harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e_{+}(\alpha)=\int_{\cal E}\log\left(\frac{\cosh((\beta_l(1-\alpha)+\beta_r\alpha)E/2)\cosh((\beta_r(1-\alpha)+\beta_l\alpha)E/2)}{\cosh(\beta_l E/2)\cosh (\beta_r E/2)}\right)\frac{\d E}{2\pi},
-2p\lambda_{1;p,f}\int_{\Omega}|u|^{\alpha-p+1}|\nabla u|^{2p-3}\langle\nabla u,\nabla|\nabla u|\rangle\,d\mu = 2(\alpha-p+1)\lambda_{1;p,f}\int_{\Omega}|u|^{\alpha-p}|\nabla u|^{2p}\,d\mu-2\lambda_{1;p,f}^2\int_{\Omega}|u|^{\alpha}|\nabla u|^{p}\,d\mu.
\P(T_1^z>\varphi_z^{-1}(z_2)) =\exp\left(-\int_0^{\varphi_z^{-1}(z_2)}\beta(\varphi_z(s))ds\right) &=\exp\left(-\int_z^{z_2}\beta(u)(\varphi_z^{-1})'(u)du\right) &\geq\exp\left(-(z_2-z_0)\sup_{[z_0,z_2]}\left(\beta(\varphi_z^{-1})'\right)\right) &\geq\exp\left(-(z_2-z_0)\left(\sup_{[z_0,z_2]}\beta\right)\left(\inf_{[z_0,z_2]}\tau\right)^{-1}\right),
\bar{P}_{l}\leq P_{s_l}l=1,\cdots,M &\bar{P}_{M+i}\leq P_{r_i}i=1,\cdots,M &f_{l}\bar{R}_{l}=f_i\bar{R}_i \forall l,i &\sum_{i=1}^{2M}f_i \leq 1
\bigl\langle(I\otimes J)W^*(I\otimes J)(v\otimes p),w\otimes q\bigr\rangle =\overline{\bigl\langle W^*(I\otimes J)(v\otimes p),Iw\otimes Jq\bigr\rangle}&=\overline{\bigl\langle(\omega_{Iv,Iw}\otimes\operatorname{id})(W^*)Jp,Jq\bigr\rangle},
\zeta_\delta(x)=O\left(\frac{\delta^\frac{4-n}{2}}{(1+\frac{|x- e_n|}{\delta})^{n-3}}+ \frac{\delta^\frac{4-n}{2}}{(1+\frac{|x+ e_n|}{\delta})^{n-3}}\right) B_1(0),
T^a_{\phantom{a}a}=-{1\over2\alpha^\prime}\beta_{\mu\nu}^Gg^{ab}\partial_aX^\mu\partial_bX^\nu-{i\over2\alpha^\prime}\beta_{\mu\nu}^B\epsilon^{ab}\partial_aX^\mu\partial_bX^\nu-{1\over2}\beta^\Phi R\ .
(\mathbf{x}_{n+1}) = \left( \left\{\min_{i=1,\ldots,n}f(\mathbf{x}_i) - f(\mathbf{x}_{n+1}) \right\}^+ \mid \mathbf{x}_1,\ldots,\mathbf{x}_n,f(\mathbf{x}_1),\ldots,f(\mathbf{x}_n) \right),
\left[ \left( s-\frac{t}{2}\right) \tilde{X}_{j}^{2}+\frac{t}{2}\tilde{Y}_{j}^{2}-u\tilde{X}_{j}\tilde{Y}_{j}\right] +\tau\partial_{j}^{2}+\bar{\tau}\bar{\partial}_{j}^{2}=s\tilde{X}_{j}^{2}.
\Gamma_{l , s}^{\alpha , \alpha_{0}}( p ) = \underbrace{\Gamma_{l , s}^{\alpha , \alpha_{0}}( 0 )}_{\mathrm{{a s \i n \ ( \r e f{g a o}})}}+ \frac{1}{2}\sum_{\mu , \nu = 0}^{3}p_{\mu}p_{\nu}\partial_{\mu}\partial_{\nu}\Gamma_{l , s}^{\alpha , \alpha_{0}}( 0 )
\begin{array}{r c l r c l}{{\left[ L_{1}, L_{- 2}\right]}}&{{=}}&{{L_{- 1},}}&{{\quad \left[ L_{1}, L_{2}\right]}}&{{=}}&{{0 ,}}\\{{\left[ L_{2}, L_{- 2}\right]}}&{{=}}&{{N + \frac{D + 1}{2},}}&{{\quad \left[ L_{0}, L_{n}\right]}}&{{=}}&{{0 ,}}\\{{\left[ L_{1}, L_{- 1}\right]}}&{{=}}&{{L_{0},}}&{{\quad \left[ N , L_{n}\right]}}&{{=}}&{{{}- n L_{n}, \quad n = 0 , \pm 1 , \pm 2 ,}}\\ \end{array}
\begin{aligned}\sum_{h=0}^{n-1}\|f(s_{h+1}) - f(s_h)\|_E^q \lesssim \sum_{N}N^\epsilon \sum_{h=0}^{n-1}\sum_{I\in J_h^N}\|f(\max I) - f(\min I)\|_E^q \leq \sum_{N}N^\epsilon \sum_{k=0}^{N-1}\|f(t_{k+1}^N) - f(t_k^N)\|_E^q. \end{aligned}
\mathcal{F}\left\{e^{-\pi t^2}|t|^\mu \right\}(\nu) = \left( \mathcal{F}\left\{e^{-\pi t^2}\right\}\ast\mathcal{F}\left\{|t|^\mu \right\}\right) (\nu) &= \left( e^{-\pi \nu^2}\ast \frac{-2\sin\left( \frac{\pi\mu}{2}\right)\Gamma(\mu +1)}{|2\pi\nu|^{1+\mu}}\right) (\nu),
2 B_{I J}= \left( \begin{array}{c c c c}{{0}}&{{d}}&{{a}}&{{c}}\\{{- d}}&{{0}}&{{- a - c}}&{{a}}\\{{- a}}&{{a + c}}&{{0}}&{{b}}\\{{- c}}&{{- a}}&{{- b}}&{{0}}\\ \end{array}\right) ~ , G_{I J}= \left( \begin{array}{c c c c}{{2 u}}&{{- u}}&{{f}}&{{h}}\\{{- u}}&{{2 u}}&{{- f - h}}&{{f}}\\{{f}}&{{- f - h}}&{{2 v}}&{{- v}}\\{{h}}&{{f}}&{{- v}}&{{2 v}}\\ \end{array}\right) ~ .
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{M}J_3^{(k),2}|\leq C\big(h^{2r}+(\Delta t)^{4}\big)+C \|\theta_{\Psi}^{M}\|_{\mathcal{L}^2}^{2}+C\Delta t \sum_{k=1}^{M-1}{\|\nabla\theta_{\Psi}^{k}\|_{\mathbf{L}^2}^{2}}.}\end{array}
f^{(k-1)}(X) = b_1X^{e_1}+ \ldots +b_tX^{e_t}; t > 1, \: d^{k-1}=e_1> \ldots >e_t \geq 0, \: b_1, \ldots ,b_t \in \mathbb{F}\setminus \left\lbrace 0 \right\rbrace.
\begin{aligned}J_1 \sum_{i\geq0}( ({-D_r})^{i+1}f'(J_{k+1}) ) E^{(i)}_S(J_k) = -D_r f'(J_{k+1})\big( J_{k+1}+ D_r E_{\tilde\rho}(J_k) \big) &+ \sum_{i\geq1}( ({-D_r})^{i+1}f'(J_{k+1}) ) \big(D_r E^{(i)}_{\tilde\rho}(J_k) - E^{(i-1)}_{\tilde\rho}(J_k)\big)\end{aligned}
\frac{1}{\rho}\frac{\partial\rho}{\partial t}=\frac{1}{a^{2}\rho}\frac{\partial p}{\partial t}=\frac{1}{a^{2}}\frac{\partial}{\partial t}\int_{p_{1}}^{p}\frac{d\tilde{p}}{d\rho(\tilde{p})}=-\frac{1}{a^{2}}\frac{\partial}{\partial t}[\frac{\partial\Phi}{\partial t}+\frac{\nabla\Phi\cdot\nabla\Phi}{2}]
g(x)-T^{\ell,x_0}[g](x) =g(x)-T^{\ell-1,x_0}[g](x)- \sum_{|\nu|= \ell}\frac{\partial^{\nu}g(x_0)}{\nu!}(x-x_0)^\nu &=\sum_{|\nu|= \ell}\frac{\partial^{\nu}g(x_0+\theta(x-x_0))}{\nu!}(x-x_0)^\nu - \sum_{|\nu|= \ell}\frac{\partial^{\nu}g(x_0)}{\nu!}(x-x_0)^\nu
\gamma_k(t)=\frac{p\left|\mathbb{E}\left[{\bf g}^H_k(t){\bf f}_k\right]\right|^2}{\sum\limits_{l=1}^K p\mathbb{E}\left[\left|{\bf g}^H_k(t){\bf f}_l\right|^2\right]-p\left|\mathbb{E}\left[{\bf g}^H_k(t){\bf f}_k\right]\right|^2+q\mathbb{E}\left[{\bf g}^H_k(t){\bf A}{\bf A}^H{\bf g}_k(t)\right]+\xi^{\rm DL}}
\left|r\frac{d}{dr}\left[{}_2\!F_{1}\left(\lambda,\lambda;2\lambda+1;A\right)\right]\right|\lesssim \frac{1}{|1-zw|}\ln\left(\frac{|1-zw|^2}{|1-z\bar{w}|^2}+2\right)\lesssim \frac{1}{|1-z\overline{w}|},z=re^{i\theta},
I_{n}( p_{\phi}) ={\frac{1}{( n - 1 ) !}}{\frac{\Gamma ( n \epsilon )}{\epsilon^{n - 1}}}{\frac{\Gamma ( 1 - \epsilon )^{n + 1}}{\Gamma ( 2 - ( n + 1 ) \epsilon )}}\times{\frac{1}{( 1 - 2 \epsilon ) \cdots ( 1 - n \epsilon )}}\times{\frac{1}{[ p_{\phi}^{2}]^{n \epsilon}}}
e^{-t}B_1= \prod_{m=1}^5\frac{(5P_B+mz)}{(P_B+z)}\ \P_B^4= 5\sum_{m=0}^4(-P_Bz^{-1})^m \sum_{k=0}^4(-1)^k 5^{4-k}{\o{Sym}}_k(1, 2, 3, 4)\times \sum_{(c_1\geq\dots \geq c_k) \vdash m}|Orbit_{S_k}(c_1, \dots, c_k) |\ \P_B^4.
\mathcal{L}^{\mathrm{KL}}= \min_{\overline{\theta}}\mathbb{E}_t\Big[\frac{\pi_{\overline{\theta}^{*}}(S_t, A_t)}{\pi_{\overline{\theta}}(S_t, A_t)}\hat{A}_t - \tilde{\beta}\mathrm{KL}(\pi_{\overline{\theta}^{*}}(S_t), \pi_{\overline{\theta}}(S_t))\Big],
\hat{\chi}_{r , s}^{( p , p^{\prime})}( q ) = \hat{\chi}_{p - r , p^{\prime}- s}^{( p , p^{\prime})}( q ) = \frac{1}{( q )_{\infty}}\sum_{j = - \infty}^{\infty}\left( q^{j ( j p p^{\prime}+ r p^{\prime}- s p )}- q^{( j p + r ) ( j p^{\prime}+ s )}\right) .
\widehat{1_{T_{0,K}}}(\xi) = \int_{M_a^{-T}T_{\delta}}\chi(-x \cdot \xi) dx &= \det(M_a)^{-1}\int_{T_{\delta}}\chi(-M_a^{-T}y \cdot \xi) dy = \det(M_a)^{-1}\delta^{-k(k+1)/2}1_{\theta_{\delta}}(M_a^{-1}\xi)
\int_{-\infty}^\infty\!\!{\rm d}x\,\frac{-\nu_2 +\alpha\left[\nu_{1+}^2-(u^{(0)}(x))^2\right]-\mu_2\left[u^{(0)}(x)-v^{(0)}(x)\right]}{\cosh^2\left[\nu_{1+}\,m\,(x-z_u)\right]}=0
f(1,W_{1}, W_{\tau_2},\ldots, W_{\tau_K}) - f(0,W_{0}, W_{\tau_2},\ldots, W_{\tau_K}) =\int_{0}^{1}\dfrac{\partial}{\partial x_1}f^{\diamond}(t, \tau_2\ldots, \tau_K,W_{t}, \ldots, W_{\tau_K}) dW_{t}+ \int_{0}^{1}\mathcal{L}f^{\diamond}(t, \tau_2\ldots, \tau_K,W_{t}, \ldots, W_{\tau_K})dt,
-\displaystyle\limsup_{\varepsilon\to 0}\left(\varepsilon\displaystyle\sum_{j=1}^{d}\displaystyle\lim_{\delta\to 0}\int_{\Omega_{T}}G_{\delta}^{''}(u^{\varepsilon})\phi B(u^{\epsilon})\left( \frac{\partial u^{\varepsilon}}{\partial x_{j}}\right)^{2}\,dx dt\right)\leq 0.
\langle S^{(k)}_e u_{B_e}^{(k)},u_{B_e}^{(k)}\rangle = \min_{w^{(k)}= (w_I^{(k)},w_{B_e}^{(k)})\in V_{h,e}^{(k)}}a^{(k)}_e(w^{(k)},w^{(k)}),
\mathrm{MDTCov}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{X})=\mathrm{E}\left[(\mathbf{X}-\mathrm{MDTE}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{X}))(\mathbf{X}-\mathrm{MDTE}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{X}))^{T}|\boldsymbol{a}<\mathbf{X}\leq\boldsymbol{b}\right].
\begin{array}{l}f^{(+)}_{lk}=\exp \{\chi^{R}_{lk}-i\chi^{J}_{lk}\}, \quad f^{(-)}_{kl}={(f^{(+)}_{lk})}^{*}, \chi^{R}_{lk}=\chi_{R}(\theta'_{+l})-\chi_{R}(\theta_{+k}),\quad\chi^{J}_{lk}=\chi_{J}(\theta'_{+l})-\chi_{J}(\theta_{+k}).\end{array}
\Xi^{\lambda}(K_{a}\otimes F_{a}\otimes E_{a}K_{a}^{-1}\otimes K_{a}) = \sum_{i,j,n,o}(2c_{j}^{i}- \delta_{j}^{i}) c_{n}^{j}\pi(K_{a}F_{a}E_{a}K_{\lambda})_{o}^{n}c_{i}^{o}.
\widetilde{\beta^\prime}_1 := (\widetilde{B}_0 \widetilde{B}_1 \widetilde{B}_2 \widetilde{C})^{-1}(\widetilde{\beta}_1), \widetilde{\beta^\prime}_2 := (\widetilde{B}_0 \widetilde{B}_1 \widetilde{B}_2 \widetilde{C})^{-1}(\widetilde{\beta}_2), \widetilde{c^\prime}:= (\widetilde{B}_0 \widetilde{B}_1 \widetilde{B}_2 \widetilde{C})^{-1}(\widetilde{c}).
\frac{(2^{-k}t )^n}{\psi (t 2^{-k})^{n-1}}= \frac{(2^{-k}t )^n}{(2^{-k}t )^{\alpha(n-1)}}\cdot \frac{(2^{-k}t )^{\alpha(n-1)}}{\psi (t 2^{-k})^{n-1}}\le (2^{-k}t )^{n- \alpha(n-1)}\frac{t^{\alpha(n-1)}}{\psi(t)^{n-1}}= 2^{-k(n-\alpha(n-1))}\frac{t^n}{\psi(t)^{n-1}},
\frac{\mathbf{P}(X_n /\sqrt{\langle X\rangle_n}>x)}{1-\Phi \left( x\right)}= 1+ O \bigg( x^3 (\epsilon_n + \delta_n)+ (1+ x ) \big( \delta_n|\ln \delta_n| + \epsilon_n|\ln \epsilon_n| \big) \bigg).
\big\|Q_K\big(Q_K^{p+1}u|_K-\widetilde{u}_h|_K\big)\!\big\|_K\!=\!\big\|Q_K\big(Q_K^{p-1}u|_K-u_h|_K\big)\!\big\|_K\!\leq\!\big\|Q_K^{p-1}u|_K-u_h\big\|_K\,.
\langle Hw, w \rangle_{L^2(\mathbb{R}^-)}= - [w(0)]^2 + \int_{-\infty}^0 \left[ z (\partial_z w)^2 + \frac{1}{4}z w^2 \right] dz\leq -\frac{1}{2}\| w \|_{L^2(\mathbb{R}^-)}^2.
\begin{array}{l l l l}{{\nonumber \phi ( o u t )_{g}}}&{{=}}&{{T r ( W_{o u t}U_{g^{- 1}})}}&{{}}\\{{}}&{{=}}&{{T r ( S W_{i n}S^{- 1}U_{g^{- 1}})}}&{{}}\\{{}}&{{=}}&{{T r ( W_{i n}S^{- 1}U_{g^{- 1}}S )}}&{{}}\\{{}}&{{=}}&{{T r ( W_{i n}U_{g^{- 1}})}}&{{= \phi ( i n )_{g}\quad .}}\\ \end{array}
{\cal G}_t =\big\{\mbox{\rm prime $p$}\,\big|\, q<p\le t,~ \mu_q(p)\ge(\log_q t)^2\big\}; \overline{\cal G}_t =\big\{\mbox{\rm prime $p$}\,\big|\, q<p\le t,~\mu_q(p)<(\log_q t)^2\big\}.
\sup_{t\in[0,T]}\int_0^L\Big(\frac{\rho u^2}{2}+\frac{\epsilon}{2}\chi_x^2+\Phi(\rho)+\frac{\rho(\chi^2-1)^2}{4\epsilon}\Big)dx+\int_0^T\int_0^L\Big(\mu^2+\nu u^2_x\Big)dxd\tau=E_0,
f(y,x) =\Big\langle \Big(\frac{3}{2}-\|y\|\Big)y,x-y\Big\rangle &\leq \Big(\frac{3}{2}-\|y\|\Big)(\langle y,x-y\rangle-\langle x,x-y\rangle) &=-\Big(\frac{3}{2}-\|y\|\Big)\|x-y\|^2 &\leq -\frac{1}{2}\|x-y\|^2.
\int\limits_{\mathbb{R}^{N}}\frac{f\left( \left( \frac{1}{1+dq}\right)^{\frac{p-1}{p}}u\left(x\right) \right)}{C\left( x\right)^{t}}dx=\left( 1+dq\right)\int\limits_{\mathbb{R}^{N}}\frac{f\left( D_{N,d,p}u\left( x\right) \right)}{C\left( x\right)^{t\left( dq+1\right) -Nd}}dx.
e^{\beta Y^{b,n}_\rho}= \e^{\beta \xi^b}-\int_\rho^T \beta e^{\beta Y_s^{b,n}}\left( g^b(s,Y_s^{b,n},Z_s^{b,n}) +\lambda_s^n \frac{1-e^{\alpha (\xi^a_s-Y_s^{b,n})}}{\alpha}\right)ds -\int_\rho^T \beta e^{\beta Y_s^{b,n}}Z_s^{b,n}\cdot dW_s -\frac{\beta^2}{2}\int_\rho^T e^{\beta Y_s^{b,n}}\|Z_s^{b,n}\|^2 ds.
\overline{\phi_1(u, p|_{P(r)s})}, \ \overline{\phi_1(u, p|_{P(rs)})}, \ \overline{\phi_1(u, p|_{P(rd(s))})}<_n\overline{\phi_1(u, p|_{\overline{\phi_2(r,s)}})}=\overline{\phi_1(u,v)}=w,
|\langle\tilde{\nabla}\psi,\psi\rangle|^2|\psi|^{4p-4}= \frac{1}{4}\big|d|\psi|^2\big|^2|\psi|^{4p-4}=\frac{1}{4p^2}\big|d|\psi|^{2p}\big|^2, |\langle\nabla d\phi,d\phi\rangle|^2|d\phi|^{2p-4}= \frac{1}{4}\big|d|d\phi|^2\big|^2|d\phi|^{2p-4}=\frac{1}{p^2}\big|d|d\phi|^p\big|^2.
a^{1342}=a^{1432},a_{1}^{1342}=a_{1}^{1432},a_{2}^{1342}=a_{2}^{1432},a_{3}^{1342}=a_{3}^{1432},a_{4}^{1342}=a_{4}^{1432}, a_{12}^{1342}=a_{12}^{1432},a_{13}^{1342}=a_{13}^{1432},a_{14}^{1342}=a_{14}^{1432},a_{23}^{1342}=a_{23}^{1432},a_{24}^{1342}=a_{24}^{1432}, a_{123}^{1342}=a_{123}^{1432},a_{124}^{1342}=a_{124}^{1432}
\vec{A}=T_1^{\frac{1}{4}-\delta_0}\big(z_{k_1}(T_1),\ldots,z_{k_{n_0}}(T_1)\big)\in\mathbb{B}_0,\;\;\vec{B}=T_1^{\frac{3}{2}}\big(a_1^-(T_1),\ldots,a_n^-(T_1)\big)\in\mathbb{B}_1.
{1 \over \beta_c}= b ( d-2 )(1-\kappa){\int}_{-1}^{1}\; d\xi \; H_c(\xi) \;{\sqrt{1-\xi^2}\over \sqrt{(\xi + \frac{3+\kappa}{1-\kappa}) (\xi +\frac{1+3 \kappa}{1-\kappa})}}\;.
\sum_{j = 1}^{n+1}\varphi_j^{(n)}(x)\,V_{jl}^{(n)}(u,z) := \kern 4em\prod_{r = 0}^{l-2}\Bigl[ 2\cos 2 \pi \textup{i}b x - 2\cos \pi b ( 2\textup{i}z - u - 2\eta - g_{n,0}+ 2 b r) \Bigr] \times & \kern 4em \prod_{r = 0}^{n-l}\Bigl[ 2\cos 2 \pi \textup{i}b x - 2\cos \pi b ( -2\textup{i}z - u - 2\eta - g_{n,0}+ 2 b r) \Bigr].
\delta_r^{-1}\widetilde{T}_r = \mu[U]\left(x^{(1)}\otimes \ldots \otimes t^{(i_r)}_r\otimes \ldots \otimes x^{(m)}\right), &\delta_s^{-1}\widetilde{T}_s = \mu[U]\left(x^{(1)}\otimes \ldots \otimes t^{(i_s)}_s\otimes \ldots \otimes x^{(m)}\right).
P_{\mathrm{m a g}}( \epsilon ) \sim \int \prod_{x , i}d \mu ( U_{x , i}) \, \exp \left( -{\bar{H}_{\mathrm{m a g}}/ T_{\mathrm{G}}}\right) \, \delta ( F ( U_{x , i}) ) \, \delta \left( \epsilon - 4 ( 1 -{\textstyle{\frac{1}{2}}}\mathrm{t r}\, U_{x_{0}, i_{0}\, j_{0}}) \right) \, .
+\cos\psi\cot\eta\cdot\left(\frac{F[-q,p](i\mu_{0})w_{2}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{1}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
T_{\mu\nu}^{eff}=\phi_{,\mu}\phi_{,\nu}-K\tilde{g}_{\mu\nu}+\frac{b_{g}M^{4}}{(\zeta +b)^{2}}e^{-2\alpha\phi/M_{p}}\tilde{g}_{\mu\nu}+T_{\mu\nu}^{(f,canonical)}-mF(\zeta)\overline{\Psi}^{\prime}\Psi^{\prime}\tilde{g}_{\mu\nu},
F_{{\mathbf i}{\mathbf j}}{\bar D}^{{\mathbf k}{\mathbf l}}F_{{\mathbf k}{\mathbf l}}={\bar D}_{{\mathbf i}{\mathbf j}}(F^{{\mathbf k}{\mathbf l}}F_{{\mathbf k}{\mathbf l}}) + \frac{3}{10}{\bar D}^{{\mathbf k}{\mathbf l}}( F_{({\mathbf i}{\mathbf j}}F_{{\mathbf k}{\mathbf l})})\ ,
{[m(T|_{H_0})]}^2 =\{{\|Tx\|}^2\colon x\in H_0, \|x\|=1 \}& \leq \{{\|Te_n\|}^2\colon n\in \mathbb{N}\}& \leq \{{t_n}^2{a_n}^2 + (1-t_n^2){b_n}^2\colon n\in \mathbb{N}\}& \leq \{{c_n}^2\colon n\in \mathbb{N}\}& \leq{a}^2.
\left. \mathcal{O}_{\widetilde{\mathcal{C}^{2}}}\left( D\right) \right\vert_{\widehat{C}}=\mathcal{O}_{\widehat{C}}\left( \sum_{k=1}^{p}a_{k}\widehat{C_{k}}+\sum_{S \in \mathcal{N}(C)}b_{S}E_{S}\right),
E[e^{uX^x_t}]= \big(1-\frac{\sigma^2}{2a}u(1-e^{-at})\big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big (\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big) & \cdot \exp \Big ( \int_0^t \int_0^{\infty}\Big(e^{\xi\psi(s,u)}-1\Big)\nu(d\xi)ds\Big)
\widetilde z''(\omega_\beta(z))=-F''_{\mu_\alpha}(\omega_\beta(z))+\frac{1}{F'_{\mu_\beta}(\omega_\alpha(z))}F''_{\mu_\alpha}(\omega_\beta(z))-\frac{1}{(F'_{\mu_\beta}(\omega_\alpha(z)))^3}F''_{\mu_\beta}(\omega_\alpha(z))\cdot( F'_{\mu_\alpha}(\omega_\beta(z)))^2\,.
T_\lambda T_\nu(f) = T_\lambda(f_\nu \cdot (f \circ \tau^{d(\nu)})) = f_\lambda \cdot ( f_\nu \circ \tau^{d(\lambda)}) \cdot (f \circ \tau^{d(\lambda)}\circ \tau^{d(\nu)}); T_{\lambda\nu}(f) = f_{\lambda\nu}\cdot (f \circ \tau^{d(\lambda \nu)}),
\frac{d \overline{a_1}}{\overline{a_1}}\wedge \frac{d \overline{a_2}}{\overline{a_2}}+ \ldots + \frac{d \overline{a_{2n-1}}}{\overline{a_{2n-1}}}\wedge \frac{d \overline{a_{2n}}}{\overline{a_{2n}}}=0
\chi^{4}= : \! \chi^{4}\! : + 6{\frac{1}{2 \pi}}\ln{\frac{\Lambda_{0 , \mathrm{e f f}}}{\Lambda}}\chi^{2}+ 3 \bigl ({\frac{1}{2 \pi}}\ln{\frac{\Lambda_{0 , \mathrm{e f f}}}{\Lambda}}\Bigr )^{2}.
(a)=(1)+(2)\leq -2\pi p_{0}I\left(\frac{1}{\varepsilon_{n}}\left(\log\frac{1}{\varepsilon_{n}}\right)^{-\frac{1}{s_{k}}}\right)-2\pi p_{0}\frac{1}{s_{k}}\log\log\frac{1}{\varepsilon_{n}}+2\pi p_{0}\log\frac{1}{\varepsilon_{n}}+ O(1).
\alpha_{k+1}+1=a_{k+1}-a_{k}=c_{k-1}+2b_{k}3^{k}\alpha_{k}+1=a_{k}-a_{k-1}=\upsilon_{k-1}+2b_{k-1}3^{k-1}\vdots \alpha_{2}+1=a_{2}-a_{1}=\upsilon_{1}+2b_{1}3^1 \alpha_{1}+1=a_{1}-a_{0}=\upsilon_{0}+2b_{0}3^0.
{\frac{d}{d t}}X = i \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho^{\vee}\! \! \cdot \hat{H}) y_{| \rho |}( \rho \cdot q , ( \rho^{\vee}\! \! \cdot \hat{H}) \xi ) ( \rho \cdot \dot{q}) \hat{s}_{\rho}
\widetilde\Omega_N=\bigcap_{1\leq l,l'\leq N^C, \xi=\pm1, \xi'=\pm1, 0<|k|\leq 2N}\left\{\omega\in\Omega:\ |k\cdot\omega+\xi\sqrt{\zeta_l}-\xi'\sqrt{\zeta_{l'}}|>2 e^{-\frac 14 N_1^{\rho_2/2}}\right\}.
\frac{\partial}{\partial x_i}\phi_{i,k}= \begin{cases}0 \mbox{if}d_{i,k}-x_i\leq 0 \frac{x_i-d_i(\omega_k)}{\bar{y}}\mbox{if}0\leq d_{i,k}-x_i\leq \bar{y}-1 \mbox{if}d_{i,k}-x_i \geq \bar{y}\end{cases}
\lim_{t\to0}\int_{0}^{L}\mu_1 \left(\frac{\left(z+tw\right)^{+}-z^{+}}{t}\right)\left(\alpha-z\right) \varphi=\int_{0}^{L}\mu_{1}w \mathbf{1}_{z>0}\left(\alpha - z\right) \varphi.
S_0 = \int d^2 x \,\Big( i\overline\psi \gamma^\mu \partial_\mu \psi - g_0 \overline\psi \gamma^\mu \gamma_5 \psi \partial_\mu \phi +{1\over 2}\partial_\mu \phi \partial^\mu \phi -{m^2\over 2}\phi^2 \Big)
{\cal Z}^{p B}( x_{1}, . . . x_{M}; p \geq M ) = \sum_{\lambda}s_{\lambda}( x_{1}, . . . x_{M}) = \prod_{i = 1}^{M}\frac{1}{( 1 - x_{i})}\prod_{i < j}^{M}\frac{1}{( 1 - x_{i}x_{j})}.
\frac{\gamma_i(\Gamma_{n,c})}{\gamma_{i+1}(\Gamma_{n,c})}=\frac{\gamma_i(\Gamma_n)/\gamma_{c+1}(\Gamma_n)}{\gamma_{i+i}(\Gamma_n)/\gamma_{c+1}(\Gamma_n)}\simeq \frac{\gamma_i(\Gamma_n)}{\gamma_{i+i}(\Gamma_n)}=\gamma_i/\gamma_{i+1},
G_n(1;a) = \begin{cases}t\,q^{n-1}\,\sum\limits_{b=1}^{k-1}G_{n-1}(1;b), a=1 G_{n-1}(1;a-1) + t\,q^{n-1}\,\sum\limits_{b=a}^{k-1}G_{n-1}(1;b), 2 \leq a \leq k-1 0, a \geq k.\end{cases}
\int \frac{d w}{2 \pi}e^{- i w ( t^{\prime}- t )}\frac{1}{2 \bar{k}}\left( e^{i \bar{k}| x^{\prime}- x |}+ K_{a}( w ) e^{- i \bar{k}( x^{\prime}+ x )}\right) , ~ \bar{k}= \sqrt{w^{2}- m_{a}^{2}}
\|I_1\|_{L^2}\leq \;C\int_0^{t-h}h^{\beta'}(t-\tau)^{-(1+\beta')}\frac{(t-\tau)^\beta}{\tau^{\beta+1/2}}R_1R_2\,d\tau = \;Ch^{\beta'}R_1R_2\int_0^{t/2}+\int_{t/2}^{t-h}(t-\tau)^{-(1+\beta'-\beta)}\tau^{-(\beta+1/2)}\,d\tau \leq \;CR_1R_2h^{\beta}t^{-(\beta+1/2)},
\begin{array}{ccl}x^{1}= y^{1}x^{2}= y^{2}x^{3}= y_{[1]}^{2}\left(1-y^{1}+y_{[1]}^{1}\right) u^{1}= y_{[1]}^{1}-y^{1}u^{2}= y_{[2]}^{2}\left(1-y_{[1]}^{1}+y_{[2]}^{1}\right)\,,\end{array}
\int e^{R^+(\psi)}d\mu_{C_{\le 0}^+}(\psi) =\int\int e^{-V(\psi+\psi^1)}d\mu_{C_{>0}^+}(\psi^1)d\mu_{C_{\le 0}^+}(\psi) &=\int e^{-V(\psi)}d\mu_{C}(\psi).
{\cal E}( r ){\cal T}= - 2 \kappa_{( 1 0 )}^{2}\int d^{1 0}x \int d^{1 0}{\tilde{x}}\ \Bigl [ 4 j_{\Phi}\Delta{\tilde{j}_{\Phi}}- j_{C}\Delta{\tilde{j}_{C}}+ T_{\mu \nu}\Delta^{\mu \nu , \rho \tau}{\tilde{T}}_{\rho \tau}\Bigr ]
\sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}Q^2/2<\mathcal{N}(q)\le Q^2 \mathcal{N}(q)=\Box}}\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left|\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_n \cdot e\left(\mbox{\rm Tr}\left(\frac{nr}{2q}\right)\right)\right|^2 = \Sigma\left(Q^2,N;\mathcal{S}\right).
y^{(m)}_{r_1,\ldots,r_k}=\left(\prod_{i=1}^k\frac{g(r_i)r_i}{\varphi(r_i)^2}\right)\sum_{a_m}\frac{y_{r_1,\ldots,r_{m-1},a_m,r_{m+1},\ldots,r_k}}{\varphi(a_m)}+O\left(\frac{y_{max}\varphi(VP_f)\log X}{VP_fD_0}\right).
\psi^{t}(h)_{\alpha}=\frac{1}{8}\sum_{s}\Gamma_{\alpha t}^{(s)}\chi(2h+s),\overline{\psi}^{t}(h)_{\alpha}=\frac{1}{8}\sum_{s}\overline{\chi}(2h+s)(\Gamma^{(s)\dagger})_{t\alpha}.
m_{e}^{+}= \inf\left\{E^{+}(v):v\in\mathcal{N}(E^{+})\right\}E^{+}(v)= \int_{\mathbb{R}_{+}^{n}}\frac{1}{2}|\nabla v|^{2}+\frac{(a-\omega^{2})}{2}|v|^{2}-\frac{1}{p}|v^{+}|^{p}dx; \mathcal{N}(E^{+})= \left\{v\in H^{1}(\mathbb{R}_{+}^{n})\smallsetminus\{0\}\ :\E^{+}(v)v=0\right\};
\frac{\left| u'_{N,\varepsilon_1 \ldots \varepsilon_n}(t) \right|}{\left| u'_{N,\varepsilon_1 \ldots \varepsilon_n}(r) \right|}=\left(\frac{q_{N,n}+r q_{N,n-1}}{q_{N,n}+t q_{N,n-1}}\right)^2 \leq\left(\frac{q_{N,n}+ q_{N,n-1}}{q_{N,n}}\right)^2 \leq \left( \frac{N+1}{N}\right)^2.
\frac{\partial{\cal V}}{\partial \varphi}={\textstyle \frac{2}{n}}\frac{1}{\mathrm{T r}({\cal M}^{2})}\,{\cal M}_{m n}\frac{\partial{\cal V}}{\partial{\cal M}_{m n}}\, , \, \, \, \, \Rightarrow{\cal M}^{m n}{\cal M}_{p q}\frac{\partial{\cal V}}{\partial{\cal M}_{p q}}- \mathrm{T r}({\cal M}^{2}) \frac{\partial{\cal V}}{\partial{\cal M}_{m n}}= 0 \, .
\left|\log\left(\frac{zF_1'(z)}{F_1(z)}\right)\right| =\left|\log\left(1-\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right)\right| =\left|\log\left(\frac{1}{e}\right)\right|=1.
L_{1}(C)=\{x\in V(C) \mid \exists\, (x,y)\in E(D_{G})\,\, \,\, y\not\in C\}& L_{2}(C)=\{x\in C\mid \mathcal{N}(x)\subseteq C\}& L_{3}(C)=C\setminus (L_{1}(C)\cup L_{2}(C))
\frac{\beta(4- a \beta)}{4(1-a(N+1))}= \lim_{k\to+\infty}\frac{1}{2\pi}\int_{\mathbb{R}^2}\varepsilon_{1,k}|x|^{2N}e^{v_k}\, dx=4(N+1), \lim_{R \to +\infty}\lim_{k \to +\infty}\frac{1}{2\pi}\int_{\vert x \vert \geq R}e^{a v_k}\, dx = \frac{a \beta(\beta-4(N+1))}{4(1-a(N+1))}
\left(\frac{A_n^{(2)}(z) - A_n^{(2)}(0)}{n^6z}\right)^2 = \mathcal{O}\left(n^{-1}\right) \frac{A_n^{(2)}(z) - A_n^{(2)}(0)}{n^6z}\frac{A_n^{(1)}(0)A_n^{(1)}(z) A_n^{(1)}(0)}{n^9z^3}= \mathcal{O}\left(n^{-\frac{3}{2}}\right)
\alpha_\phi^{ij}(x;y)=\alpha^{ij}(x)+\nabla_r\alpha^{ij}(x)\,y^r+\left(\frac12\,\nabla_r\nabla_s\alpha^{ij}(x)-\frac16\,R^{[i}_{rks}(x)\alpha^{j]k}(x)\right)\,y^ry^s+\cdots,
\varphi_m(x_k) \stackrel{\bigtriangleup}{=}\exp\left( - \left(\Re ({\bf w}_m^{(k)}) \atop \Im({\bf w}_m^{(k)}) \right)^T{\bf \Lambda}_k^{-1}\left(\Re ({\bf w}_m^{(k)}) \atop \Im({\bf w}_m^{(k)}) \right)\right),
\frac{\partial}{\partial t}g_{ij}=-\alpha'\frac{R}{2}g_{ij}-\alpha'^2\frac{R^2}{8}g_{ij}-\alpha'^3\big(\frac{1}{16}|\nabla R|^2g_{ij}-\frac{1}{16}\nabla_iR\nabla_jR+\frac{5}{32}R^3g_{ij}\big).
G_c(h^{H}+h^K,k^{H}+k^K) &\qquad=G_c(h^{H},k^{H})+G_c(h^{H},k^K)+G_c(h^K,k^{H})+G_c(h^K,k^K) &\qquad=G_c(h^{H},k^{H})+0+G_c(h^{K},k^{K})\;,
J_{151}= \int \partial^\zeta_x(\nabla \phi^2)_t \cdot Q(u)\cdot \partial^\zeta_x u_t \leq C \|\nabla \phi\|_2\|\phi_t\|_2\|u\|_3|u_t|_{D^2}+\int \phi \partial^\zeta_x \nabla \phi_t \cdot Q(u)\cdot \partial^\zeta_x u_t.
d_H(gA, hB) =d_H(A, g^{-1}hB)\leq d_H(A,g^{-1}hA)+d_H(g^{-1}hA,g^{-1}hB) &=d_H(A,g^{-1}hA)+d_H(A,B)<\varepsilon/2+\varepsilon/2=\varepsilon.
\left|D_{x^{i}}\left(|x|^{-d}\textnormal{H}_{k,0}^{p}(R)\right)\right| =\left|-dx^{i}|x|^{-d-2}\textnormal{H}_{k,0}^{p}(R)-2x^{i}|x|^{-d-2}\textnormal{H}_{k,1}^{p}(R)\right| =\left|-x^{i}|x|^{-d-2}[d\textnormal{H}_{k,0}^{p}(R)+2\textnormal{H}_{k,1}^{p}(R)]\right| &\leq N |x|^{-d-1}\left|d\textnormal{H}_{k,0}^{p}(R)+2\textnormal{H}_{k,1}^{p}(R)\right|,
b_8 = \frac{1}{1260 \zeta_q(2)}\Big[ 74 A + \frac{15}{\log g}\Big(A_1 +A_2 +A_3 +A_4 \Big) + \frac{2}{( \log q)^2}\Big( A_{12}+ A_{13}+A_{14}+A_{23}+A_{24}+A_{34}\Big)\Big].
\left[\alpha(1-\alpha)\frac{\partial^2}{\partial\alpha^2}+(1-2\alpha)\frac{\partial}{\partial \alpha}+\nu(\nu+1)\right]\left.\frac{\partial}{\partial \varepsilon}\right|_{\varepsilon=0}{_2}F_1\left(\left. \begin{array}{c}-\nu,\nu+1\ 1+\varepsilon\ \end{array}\right| \alpha\right)=-\frac{\partial P_{\nu}(1-2\alpha)}{\partial \alpha}.
\Psi_{2 2}( r , \theta^{\prime}\pm{\frac{\pi}{\alpha}}, t ) = -{\frac{1}{2}}\sqrt{{\frac{i}{r}}}{\frac{1}{\sqrt{2 \pi k_{0}}}}\cot{\frac{\pi}{\alpha}}\Psi_{\mathrm{{f r e e}}}( r , \alpha^{2}t ) ,
H_{2,1}= U(N_{a,1}+N_{a,2}-N_{b,1})^2+\mu(N_{a,1}+N_{a,2}-N_{b,1}) &\quad+t_{1,1}(a_{1}b_{1}^\dagger+a_{1}^\dagger b_{1})+t_{2,1}(a_{2}b_{1}^\dagger+a_{2}^\dagger b_{1})
iD_{\mu\nu}(p)=-i\frac{1}{p^2-\mu^2}\left(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)+\mu\frac{1}{p^2-\mu^2}\frac{1}{p^2}\varepsilon_{\mu\nu\rho}p^{\rho}-i\alpha\frac{p_\mu p_\nu}{p^4}.
\Pi^{\alpha \dot{\alpha}}\ = \ \dot{x}^{\alpha \dot{\alpha}}- 2 \, \dot{\theta}^{\alpha}\tilde{\theta}^{\dot{\alpha}}+ 2 \, \theta^{\alpha}\dot{{\tilde{\theta}}^{\dot{\alpha}}}+ \omega \psi^{\alpha \dot{\beta}}\psi_{\beta \dot{\beta}}\psi^{\beta \dot{\alpha}}+ \chi \psi^{\alpha \dot{\alpha}}+ 2 \, \rho^{\alpha}\tilde{\mu}^{\dot{\alpha}}\quad .
q_{-}^{0}( \tau ) = - \frac{\Omega \vartheta_{1}^{\prime}( 0 )}{2 \pi}\frac{e^{\Omega \tau \frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}( \Omega y )}\vartheta_{1}( \Omega ( y - \tau ) ) + e^{- \Omega \tau \frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}( \Omega y )}\vartheta_{1}( \Omega ( y + \tau ) )}{\vartheta_{1}( \Omega \tau ) \vartheta_{1}( \Omega y )},
\left( q\left( x,\partial_{x}\right) Su\right) \left( x\right)=\int_{\mathbb{R}}e^{i\xi\cdot x}q\left( i\partial_{\xi},\partial_{x}+i\xi\right) \left[ \Gamma\left( x,\xi\right) \hat{u}\left( \xi\right)\right] d\xi,
( G_{{\cal L}}^{T}G_{{\cal R}})^{i j}= H_{L}^{i k}( - l^{k \underline{{{m}}}}r_{\underline{{{m}}}}^{l}+{\frac{2 ( r^{( + + ) \underline{{{n}}}}l_{\underline{{{n}}}}^{k}) ( l^{( -- ) \underline{{{n^{\prime}}}}}r_{\underline{{{n^{\prime}}}}}^{l})}{( l^{( -- ) \underline{{{n^{\prime \prime}}}}}r_{\underline{{{n^{\prime \prime}}}}}^{( + + )})}}) H_{R}^{j l}~ ,