_1) (1-\alpha)\delta_2\mu_2 + \mu_1(\delta_2-\alpha\delta_1)-\gamma\end{pmatrix}.
e( n+1) = e( \overline{( 01)^{k-1}10}) = e( \overline{( 01)^{k-1}1}) +1=e( \overline{( 01)^{k-2}011}) +1 &= e( \overline{( 01)^{k-2}1}) +2) =e( \overline{11}) +k-1= k,
\big( (g_{1},\chi_{1}^{-1}), \dots, (g_{n}, \chi_{n}^{-1}), \varphi_{1}, \dots,\varphi_{n}\big), \mbox{with}\varphi_{i}\in\widehat{\Sigma},\,\varphi_{i}(g, \chi) = \chi_{i}(g),\, i=1,\dots,n.
O(z^{-1})\sim(e^{-zU}z^{B}dC_{\nu}\cdot C_{\nu}^{-1}z^{-B}e^{Uz})^{\alpha}_{\beta}=e^{u_{\beta}-u_{\alpha}}z^{b_{\alpha}-b_{\beta}}(dC_{\nu}\cdot C_{\nu}^{-1})^{\alpha}_{\beta}
\widetilde G(T,Y)=Y^\delta+(\xi_1T)Y^{\delta-1}+\cdots+\xi_\delta T^\delta, \widetilde F_{i}(T,Y)=(\eta_{i,1}T^{n_i-\delta+1})Y^{\delta-1}+\cdots+\eta_{i,\delta}T^{n_i}.
y_k^s(x)=\sin (kx)+R_k(x), \quad\|R_k\|_{L^\infty}= O\Big(\frac1k\Big) k\to+\infty. \lambda_k^s =k^2+\widetilde a -\frac{1}{2}a_{c,2k}+O\Big(\frac1k\Big) k\to+\infty.
T_{1,C_1}(V=0) = \frac{q^{2g+1}}{9! \pi \zeta_q(2)}\sum_{r=0}^9 c_r \Bigg[ \sum_{j=0}^{\alpha-1}\sum_{m=0}^{2g}\frac{\sin (2 \pi \theta_0(2m-j))}{2m-j}(2g-m)^r + \sum_{j=0}^{\alpha-1}\sum_{m=1}^{2g}\frac{\sin (2 \pi \theta_0(2m+j))}{2m+j}(2g-m)^r \Bigg] +O(q^{2g+1}\alpha).
c_{n}= \prod_{k=-\kappa+1}^{-\kappa+n}\lambda_{k}^{2}, n\in\mathbb{N}\cap [0,\kappa], c_{n}= \prod_{k=0}^{\kappa-1}\lambda_{k}^{2}\sum_{i=1}^{\eta}\prod_{j=1}^{n-\kappa}\lambda_{i, j}^{2}, n\in\mathbb{N}\cap [\kappa+1,\kappa+p]
(A_\omega m)(Z(v,k)) = \sum_{e \in vE^1, \tau \in s(e)E^{n_k-1}}\frac{1}{n_k}\rho(A_E)^{-|\tau|}x^E_{s(\tau)}= \frac{1}{n_k}\sum_{w \in E^0}\sum_{\lambda \in vE^{n_k}w}\rho(A_E)^{-|\lambda|+1}x^E_w = \frac{1}{n_k}(A^{n_k}\rho(A_E)^{-n_k+1}x^E_w)_v = \frac{1}{n_k}\rho(A_E)x^E_v.
Z_4 = \frac{(m L)^4}{4!}I_1^4 - (m L)^3 I_1^2 I_2 + \frac{(M L)^2}{2}\left[\frac{2}{3}I_1 I_3 + \left(\frac{I_2}{2}\right)^2\right] - \frac{(m L)}{4}I_4 \,\,\, .
\mathcal{I}_2(\bar X, t)= \int_{S^{N+1}(t)}d y \int_{}d w\, \frac{f_0^{N+1}( y, w)}{|\bar X - Y^{N+1}(t)|^{2}}\chi\left(N\leq |w|\leq N+1\right) \leq & C_0\,e^{-\lambda N^2}\int_{S^{N+1}(t)}dy \int_{}dw\,\, \frac{g(|y|)}{|\bar X - Y^{N+1}(t)|^{2}}\chi\left(|w|\leq N+1\right).
\frac{\partial \widetilde{x}^b}{\partial x^a}=\frac{\partial \widetilde{y}^b}{\partial y^a}\,,\,\frac{\partial \widetilde{x}^b}{\partial y^a}=\frac{\partial \widetilde{y}^b}{\partial x^a}\,,\,a,b\in\{1,\ldots n\}.
\phi_1(u,v)P(w) = P(\phi_1(u,v)w) + \phi_1(u,P(v)w + vP(w) + \lambda vw ) &= P(\phi_1(u,v)w) + \phi_1(u,P(v)w) + \phi_1(u, vP(w)) +\lambda \phi_1(u, vw).
\left| \frac{{\partial \varphi}_{i}}{{\partial y}_{j}}\right|\le \frac{l_{i}^{x}}{l_{1}^{x}}\frac{\left( M_{1}^{x}M_{i}^{x}-\mu_{1}^{x}\mu_{i}^{x}\right)}{{(\mu_{i}^{x})}^{2}}\frac{l_{1}^{y}}{\mu_{1}^{y}\sum\limits_{j=1}^n \frac{l_{j}^{y}}{M_{j}^{y}}},\thinspace i=1,\thinspace \mathellipsis ,\thinspace m;\thinspace j=1,..,n.
\sum_{b=1}^{j-i+1}\sum_{k: M_k\in A_{j+1}^{i+b}}\lambda_{k}|A-M_{k}|\lesssim\sum_{b=1}^{j-i+1}\sum_{k: M_k\in A_{j+1}^{i+b}}\lambda_{k}(i+b)= \sum_{b=1}^{j-i+1}\nu(A_{j+1}^{i+b})(i+b) &\lesssim \sum_{b=1}^{j-i+1}j^{2(m_1'+m_2'-n')}\frac{i^{m_1'}}{(i+b)^{m_1'+1}}\lesssim j^{-2}.
c_3 = 2\Gamma^2 + 2\Gamma\Lambda + 2\Theta^2 + 2\Gamma cov\{\hat{m}_{2y}(\lambda),\hat{m}_{2y}(-\beta)\}&\ + \Delta(\Gamma + \Lambda) + 2\Theta cov\{\hat{m}_{2y}(-\lambda),\hat{m}_{2y}(0)\}+ 2\Upsilon^2 &\ + 2\Upsilon cov\{\hat{m}_{2y}(0),\hat{m}_{2y}(-\beta)\}+2\big[cov\{\hat{m}_{2y}(0),\hat{m}_{2y}(0)\}\big]^2,
\begin{aligned}(\boldsymbol{D}^{-s}\psi, \boldsymbol{D}^{s*}\psi) =(\widehat{\boldsymbol{D}^{-s}\psi}, \overline{\widehat{\boldsymbol{D}^{s*}\psi}}) =(\mathcal{F}(\boldsymbol{D}^{-s}\psi), \overline{\mathcal{F}(\boldsymbol{D}^{s*}\psi)}) =((2\pi i\xi)^{-s}\widehat{\psi}, \overline{(-2\pi i \xi)^{s}\widehat{\psi}}). \end{aligned}
\Omega_1(|s|-s) =A_{1}\tau(|s|+s) + q (A_{1}\tau+\Omega_1)s =(\Omega_{1}-A_{1}\tau)|s|-q ((M_{1}+I- L_1)\tau+\Omega_1)s =(N_1+I- L_1)\tau s+(\Omega_1-A_{1}\tau)|s|-q.
|S| \leqslant \int_{\|u-y\|\leqslant t}|f_t(x,u)-f_t(x,y)| \cdot |g_s(u,y)|d\omega(u) \quad+\int_{\|u-y\|> t}\big(|f_t(x,u)|+|f_t(x,y)|\big) \cdot |g_s(u,y)|d\omega(u) =: I+I\!I,
|c_i(t)e^{-t}-\tilde{h}^{-}(t)| \le \int_{-\frac{1}{i}}^0|\tilde{h}(t+y)-h^{-}(t)|g_i(y)dy &+\int_{0}^{\frac{1}{i}}\tilde{h}(t+y)g_i(y)dy.
\begin{aligned}E_n =\int_{x_0}^{x^*}(f(x)-p_n(x))\dx=\frac{f^{(n+1)}(\xi_1)}{(n+1)!}h^{n+2}\int_{0}^{\frac{x^*-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)\d\mu &+\frac{f^{(n+2)}(\xi_2)}{(n+2)!}h^{n+3}\int_{0}^{\frac{x^*-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)(\mu-\frac{x^*-x_0}{h})\d\mu.\end{aligned}
\textbf{A}=\textrm{tri}\bigg[\frac{s-ph}{2(phc-s)},1,\frac{s-ph}{2(phc-s)}\bigg], &\textbf{K}_1=\textrm{tri}\bigg[\frac{p(1-c)}{2(phc-s)},0,-\frac{p(1-c)}{2(phc-s)}\bigg], &\textbf{K}_2=\textrm{tri}\bigg[\frac{p^2s}{2(phc-s)},-\frac{p^2s}{phc-s},\frac{p^2s}{2(phc-s)}\bigg],
G_4^{\mu\nu\alpha\beta}(x)=\int{dp^4\over (2\pi)^4}\left ({1\over2}(g^{\mu\alpha}g^{\nu\beta}+ g^{\mu\beta}g^{\nu\alpha}) -{1\over3}g^{\mu\nu}g^{\alpha\beta}+{\cal O}(p) \right )D(p^2,m_0^2, \Gamma)e^{-ip(x)},
c_0=\alpha^2_{+}\;,\quad c_1=\frac{\epsilon\left(\alpha_{+}+\epsilon\alpha_{-}\right)^2}{\epsilon-\sin\frac{n\gamma}{16}}+2\epsilon\;,\quad c_2=\frac{\left(\alpha_{+}+\epsilon\alpha_{-}\right)^2}{\left(\epsilon-\sin\frac{n\gamma}{16}\right)^2}\;,
(\kappa^*(t), R^*(t)) & = \arg \max_{\substack{\kappa(t) \in \mathcal{P}(\mathcal{F}) R(t) \in \mathcal{R}(t)}}\ \sum_{u \in \mathcal{K}_{R(t)}\cap \tau_{\kappa(t)}}\log \left(\frac{R(t)}{\epsilon_u(R,R_u)N}\right),
f(\xi_1,\xi_2,\dots,\xi_N) =\xi^{\top}\Bigg(\int_0^tA_\nu(\tau)d\tau\Bigg)\xi =\sum\limits_{j=1}^N\Bigg(2\int_0^tO_\tau^{(j)}d\tau-2Lt-2\nu{t}\Bigg)\xi_j^2+2\nu{t}\sum\limits_{j=1}^{N}\xi_j\xi_{j-1}\leq\sum\limits_{j=1}^N\Bigg(2\int_0^tO_\tau^{(j)}d\tau-Lt\Bigg)\xi_j^2-Lt\sum\limits_{j=1}^N\xi_j^2
H ( r , z_{i}) = 1 \, + \, f_{0}( N_{I}) \, \frac{Q}{r^{4}}+ \frac{K^{2}}{4 \, r^{4}}\sum_{i}f_{i}( N_{I})^{2}\left[ \log \left( \frac{r^{4}}{\epsilon^{2}( r^{2}- \rho_{i}^{2})}\right) - 1 + \frac{\rho_{i}^{2}}{r^{2}- \rho_{i}^{2}}\right]
\lim_{R\nearrow\infty}\lim_{\epsilon\downarrow 0}I_\pm(\epsilon,R)=\lim_{\epsilon\downarrow 0}\lim_{R\nearrow\infty}I_\pm(\epsilon,R)=\frac{1}{2}{\rm Tr}\; SD_a(P_\pm -D_aP_\pm D_a)^{2n+2}.
\Omega = - \frac i 2 V \int \frac{d^{3}p}{( 2 \pi )^{3}}\int \frac{d p_{0}}{2 \pi}\mathrm{T r}\left[ M_{1 1}^{2}\mathrm{L n}\gamma^{0}S^{+}( p ) - M_{1 2}M_{2 1}\mathrm{L n}\gamma^{0}S^{-}( p ) \right] - \Delta \Omega .
\begin{array}{rcl}s A^a_\mu = -\,(D_\mu c)^a \quad , begin{align*}2mm] s \phi^a = f^{abc}c^b \phi^c \quad , begin{align*}2mm] s c^a = \frac{1}{2}\, f^{abc}c^b c^c \quad , begin{align*}2mm] s \bar{c}^a = b^a \quad , begin{align*}2mm] s b^a = 0 \quad , begin{align*}2mm] s^2 = 0 \quad .\end{array}
1+ \frac{1}{r}< 1 + \frac{2}{r-1}\leq \bigg(1 + \frac{1}{r-1}\bigg)^2 &= \frac{1}{(1 - 1/r)^2}= \bigg(\frac{n_1'}{\phi(n_1')}\cdot \frac{\phi(n_1)}{n_1}\bigg)^2 &\leq 1 + \frac{\frac{3}{2}(16+16)}{y (\log y)^{1/6}}= 1 + \frac{48}{y (\log y)^{1/16}}.
h_q(x) \leq h_q(x) + h_q(-x) &=\frac{2}{\pi}\int^\infty_0 \Re\left( \frac{1-\cos(ux)}{q-\eta(u)}\right)du &\leq \frac{2}{\pi}\int^\infty_0 \frac{1-\cos(ux)}{|\eta(u)|}du.
H(\{q_{N_s}\})=(u_1-u_2)(v_1-u_2)\left[1-\frac{\mu_1\mu_2}{2(u_1-u_2)}-\frac{\nu_1\mu_2}{2(v_1-u_2)}\right]\prod_{s=1}^n(u_s-v_s-\mu_s\nu_s)^{-1}
Z = \sum_{i , \bar{\imath}}Z_{i , \bar{\imath}}^{( 9 , 9 )}Z_{i , \bar{\imath}}^{( 6 , 6 )}B_{i}^{( - 2 )}\left( B_{\bar{\imath}}^{( E_{8}^{\prime}\times S O ( 6 ) )}\right)^{*}.
\mathcal{E}(t) \leq \tfrac{1}{\alpha_{1}}\mathcal{F}(t)\leq \tfrac{1}{\alpha_{1}}\mathcal{E}(0) e^{-\tfrac{C}{\alpha_{2}}t}=: C \mathcal{E}(0) e^{-2\alpha t}t \geq 0
\begin{array}{llccl}J_L : T_L^* ({\cal M}_n) \rightarrow T_L({\cal M}_n) & U \mapsto \ (LU)_+\,L \, -\, L\,(UL)_+ \, +\, L\,\Phi_U(L) \, +\, \bar{\Phi}_U(L)\, L\end{array}
\sum_{n=1}^{\infty}\frac{e^{x^{\frac{1}{n}}}}{n!}+x^{3}+logx+loglogx+\sum_{n=0}^{\infty}x^{-n}+\sum_{i=1}^{\infty}e^{-\sum_{j=1}^{\infty}e^{ix^{2}-jx}}.
\Sigma_{A}( \bar{p}) \simeq \frac{i e^{2}}{( 2 \pi )^{3}}| e H | \int d q_{0}d q_{3}\int_{0}^{\infty}d r^{2}\mathrm{e}^{- r^{2}}\frac{G_{1}}{q^{2}}\frac{\Sigma_{A}( \bar{p "})}{\bar{p "}^{2}+ \Sigma_{A}( \bar{p "})}
\delta{\cal I}_{U ( 1 )}= - \lim_{\beta \rightarrow 0}2^{N / 2 - 1}\int [ d X ] \frac{e^{- ( X + X )^{2}/ 2 \beta}}{( 2 \pi \beta )^{( D - 1 ) / 2}}= - \frac{2^{N / 2 - 1}}{2^{D - 1}}= - \frac{1}{4},
B = \overline{\operatorname{span}}\, \left\{\tau^\infty(\xi_1) \cdots \tau^\infty(\xi_k) \pi^\infty(a) \tau^\infty(\eta_l)^* \cdots \tau^\infty(\eta_1)^* u^m : \right. \, \left. \eta_i, \xi_i \in H^\infty, a \in A^\infty, k, l, m \in \mathbb{Z}\right\}.
\left|\tau(\sum_n B^\delta_{R_n}(x)y_n)\right| =|C_{\alpha,\beta}|\left|\tau(\sum_n\left(\int^1_0\varphi(t)M^\alpha_{R_nt}(x)dt\right)y_n)\right| &\leq |C_{\alpha,\beta}|\int^1_0|\varphi(t)|\left|\tau(\sum_nM^\alpha_{R_nt}(x)y_n)\right|dt &\lesssim \|{\sup_{R>0}}^+M^\alpha_{R}(x)\|_2.
T^{{\underline a}}={\cal D}E^{{\underline a}}= -{i \over 2}\hat{E}^{\hat{\underline \alpha}}\wedge \hat{E}^{\hat{\underline \beta}}\Gamma^{{\underline a}}_{{\underline{\hat{\alpha}\hat{\beta}}}}=-{i \over 2}(\hat{E}\Gamma^{{\underline a}}\hat{E})