harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u_i(r,\phi,0)=\sqrt{\frac{2}{\pi|i|!}}\frac{1}{w_0}\left(\frac{\sqrt{2}r}{w_0}\right)^{|i|}L^i_0\left(\frac{2r^2}{w_0^2}\right)\mathrm{exp}\left(\frac{-r^2}{w_0^2}\right)\mathrm{exp}\left(-ji\phi\right),
\big( \int_0^te^{-i(t-s) H_0}(\tau_{x_n}F(s) \star\rho_q))(-x) ds\big)\cdot1_{x\leq 0}(x) &=\tau_{x^n}\big( \int_0^t(e^{-i(t-s) H_0}(F(s) \star\rho_q))ds\big)(-x) \cdot 1_{x\leq 0}(x) .
G(\kappa,\lambda_1,\lambda_2;x):=\begin{array}{ccccccc}c \kappa_0 \kappa_{k-2}\kappa \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7}\Phi_0(\nu;x) + x^2+\lambda_{0,1}x+\lambda_{0,2}+\cdots+ x^2+\lambda_{k-2,1}x+\lambda_{k-2,2}+ x^2+\lambda_1 x+\lambda_2\end{array}.
( 0 ~ 0 ~ 0 ~ 4 ) \longrightarrow ( 0 ~ 0 ~ 4 ) \oplus ( 0 ~ 0 ~ 3 ) , ~ ~ \, \, ( 0 ~ 0 ~ 0 ~ 2 ) \longrightarrow ( 0 ~ 0 ~ 2 ) \oplus ( 0 ~ 0 ~ 1 ) .
\|w_n\|_{L^2(B_{R})}^2 = \Theta_n(r_n)^{-1}r_n^{-N+2\varrho}\|u_n\|_{L^2(B_{r_nR}(x_n) )}^2 &\leq \Theta_n(r_n)^{-1}r_n^{-N+2\varrho}\Theta_n(r_nR) (r_nR)^{N-2\varrho}\leq R^{N-2\varrho},
e^{2(x+{\frac{i}{2}}\overrightarrow{\partial}_{p})}\,K_{i\left(P-p\right)}(e^{x+X})\,K_{i\left( P+p\right)}(e^{x-X})=e^{2x}\,K_{1+i\left(P-p\right)}(e^{x+X})\,K_{-1+i\left( P+p\right)}(e^{x-X})\,\;.
\hat{\sigma}(\omega)=\sigma_{1}(\omega)+i\sigma_{2}(\omega)=\frac{Ne^{2}}{m}\frac{\frac{\omega^{2}}{\tau}}{(\omega_{0}^{2}-\omega^{2})^{2}+\omega^{2}/\tau^{2}}-i\frac{Ne^{2}}{m}\frac{(\omega_{0}^{2}-\omega^{2})\omega}{(\omega_{0}^{2}-\omega^{2})^{2}+\omega^{2}/\tau^{2}}
\int_{(-\infty; 0]}|x|dm^{N,j}_t(x) = \int_{(-\infty; \tau^{-1}_{n,N}(t)]}|\Phi^{e_j}_{t}(x,t_{n - 1})|dm^{N,j}_{t_{n-1}}(x) &\leq \int_{(-\infty; \tau^{-1}_{n,N}(t)]}|x|dm^{N,j}_{t_{n-1}}(x) + m^N_{t_{n-1}}((-\infty; \tau^{-1}_{n,N}(t)])V_{max}\Delta t^N &\leq \int_{(-\infty; 0]}|x|dm^{N,j}_{t_{n-1}}+ m^{N,j}_{t_{n-1}}(e_j)\Delta t^N V_{max}.
t^{(q)}_l = \underbrace{\sum_{q_0=0}^l \sum_{q_1=0}^l \cdots \sum_{q_l=0}^l}_{q_0 +\dots+ q_l = q \atop 1\cdot q_1 + 2\cdot q_2 +\dots+ l\cdot q_l = l+1-q}{n_0 + q_0 - 1 \choose q_0}{n_1 + q_1 - 1 \choose q_1}\cdots{n_l + q_l - 1 \choose q_l}.
\zeta^{\top}= \pi_t\left(X_t d\mathbf{Y}_t^{\top}\right) - \pi_t\left(X\right) \pi_t \left(d\mathbf{Y}_t^{\top}\right) + \pi_t \left(\left[\mathbf{L}^{\dagger}_t,X_t\right]\mathbf{S}_td\mathbf{A}d\mathbf{Y}_t^{\top}\right), \Sigma = \pi_t \left(d\mathbf{Y}_td\mathbf{Y}_t^{\top}\right).
{\|}\,{( \cal{A}}_{1})^{n}\,{\|}= ( i g )^{n}\int{\sf D}( 1 , \cdots , n ) \, \psi_{( 1 ) i_{1}}^{{\alpha_{1}}}( 1 ) \; \psi_{( 1 ) i_{2}}^{{\alpha_{2}}}( 2 ) \; \cdots \psi_{( 1 ) i_{n}}^{{\alpha_{n}}}( n ) \; \Pi_{i_{1}}^{\alpha_{1}}( 1 ) \; \Pi_{i_{2}}^{\alpha_{2}}( 2 ) \; \cdots \; \Pi_{i_{n}}^{\alpha_{n}}( n ) \; ,
\int_{0^-}^{\infty}e^{-s x}dG_1(x)=e^{\int_{0}^{\infty}(e^{-s x}-1) \Pi_1(dx)}&=e^{-\Pi_1(0,\infty)}\sum_{n=0}^{\infty}\frac{\int_{0}^{\infty}e^{-sx}d\Pi_1^{*n}(0,x)}{n!},
d s^{2}= ( 1 -{\frac{q c o s \theta}{r}})^{\frac{2}{1 + \alpha^{2}}}\lbrace d r^{2}+ r^{2}d \theta^{2}+ r^{2}s i n^{2}\theta d \varphi^{2}\rbrace -{\frac{d t^{2}}{( 1 -{\frac{q c o s \theta}{r}})^{\frac{2}{1 + \alpha^{2}}}}}.
y_{m,j}^{<i>}=(-1)^{m-1}(-[i])^{d_m-j}\underset{0\leq i_1\leq\cdots\leq i_{m-1}<i}{\sum}\frac{u_1^{(i_1)}\cdots u_{m-1}^{(i_{m-1})}}{L_{i_1}^{s_1}\cdots L_{i_{m-1}}^{s_{m-1}}L_i^{d_m}}.
\psi_{\scriptscriptstyle{l}}( k , r ) = \frac{e^{- i \pi W / 2}e^{i \pi | l |}}{2 \sqrt 2}\left( H_{\scriptscriptstyle{W}}^{( - )}( k r ) + e^{i \pi ( W - | l | )}e^{2 i \delta_{l}( k )}H_{\scriptscriptstyle{W}}^{( + )}( k r ) \right) ,
\frac{d}{d\rho}h_{12,\mathsf{P}}(\rho) = \frac{-2\sqrt{h_1h_2}\rho^2 + 2(h_1+h_2)\rho-2\sqrt{h_1h_2}}{\left(1-\rho^2 + \mathsf{P}(h_1+h_2-2\rho\sqrt{h_1h_2})\right)^2}
{}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \tilde A_1^\dagger \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q =\frac{\sqrt{J_1}}{E_q(J_1,J_2)}F_q(J_1,J_2,\gamma_1);
-\int_{0}^{\infty}e^{-\lambda s}\mathbb{E}\left[\left(-2 \int_{0}^{s}B(u)du \right) \vert B(s)=-x\right]\frac{x}{\sqrt{2\pi s^3}}e^{-\frac{x^2}{2s}}ds=\frac{\mathrm{Ai}(2^{-\frac{1}{3}}\lambda -4^{\frac{1}{3}}x)}{\mathrm{Ai}(2^{-\frac{1}{3}}\lambda)}
\sum_{j = 0}^\infty \binom{m}{j}\big(a_{(n+j)}b\big)_{(m+k-j)}c = \sum_{j=0}^\infty (-1)^j \binom{n}{j}a_{(m+n-j)}b_{(k+j)}c -(-1)^{p(a)p(b)}\sum_{j=0}^\infty(-1)^{j+n}\binom{n}{j}b_{(n+k-j)}a_{(m+j)}c
{}_4\phi_3\left[{q^{-m}, abcdq^{m-1},ax,a/x\atop ab,ac,ad};q,q \right]=a^m x^{-m}{(abcdq^{m-1};q)_m \over (ab,ac,ad;q)_m}\Psi(x;q^{-m}|a,b,c,d|q).
\begin{array}{l}{{p^{0 0}= 0 ,}}\\{{p^{0 k}= - g^{i k}\partial_{i}( a e^{- \Phi}) ,}}\\{{p^{i k}= \frac{1}{2}a e^{- \Phi}\left[ ( \dot{h}_{l m}g^{l m}- 2 \dot{\varphi}) g^{i k}+ \dot{h}^{i k}\right] ,}}\\{{p_{\varphi}= a e^{- \Phi}( 2 \dot{\varphi}- \dot{h}_{l m}g^{l m}) .}}\\ \end{array}
\chi_{Ra_!b^*E^{\theta},\varphi}(g) =\sum_{i\in\mathbb{Z}}(-1)^i\cdot\mathrm{Tr}\left(\varphi_g,\mathcal{H}^i(Ra_!(b^*E^{\theta}))_g\right) &=\sum_{i\in\mathbb{Z}}(-1)^i\cdot\mathrm{Tr}\left(\varphi,{H}_c^i(a^{-1}(g),b^*E^{\theta})\right).
x\wedge y=\left(\chi_1 \zeta_2 - \chi_2\zeta_1\right)(\ell_1\wedge\ell_2)=\det\left(\begin{array}{cc}\chi_1 \chi_2 \zeta_1 \zeta_2\end{array}\right)\ell_1 \ell_2=\det\left(\begin{array}{cc}\chi_1 \chi_2 \zeta_1 \zeta_2\end{array}\right)\mathbb{I}_2\ ,
\mathbb{C}_{a,p-1}Q(\xi_{a}^{(0)})=\prod_{b=1}^{(p-1)\mathsf{N}}\frac{X_{a}^{(p-1)}-w_{b}}{w_{(p-1)\mathsf{N}+1}-w_{b}}Q(\xi_{(p-1)\mathsf{N}+1})+\sum_{n=1}^{\mathsf{N}}\mathbb{D}_{a,n}Q(\xi_{n}^{(0)}),
v^j_k(x)=\sqrt{k}(z_j+\beta_j(x))v_k(x)\in C^\infty_0(D_0),\ \j=1,\ldots,n-1, &v^n_k(x)=k(x_{2n-1}+\beta_n(x))v_k(x)\in C^\infty_0(D_0).
\begin{cases}kx(x-1)\prod_{i=1}^{\frac{D}{2}-1}(x-\alpha_i)^2-(x-\lambda)\prod_{i=1}^{\frac{D}{2}-1}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\gamma_i)^2 kx(x-1)\prod_{i=1}^{\frac{D}{2}-1}(x-\alpha_i)^2-\lambda(x-\lambda)\prod_{i=1}^{\frac{D}{2}-1}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\delta_i)^2\end{cases}
\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) =\mathrm{dim\,}\mathrm{Ker\,}\widetilde{\Box}^+_{b,m}-\mathrm{dim\,}\mathrm{Ker\,}\widetilde{\Box}^-_{b,m}.
L ( \phi_{\mathrm{m}}, - \phi_{\mathrm{m}}) = 2 \ln \left[ \sinh \chi_{\mathrm{m}}\, \sin \phi_{\mathrm{m}}+ ( \sinh^{2}\chi_{\mathrm{m}}\, \sin^{2}\phi_{\mathrm{m}}+ 1 )^{1 / 2}\right] .
LHS= \Bigg( \underbrace{\sum_{j=1}^{i}\int_{\Lambda_j}u n_j \int_{x_{i+ 1/2}-x_j}^{x_{}}\beta(x_j,v)n(t,v)dv du}_{I_1}- \underbrace{\sum_{j=1}^{i-1}\int_{\Lambda_j}u n_j \int_{x_{i-1/2}-x_j}^{x_{}}\beta(x_j,v)n(t,v)dv du}_{I_2}\Bigg).
\begin{cases}-\Delta v_\lambda (0,c_0) + \frac{1}{|\Omega|}\int_{\partial \Omega}b g(c_0) = Q[f(x, c_0)] \mbox{in $\Omega$}, \dfrac{\partial v_\lambda (0,c_0)}{\partial \mathbf{n}}= b g(c_0) \mbox{on $\partial \Omega$}, \end{cases}
trln{\left(\partial\!\!\!\slash+\bar{M}\right)}=\frac{2}{\beta}\sum_{n=-\infty}^{\infty}\int \frac{d^{d-1}{\bf k}}{(2\pi)^{d-1}}\left\{2\ln{\left({\bf k}^{2}+\omega_{n}^{2}\right)}+ln{\left( 1+\frac{\bar{m}_{1}^{2}}{{\bf k}^{2}+\omega_{n}^{2}}\right)}+ln{\left( 1+\frac{\bar{m}_{2}^{2}}{{\bf k}^{2}+\omega_{n}^{2}}\right)}\right\}.
{R^*}= \mathop{\arg \max}\limits_{{W_s}\in \left[{{W^{low}},{W^{up}}}\right]}\;\left({W -{W_s}}\right){\log_2}\left({1 + \frac{{\left[{P - \overline{{p_K}\left({\overline S,{W_s}}\right)}}\right]{{\left|{{h_b}}\right|}^2}}}{{\left({W -{W_s}}\right){N_0}}}}\right),
\phi(a_1,\dots,a_{m-1},a_m)=\mathbf{1}_{\{|a_m|\leq q^{-N'}\}}f(a_1,\dots, a_{m-1}, a_m)+\mathbf{1}_{\{q^{-N'}\leq |a_m|\leq 1\}}\phi(a_1,\dots,a_{m-1},a_m).
\int_{y^{( + )}}^{y^{( - )}}{\frac{d y}{a^{2}}}u_{0}( y ) h_{\mu \nu}^{( K K )}( y ) \propto \int_{y^{( + )}}^{y^{( - )}}d y \, h_{\mu \nu}^{( K K )}( y ) = 0 .
\widehat \alpha_{0,0}(\theta) - \alpha_{0,0}(\theta) = \widetilde \alpha_{0,0}(\theta) + \overline \alpha_{0,0}(\theta), \widehat \beta_{j,k}(\theta) - \beta_{j,k}(\theta) = \widetilde \beta_{j,k}(\theta) + \overline \beta_{j,k}(\theta),
\frac{\partial^{||\beta(\alpha(J)||}}{\partial z^{\beta(\alpha,J)}}=\frac{\partial^{m_{1j_{1}}(\alpha_{j_{1}}+1)-1+\ldots+m_{nj_{n}}(\alpha_{j_{n}}+1)-1}}{\partial z_1^{m_{1j_{1}}(\alpha_{j_{1}}+1)-1}\ldots \partial z_n^{m_{nj_{n}}(\alpha_{j_{n}}+1)-1}}.
\begin{cases}x_1^{\frac{2^\ast}{2}-1}+\nu\alpha x_1^{\frac{\alpha}{2}-1}x_2^{\frac{\beta}{2}}=1, &x_2^{\frac{2^\ast}{2}-1}+\nu\beta x_1^{\frac{\alpha}{2}}x_2^{\frac{\beta}{2}-1}=1,\end{cases}
P_{\mathcal{F}_{\mathcal{G}}}(\mathbb{S}^n_{+}) \subseteq \mathbb{S}^n_{+}, P_{\mathcal{F}_{\mathcal{G}}}(Y + \mathcal{L}) \subseteq Y+\mathcal{L}, P_{\mathcal{F}_{\mathcal{G}}}(C)=C,
\wp(p_{C}^{(0)}(\tau)|\tau)= & \frac{2g_{3}(\tau)(C_{1}-\tau)^{3}-4(C_{1}\eta_{1}(\tau)-\eta_{2}(\tau))^{3}-g_{2}(\tau)(C_{1}\eta_{1}(\tau)-\eta_{2}(\tau))(C_{1}-\tau)^{2}}{(C_{1}-\tau)[12(C_{1}\eta_{1}(\tau)-\eta_{2}(\tau))^{2}-g_{2}(\tau)(C_{1}-\tau)^{2}]}
\times\bigg(\left(t_{C_1}\right)_{R_2r_2S_2s_2}\left(t_{C_2}\right)_{V_2v_2W_2w_2}\left(\left(t_{E_2\alpha_2}\right)_{S_2s_2V_2v_2}\delta_{W_2R_2}\delta_{w_2r_2}-\delta_{S_2V_2}\delta_{s_2v_2}\left(t_{E_2\alpha_2}\right)_{W_2w_2R_2r_2}\right)\bigg)\times
\|K\tilde{D}^{-1}K^T-K_{\varepsilon}\tilde{D}^{-1}K_{\varepsilon}^T\|=\left\| \sum_{j=1}^k \sum_{d_i\in S_j}d_i^{-1}(\Re(P_M(d_i))-\Re(P_{M_{\varepsilon}}(e_i)))\right\|
r_{\eta}=r(\eta)e^{2itg(\eta)}D^{-2}_{0}(\eta,\xi)\exp\left[i\nu(\eta)\log(2tg''\left(\eta\right))\right], r_{-\eta}=r(-\eta)e^{2itg(-\eta)}D^{-2}_{0}(-\eta,\xi)\exp\left[i\nu(-\eta)\log(-2tg''\left(-\eta\right))\right].
\frac1{2h}\int G_h(z) \int u_j (u_i^z-u_i) \zeta - u_j^z(u_i^z-u_i ) \zeta^z - u_i(u_j^z-u_j)\zeta + u_i^z (u_j^z-u_j) \zeta^z \,dx\,dz = \frac1{2h}\int G_h(z) \int (u_i^z-u_i) (u_j\zeta - u_j^z \zeta^z) -(u_j^z-u_j) (u_i\zeta - u_i^z \zeta^z) \,dx\,dz.
\sup\limits_{\|f_1\|_{\widetilde{H}^{s}(W)}=1=\|f_2\|_{\widetilde{H}^s(W)}}\int\limits_{W}((\Lambda_{q_1}-\Lambda_{q_2}) f_1, f_2) dx \leq \|\Lambda_{q_1}-\Lambda_{q_2}\|_{L^2(W) \rightarrow L^2(W)}\|f_1\|_{L^2(W)}\|f_2\|_{L^2(W)}&\leq C_p\|\Lambda_{q_1}-\Lambda_{q_2}\|_{L^2(W)\rightarrow L^2(W)},
\mathfrak{M}_{\mathbf{X}}(\mathbf{z}) := \mathbf{z}M_{\mathbf{X}}(\mathbf{z})=(z_1M_{\mathbf{X}}(\mathbf{z}),\ldots,z_r M_{\mathbf{X}}(\mathbf{z})), \mathfrak{H}_{\mathbf{X}}(\mathbf{z}) := \mathbf{z}H_{\mathbf{X}}(\mathbf{z})=(z_1H_{\mathbf{X}}(\mathbf{z}),\ldots,z_r H_{\mathbf{X}}(\mathbf{z})).
\begin{aligned}F^{ii}\nabla_{l}u \nabla_l U_{ii}= \, \nabla_{l}u \nabla'_{l}\psi + \psi_u |\nabla u|^2 + \psi_{p_k}\nabla_{l}u \nabla_{lk}u = \, \nabla_{l}u \nabla'_{l}\psi + \psi_u |\nabla u|^2 + \frac{a w^2}{\phi}\psi_{p_k}\nabla_k \phi. \end{aligned}
\mathrm{APEP}\approx \frac{1}{12}\prod_{n=1}^{N}\left[\mathcal{M}_{|\Delta_n|^2}\left(\frac{\mu}{8}\right)\right]+\frac{1}{4}\prod_{n=1}^{N}\left[\mathcal{M}_{|\Delta_n|^2}\left(\frac{\mu}{6}\right)\right].
P =\{\lambda\in\mathfrak{h}^*\mid \langle\lambda,\alpha^\vee\rangle\in\mathbb{Z},\ \forall \alpha\in\Pi\}P^+ =\{\lambda\in\mathfrak{h}^*\mid \langle \lambda,\alpha^\vee\rangle\in\mathbb{N},\ \forall \alpha\in\Pi\}P^\vee =\{\lambda\in\mathfrak{h}^*\mid \langle\lambda,\alpha\rangle\in\mathbb{Z},\ \forall \alpha\in\Pi\}
\times \left[ \begin{array}{r r r}{{p_{1 1}^{j}}}&{{p_{1 2}^{j}}}&{{p_{1 3}^{j}}}\\{{p_{2 1}^{j}}}&{{p_{2 2}^{j}}}&{{p_{2 3}^{j}}}\\{{p_{3 1}^{j}}}&{{p_{3 2}^{j}}}&{{p_{3 3}^{j}}}\\ \end{array}\right] , \verb + + \scriptscriptstyle{( j = 1 , 2 , 3 )},
\bar{b}^2(y) = \left(\int -\kappa(y - \theta(x)) \mu^y(dx) \right)^2 \leq{\kappa}^2 \int F(x, y) \mu^y(dx) \leq -\frac{\kappa \varepsilon}{\lambda_\theta}\int L^X F(x, y) \mu^y(dx) + \frac{\kappa}{\lambda_\theta}\int G(x) \mu^y(dx) =\frac{\kappa}{\lambda_\theta}\int G(x) \mu^y(dx)
p'\theta^{(\ell)}-p'\theta^* = (p'\theta^{(\ell)}-p'\Pi_{\mathcal C}\theta^{(\ell)})+(p'\Pi_{\mathcal C}\theta^{(\ell)}-p'\theta^*) &\le \|p\|\|\theta^{(\ell)}-\Pi_{\mathcal C}\theta^{(\ell)}\|+(p'\Pi_{\mathcal C}\theta^{(\ell)}-p'\theta^*)\le d(\theta^{(\ell)},\mathcal C),
\varphi_T(a) \ =\ \left<1\otimes 1, W^*W\, a \otimes 1\right>_\varphi \ =\ \left<W\, 1\otimes 1, W\, a\otimes1\right>_\psi \ = \ \left<1\otimes1, a\otimes1\right>_\psi \ = \ \psi(a),
\frac{1}{(q)_N}\sum_{n=1}^N\frac{(q)_{n-1}q^n}{(1-q^n)}=\sum_{n=1}^{\infty}\textup{spt}(n, N)q^n, \frac{1}{(q)_N}\sum_{n=1}^N\frac{q^n}{(1-q^n)^2}=\frac{1}{2}\sum_{n=1}^{\infty}M_{2,N}(n)q^n.
D_{pert}(\psi+s\mathfrak{c}_0-s\mathfrak{h}_0) = sT^s(\mathfrak{c}_0-\mathfrak{h}_0) &=D_{pert}(\psi+s\mathfrak{c}^g_0-s\mathfrak{h}^g_0)+D_{pert}(s\mathfrak{c}^b_0-s\mathfrak{h}^b_0) &=D_{pert}(\psi+s\mathfrak{c}^g_0-s\mathfrak{h}^g_0)+D|_{N_R}(s\mathfrak{c}^b_0-s\mathfrak{h}^b_0) \mbox{}mod(\ker(D|_{L^2_1})),
\left[\begin{array}{cc}\!\!I_{M}\!\!\!\!I_{M}\!\!\! \end{array}\right]P_{\operatorname{Box}[\boldsymbol{0}, \boldsymbol{1}]}\biggl(\boldsymbol{\rho}\left(i+\frac{1}{2}\right)-\begin{bmatrix}\boldsymbol{\lambda}^{\star}\boldsymbol{\lambda}^{\star}\end{bmatrix}\biggl)=\boldsymbol{1},
E_{N,M,1}:=\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}\right)^{1+s}\widetilde{\lambda}_{N,k}, E_{N,M,2}:=\left(\left(\frac{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}{N}\right)^{1+s}-1\right) \sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}\right)^{1+s}\widetilde{\lambda}_{N,k}.
\widehat{\deg}(f_*(\overline L^{\otimes n}))\geqslant\sum_{j=1}^n\sum_{i=1}^r\widehat{\deg}(\overline{I_{i,j}/I_{i-1,j}})+\widehat{\deg}(s)\sum_{k=0}^{n-1}h^0(L^{\otimes k}).
\begin{array}{ll}K(x,y) : = (1- \chi_{\mu_{5}^{2}}(x)) \mu_{2}^{n}\int e^{i ( (t-t_{0}) |\xi|^{4}+ (x-y) \cdot \xi )}\phi \left(\mu_{2}\xi \right) \, d \xi \, \tilde{\chi}_{\mu_{5}}(y)\end{array}
\mathcal{S}\Lambda_\iota+\mathcal{T}_\iota\Lambda_\iota^2=O \Big( N^{\varepsilon}\frac{|\mathcal{S}|+ \hat{\Lambda}}{(N\eta)^{\frac13}}+N^{-\frac{\gamma}{4}}\hat{\Lambda}^2\Big),\iota=A, B.
L_{C}=-\frac{g}{\sqrt{2}}[\overline{u}_{i}\gamma^{\mu}\frac{1-\gamma^{5}}{2}M_{ij}^{CKM}d_{j}+\overline{\nu}_{i}\gamma^{\mu}\frac{1-\gamma^{5}}{2}e_{i}]W_{\mu}^{+}+h.c.,
H_1 \geq (3)(64)\left(\frac{\sigma}{2}\right)^{-\frac{d}{p}}, H_2 \geq \left(2^{100}C_0\left(\frac{\sigma}{2}\right)^{-\frac{d}{p}(k+1)}\right)^{\max\{1,\frac{1}{p-1}\}}H_3 \geq 2^{100}C_0\left(\frac{\sigma}{2}\right)^{-\frac{d}{p}(k+1)}
a_{1}=20a^{2}-\left( a+b\right)^{2},a_{2}=a\left( b-a\right)^{2}\left( b-5a\right) , a_{3}=a\left( b-3a\right) \left( 3a+b\right) \left( b-a\right)^{2}\left( b-5a\right) .
\sum_{i=1}^{N}\int_{G_{N}}f(x)M_{\epsilon,i}^{2}(x)dx\le \sum_{i=1}^{N}\int_{G_{N}}f(x)M_{\epsilon,i}(x)dx+\epsilon N\int_{G_{N}}f(x)\sum_{l=1}^{N}M_{2\epsilon,l}(x)dx -\sum_{i=1}^{N}\int_{G_{N}}f(x)\nabla_{i}K_{\epsilon,i}(x)dx.
Z = \sum_{\{{\sigma}_{0}\}}e x p \left\{J_{3}\sum_{\triangle}{\sigma}_{0}^{( 1 )}{\sigma}_{0}^{( 2 )}{\sigma}_{0}^{( 3 )}+ h \sum_{j}\sigma_{0}^{( j )}\right\}{[ g_{n}({\sigma}_{0}^{( 1 )}) ]}^{\gamma - 1}{[ g_{n}({\sigma}_{0}^{( 2 )}) ]}^{\gamma - 1}{[ g_{n}({\sigma}_{0}^{( 3 )}) ]}^{\gamma - 1},
\sup_{\lambda/n\leq s\leq1-\lambda/n}\frac{n^{\nu}\left\vert \alpha_{n}\left(s\right) -B_{n}\left( s\right) \right\vert}{\left[ s\left( 1-s\right)\right]^{1/2-\nu}}=O_{\mathbb{P}}\left( 1\right) ,
\begin{pmatrix}T_i^* T_i T_j^* T_i T_i^* T_j T_j^* T_j\end{pmatrix}-\begin{pmatrix}T_i T_i^* T_i T_j^* T_j T_i^* T_j T_j^*\end{pmatrix}\succcurlyeq 0.
\sum_{\pi\in \Sigma_2}\ [\bar{x}^{\pm}_{i,r_{\pi(1)}}, [\bar{x}^{\pm}_{i,r_{\pi(2)}}, \bar{x}^{\pm}_{i\pm 1,s}]]=0 \ \mathrm{and}\ [\bar{x}^{\pm}_{i,r}, \bar{x}^{\pm}_{j,s}]=0\ \mathrm{for}\j\ne i, i\pm 1.
\bar{\Psi}_{\rm CT}= - \frac{i}{2 e^2}\int d^4x\,\sqrt{g}\, \Bigr\{\xi^\nu_- ( \nabla^\mu T_{\mu\nu}^+ - i V^\mu T_{\mu\nu}^+ - Y_\nu^+ ) -( \nabla^\mu T_{\mu\nu}^- - i T_{\mu\nu}^- V^\mu - Y_\nu^- ) \xi^\nu_+ \Bigr\}.
u=A_0+\sum\limits_{k\in \mathbb{N}_+}\left[\frac{\left(\sqrt{\Delta}+a\right)k\pi A_k-2lB_{k}}{2k\pi \sqrt{\Delta}}\cos h_{1k}+\frac{\left(\sqrt{\Delta}-a\right)k\pi A_k+2lB_{k}}{2k\pi \sqrt{\Delta}}\cos h_{2k}\right],
(\lambda_{k}(t)-(2p\pi i+it)^{n})\left( \Psi_{k,t},e^{(2p\pi i+i\overline{t})}\right) =\left( p_{2}\Psi_{k,t}^{(n-2)}+\cdots+p_{n}\Psi_{k,t},e^{(2p\pi i+i\overline{t})x}\right)
\max \sum_{c,u,b,z}\pi_{cubz}X_{cubz}{\rm s.t.\}Y_{cub}= \min\bigg(\sum_{z}X_{cubz},1\bigg), \forall \ (c,u,b) , & \sum_{c,b}Y_{cub}\leq 1,\forall \u\in\mathcal{U}, & \sum_{u}X_{cubz}=1, \forall \ (c,b,z)\in \mathcal{C}\times \mathcal{B}\times\mathcal{Z}, & X_{cubz},Y_{cub}\in \{0,1\}, \forall \ (c,u,b,z),
\sum_{n=1}^{\infty}\frac{e^{2\pi n}}{\left(e^{2\pi n}-1\right)^{2}}=\frac{1}{24}-\frac{1}{8\pi},\,\,\,\,\,\,\,\,\,\,\,\,\sum_{n=1}^{\infty}\frac{e^{\pi(2n-1)}}{\left(e^{\pi(2n-1)}+1\right)^{2}}=\frac{1}{8\pi}.
\lim_{\varepsilon \to 0}\Big( \int_{\{\vert f\vert \geq \varepsilon \}}\vert f\vert^{2\alpha}f^{-N}\xi + 2i\pi \sum_{m = m'+N}^{\alpha+r +m' < 0}\tilde{T}_{m,m'}^{r, j}(\xi) \frac{\varepsilon^{2(\alpha + r + m')}}{2(\alpha+r+m')}(-Log\varepsilon)^j\Big)
\int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i}(\xi(t)) dt \geq \int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i,\lambda}(u_{\lambda}(t)) dt + \int_0^T \left\langle -{\dot{u}}_{\lambda}(t) -{\eta}_{\lambda}(t), \xi(t) - u_{\lambda}(t) \right\rangle dt.
P(x_{m})=|\langle x_{m}|U|0\rangle^{n}|^{2}=|\sum_{x_{1},x_{2},x_{3},...,x_{m-1}\in\{0,1\}^{n}}\prod_{j=1}^{m}\langle x_{j}|U_{j}|x_{j-1}\rangle|^{2}
\Pi_{4}= \frac{g^{4}}{4}\int \frac{d^{3}p}{( 2 \pi )^{3}}\frac{N ( \omega )}{2 \omega}\left[ \frac{1}{\hbar^{2}k^{2}+ 2 \hbar k \cdot p}+ \frac{1}{\hbar^{2}k^{2}- 2 \hbar k \cdot p}\right] .
\partial_t c = \underbrace{D_c \triangle c}_{}- \operatorname{div}(\underbrace{\chi_u c \nabla u}_{}+ \underbrace{\chi_p c \nabla p}_{}+ \underbrace{\chi_v \nabla v}_{}) + \underbrace{\mu_1 c\left(1-\frac{c}{c_0}\right)}_{}+ \underbrace{\phi_{1,3}~cu}_{}.
I_H^*= \begin{cases}(x_3x_4, x_2^2, x_3^2, x_4^2-x_2x_3, x_2x_4), a\neq 1 b\neq 1, (x_3x_4, x_2^2-x_1 x_4, x_3^2, x_4^2-x_2x_3, x_3x_1-x_2x_4), a= 1, b\neq 1, (x_3x_4, x_2^2, x_3^2-x_1x_2, x_4^2-x_2x_3, x_2x_4), a\neq 1 b = 1.\end{cases}
U(z)=\frac{1-z^{2}}{1-4z^{2}+z^{4}}=\frac{1}{3-\sqrt{3}}\cdot\frac{1}{1-(2+\sqrt{3})z^{2}}+\frac{1}{3+\sqrt{3}}\cdot\frac{1}{1-(2-\sqrt{3})z^{2}}.
{\cal S}\equiv \sum_{\{n_{\bf k}\}}\left( \prod_{\bf k}N_{j_{\bf k}j_{\bf k}'}^{m_{\bf k}n_{\bf k}'}N_{i_{\bf k}i_{\bf k}'}^{n_{\bf k}n_{\bf k}'}i_{\bf k}! i_{\bf k}' ! j_{\bf k}! j_{\bf k}' ! \right) \langle \{n_{\bf k}\}\mid \, \rho \, | \{m_{\bf k}\}\rangle \, .
\Phi_{t}^{B}{u}={}\int_{0}^{t}T_{-1}(s)Bu(s)\, ds ={}-\int_{0}^{t}T_{-1}(\tfrac{s}{2})(-A_{-1})^{\frac{1}{2}}u( s)(-A)^{\frac{1}{2}}T(\tfrac{s}{2})x_{0}\,ds ={}-\frac{1}{2}\int_{0}^{\frac{t}{2}}T_{-1}(s)(-A_{-1})^{\frac{1}{2}}u(2s)f(s)\,ds ={}-\frac{1}{2}\Phi_{\frac{t}{2}}^{(-A_{-1})^{\frac{1}{2}}}\left(u(2\cdot)f\right)\in X.
\phi^{\prime}( x^{5}) \ = \c \, e^{- 4 [ A ( x^{5}) - A ( 0 ) ]}\qquad \phi ( x^{5}) \ = \c \int_{0}^{x^{5}}\! \! \! d y \, e^{- 4 [ A ( y ) - A ( 0 ) ]}\, + \, d \ ,
\int_{0}^{\infty}sn^{2}(t)e^{-tz}dt=\frac{2z^{-1}}{2^{2}(1+k^{2})+z^{2}-}\frac{2\cdot3^{2}\cdot4k^{2}}{4^{2}(1+k^{2})+z^{2}-}\frac{4\cdot5^{2}\cdot6k^{2}}{6^{2}(1+k^{2})+z^{2}-}\cdot\cdot\cdot
w_s(x,\omega,x^\ast,\omega^\ast) := w_s(\omega,\omega^\ast) := \max\left\{\frac{v_s(\omega^\ast)}{v_s(\omega)},\frac{v_s(\omega)}{v_s(\omega^\ast)}\right\}& = \max\left\{\left(\frac{1+\left|\omega^\ast\right|}{1+\left|\omega\right|}\right),\left(\frac{1+\left|\omega\right|}{1+\left|\omega^\ast\right|}\right)\right\}^{\left|s\right|}.
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda=\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\partial\Omega}(x\cdot \nu)d\sigma(x)\int_0^\infty t^{n-1}w(\lambda tx)w\left(\frac{tx}{\lambda}\right)dt.
D_{s}\phi(X_{t}^{x;l^\epsilon})=\nabla\phi(X_{t}^{x;l^\epsilon})D_{s}X_{t}^{x;l^\epsilon}=\nabla\phi(X_{t}^{x;l^\epsilon})\nabla_{x}X_{t}^{x;l^\epsilon}(\nabla_{x}X_{\gamma^{\epsilon}_{s}}^{x;l^\epsilon})^{-1}.
{\cal L}= \mu \int_{S^{1}\times R^{3 + 1}}\, t r \left\{- \frac{1}{2}{\cal F}_{M N}{\cal F}^{M N}+ \sum_{P}{\cal D}_{M}\Phi_{P}{\cal D}^{M}\Phi_{P}+ \frac{1}{2}\sum_{P , Q}[ \Phi_{P}, \Phi_{Q}]^{2}\right\}
h^W\left({\bf R}^{(pq)}_{{\bf i}}\right) \;:=\;{2k \choose k}^{-1}\sum_{\substack{{\bf j}\subset{\bf i}{\bf j}\in \mathcal{P}(n, k)}}h\left({\bf R}^{(pq)}_{{\bf j}}\right) h\left({\bf R}^{(pq)}_{{\bf i}\setminus{\bf j}}\right)
( \delta_{V}\delta_{V}\mathcal{L}_{0}) ( \gamma ) =\left.\big\langle\nabla_{V(t)}^{g(t)}V(t),\dot{\gamma}(t)\big\rangle_{g(t)}\right\vert_{t=t^{\prime}}^{t=t^{\prime\prime}}+ \mathcal{L}_0 I_\gamma ( V , V ).
s^c_{(1;4)}=m_{(1;4)}+m_{(1;3,1)}+m_{(1;2,2)}+m_{(1;2,1,1)}+m_{(1;1,1,1,1)}\phantom{s^c_{(1;4)}=}{}+m_{(2;3)}+m_{(2;2,1)}+m_{(2;1,1,1)}+m_{(3;2)}+m_{(3;1,1)}.
Z_{1}( 0 ) = - 2 \zeta ( - 1 ) \int_{0}^{\infty}d y \ln \frac{y}{\sqrt{1 + y^{2}}}= - 2 \left( - \frac{1}{1 2}\right) \left( - \frac{\pi}{2}\right) = - \frac{\pi}{1 2}.
\tilde{\eta}_{1}^{\exp}:=\tilde{\eta}_{1}^{\exp}\left( \mathbf{X}_{h}\right):=\sup_{\mathbf{w}\in\mathbf{X}\backslash\left\{0\right\}}\inf_{\mathbf{z}_{h}\in\mathbf{X}_{h}}\frac{\left\Vert \mathcal{N}_{1}^{\mathcal{A}}\mathbf{w}-\mathbf{z}_{h}\right\Vert_{\operatorname*{curl},\Omega,k}}{\left\Vert \mathbf{w}\right\Vert_{\operatorname*{curl},\Omega,k}}.
\mathcal{P}_i= \left\lbrace \mathbf{P}_i=(P_i^{(1)},\ldots,P_i^{(M)}): P_i^{(k)}\in \{p_i^{(1)},\ldots,p_i^{(M)}\}, \sum_{j=1}^M \alpha_i(j)P_i^{(j)}\leq \overline{P}_i \right\rbrace.
\left| \partial_{y_d}^2 r'_\lambda(y',y_d,z_d)\right|+\left| \partial_{y_d}^2 r_{d,\lambda}(y',y_d,z_d)\right| \leq \frac{C e^{-c|\lambda|^{\frac 12}z_d}}{(y_d + z_d +|y'|)^{d}\big (1+|\lambda|^\frac12 (y_d +z_d)\big )}.
j(u)(z) := \mathbb{E}[ \int_0^T (\int_{D}h(t, Y(t,x,z),u(t,x,z),z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_t]dx)dt +\int_D k(x,Y(T,x,z),z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]dx.
\frac{\partial}{\partial_t}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}=-\operatorname{ad}^*_{\mathcal{U}}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}+\operatorname{div}^\Gamma\frac{\delta l^\mathcal{L}}{\delta\Gamma}.
\mathrm{I}(\xi_{1}^{x},\mathcal{T}_{x})=\int_{(L_{x}\backslash G)^{\mathbb{N}}}\log\frac{d\overline{\mathbb{P}}_{\mu,x,\mathrm{\xi_{1}^{x}}}|_{\mathcal{T}_{x}}}{d\overline{\mathbb{P}}_{\mu,x,e}|_{\mathcal{T}_{x}}}d\overline{\mathbb{P}}_{\mu,x,e},
g^{( 2 )}( y ) = \frac{\hbar c \xi^{2}}{4 \pi}\sum_{n = - \infty}^{\infty}\, y^{2}\partial_{y}\left[ y{{\partial}_{y}}( I_{n}( a y ) K_{n}( a y ) ) \right]^{2}= \frac{2 \hbar c \xi^{2}}{\pi^{2}}\int_{0}^{1}d u \, \frac{u^{2}}{\sqrt{1 - u^{2}}}y^{2}{{\partial}_{y}}( a y K_{1}( 2 a y u ) )^{2},
O_P\left(\sqrt{n}2^{\sum_{l=1}^d(j_l/2+J_{n,l})}(\log{n}/n)^{\alpha^{*}/(2\alpha^{*}+d)}+n\prod_{l=1}^d2^{-j_l/2}\sum_{l=1}^d2^{-\alpha_lJ_{n,l}}\right) &=O_P\left(\sqrt{n}n^{1/4}2^{\sum_{l=1}^dJ_{n,l}/2}\epsilon_n^{1/2}\sum_{l=1}^d2^{-\alpha_lJ_{n,l}/2}\right),
\widetilde{Y}^m_t =\sup_n\inf_k Y_{\tau_m}^{n, k}+ \int_{t\wedge \tau_m}^{\tau_m}\big(F(s,\widetilde{Y}^m_s, \widetilde{Z}^m_s ) dA_s + g_s \langle \widetilde{N}^m \rangle_s \big) -\int_{t\wedge \tau_m}^{\tau_m}\big(\widetilde{Z}_s^m dM_s + d\widetilde{N}_s \big).
\left(N_{\varXi_{\iota}}\left(r,\,\nu,\,\omega\right)\cdot\mathbf{e}\right)^{-1}=\frac{\sqrt{1+\left(\partial_{r}\hat{h}\left(r,\,\iota\right)\right)^{2}}}{\left(\vec{0},\,\omega\right)\cdot\mathbf{e}}
V_{1 ( - 1 / 2 )}^{f}= e^{- c / 2}S_{\alpha}e^{- i / 2 \H_{2}}\Sigma_{3}^{+}\Sigma_{4}^{+}e^{i / 2 \K_{1}X}\bar{G}_{1}e^{i / 2 \K_{1}\bar{X}}
\prod_{n=m^3}^{N}\left|\frac{\sin(\frac{z}{n^2})}{\frac{z}{n^2}}\right| \leq \prod_{n=m^3}^{N}\exp\left({\left|\frac{z}{n^2}\right|}\right)= \exp({|z|\sum_{n=m^3}^{N}\frac{1}{n^2}})\leq \exp(C|z|),
\partial_t F_{i\alpha}-\partial_{\alpha}v_i =0 \partial_t v_i-\partial_{\alpha}\Sigma_{i\alpha}=\partial_{\alpha}Z_{i\alpha}\partial_t\left(\frac{1}{2}|v|^2+e\right)-\partial_{\alpha}(\Sigma_{i\alpha}v_i) =\partial_{\alpha}(Z_{i\alpha}v_i)+\partial_{\alpha}Q_{\alpha}+r \, ,
\nabla [\L^{-s}, a] f(x) =c_s\int_0^\infty t^{-1+\frac s2}\int_\Omega \nabla_xH(x, y, t)[a(y)-a(x)]f(y)dt &-c_s\int_0^\infty t^{-1+\frac s2}\int_\Omega H(x, y, t)\nabla a(x)f(y)dt &=:I +II.
|v_{4}^{\lambda_{1}}-v_{4}^{\lambda_{2}}|(t,r) \leq \begin{cases}\frac{C ||e_{1}-e_{2}||_{X}}{t^{2}\sqrt{\log(\log(t))}\log^{N+3b-1}(t)}, r \leq \frac{t}{2}\frac{C ||e_{1}-e_{2}||_{X}}{t \log^{3b+2N}(t) \sqrt{\log(\log(t))}}, r \geq \frac{t}{2}\end{cases}