harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{{\bf E}}{}={}\frac{1}{2}\,\left({\bf u}\,{\rm e}^{{\rm i}\Theta}{}+{}{\bf u}^{*}\,{\rm e}^{-{\rm i}\Theta}\right)\quad,\qquad\tilde{{\bf B}}{}={}\frac{1}{2}\,\bar{\bf k}{}\times{}\left({\bf u}\,{\rm e}^{{\rm i}\Theta}{}+{}{\bf u}^{*}\,{\rm e}^{-{\rm i}\Theta}\right)\quad,
\sum_{\substack{1\leq j\leq n \epsilon \in \{1,-1\}}}V^{}_{\epsilon j}(\xi;g) \Bigl( \Phi^{}_{\xi +\epsilon e_j}(x;g)-\Phi^{}_\xi (x;g)\Bigl) = E^{}_1(x) \Phi^{}_\xi (x;g)
\int\limits_0^\infty F(\sqrt{p^2+t^2})t^{n-2}dt = \int\limits_{|p|}^\infty F(r)(r^2-p^2)^{(n-3)/2}r dr =\frac{1}{|S^{n-1}|}\int\limits_{S^{n-1}}\int\limits_{|p|}^\infty f(r\omega) (r^2-p^2)^{(n-3)/2}r dr d\omega.
\hat{H}=-\frac{\hbar^{2}}{2m}\sum_{i}{\nabla^{2}}_{r_{i}}-\sum_{i}\frac{Ze^{2}}{4\pi\epsilon_{0}|r_{i}|}+\frac{1}{2}\sum_{i\ne j}\frac{e^{2}}{4\pi\epsilon_{0}|r_{i}-r_{j}|}
\P(G_0v (d, 0, 0)) (1 - \theta_1) \leq \P(G_0v (d, 0, E)) + \mathcal{O}_{d, \upsilon}\left( \sum_{k=2}^{\infty}N^{-3/2}(1+ k^3) e^{- \frac{(k-1)^2}{4 d^2 ||E||_1^{1/3}}}\right).
\begin{array}{rl}1= \langle x\otimes w|y\otimes z\rangle = \langle x'\otimes w|z\rangle = \langle y\otimes x'\otimes w|y\otimes z\rangle = \langle y\otimes x'\otimes w|x\otimes w\rangle = \langle y\otimes x'|x\rangle. \end{array}
\|u(t)\|_{\dot{B}^{\ell+2}_{r,1}}+\|(v(t),w(t))\|_{\dot{B}^{\ell+1}_{r,1}}\geq C(\|u(t)\|_{\dot{B}^{\ell}_{r,1}}+\|(v(t),w(t))\|_{\dot{B}^{\ell-1}_{r,1}})^{1+\frac{2}{\ell+s}}=C\mathcal{F}(t)^{1+\frac{2}{\ell+s}}.
\int\limits_{[0,1]^N}\prod\limits_{i=1}^N t_i^{\lambda_1}(1-t_i)^{\lambda_2}\prod\limits_{k<p}^N |t_p-t_k|^{2\lambda}\,dt = \prod_{k=0}^{N-1}\frac{\Gamma(1+(k+1)\lambda)\Gamma(1+\lambda_1+k\lambda)\Gamma(1+\lambda_2+k\lambda)}{\Gamma(1+\lambda)\Gamma(2+\lambda_1+\lambda_2+(N+k-1)\lambda)},
\kappa:\lambda(\psi)(s) = \int_0^s\int_{S_2}\Big[\kappa:v \otimes v \sum_{q=1}^{+\infty}\psi_q(v)H_q(s')\Big]dvds' = \sqrt{2}\sum_{q=1}^{+\infty}\Big\{\dfrac{1}{q\pi}\big[1-cos(q \pi s)\big]\int_{S_2}\psi_q(v)\kappa:v \otimes vdv\Big\}
{\bf f}( x ) = \left( \begin{array}{c}{{u_{-}( x )}}\\{{u_{+}( x )}}\\ \end{array}\right) , \; \; \hat{x}^{\mu}= x^{\mu}{\bf I}\; , \; \; \hat{\pi}_{\mu}= - \imath \partial_{\mu}{\bf I}\; , \; \; \hat{\psi}^{a}= \frac{\imath}{2}\gamma^{a},
\Delta^{+}(\omega)=\ln\left(\frac{\Lambda^{2}}{m^{2}}\right) - \int_{-\Lambda}^{+\Lambda}\frac{d{\bf k}}{[{\bf k}^{2}+m_{\mbox{\scriptsize eff}}^{2}]^{1/2}}\frac{{\bf k}^{2}}{[{\bf k}^{2}+m_{\mbox{\scriptsize eff}}^{2}-\omega^{2}/4+i\epsilon]}\;\;\;,
\Vert B_\varepsilon^{1/2}(B_\varepsilon -\zeta Q_0^\varepsilon )^{-1}\Vert_{H^{-1}\rightarrow L_2}=\Vert f^\varepsilon \widetilde{B}_\varepsilon^{1/2}(\widetilde{B}_\varepsilon -\zeta^* I)^{-1}\Vert_{L_2\rightarrow H^1}.
( W_{0}( n ) \overline{{P}}_{\beta}^{k}W_{0}( n )^{- 1})_{-}{}= \overline{{B}}_{k \beta}^{K P}( n ) |_{t = ( x , 0 , \cdots )}\mathrm{f o r}^{\forall}k \in \mathrm{\bf{Z}}_{\geq 1}.
\frac{d^{n-2}M}{dv^{n-2}}=\frac{d^{n-2}}{dv^{n-2}}\left\{F_\alpha\frac{d\alpha_+}{dv}+F_\beta\frac{d\beta_+}{dv}-F_t\frac{df}{dv}-F_w\frac{dz}{dv}\right\}.
\lim_{n\to\infty}\int_0^T\left(1+\|v^n(t)\|_{\L^{2\delta}}^{4\delta}+\|u(t)\|_{\L^{2\delta}}^{4\delta}\right)\|v^n(t)-u(t)\|_{\L^2}^2\d t &= \int_0^T\left(1+\|v(t)\|_{\L^{2\delta}}^{4\delta}+\|u(t)\|_{\L^{2\delta}}^{4\delta}\right)\|v(t)-u(t)\|_{\L^2}^2\d t.
Z(\alpha)=\int{\cal D}A^{\omega}_{\mu}{\cal D}B{\cal D}\bar{c}{\cal D}c e^{-S_{0}(A^{\omega})-\frac{\alpha}{2}\int B^{2}dx +\int[-iB\frac{1}{\sqrt{-\partial_{\mu}\partial_{\mu}}}\partial_{\mu}A^{\omega}_{\mu}+\bar{c}\sqrt{-\partial_{\mu}\partial_{\mu}}c]dx}
\Delta_{f}\left\vert \mathrm{Rm}\right\vert^{2}=2\left\vert \nabla \mathrm{Rm}\right\vert^{2}+2\left\vert \mathrm{Rm}\right\vert^{2}-8R_{ijkl}R_{piqk}R_{pjql}-2R_{ijkl}R_{ijpq}R_{pqkl}.
\frac{d}{dt}|A|^2= \Delta |A|^2 -2|\nabla^{\bot}A|^2 + 2\sum_{\alpha,\gamma, i,m}(\sum_k h_{\alpha ik}h_{\gamma mk}-h_{\alpha mk}h_{\gamma ik})^2+2\sum_{i,j,m,k}(\sum_{\alpha}h_{\alpha ij}h_{\alpha mk})^2
\varphi_{l,j}(x)=e^{i\left\langle a_{l,j}+n_{l}v_{k}+t,x\right\rangle}+{\textstyle\sum\limits_{n=n_{l}+1}^{\infty}}\left({\textstyle\sum\limits_{a\in\Gamma(k)}}c(a,n)e^{i\left\langle a+nv_{k}+t,x\right\rangle}\right)
Z = \int \prod_{n = 1}^{\mathcal{N}}D [ P_{n}] D [ \pi^{i j}] D [ h_{i j}] \delta ( \chi ) \delta ( \chi^{1}) \delta ( \chi^{2}) | \mathrm{D e t}\{\chi^{\mu}, H_{\nu}\}| \delta ( \frac{H_{i}}{\sqrt{h}}) \delta ( \frac{H}{\sqrt{h}}) e^{i S}.
(d\alpha,\beta)=\int d\alpha\wedge*\overline{\beta}=\int d\alpha\wedge*\overline{\beta}-\int d(\alpha\wedge*\overline{\beta})=(-1)^{\partial\alpha}\int\alpha\wedge d*\overline{\beta}=\int\alpha\wedge*\overline{\delta\beta}=(\alpha,\delta\beta).
\sum_{\substack{\xi_1+ \xi_2+ \xi_3-\xi_1'- \xi_2'-\xi_3'=0}}|a_{\xi_1}\overline{a_{\xi_1'}}a_{\xi_2}\overline{a_{\xi_2'}}a_{\xi_3}\overline{a_{\xi_3'}}| \leq (u(N)/N)^3 \cdot N^{\gamma 6}= O(1).
\begin{array}{c c}{{\phi^{0}:}}&{{\eta , \eta^{'}}}\\{{\phi^{i}:}}&{{~ \pi^{i}( i = 1 , 2 , 3 )}}\\{{V_{\mu}^{0}:}}&{{\omega_{\mu}, \phi_{\mu}}}\\{{V_{\mu}^{i}:}}&{{\rho_{\mu}^{i}}}\\{{\chi^{0}:}}&{{\epsilon ( f_{0})}}\\{{\chi^{i}:}}&{{\delta^{i}( a_{0}^{i})}}\\{{W_{\mu}^{0}:}}&{{E_{\mu}}}\\{{W_{\mu}^{i}:}}&{{~ A_{\mu}^{( 1 ) i}( a_{\mu}^{( 1 ) i})}}\\ \end{array}
|F(p)| e^{-\epsilon |p|}\leq\int_0^1 \frac{{\rm d}L}{L}L^{-\sigma}|G(Lp)| e^{-\epsilon |p|}\leq\int_0^1 \frac{{\rm d}L}{L}L^{-\sigma}e^{-(1-L)\epsilon |p|}\| G\|_{\infty,\epsilon}
\mathbf{D}_{i,j}=\left(\begin{array}{ccccccc}2(i-j^{'}) c_{1}c_{1}\ddots \ddots & \ddots 2(i-j^{'}) c_{s-1}& c_{s-1}2(i-j^{'})-2e c_{s}& c_{s}2(i-j^{'}) \ddots & \ddots \ddots c_{m-1}& c_{m-1}2(i-j^{'})\end{array}\right).
K := \Vert F \Vert_{L^{p}}^{p}\big{(}\Vert G \Vert_{L^{2}}+ \Vert G \Vert_{L^{\infty}}\big{)}^{p}\big{(}1 + \Vert V \Vert_{\infty}\big{)}^{d + \frac{p}{2}+ \frac{3}{2}+ \varepsilon},
S_{0T}^{\mathcal{K}}(\varphi(t)) = \frac{1}{2}\int_{0}^{T}\biggl\Vert\frac{d\varphi(t)}{dt}- \Bigl(A \varphi(t) + \bigl(B, \mathcal{K}\bigr)\varphi(t) \Bigr) \biggr\Vert^2dt,
\begin{aligned}x_1 = 2m^2-6t(t-2)m+6(t-1)(t+2)(3t-2), x_2 = m^2t+16(t-1)m-3t(t+2)(3t-2), x_3 = (2t-2)m^2-2(3t^2-4t+4)m-6(t+2)(3t-2), x_4 =-x_3, x_5=-x_2, x_6=-x_1, y_1 = (2t-2)m^2-6t(t-2)m+6(t+2)(3t-2), y_2 = m^2t-16(t-1)m-3t(t+2)(3t-2), y_3 = 2m^2+2(3t^2-4t+4)m-6(t-1)(t+2)(3t-2), y_4 =-y_3, y_5=-y_2, y_6=-y_1,\end{aligned}
\rho_c''(x) ={}\rho''(x) - a \left( \frac{\zeta''(x/c)}{c^2}x^4 - 12\zeta(x/c)x^2 - \frac{8 \zeta'(x/c)}{c}x^3 \right) \ge{}2A^2 - C c^2\ge A^2>0,
I(t):=I(u(t))=\left \| \Delta u_{t}\right \|_{2}^{2}+(1-\int_{0}^{t}b(s)ds)\left \| \Delta u \right \|_{2}^{2}+\left \| u \right \|_{2}^{2}+(b\circ\Delta u)-\int_{\Omega}u^{2}\ln \left | u \right |^{k},
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)| \geq \frac{\frac{1}{d}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^2}{n^2-n}=\frac{\frac{1}{(\mathcal{X})}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^2}{n^2-n},
\tilde{S}= \oplus^{\operatorname{s}}_i \begin{pmatrix}\tilde{S}_{\alpha_i \alpha_i}\tilde{S}_{\alpha_i \beta_i}-\tilde{S}_{\alpha_i \beta_i}\tilde{S}_{\alpha_i \alpha_i}\end{pmatrix}+\mathcal{O}(\|H\|).
\text{LHS of (\ref{eqn:final})}= \sum_{m=0}^1 (-1)^m \binom{n-1 + 1-m}{n-1}\binom{n}{m}= \binom{n}{n-1}\binom{n}{0}- \binom{n-1}{n-1}\binom{n}{1}= \binom{n}{n-1}- \binom{n}{1}= 0
\Gamma_{c . t .}^{\prime}= -{\frac{i}{{( 2 \pi )}^{3}}}\int B_{0}^{( 9 )}\; X_{6}^{( 9 )}+{\frac{i}{{( 2 \pi )}^{3}}}\int B_{0}^{( 5 )}\; X_{6}^{( 5 )},
\frac{\sinh\frac{\pi}{\zeta}\left(\theta+i\alpha\frac{2}{3}\pi\right)}{\sinh\frac{\pi}{\zeta}\left(\theta-i\alpha\frac{2}{3}\pi\right)}=\exp\left( -i\int\limits_{-\infty}^\infty\frac{dk}{k}\sin k\theta \frac{\sinh\left(\frac{1}{2}\zeta-\alpha\frac{2}{3}\pi\right) k}{\sinh\frac{1}{2}k\zeta}\right)
R_0[U] = U, \frac{\partial R_{i+1}}{\partial q_0}= \frac{1}{2i + 3}\Big(\frac{\partial U}{\partial q_0}+ 2U \frac{\partial}{\partial q_0}+ \frac{\epsilon^2}4 \frac{\partial^3}{\partial q_0^3}\Big)R_i.
\kappa_{2k;\alpha}^{(\ell)}:= \kappa_{(\alpha\alpha)\cdots(\alpha\alpha)}^{(00)\cdots (00),(\ell)}=\frac{1}{(2k-1)!!}\kappa\bigg(\underbrace{z_{i;\alpha}^{(\ell+1)},\ldots, z_{i;\alpha}^{(\ell+1)}}_{2k}\bigg)
H({\lambda}) = - \frac{1}{sin \gamma \;\;d_0^2 d_1^2}\sum^{L}_{k=1}\frac{{\lambda}q+{\lambda}^{-1}q^{-1}}{2}( S^X_k S^X_{k+1}+ S^Y_k S^Y_{k + 1}) - \frac{1}{2}S^Z_k S^Z_{k + 1}
\mathcal{L}_{0}= - \sum_{n}\left( \frac{n + \omega}{R}\right)^{2}\bar{A}^{( n )}A^{( n )}- \sum_{m , n}\bar{A}^{( m )}\left[ \delta ( 0 ) \ \frac{W^{2}}{\pi R}+ i \, \frac{W}{\pi}\ \frac{m - n}{R^{2}}\right] A^{( n )}\, .
(\mathcal{M}_{\Delta}\times E,\Lambda|_{\mathcal{M}_{\Delta}\times E})=(\bigcup_{j=1}^l U_j\times \Delta\times E, \sum_{\alpha,\beta=1}^n g_{\alpha\beta}^j(z_j,t)\frac{\partial}{\partial z_j^{\alpha}}\wedge\frac{\partial}{\partial z_j^{\beta}})
f_l(v)-f_l(u)\geq -M\,(v_l-u_l)\ \ (l=1,2,\cdots,m) &\ \{\rm for\any}\ \u,\,v\in[p^-,p^+]\ \{\rm with}\ \u\preceq v\ \ \ \ (M:=\max_{w\in[p^-,p^+]}|DF(w)|).
\mathbb{P}_\eta\left(\left. \mathcal{T}_t(x) < \frac{1-p}{4}t\right|\mathcal{F}_{k}\right) \leq \mathbb{P}_\eta\left(\left.\int_{t/2}^{t}\mathbb{1}_{\{\eta_s(x)=0\}}\mathrm{d}s < \frac{1-p}{4}t\right|\mathcal{F}_{k}\right)
X_{n,p_k}^2 = \frac4{n^2\delta_n^2(k)}\sum_{\substack{\mathbf{p}_{k,1}\in \mathcal P(d,k): p_{k,1}=p_k \mathbf{p}_{k,2}\in \mathcal P(d,k): p_{k,2}=p_k}}\sum_{\mathbf{i}\in \mathcal{J}}\Big(\prod_{\ell=1}^{k-1}\tilde I^{(p_{\ell,1})}_{i_1,i_2}\tilde I^{(p_{\ell,2})}_{i_3,i_4}\Big) \cdot \tilde I^{(p_{k})}_{i_1,i_2}\tilde I^{(p_{k})}_{i_3,i_4}.
K(x/v,v)^2A(v/x,v) =K(x/v,v)-\frac{x^2}{v(v-x)}\big(K(x/v,v)-1\big) &-\frac{x}{1-v}K(x/v,v)A(x,1)-\frac{x^3}{v(1-v)(v-x)}C(x),
g_{NN}^{2}=\frac{G_{11}G_{22}-G_{12}^{2}+B_{12}^{2}}{\sqrt{\mathrm{det}G}}\hspace{2cm}g_{DN}^{2}=\frac{G_{22}}{4\sqrt{\mathrm{det}G}}\hspace{2cm}g_{DD}^{2}=\frac{1}{4\sqrt{\mathrm{det}G}}\ .
(a_1,a_2)\wedge (b_1,b_2) = (a_1\wedge b_1,a_2\vee b_2) (a_1,a_2)\vee (b_1,b_2) = (a_1\vee b_1,a_2\wedge b_2) \neg(a_1,a_2) = (a_2,a_1) (a_1,a_2)\leq (b_1,b_2) \mbox{iff}a_1\leq_{[0,1]}b_1 \mbox{and}b_2\leq_{[0,1]}a_2 (a_1,a_2)\leq_i (b_1,b_2) \mbox{iff}a_1\leq_{[0,1]}b_1 \mbox{and}a_2\leq_{[0,1]}b_2
\frac{d[M_{1}]}{d[M_{2}]}=\frac{[M_{1}]}{[M_{2}]}(\frac{k_{11}\frac{\sum[M_{1}^{*}]}{\sum[M_{2}^{*}]}+k_{21}}{k_{12}\frac{\sum[M_{1}^{*}]}{\sum[M_{2}^{*}]}+k_{22}})
\begin{array}{r l}{\delta e}{= - \partial_{\tau}( a e ) -{i}\alpha ( \tau ) \psi ,}{\delta \psi}{= - a ( \tau ) \dot{\psi}-{\frac{3}{2}}\dot{a}\psi -{\frac{1}{4}}\alpha ( \tau ) \dot{e}-{\frac{1}{2}}\dot{\alpha}( \tau ) e .}\end{array}
z(1-z) = \frac{\big[(t-b)^2-(y-x)^2\big]\big[(t+b+2)^2-(t-b)^2\big]}{\big[(t+b+2)^2-(y-x)^2\big]^2}= \frac{4(1+t)(1+b)\big[(t-b)^2-(y-x)^2\big]}{\big[(t+b+2)^2-(y-x)^2\big]^2},
K\times\mathcal{O}\ni (x,p)\mapsto (t\mapsto \frac{d}{d\tau}\big|_{\tau=s}f\circ\Phi^{p_0}_{\tau,t_0}\circ (\Phi^{p}_{s,t_0})^{-1}(x))\in L^1_{}(|t_0,t_1|;\mathbb{R})
\int_{H}\left((u')^2 - (\tilde{l}')^2\right) \geq \int_{H}2\tilde{l}'(u' -\tilde{l}') = 2\tilde{l}' \int_{H}( u' - \tilde{l}') = 2\tilde{l}' [ u - \tilde{l}]_{-d}^d = 2 \tilde{l}' [u -\tilde{w}]_{- d}^{d}& = 2(I_{+}- I_{-}).
f\circ_{\alpha}g(x,y,z)= f(\alpha(x),g(y,z))-(-1)^{|x||y|}f(\alpha(y),g(x,z))+(-1)^{|x||y|}f(g(y,x),\alpha(z)) &-(-1)^{|y||z|}f(g(x,z),\alpha(y)).
\lambda_k = \int_0^1 \log( \langle a_k(x); a_{k-1}(x) \dots, a_1(x) \rangle ) \d\mu(x) =\sum_{s}g(s) \mu(C_s) &= \sum_s g(s)\mu(C_{\overline{s}}) = \int_0^1 \log ( \langle a_1(x) ; a_2(x), \dots a_k(x) \rangle) \d\mu(x),
y_{N(\hat{k}-1)+\hat{\imath}}\,x_{N(\hat{\ell}-1)+\hat{\jmath}}=\min_{\substack{1\leq i,j\leq N begin{align*}1mm]1\leq k,\ell\leq L begin{align*}0.5mm]w_{ij}^{(k,\ell)}>0}}y_{N(k-1)+i}\,x_{N(\ell-1)+j}.
\delta_f L = \frac{d}{dt}\left[\frac{\kappa}{2 \pi}\int_{D}d^2 x \epsilon^{ij}A_i^a D_j(f^kA_k)^a\right]+\frac{\kappa}{2 \pi}\oint_{\partial{D}}d \varphi f^iA^a_i{\dot A}^a_\varphi.
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E)=\int_X\Bigr(\mathrm{Tr\,}e^{-t\widetilde{\Box}^{+}_{b,m}}(x,x)-\mathrm{Tr\,}e^{-t\widetilde{\Box}^{-}_{b,m}}(x,x)\Bigr)dv_X(x).
\nu^{2}_{n}=\left(\frac{32V_0}{b^2}\right)^{\frac{1}{2}}\left[n+\frac{1}{2}+ \frac{1}{4}\left(\sqrt{8V_{0}a^{2}+1}-\sqrt{8V_{0}a^{2}}\right)\right], ~~~~ n=0,1,2,\dots
(E \otimes F_1) \otimes w_{p_1 - p_2}\in (p_2 - p_1 + 1)(1 \otimes F_2) \otimes w_{p_1 - p_2}- (1 \otimes F_2) \otimes w_{p_1 - p_2}+ Z & = (p_2 - p_1)(1 \otimes F_2) \otimes w_{p_1 - p_2}+ Z
w := \sum_{x \in A^G_{}}\vert x \vert_k = \sum_{x \in A^G}\vert x \vert_k - \sum_{x \in A^G_{}}\vert x \vert_k & = n(q-1) q^{n-1}- n(q-1) = n(q-1) ( q^{n-1}- 1) .
\Omega^{(1)}_{\bar 1}(x,y) = \int d^2 y \Sigma^{(1)}(x,y) L^{(1)}_{i_1}(y) = - \kappa \epsilon_{ij}\partial^i \dot A^j - \kappa \partial_1 \dot \Phi^1 -{\sqrt \kappa}\partial_2 \dot \Phi^2 =0.
u\left( t\right) \leq v\left( t\right) +\int_{a}^{t}\overset{\infty}{\underset{k=1}{\sum}}\frac{\left[ g\left( t\right) \Gamma \left( \alpha\right) \right]^{k}}{\Gamma \left( \alpha k\right)}\psi^{\prime}\left(\tau \right) \left[ \psi \left( t\right) -\psi \left( \tau \right) \right]^{\alpha k-1}v\left( \tau \right) d\tau,
|k(t , s)| \le \prod_{l = 1}^d \left \| K_{h_l^*}\right \|_{\infty}\left \| P_{\tilde{w}_1- w_d}g(X_{w_d}) - \pi(g) \right \|_{L^1}& \le \frac{c}{\prod_{l = 1}^d h_l^*}e^{- \rho (\tilde{w}_1- w_d)}\left \|g \right \|_\infty & \le \frac{c}{(\prod_{l = 1}^d h_l^*)^2}e^{- \rho (\tilde{w}_1- w_d)}.
{\cal L}_{eff}=-\mbox{\boldmath $\pi$}\cdot \dot{\bf A}\rm + p_{\it a}\dot{\theta_{\it a}}- H^{(2)}_{0}-H^{(3)}_{0}-A_{0}(\rho^{(2)}+\rho^{(3)}-\nabla\cdot\mbox{\boldmath $\pi$})+O(\theta^4)\; \; ,\vspace{1em}
W_{n+1}(x)=(x-\beta_{n})W_{n}(x)-\alpha_{n}W_{n-1}(x)-\gamma_{n-1}W_{n-2}(x),\:\: n \geq 2, W_{0}(x)=1,\:\: W_{1}(x)=x-\beta_{0},\:\:W_{2}(x)=(x-\beta_{1})W_{1}(x)-\alpha_{1}.
E_{1 , 2}^{2}={\bf p}^{2}+ m^{2}+ \frac{g^{2}( \lambda_{1}+ \lambda_{2})}{4}\pm g \sqrt{\lambda_{1}p_{1}^{2}+ \lambda_{2}p_{2}^{2}+ \frac{( H_{z}^{3})^{2}}{4}}
k \underset{m=1}{\overset{n}{\sum}}(\Vert (\delta_t{u}^m_h, \delta_t{\boldsymbol \sigma}^m_h)\Vert_{0}^2 + \Vert (\hat{u}^m_h,{\boldsymbol \sigma}^m_h)\Vert_{W^{1,6}}^2)\leq K_1, \ \ \ \forall n\geq 1,
\begin{array}{l}\dot J_1=[J_1,H]_{GPB}=\displaystyle\frac{I-I_3}{II_3}J_2J_3~, begin{align*}3mm]\dot J_2=[J_2,H]_{GPB}=\displaystyle\frac{I_3-I}{II_3}J_1J_3~, begin{align*}3mm]\dot J_3=[J_3,H]_{GPB}=0~.\end{array}
\alpha=c_{0}^{-\frac{n-2}{2n}}+\frac{6-n}{2(n-2)}c_{0}^{-\frac{n+2}{2n}}\int \varphi_{a, \lambda}^{\frac{4}{n-2}}v^{2}+o_{\frac{1}{\lambda}}(1)+o(\Vert v \Vert^{2})
\psi\left(f(\zeta )\right) =-\sum_{n=0}^{+\infty}h\big( f^{\circ(n+1)}(\zeta )\big) &=-\sum_{n=1}^{+\infty}h\left( f^{\circ n}(\zeta )\right) & =-\big(-\psi(\zeta )- h(\zeta )\big) & =\psi(\zeta )+ h(\zeta ),\zeta \in D.
g_{1}(\nabla^{^{M_1}}_{X}Y,V) =-g_{1}(V,\mathcal{V}\nabla^{^{M_1}}_{X}\phi \mathcal{B}Y+\mathcal{A}_{X}\omega \mathcal{B}Y)+g_{2}(\pi_*(CY),(\nabla \pi_*)(X,\varphi V)),
\gamma_{n}(a)=lim_{m\rightarrow\infty}\{\sum_{k=0}^{m}\frac{(ln(k+a))^{n}}{k+a}-\frac{(ln(m+a))^{n+1}}{n+1}\},\begin{matrix}n=0,1,2,...\\a\ne0,-1,-2,...\end{matrix}
\hat{q}(t):= q(t) - e^{\gamma_* t}\bigg(\sum_{i\in I_1}\textup{Re}\big( w_i e^{\mathbf{i}\beta_i t}\big) \psi_{i,1}+\textup{Im}\big( w_i e^{\mathbf{i}\beta_i t}\big) \psi_{i,2}+ \sum_{i\in I_2}c_{i,3}\psi_{i,3}+(tc_{i,3}+c_{i,4})\psi_{i,4}\bigg)
c_{(a^i)(b^j)}^{\rho}= c_{(a^i)(b^j)}^{(a+1,a^{i-1},b^{j-1},b-1)}\ge c_{(a^i)(b)}^{(a+1,a^{i-1},b-1)}\ge c_{(a)(b)}^{(a+1,b-1)}=1
F^{(4)}_{j_1 j_2 j_3 j_4}:=\begin{cases}\dfrac{H^{(3)}_{4,\,j_1 j_2 j_3 j_4}}{\mathrm{i}(j_1^3+j_2^3+j_3^3+j_4^3)}\mbox{if}\,\,(j_1, j_2, j_3, j_4)\in\mathcal{A}_4, begin{align*}2mm]0 \mbox{otherwise}, \end{cases}
\exp_{\*}\left( A\right) \* \exp_{\*}\left( B\right)=\exp_{\*}\left( A+B+\frac{1}{2}\left[ A,B\right]_{\*}+\frac{1}{12}\left[ A,\left[ A,B\right]_{\*}\right]_{\*}+\frac{1}{12}\left[ \left[ A,B\right]_{\*},B\right]_{\*}+ C \right),
\left. \mathrm{T r}V_{1 \nu}^{~ \mu}( \nabla ) \right|^{\mathrm{d i v}}= \frac{2 i}{4 - d}\, \frac{\kappa^{2}}{2 4 \pi^{2}}\int d^{4}x \sqrt{- g}\left[ - 5 R_{\mu \nu}{\cal E}^{\mu \nu}+ \frac{5}{2}R g_{\mu \nu}{\cal E}^{\mu \nu}\right] .
\mathcal{D}(A)=\{U=\mathcal{P}\Xi\,:\,\Xi\in\mathcal{H}_{\boldsymbol\Gamma}^m,a_i\xi_{i,x}+ b_i\xi_{i,y}\in\mathcal{H}_{\boldsymbol\Gamma}^m(\Omega),\forall\, i\in\{1,2,3\}\}.
\left\langle \Psi, \sum_{i=1}^N (-\Delta_{i})^s \Psi \right\rangle \ge \sum_Q \left[\frac{1}{C}\frac{\int_Q \rho_\Psi^{1+2s/d}}{\Big(\int_Q \rho_\Psi \Big)^{2s/d}}- \frac{C}{|Q|^{2s/d}}\int_Q \rho_\Psi \right],
\|\overline{u}\|_{L^{m\chi^{n+1}}(\Omega_{R_{n+1}})}\le C^{1/\chi^n}\left[\frac{2^{n+1}}{R_{1}-R_{2}}+\left(\frac{2^{n+1}}{R_{1}-R_{2}}\right)^{1/m}+\left(\frac{\|f\|_q}{k^{m-1}}\right)^{\beta}\right]^{1/\chi^n}\|\overline{u}\|_{L^{m\chi^n}(\Omega_{R_n})},
{\gamma\cdot D\Lambda ={e_{{\hat L}}}^{M}\gamma^{\hat{L}}\left( \partial_{M}+\frac{1}{4}\omega^{\hat{M}\hat{N}}_{M}\gamma_{\hat{M}\hat{N}}\right)\Lambda = (\rho \gamma^{\hat 5}\partial_{5}+ \rho \gamma^{\hat\mu}\partial_{\mu}- 2 \gamma^{\hat 5})\Lambda \; ,}
\begin{array}{lcl}x_{a}\star x_{b}= \sum\limits_{\alpha = 0}^{\infty}(x_a \star x_b)_{\alpha}&= x_{a}x_{b}+ \sum\limits_{{\alpha}= 1}^{\infty}\frac{1}{\alpha!}(\frac{i}{2})^{\alpha}\prod\limits_{i=1}^{\alpha}\theta^{\mu_i \nu_i}(\prod\limits_{j=1}^{\alpha}\partial_{\mu_{j}}x_a)(\prod\limits_{k=1}^{\alpha}\partial_{\nu_k}x_b)\end{array}
\mathbf{osc~}{(u^*-u^\infty)(\cdot,t)}= \lim\limits_{i \to \infty}\mathbf{osc~}((u(\cdot,t+t_i)-s^\infty t_i-u^\infty(\cdot,t)) = \lim \limits_{i \to \infty}\mathbf{osc~}((u(\cdot,t+t_i)-u^\infty(\cdot,t+t_i)) = \lim \limits_{i \to \infty}\mathbf{osc~}(w(\cdot,t+t_i)) = \delta.
\prod_{A = 1}^{N^{2}- 1}\frac{d X_{D}^{A}}{\sqrt{2 \pi}}= \frac{2^{\frac{N ( N - 1 )}{2}}}{\prod_{i = 1}^{N}i !}\sqrt{N}\prod_{k = 1}^{N}\frac{d \lambda_{k}}{\sqrt{\pi}}\Delta^{2}( \lambda_{i}) \sqrt{\pi}\delta ( \lambda_{1}+ \ldots + \lambda_{N})
\frac{\partial^{k+1}F}{\partial v^{k+1}}(a,v)=\int_a^v\frac{\partial^{k+1}f}{\partial u^{k+1}}(v,v')dv'+\sum_{l=0}^k\binom{k+1}{l+1}\frac{\partial^kf}{\partial u^{k-l}\partial v^l}(v,v),
S_{c}^{<}( x^{0}- \tau^{0}+ i \beta , \vec{p}) \, S_{c}^{>}( \tau^{0}- i \beta - y^{0}, \vec{p}) = S_{c}^{>}( x^{0}- \tau^{0}, \vec{p}) \, S_{c}^{<}( \tau^{0}- y^{0}, \vec{p})
rarctan(\frac{y}{x})=\frac{1}{1}\cdot\frac{ry}{x}-\frac{1}{3}\cdot\frac{ry^{3}}{x^{3}}+\frac{1}{5}\cdot\frac{ry^{5}}{x^{5}}-\cdot\cdot\cdot,wherey/x\le1.
dJ_t = \sum_{i=1}^r \nabla V_i ( x_t) J_t dw_t^i + \nabla V_0 ( x_t) J_t d\lambda_t \mbox{with $J_0 ={\rm Id}_d \in{\rm Mat}(d,d)$,}dK_t = - \sum_{i=1}^r K_t \nabla V_i ( x_t) dw_t^i - K_t \nabla V_0 ( x_t) d\lambda_t \mbox{with $K_0 ={\rm Id}_d \in{\rm Mat}(d,d)$.}
~\frac{r!}{[(r+1)!]^2}\sum^r_{i=0}\sum^\mu_{j=0}{\mu\choose j}{\nu\choose i-j}i!(r-i)!\int^\infty_0(\cosh t)^{2j-2\mu-2\beta-4}D(t)dt \|\phi\|^2 = ~\frac{r!}{[(r+1)!]^2}\sum^\mu_{j=0}{\mu\choose j}\sum^r_{i=0}{\nu\choose i-j}i!(r-i)!\frac{1}{(\mu-j+\beta)(\mu-j+\beta+1)}\|\phi\|^2.
\frac{\omega(\theta_j(\zeta_1),\gamma_1)}{\omega(\theta_j(\zeta_2),\gamma_2)}=\frac{\beta_j(\gamma_1,\gamma_2)}{\beta_j(\zeta_1,\zeta_2)}\left(\frac{1-\bar\delta_j\zeta_2}{1-\bar\delta_j\zeta_1}\right)\frac{\omega(\zeta_1,\gamma_1)}{\omega(\zeta_2,\gamma_2)},
\Psi [ A_{i}] = \int D \phi \exp \left\{- \frac{1}{2}\int d^{2}x d^{2}y \left[ A_{i}( x ) - \frac{1}{g}\partial_{i}\phi ( x ) \right] G^{- 1}( x - y ) \left[ A_{i}( y ) - \frac{1}{g}\partial_{i}\phi_{i}( y ) \right] \right\}
\tiny (k-1)!\Biggl(\theta_{n-m+l}^{n-m+k-1}-\sum_{p=1}^{n-m}\sum_{q=1}^{n-m}\left[\mathbf{\Psi}^{-1}\right]_{p,q}& \times \theta_{n-m+l}^{p-1}\theta_{q}^{n-m+k-1}\Biggr).
U_n(t):=(n+1){\mbox{$_2$F$_1$}(-n,\n+2;\{{3}\over{2}};\{{1-t}\over{2}})}={{P_n^{{{1}\over{2}},{{1}\over{2}}}(t)}\over{P_n^{{{1}\over{2}},{{1}\over{2}}}(1)}}, \ \forall n\in\mathbb{N},
\begin{aligned}\varphi:(0,\infty)\to (0,\infty) \,\,\rm{is\,\, increasing \, \,with}\,\,\lim_{x\to +0}\varphi(x)=0; \displaystyle{\int_1^{\infty}\varphi(a\nu^{x})\,dx <\infty}\,\,\rm{for\,\, some\,\, constants}\,\,a>0 \,\, \rm{and}\,\, 0<\nu<1. \end{aligned}
{\mathbb{R}}\Gamma_{rig}(M/{\mathfrak S}^{0})={\mathbb{R}}\Gamma(]M_{\bullet}[_{{\mathfrak Y}_{\bullet}},{\omega}_{{\mathfrak Y}_{\bullet}}^{\bullet})\longrightarrow{\mathbb{R}}\Gamma(]M_{\bullet}[_{{\mathfrak Y}_{\bullet}},{\omega}_{{\mathfrak Y}_{\bullet}}^{\bullet}\otimes{\cal O}_{{\mathfrak M}_{\bullet}})
\pi_{[a,i]}\cdot \pi_{[i+1,b]}= \binom{b-a+1}{i-a+1}\pi_{[a-1,b-1]}+\binom{b-a}{i-a}\pi_{[a,b-1]}y_b +\binom{b-a}{i-a+1}\pi_{[a,b-1]}y_{b}&= \binom{b-a+1}{i-a+1}\pi_{[a-1,b-1]}+\binom{b-a+1}{i-a+1}\pi_{[a,b-1]}y_{b}&= \binom{b-a+1}{i-a+1}\pi_{[a,b]},
\frac{1}{\left|\Gamma(u+ix)\right|^{2}}=\left(u^{2}+x^{2}\right)e^{2\gamma u}\prod_{n=1}^{\infty}\left[\left(\left(1+\frac{u}{n}\right)^{2}+\left(\frac{x}{n}\right)^{2}\right)e^{-2u/n}\right]
|f(x)-f(y)| \leq \sum_{k=0}^\infty |f(z_k)-f(z_{k+1})| &\leq \sum_{k=0}^\infty [f]_{\mathrm{Lip},\mathrm{cl}(\Omega_{r_k})}|z_{k}-z_{k+1}| &\leq \left(\max_{r=0,1\ldots}[f]_{\mathrm{Lip},\mathrm{cl}(\Omega_r)}\right) \sum_{k=0}^\infty |z_{k}-z_{k+1}| &= \left(\max_{r=0,1,\ldots}[f]_{\mathrm{Lip},\mathrm{cl}(\Omega_r)}\right) |x-y|
\begin{array}{r c l}{{I_{2}( \rho^{2}, m )}}&{{=}}&{{\displaystyle \mu^{4 - n}\int \frac{d^{n}k}{( 2 \pi )^{n}}[ ( k + p )^{2}+ m^{2}]^{- 2}\cos ( k \cdot \rho )}}\\{{}}&{{=}}&{{\displaystyle I_{1}( \rho^{2}, m ) \cos ( p \cdot \rho ) .}}\\ \end{array}
\eta_{2,1}(x_1,t)= \frac{h_{2,1}\, \partial_{x_2}v^{(2,1)}_0(x_1,-(l_1+l_2),t)}{h_{2,1}\,\partial_{x_2}v^{(2,1)}_0(x_1,-(l_1+l_2),t) + h_{2,2}\,\partial_{x_2}v^{(2,2)}_0(x_1,-(l_1+l_2),t)},
\varphi_i(x_i) = \int_F \varphi'_i(x_i') \psi \left( \frac{1}{2}(x_i' y_i' - x_i y_i) \right) dy_i = \left| \frac{2a_i}{b_i}\right| \psi \left( \frac{a_i}{2}x_i^2 \right) \int_F \varphi'_i(x_i') \psi \left( \frac{a_i}{b_i^2}(x_i')^2 - \frac{2a_i}{b_i}x_i x_i' \right) dx_i'.
\begin{aligned}\Theta_P(\alpha) = \theta(g) \cdot \Theta(\beta^{\bullet}) \cdot \theta(C_S(P)) = = \Theta(\beta^{\bullet}) \cdot \theta(\beta^{\bullet}(g)) \cdot \theta(C_S(P)) = \Theta(\beta^{\bullet}) \cdot \theta(C_S(P)).\end{aligned}
\lesssim 2^{4\epsilon M_t}\| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty}\big( \min\{2^{n+4j/3}, 2^{k+ n}\}\mathbf{1}_{l> -j}+ \min\{1+ 2^{n+j}, 2^{k/2+n}\}\mathbf{1}_{l=-j}\big)
y \mapsto\Vert n(x,x-y)\Vert\,w(y) &=\Vert n(x,x-y)\Vert\,\sum_{i=1}^\infty\Vert d_i(y)\Vert\cdot|u(y-h_i)| &=\sum_{i=1}^\infty\Vert n(x,x-y)\Vert\cdot\Vert d_i(y)\Vert\cdot|u(y-h_i)|
F_{N, M}\Xi (\eta_{1}, \cdots, \eta_{M}){=}\sum_{\begin{subarray}N-M=N_{1}+\cdots+N_{g}, |\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|<\kappa\end{subarray}}c_{\overrightarrow{\boldsymbol{N}}}d_{\overrightarrow{\boldsymbol{N}}},
\Psi^{\prime}= \Psi - \frac{\delta \Psi}{\delta \Phi^{m}}\left( \frac{\delta F}{\delta \Phi_{m}^{*}}\right)_{\Phi^{*}= \delta \Psi / \delta \Phi + K}+ ( F )_{\Phi^{*}= \delta \Psi / \delta \Phi + K}.
\log_2 \frac{d^d \big(\lfloor\frac{d}{n}\rfloor !\big)^l\big(\lceil\frac{d}{n}\rceil !\big)^{n-l}}{d!\big(\lfloor\frac{d}{n}\rfloor\big)^{l\lfloor\frac{d}{n}\rfloor}\big(\lceil\frac{d}{n}\rceil\big)^{(n-l)\lceil\frac{d}{n}\rceil}}\le \eta_{sym}(d,n)\le \log_2{n+d-1\choose n-1}, \;l=n\lceil\frac{d}{n}\rceil -d.
\Sigma_1^i(z-v)=\frac{-3f_l^2}{16(4\pi^2)}\left(2\frac{Y_L^{(e)}}{Y_R^{(e)}}L\delta_{i(e)}+\frac{Y_R^{(e)}}{Y_L^{(e)}}R\right)/\hspace{-2mm}\partial\delta^4(z-v)\, .
\left(P\, S, S\, P\right)_2=\sum_{i, j=1}^N\lambda_i\, a_{ij}\, \lambda_j\, a_{ji}=\sum_{i, j=1}^N\lambda_i\, \lambda_j\, a_{ij}^2\ge \frac{1}{\beta}\sum_{i, j=1}^N\lambda_i^2\, a_{ij}^2=\frac{1}{\beta}\, |P\, S|_2^2,
\varphi_1 \cdots \varphi_{m_k}.(u \otimes (v_q)_z) = \tau_1 \cdots \tau_{b-1}(\tau_b (x_b - x_{b+1}) +1) \tau_{b+1}\cdots \tau_{c-1}(\tau_c (x_c - x_{c+1}) +1) Q(z)
z(t)= S(t)z_0-\int_0^t S(t-s)(i \nu z(s) +\epsilon(\gamma z(s) -\mu \overline{z}(s)))\d s-\tfrac{1}{2}\int_0^t S(t-s)(z(s)F_\Phi )\d s +i \kappa \int_0^t S(t-s)(|z(s)|^2z(s))\d s-i\int_0^t S(t-s)(z(s)\d W(s))
\textbf{err}^{(5)}= \mathcal{O}\Bigg(\varepsilon^6\sqrt{\mathcal{K}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-z|^{4-2\mathrm{r}}}\Big\Vert|\mathrm{E}|^{2}\Big\Vert_{\mathrm{L}^2(\Omega)}\Bigg).