harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[\mathbf{A}\mathbf{b}\right]_{ij}^{\,\,\,k}=A_{ij}b^{k}\,\,\,\,\,\,\,\,\,\,\underrightarrow{\mathrm{contraction\,\,on}\,\,jk\,\,}\,\,\,\,\,\,\,\,\,\,\left[\mathbf{A}\cdot\mathbf{b}\right]_{i}=A_{ij}b^{j}
\sum_{i=0}^{n-1}\psi_\ell(x_i,k+i)=\sum_{i=\ell-k}^{\ell-k+N-1}\psi_\ell(x_i,k+i)=\sum_{j=0}^{N-1}\psi_\ell(x_{\ell-k+j},\ell+j).
(\{\lambda_{j}\},\{\Gamma_{jk}\})=\sum_{u=1}^m h_{u}(\{\Lambda_{uj}'\},\{\theta_{ujk}\})-\delta_{HP}(\{g_{j}\}),\,\,\,\,\,\,\,\, h_u\in \mathbb{C}, \{g_j\}\in C^0(\mathcal{U},\Theta_0)
\overline{f}=\begin{pmatrix}f_{\star}0 0 (f^*)^{-1}\end{pmatrix}: X+\alpha\longmapsto f_{*}X+(f^*)^{-1}(\alpha), e^B =\begin{pmatrix}Id 0 B Id \end{pmatrix}: X+\alpha\longmapsto X+\alpha+i_XB .
\omega_{ne_i}^{g,0}(z_0, I)=\sum_{\pm 1}\underset{z\pm 1}{Res}{}^iK(z,z_0)\Bigl[ \omega_{(n+1)e_i}^{g-1,0}(z,\iota(z), I)+\sum_{\substack{J\subseteq I 0\le h \le g}}^{'}\omega_{(|J|+1)e_i}^{g,0}(z,J)\otimes\omega_{(|I-J|+1)e_i}^{g,0}(\iota(z),I-J)\Bigr].
R_0(z) (x,y) = \sum_{k=0}^{\frac{d-3}{2}}\frac{\tilde{c_k}}{|x-y|^{d-1-s_k}}\partial_s^{s_k}\Big\{e^{izs}E_i( - izs)+e^{-izs}E_i ( -izs) + 2 \pi i e^{iz s}\Big\}\Big|_{(s=|x-y|)}
\frac{1}{\sqrt{\pi}}\int_{x_0}^{0}\frac{B'(0,u)}{\sqrt{-\ln (u/x_0)}}du=\frac{1}{\pi \sqrt{\pi}}\int_{x_0}^0 \int_0^{\pi}\frac{e^{2 \sqrt{u}\cos \theta}\cos \theta}{\sqrt{u}\sqrt{-\ln(u/x_0)}}d\theta du
N_{\ell,0}(\Gamma)^\circ =\bigoplus_{\substack{\ell'=2 \ell'\equiv\ell\bmod2}}^\ell\; \bigoplus_{\substack{m'=0 m'\equiv0\bmod2}}^{\ell-\ell'}\;D_+^{(\ell-\ell'-m')/2}\,{U}^{m'/2}\,S_{\ell',m'}(\Gamma)\;\oplus\;N_{\ell,0}(\Gamma)^\circ_1,
\int_{M_{\delta}}d p \; p^{\alpha}\ \leq \ \frac{1}{\delta^{2}}\int_{0}^{\infty}d p \; p^{\alpha}\left| \chi \left( p \right) - \phi_{0}\right|^{2}\ < \ \frac{q_{0}^{2}\varepsilon \left( q_{0}\right)}{\delta^{2}}\ .
\varphi^1_{z\bar z}+\frac{1}{2}(\varphi^2_{z}\varphi^3_{\bar z}+\varphi^2_{\bar z}\varphi^3_z)=0,\;\varphi^2_{z\bar z}-\frac{1}{2}(\varphi^1_{z}\varphi^3_{\bar z}+\varphi^1_{\bar z}\varphi^3_z)=0,\; \varphi^3_{z\bar z}=0.
S_{BF}[A,\phi] = \int d^{2}x\, tr\left( \phi^{0}F^{0}+ \phi^{a}F^{b}\right) = \int d^{2}x\, \left( - \phi^{0}F^{0}_{\alpha\beta}\epsilon^{\alpha\beta}+ \Lambda \delta_{ab}\phi^{a}F^{b}_{\alpha\beta}\epsilon^{\alpha\beta}\right).
L_1^{\mu}(\beta b(X),\xi) = - b(X)\mathrm{sech}(\sqrt{\mu}|\xi|) - \frac{1}{6\beta}\Big(\beta^3 b(X)^3 \int_0^1 \cosh(t\beta b(X) \sqrt{\mu}|\xi|) (1-t)^2 \: \mathrm{d}t\Big) (\sqrt{\mu}|\xi|)^2\mathrm{sech}(\sqrt{\mu}|\xi|).
s^{N-1}(\zeta_{[a,b]}''(s)\zeta_{[a,b]}(s)-\zeta_{[a,b]}'(s)^2) = \zeta_{[a,b]}''(b)\zeta_{[a,b]}(b) - (N-1) \int_s^b r^{N-3}\zeta_{[a,b]}'\zeta_{[a,b]}\,dr.
y_1=\frac{1}{\sqrt{\mu}}z, \, y_2=x, \, y_3=\frac{1}{\sqrt{1-\tau}}(x-y); &[y_1,y_2]=\frac{1}{\sqrt{\mu}}y_2- \frac{\sqrt{1-\tau}}{\sqrt{\mu}}y_3, \,[y_2,y_3]=0, \,[y_3,y_1]=\frac{1-c}{\sqrt{(1-\tau)\mu}}y_2-\frac{1}{\sqrt{\mu}}y_3.
_{a}^{C}\mathcal{D}_{x}^{\beta}f(x) &:=\frac{1}{\Gamma^{\alpha}_{F}(n-\beta)}\int_{S_{F}^{\alpha}(a)}^{S_{F}^{\alpha}(x)}(S_{F}^{\alpha}(x)-S_{F}^{\alpha}(t))^{n\alpha-\beta-\alpha}(D_{F}^{\alpha})^nf(t) d_{F}^{\alpha}t.
g = \frac{1}{2}d\eta (\phi(\cdot), \cdot)+\eta \otimes \eta, J|_{\{r=1 \}}= \phi - \xi \otimes dr + \frac{\partial}{\partial r}\otimes \eta, \bar{\omega}= \frac{1}{2}d(r^{2}\tilde{\eta})=-\frac{1}{2}dd^{c}_{J}r^{2}.
\mathcal{A}_h\bigl( (\hat{w}_h, \hat{r}_h), (v_h, q_h) \bigr) + a_\theta (\hat{\phi}_h, \psi_h)= \mathcal{A}\bigl( (w, r), (v_h, q_h) \bigr) + a_\theta (\phi, \psi_h), \forall (v_h, q_h,\psi_h) \in V_h\times Q_h \times \Psi_h.
\overline{\theta}_{\left( M,c\right)}=\left[ -\right]_{P\left( \mathbb{A}\right)}\circ \left( H\mathbb{I}_{\mathrm{Br}}^{s}\overline{\eta}_{\mathrm{Br}}^{s}M_{0}\otimes H\mathbb{I}_{\mathrm{Br}}^{s}\overline{\eta}_{\mathrm{Br}}^{s}M_{0}\right) .
\Omega^{\alpha}_{\beta}= R^{\alpha}_{\beta;\mu\bar{\nu}}dz^{\mu}\wedge dz^{\bar{\nu}}+ S^{\alpha}_{\beta\mu;\bar{\nu}}\delta v^{\mu}\wedge dz^{\bar{\nu}}+ P^{\alpha}_{\beta\bar{\nu};\mu}dz^{\mu}\wedge \delta v^{\bar{\nu}}+ Q^{\alpha}_{\beta\mu\bar{\nu}}\delta v^{\mu}\wedge \delta v^{\bar{\nu}}.
|\Psi^{\lambda_{u},\lambda_{w}}|^2(\lambda_{v}) = \Psi^{\lambda_{u^{-1}},\lambda_{w^{-1}}}\Psi^{\lambda_{u},\lambda_{w}}(\lambda_{v}) = \gamma_{u^{-1},w^{-1}}^{\psi}(uvw)\gamma_{u,w}^{\psi}(v)\lambda_{v}= |\gamma_{u,w}^{\psi}(v)|^2\lambda_{v}.
((H^{1-s_0}_0(\Omega))^\prime,(H^1_0(\Omega))^\prime)_{\theta,\infty}= ((H^{1-s_0}_0(\Omega),H^1_0(\Omega))_{\theta,1})^\prime &\supset ((H^{1-s_0}(\Omega),H^1(\Omega))_{\theta,1})^\prime = (B^{1-s_0(1-\theta)}_{2,1}(\Omega))^\prime.
\langle \bar{\chi}_{\sigma_{1}u_{1}}( x_{1}) \chi_{\sigma_{2}u_{2}}( x_{2}) \rangle = \int \!{\cal D}z{\cal D}z^{*}\exp ( - S_{C P_{1}}) \bar{\chi}_{\sigma_{1}u_{1}}( x_{1}) \chi_{\sigma_{2}u_{2}}( x_{2}) .
\mathcal{H}^{\ast}= \int d^{2}x \left[ \frac{1}{2}\left( \xi_{5}^{\ast 2}+ \xi_{6}^{\ast 2}\right) + \frac{1}{2}\left( \partial_{1}\xi_{3}^{\ast}- \partial_{2}\xi_{2}^{\ast}\right)^{2}+ \frac{1}{2}m^{2}\left( \xi_{2}^{\ast 2}+ \xi_{3}^{\ast 2}\right) + m \left( \xi_{2}^{\ast}\xi_{6}^{\ast}- \xi_{3}^{\ast}\xi_{5}^{\ast}\right) \right] .
\left[ \frac{d^{2}}{d \varrho^{2}}+ \widetilde{E}- \mathrm{s g n}\left( \widetilde{\beta}\right) \, \widetilde{\lambda}\varrho^{\widetilde{\beta}}- \frac{\left( \widetilde{l}+ \widetilde{\nu}\right)^{2}- 1 / 4}{\varrho^{2}}\right] \widetilde{u}_{l}( \varrho ) = 0 \; ,
\mathfrak{J}(\{\pmb{F}_k^{(\kappa+1)}\}_{k=1}^K, \{\pmb{U}_k^{(\kappa+1)}\}_{k=1}^{K+1}) \leq \mathfrak{J}(\{\pmb{F}_k^{(\kappa)}\}_{k=1}^K, \{\pmb{U}_k^{(\kappa)}\}_{k=1}^{K+1})
\frac{\left(w_i'(I_d-P_{N_{(i-1)}})(I_d-P_{B,N_{(i-1)}})w_i\right)^2}{\|(I_d-P_{N_{(i-1)}})w_i\|^2\|(I_d-P_{B,N_{(i-1)}})w_i\|^2}\quad=\frac{w_i'(I_d-P_{B,N_{(i-1)}})w_i}{w_i'(I_d-P_{N_{(i-1)}})w_i}.
{\displaystyle\sum\limits_{n\geq0}}B_{n,p}\frac{t^{n}}{n!}=\frac{\left( p+1\right) e^{pt}}{\left(e^{t}-1\right)^{p+1}}\left( t+{\displaystyle\sum\limits_{s=1}^{p}}\left( -1\right)^{s}\frac{1}{s}\dbinom{p}{s}\right) +\left(p+1\right){\displaystyle\sum\limits_{k=0}^{p}}\dbinom{p}{k}\frac{H_{k}}{\left( e^{t}-1\right)^{k+1}}.
\theta_{\mu \nu}-\theta_{\nu \mu}=-\frac{{\rm i}}4\partial_\alpha (\psi^{+}\{\gamma_\alpha ,\sigma_{\mu \nu}\}\psi )=\frac 12\varepsilon_{\mu\nu \alpha \lambda}\partial_\alpha (\psi^{+}\gamma_\lambda \gamma_5\psi).
d_{2}(\mathbf{x},\mathbf{s})=\sqrt{2}\|\mathbf{w}_{xs}-\mu\mathbf{e}\|=\min_{\mathbf{D}\in\mathcal{G}}\|(\mathbf{D}^{-T}\mathbf{x})\circ(\mathbf{D}\mathbf{s})-\mu\mathbf{e}\|
J_\alpha^{\mu{\mathbf i}}= 192 \left[-3 i \sigma^\mu_{\alpha\dot\alpha}{\bar D}^{\dot\alpha}_{\mathbf j}D^{{\mathbf i}{\mathbf j}}T- 3 i{\bar\sigma}^{\mu\dot\alpha\beta}{\bar D}^{\mathbf i}_{\dot\alpha}D_{\alpha\beta}T- 12 \partial^\mu D^{\mathbf i}_\alpha T+ a \sigma_\alpha^{\mu\nu\,\beta}\partial_\nu D_\beta^{\mathbf i}T \right]| \ .
1_{F(n)}= \sum_{|\underline{i}|\leq d}a_{\underline{i}}F_{n,n}(\phi_{\underline{i}}^d) &= \sum_{|\underline{i}|\leq d}a_{\underline{i}}F_{m,n}(\phi_{\underline{i}}^d) \circ F_{n,m}(\phi_{\underline{i}}^d) ,
s_3 \ge s_3|_{a=1}=-2M^3\mu w^2+M^3\mu w+4{M}^2\mu{w}^2-3{M}^3\mu+{M}^3w-{M}^2\mu w+{M}^2{w}^2 -2M\mu w^2+3M^3-M^2\mu-5 M^2 w-2 M w^2+2 M^2+2 M\mu+4 Mw+w^2-2M-1=:s_4
\partial_1^2(\tilde{q}+ w) - a \diamond \partial_1^2 \tilde{q}- a \diamond \partial_1^2 w + (\tilde{q}+ w) = (\partial_1^2\tilde{q}- a \diamond \partial_1^2 \tilde{q}+ \tilde{q}) - (\partial_1^2 q - a \partial_1^2 q + q)^E \textrm{in}\mathbb{R}^2.
u(t) = e^{t(d \Delta -aI)}u_0+ \int_0^t e^{(t-s)(d \Delta -aI)}\big(u^pf(v)\big)(s)\,ds, &v(t) = e^{t(D\Delta -bI)}v_0- \int_0^t e^{(t-s)(D\Delta -bI)}\big(u^pf(v)\big)(s)\,ds+ \int_0^t e^{(t-s)(D\Delta -bI)}\kappa\,ds,
\Omega=\left[\begin{array}[c]{cc}0 -\mathrm{i}\mathrm{i}0\end{array}\right] ,B=\left[\begin{array}[c]{cc}1 0 0 0\end{array}\right] , A\left( \beta\right) =\Omega-\mathrm{i}\beta B,\beta\geq0,
\tilde{\Omega}^t({\bf e}_j):=\sum_{i=1}^n\tilde{\Omega}_{ji}{\bf e}_i, T^t({\bf e}_j):=\sum_{i=1}^nT_{ji}{\bf e}_i, B_{\infty}({\bf e}_j):=\sum_{i=1}^n(B_{\infty})_{ji}{\bf e}_i
( \Gamma_{\phi}^{\alpha \eta})_{a b c}= 2 e^{3}( \gamma^{\lambda}\gamma^{\phi}\gamma^{\tau})_{\alpha \eta}\int \frac{d^{4}S}{( 2 \pi )^{4}}( Q + S )_{\tau}( P + S )_{\lambda}[ \tau_{3}D ( S ) ]_{a b}[ \tau_{3}D ( P + S ) ]_{b c}[ \tau_{3}D ( Q + S ) ]_{c a}.
\|e^{s\psi}Pu\|^2 =\|P_pv\|^2+\|P_nv\|^2+2\int_{\mathbb{T}}h_1(\psi)v^2dx+2\int_{\mathbb{T}}h_2(\psi)v_{x}^{2}dx+ 2\int_{\mathbb{T}}h_3(\psi)v_{xx}^{2}dx+2\int_{\mathbb{T}}h_4(\psi)v_{xxx}^{2}dx,
a =\left(1, x_1, 0, \tfrac{351}{8}- x_1, \tfrac{351}{8}, x_1-\tfrac{117}{8}, \tfrac{351}{4}, \tfrac{351}{4}, \tfrac{351}{4}, \tfrac{117}{8}- x_1\right), aQ =\left(352, 0, 3744, 0, 0, 32 (4x_1-\tfrac{117}{2}),0,0,0,32(\tfrac{117}{2}-4x_1)\right).
\begin{aligned}P \succeq J \left( R \right)^{-1}- \frac{1}{12}\left( Tr \left( P \right) I_3 - P \right) J \left( R \right)^{-1}&-\frac{1}{12}J \left( R \right)^{-1}\left( Tr \left( P \right) I_3 - P \right) \end{aligned}
G(u,v) \leq\sum_{n\geq1}\gamma_{n}(\ln\gamma_{n}-1)=\sum_{k=1}^{\overline{n}}\bar{\gamma}_{k}^{u}(\ln\bar{\gamma}_{k}^{u}-1)+\gamma_{m}^{\prime}(\ln\gamma_{m}^{\prime}-1)+\gamma_{m+1}^{\prime}(\ln\gamma_{m+1}^{\prime}-1) & \leq f^{\ast}(u)+\varepsilon/2+\varepsilon/2=f^{\ast}(u)+\varepsilon=g^{\ast}(u,v)+\varepsilon.
\Phi_1^{-1}=\exp(-\varepsilon A_1)=\mathrm{I}_{H_S^{\perp}}+\varepsilon \check{A}_1, \,\, \check{A}_1:=\sum_{n\geq 1}\frac{\varepsilon^{n-1}}{n!}(-A_1)^n,\,\,\lvert \check{A}_1 \partial_x \rvert_s^{Lip(\gamma)}+\lvert \partial_x \check{A}_1 \rvert_s^{Lip(\gamma)}\le C(s).
H_{2,\boldsymbol{\alpha}}(\tau)=-2i\int_{\mathbb R^2}w_2\left(\mathcal G_{\alpha_1}(w_1)\mathcal F_{\alpha_2}(w_2)+\mathcal F_{\alpha_1}(w_1)\mathcal G_{\alpha_2}(w_2)\right) e^{2\pi i\tau Q(\boldsymbol{w})}\boldsymbol{dw}.
\frac{1}{u(z_{0})^{2}+v(z_{0})^{2}}\cdot(y_{0}u(z_{0})u_{x}(z_{0})+y_{0}v(z_{0})v_{x}(z_{0})+x_{0}u(z_{0})v_{x}(z_{0})-x_{0}u_{x}(z_{0})v(z_{0}))
0\leq \sum_{i=1}^{N-1}(x_{i+1}-x_i)^2= \sum_{i=1}^{N-1}x_{i+1}^2+\sum_{i=1}^{N-1}x_i^2-2 \sum_{i=1}^{N-1}x_{i+1}x_i \leq 2\left[\sum_{i=1}^{N}x_i^2- \sum_{i=1}^{N-1}x_{i+1}x_i\right].
\phi(Q(j),Q(i)) = l_j-2l_{i+1}-\Delta(j,i+1) -\Delta(i+1-|[1,i)\cap J|+|[1,j)\cap J|,i+1) = l_j-2l_i-\Delta(j,i) -\Delta(i+1-|[1,i)\cap J|+|[1,j)\cap J|,i)\,\cdot
(a\circ_i b)\circ_{i+j-1}c = a\circ_i(b\circ_j c),i\in\{1,\ldots,k\}, j\in \{1,\ldots,l\}, (a\circ_i b)\circ_{l+j-1}c = (a\circ_j c)\circ_i b, i,j\in\{1,\ldots,k\},i<j.
\frac{\partial_{xx}w(t,x)}{1 + |\partial_x w(t,x)|^2}- \frac{1}{w(t,x)}\begin{cases}\leq \frac{R T^{1/4}}{1+0}- \frac{1}{w_0(x)+R T^{1/4}}\leq -\frac{1}{w_0(x)}+ \tilde{\varepsilon}, \geq \frac{-R T^{1/4}}{1+R^2 T^{1/2}}- \frac{1}{w_0(x) - R T^{1/4}}\geq - \frac{1}{w_0(x)}- \tilde{\varepsilon}\end{cases}
\langle T_{\eta}^{\eta}\rangle^{C}= \frac 1{2 C}\phi_{c}^{^{\prime}}\phi_{c}^{^{\prime}}+ 2 \xi \frac{C^{^{\prime}}}{C^{2}}\phi_{c}\phi_{c}^{^{\prime}}+ \frac{3 \xi}C ( \frac{C^{^{\prime}}}C + k ) \phi_{c}^{2}+ \frac{m^{2}}2 \phi_{c}^{2}- 2 \kappa \phi_{c}^{4}+ \frac 3 2 \lambda^{2}\phi_{c}^{4}
\frac{\nabla V}{dt}= \frac{\nabla}{\partial t}\biggl(\frac{\nabla}{\partial s_1}[(\nabla^{k-1}\Phi^t_*)(\gimel_{s_2},\ldots,\gimel_{s_{k+1}})]\biggr) - \frac{\nabla}{dt}\biggl(\sum_{\ell= 2}^{k+1}(\nabla^{k-1}\Phi^t_*)(v_2,\ldots, \frac{\nabla\gimel_{s_\ell}}{\partial s_1}, \ldots, v_{k+1})\biggr).
H_{n,n+1}= (\cos\gamma)\,{\cal P}_1 + \frac{\sin 3\gamma}{\sin 2\gamma}\,{\cal P}_2 + \Bigl( \frac{\sin 3\gamma}{\sin 2\gamma}+ (\sin\gamma) \frac{\cos 3\gamma}{\sin 3\gamma}\Bigr) \,{\cal P}_3 \,.
\left\langle - p_{0 N}\right| h_{\mu \nu}( p_{0}) \alpha_{1}^{\mu}\tilde{\alpha}_{1}^{\nu}= \left. \frac{4}{\alpha^{\prime}}h_{\mu \nu}( p_{0}) \frac{\partial^{2}}{\partial \bar{\psi}_{1 \mu}\partial \psi_{1 \nu}}\left\langle \Omega \left( \psi_{n}, \bar{\psi}_{n}; p_{0}\right) \right| \right|_{\psi , \bar{\psi}= 0},
\underline{v}'=\overline{v}'+rq(1)e^{\lambda z}\left[\frac{1}{2}(-z)^{-1/2}-\lambda(-z)^{1/2}\right], begin{align*}0.2cm] \underline{v}''=\overline{v}''+rq(1)e^{\lambda z}\left[\lambda(-z)^{-1/2}+\frac{1}{4}(-z)^{-3/2}-\lambda^2(-z)^{1/2}\right].
\mathbf{III}\leq \frac{C_6\Lambda}{t}\int^{r_0}_0 \frac{\mathrm{Vol}_{g_0}\left( B_{g_0}(x_0,r)\right)}{\mathrm{Vol}_{g_0}\left( B_{g_0}(x_0,\sqrt{t})\right)}\exp\left(-\frac{r^2}{C_1t}\right)r \,dr &\leq \frac{C_7 r_0^{n+2}\Lambda t^{-1-\frac12}}{{\mathrm{Vol}_{g_0}\left( B_{g_0}(x_0,1)\right)}}.
\int_{\Delta_{r - 2}}e^{u_{1}x_{1}+ \cdots + u_{r - 2}x_{r - 2}}\left( \int_{0}^{\alpha}e^{u_{r - 1}x_{r - 1}+ u_{r}( \alpha - x_{r})}d x_{r - 1}\right) d x_{1}\cdots d x_{r - 2},
\frac{n}{B_n}\left( (1+\tilde b^{(n)})^2 - (1+ b^{(n)})^2 \right)= \frac{n}{B_n}\langle \Xi^{(n)},w\rangle + o_{\mathbb P}(1),\; w= \left(0,\frac{1}{\sqrt{2}},0,-\frac{1}{2},-\frac{1}{2}\right).
\mathrm{I m ~}F_{+ + + +}= - 1 6 \pi \alpha^{2}\left[ \frac{s}{u}\ln \Big | \frac{u}{m_{W}^{2}}\Big | + \frac{s}{t}\ln \Big | \frac{t}{m_{W}^{2}}\Big | \right] \, \, \, ,
u(0,x) =T^{-\frac{2}{p-1}}\psi_1( 0 , \tfrac{|x|}{T}) = u_{T_0}(0,x) + f(|x|) = u_0(x) \partial_t u(0,x) =T^{-\frac{p+1}{p-1}}\psi_1( 0 , \tfrac{|x|}{T}) = \partial_t u_{T_0}(0,x)+ g(|x|) = u_1(x)
q^{2\nu}\widehat{h}(Q) = \widehat{h}((\alpha',\beta')) \leq h(\alpha') + c = h(\alpha^{2-2p}- \zeta^2\alpha^{-2p}) + c \leq h(\alpha^{2p-2})+h(\alpha^{2p})+c+\log{2}=(4p-2)h(\alpha)+c+\log{2}\leq (4p-2)(\widehat{h}(Q)+c)+c+\log{2}\leq 4p\widehat{h}(Q)+4cp+\log{2}
\left({x^- \over x^+}\right)^2 F(x^-,x^+) ={N^2 \over k}\sum_n \left({3 (k-2 n)^2 n_f \over 4 \pi^2 k^2 r^4}+{3 n (k-n) n_b \over \pi^2 k^2 r^4}\right) ={N^2(2 n_b + n_f) \over 4\pi^2 r^4}\left(1 -{1 \over k}\right)
p(1) =p(0)=1, p(4) =p(3)+p(2)=3+2=5, p(2) =p(1)+p(0)=2, p(5) =p(4)+p(3)-p(0)=5+3-1=7, p(3) =p(2)+p(1)=3, p(6) =p(5)+p(4)-p(1)=7+5-1=11
U_C=(v_C,y_C)=\left(\frac{(2\alpha \alpha_1-\alpha^2\sqrt{y_L^2-4v_L}+\alpha^2 y_L)(\sqrt{y_L^2-4v_L}-y_L)}{4\alpha^2-2\alpha_2(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_2 v_C}{\alpha^2}-\frac{\alpha_1}{\alpha}\right);
\fbox{\chi ( f ) = \underbrace{< \chi^{( 1 )}, f_{( 1 )}> \chi^{( 2 )^{\prime}}f_{( 2 )}}_{\mathrm{f r o m r i g h t t r a n s l a t i o n}}= \underbrace{\chi^{( 1 )^{\prime}}f_{( 1 )}< \chi^{( 2 )}, f_{( 2 )}>}_{\mathrm{f r o m l e f t t r a n s l a t i o n}}}.
\frac{\partial Y}{\partial x}=\left(\frac{A_2}{x^2}+\frac{A_1}{x}+A_0\right)Y ,\frac{\partial Y}{\partial t_1}=(E_3x+B_1)Y ,\frac{\partial Y}{\partial t_2}=\left( \frac{1}{t_2}Nx+B_2 \right)Y.
\begin{array}{c}{( \dot{x}, \dot{y}) = i ( a \alpha \zeta^{\alpha - 1}, b \beta \zeta^{\beta - 1}) = ( A \zeta^{\alpha - 1}, B \zeta^{\beta - 1})}\\{( i \dot{y}^{*}, - i \dot{x}^{*}) = ( i B^{*}( \zeta^{*})^{\beta - 1}, - i A^{*}( \zeta^{*})^{\alpha - 1})}\\ \end{array}
\{\varepsilon^{\mu \nu}\}\rightarrow \{\varepsilon^{\mu \nu}\}= \{\varepsilon^{\mu \nu}\}+ \omega \left( \begin{array}{c c c c}{{0}}&{{f^{1}}}&{{f^{2}}}&{{f^{0}- f^{3}}}\\{{- f^{1}}}&{{0}}&{{0}}&{{- f^{1}}}\\{{- f^{2}}}&{{0}}&{{0}}&{{- f^{2}}}\\{{f^{3}- f^{0}}}&{{f^{1}}}&{{f^{2}}}&{{0}}\\ \end{array}\right)
{p_{1,k_1,k_2}(x,t) =\frac{(-1)^{k_2}\pi^{k_1+k_2}}{4^n (\pi\delta)^{n+k_1-1}}e^{-\frac{1}{4}d(x,t)^2}e^{-\kappa}I_{{n+k_1-1}}(\kappa) \left[1 +O(\delta)\right].}
\Bigl\{(E,V):V\subset H^0(C,E), \ \mbox{dim}(V)=4\Bigr\}\stackrel{1:1}\longleftrightarrow \Bigl\{(L,q):0\neq q\in K_{1,1}(C,L), \ \mbox{rk}(q)\leq 6\Bigr\},
\bar{\cal R}^{\frac{1}{2}\,z\,\omega_1\,,\,\frac{1}{2}\,z\,\omega_2}\,=\,c\left( \begin{array}{cccc}t_1 0 0 0 0 s^{-1}w 0 0 0 st_1t_2^{\,-1}0 0 0 0 -t_2^{\,-1}\end{array}\right)\,,
v\left( \overline{\theta}_{h}\right) =\int_{a}^{\overline{\theta}_{h}}\phi^{-1}\left( \int_{a}^{\xi}h-\int_{a}^{y}h\right) dy\geq\int_{a}^{\overline{\theta}_{h}}\phi^{-1}\left( \int_{y}^{\overline{\theta}_{h}}h\right) dy.
~\Pr(\{\mathfrak W(c)\ne \emptyset \}\cap \{W^{-\delta}(\theta_n^\prime,c)= \emptyset \}) = ~\Pr(\{\mu'g\ge 0, \forall\mu\in \mathcal M\}\cap \{\mu'(g-\delta \tau)< 0, \exists\mu\in \mathcal M\}) = ~\Pr(\{\mu'g\ge 0, \forall\mu\in \mathcal M\}\cap\{\mu'g< \delta\mu'\tau, \exists\mu\in \mathcal M\}).
\mathbf{P}\Bigl( \Delta X_{s}(y) >Q\,\bigl(y\,(t-s)\bigr)^{1/(1+\beta)}\log^{1/(1+\beta)}\bigl((t-s)^{-1}\bigr) &s\in(t-\varepsilon,t)\frac{c_{(\ref{nn.L3})}}{2}\,(t-s)^{1/2}\leq y\leq\frac{3c_{(\ref{nn.L3})}}{2}\,(t-s)^{1/2}\big|\,D_{\theta}\Bigr) =\,1
\mathcal F_{s_k,p}(u):=\mathcal{E}_{s_k,p}(u)^p+\int_\Omega F_{s_k}\, u\,dx\mbox{and}\mathcal F_{1,p}(u):=\mathcal{E}_{1,p}(u)^p+\int_\Omega F\,u\,dx,
\begin{matrix}1&5&0&6&3&4&2\\ 1&4&0&5&2&3&1\\ 1&4&0&4&2&3&1\\ 1&4&0&3&1&2&0\\ 1&4&0&3&1&2&0\\ 1&4&0&3&1&1&0\\ 1&4&0&3&1&1&0\\ \end{matrix}
\rho(t)<\epsilon -\pi<\rho'(0)<0 \textrm{supp}(\rho)\subset [0,(3r)^2) \rho'(t)\le 0 \rho''(t)<0 \ \textrm{on}\t< r^2 \rho'|_{t\in [r^2,(2r)^2]}\equiv -\frac{4}{3}\pi \ \ \Big(\Longrightarrow \min_{t\in [0,r^2]}\rho'(t)=-\frac{4}{3}\pi \Big)
\Delta H\equiv\Delta H_{L}+\Delta H_{T}=\frac{(g_{s}-1)\mu_{B}}{\hbar m_{e}ec^{2}}\frac{1}{r}\frac{\partial U(r)}{\partial r}L\cdot S\approx\frac{\mu_{B}}{\hbar m_{e}ec^{2}}\frac{1}{r}\frac{\partial U(r)}{\partial r}L\cdot S.
W:\begin{array}{ll}x_{k}' = \frac{\sum_{i,j=1}^{n,\nu}\gamma_{ijk}x_{i}y_{j}}{\left(\sum_{i=1}^{n}x_{i}\right)\bigl(\sum_{j=1}^{\nu}y_{j}\bigr)}begin{align*}4mm]y_{k}'= \frac{\sum_{i,j=1}^{n,\nu}\widetilde{\gamma}_{ijr}x_{i}y_{j}}{\left(\sum_{i=1}^{n}x_{i}\right)\bigl(\sum_{j=1}^{\nu}y_{j}\bigr)}.\end{array}
\Lambda^{b}= ( M_{\mathrm{P V}})^{b}\left( \frac{1 6 \pi^{2}}{b g^{2}}\right)^{T_{G}}\! \exp \left( - \frac{8 \pi^{2}}{g^{2}}\right) \prod_{i}Z_{i}^{- T ( R_{i})},
\sum_{\ell=0}^{n}\frac{1}{2(n-1)!}\sum_{j=0}^{\ell}\sum_{k=0}^{n-\ell-1}a_ja_{\ell-j}\frac{(n-1-k)!}{(n-\ell-1)!}B_{n-\ell-1, k}(-a_1, -2!a_2, \ldots) (2nj(\ell-j) + (\ell+1)(2n-\ell-2))
Z_{1/R}(1,0^{15}) \to \int \frac{d^2\tau}{\tau_2}\tau_2^{-5}|\eta(\tau)|^{-16}\left( Z^0_0(\tau)^{16}{Z^0_0(\tau)^*}^4-Z^0_1(\tau)^{16}{Z^0_1(\tau)^*}^4 -Z^1_0(\tau)^{16}{Z^1_0(\tau)^*}^4 \right)\ ,
A=\begin{pmatrix}\mathbf{a}_1 \vdots \mathbf{a}_m \end{pmatrix}=\begin{pmatrix}\mathbf{a}_1^1 \cdots \mathbf{a}_1^m \vdots \cdots \vdots \mathbf{a}_m^1 \cdots \mathbf{a}_m^m \end{pmatrix}, where \ \mathbf{a}_i^j=(a_{i 1}^j,\cdots,a_{i n_i}^j).
D_{\mu \mu}= - \frac{g_{\mu \mu}}{3 2 \pi^{2}}\left[ \frac{\alpha^{2}}{( 3 - d )}+ 3 2 \pi^{2}\alpha \left( \frac{\mu_{R}^{2}}{\lambda_{R}}+ \frac{1}{2}\varphi^{2}\right) - 1 6 \pi^{2}\frac{\alpha^{2}}{\lambda_{R}}\right] ,
-\mathrm{div}\left(F_*(x)^\gamma\,F(\nabla u)^{p-1}\, \nabla F(\nabla u)\right) =\beta^{p-1}\, \mathrm{div}\left(F_*(x)^{-\beta\,p+\beta-p+\gamma}\, x\right) &=\beta^{p-1}\, [N-\beta\,p+\beta-p+\gamma]\, F_*(x)^{-\beta\,(p-1)+\gamma-p},
y=A(x)=(\begin{matrix}a_{11}x_{1}+\cdot\cdot\cdot+a_{1n}x_{n}\\a_{21}x_{1}+\cdot\cdot\cdot+a_{2n}x_{n}\\ \vdots\\a_{m1}x_{1}+\cdot\cdot\cdot+a_{mn}x_{n}\end{matrix}).
\ \ \mbox{$c_i = q'_i$ for $i = 1, 2, \ldots, mu$, where $q'_i$ is defined in (\ref{q}).}\ \ \mbox{The condition EMU for $m\Delta_{\overline{t}, \overline{u}, \overline{s}}$ is not satisfied if and only if there exists $i$}\mbox{satisfying $0<i<mu$ and $e_i > i$.}
\operatorname{DET}\big | F_{n+i+j-2}^{(2)}(0) \big |_{1 \le i,j \le k}=\prod_{j=1}^{k}F_{n+j-1}^{(2)}(0) \frac{{k+1 \choose 2}!}{\prod\limits_{1 \le i\le k-1 \atop 1 \le j \le k}(n+i+j)}, D(n,k,b,1)= \prod_{j=1}^{k}F_{n+j}^{(2)}(b) \frac{\prod\limits_{1 \le i \le j \le k}(b+i+j-1)}{\prod\limits_{1 \le i \le k-1 \atop 1 \le j \le k}(n+b+i+j+1)}.
E_0(t)+E_1(t) \leq 8\delta_0\exp \big{(}CM\int_0^t \int_{\mathbb{R}^3}(|\widehat{f}(\tau,\eta)|+|\widehat{g}(\tau,\eta)| )d\eta d\tau \big{)}\leq 8\delta_0 \exp (CM^2),
b_{n,ht^2 k}= \mathbf{1}_{(n,v)=1}G \left(\frac{nf}{N}\right) \mathop{\sum_{\ell_1}\sum_{\ell_3}\sum_v}_{\substack{\ell_1 \ell_3 v = k (\ell_1\ell_3,v)=1}}\beta(\ell_1) \gamma(\ell_3) \alpha(v) G \left( \frac{vt}{V}\right)G \left( \frac{\ell_1}{L_1}\right)G \left( \frac{\ell_3}{L_3}\right)
\sum_{a=0}^{p-1}\sum_{b=0}^a\sum_{c=0}^b\binom{b}c^2\binom{a}b^2H_a\equiv_p\sum_{a=0}^{p-1}\sum_{b=0}^a\sum_{c=0}^b\binom{b}c^2\binom{a}b^2H_c.
\overline{V}_\cdot^{(m),i,n}=\overline{\Psi}_\cdot^{i}(\overline{\textbf{V}}^{(m-1),n})=\underline{\Psi}_\cdot^{i}(\overline{\textbf{V}}^{(m-1),n})=\underline{\Psi}_\cdot^{i}(\underline{\textbf{V}}^{(m-1),n})=\underline{V}_\cdot^{(m),i,n}.
\int_{\mathbb{R}^d}\chi^k (x,\omega)\partial_i \eta (x)dx =\int_{\mathbb{R}^d}U_i^k(\tau_y \omega)\int_0^1 \sum_{j\neq i}\partial_j\left (\frac{y_j}{t^d}\eta \left (\frac{y}{t}\right) \right) + \frac{y_i}{t^{d+1}}\partial_i \eta \left ( \frac{y}{t}\right) dtdy .
\| \Delta (w_m - w) \|_{L^2}= \| \beta(w) - \beta_m (w_m) \|_{L^2}&\le \| \beta(w) - \beta(w_m) \|_{L^2}+ \| \beta(w_m) - \beta_m (w_m) \|_{L^2}&\le \| \beta' \|_{L^\infty}\| w_m - w \|_{L^2}+ \| \beta_m - \beta \|_{L^\infty}.
\begin{aligned}u(t)=e^{taB}\varphi+\dfrac{1}{b-a}B^{-1}\Big[e^{tbB}-e^{taB}\Big](\psi -aB\varphi) &\dfrac{1}{1+b}B^{-2}\Big[\dfrac{1}{b-a}e^{tbB}-\dfrac{1+b}{(1+a)(b-a)}e^{taB}+\dfrac{1}{1+a}e^{-tB}\Big](\xi-(a+b)B\psi+abB\varphi),\end{aligned}
\mathcal{L}_{n_j}[4]=\sqrt{\frac{E_{n_j}}{512\, \mathcal N^2_{n_j}}}\Big(1 + W_1(n_j)^2-2W_2(n_j)^2-2W_3(n_j)^2 - 4W_4(n_j)^2+ o_\P (1) \Big).
\sqrt{\frac{2}{\pi}}e^{-in\theta}\phi_{2n}(e^{i\theta})=\sqrt{1+\frac{f_{2n}}{\ell_{2n}}}\varphi_n(\cos\theta)+i\sqrt{1-\frac{f_{2n}}{\ell_{2n}}}\sin\theta\,\psi_{n-1}(\cos\theta),-\pi\le \theta\le \pi,\,n\ge 1.
\epsilon_{RPA}({\bf{q}},\omega) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}+{\bf{q}}/2}-{\bar{n}}_{{\bf{k}}-{\bf{q}}/2}}{\omega - \frac{{\bf{k.q}}}{m}}
L^{\mathrm{DIF}}f(x) = c\sum_{1 \leq i < j \leq N}\left( \frac{\partial}{\partial x_i}- \frac{\partial}{\partial x_j}\right)^2 f(x)- (x_i-x_j)\left( \frac{\partial}{\partial x_i}- \frac{\partial}{\partial x_j}\right) f(x),
\frac{p_{\alpha}( p + k_1 )_{\beta}( p+ k )_{\gamma}}{p^2 ( p + k_1 )^2 ( p + k )^2}= \frac{p_{\alpha}( p + k_1 )_{\beta}}{p^4 ( p + k_1 )^2}\left[ ( p + k )_{\gamma}- \frac{2 ( k p ) p_{\gamma}}{p^2}\right] + \ldots \; .
q=\frac{-1}{8\pi^2a^2r^2}\sum_{l=1}^{\infty}(2l+1)\int_{0}^{\infty}{dzz\chi_\mu (z/a) \left[ \frac{e_l(z)}{s_l(z)}+\frac{e'_l(z)}{s'_l(z)}\right] \left\{s_l^{'2}(zx)-s_l^{2}(zx)\left[ 1-\frac{l(l+1)}{(-1)^iz^2x^2}\right] \right\}},
\begin{aligned}\log N(\varepsilon, \, \mathcal{F}_{n}, \, d_n) \leq \log\Big(\sum_{\lambda}N(\varepsilon, \, \mathcal{F}_{n,\lambda}, \, d_n) \Big) \leq 2\,n\,\varepsilon^2, \mbox{\ \and}\Pi_\lambda(\mathcal{F}_{n}^c) \leq \sum_{\lambda}\Pi_\lambda(\mathcal{F}_{n,\lambda}^c) \leq e^{-\frac{1}{2}\, D\,n\,\varepsilon^2}.\end{aligned}
m_2(L)=\left(\frac{1}{2}\right)^7\cdot (2^{m-l})^{\frac{1}{2}\cdot 1\cdot 1}\cdot (2^{m})^{\frac{1}{2}\cdot 2\cdot 1}\cdot (2^{l})^{\frac{1}{2}\cdot 2\cdot 1}=2^{\frac{3m+2l-14}{2}}\geq 2^9,
\begin{cases}\partial_tc_1 - d_1\Delta c_1 = -k_1c_1 + k_3c_2^{\alpha+1}, x\in\Omega, t>0, \partial_tc_2 - d_2\Delta c_2 = k_1\alpha c_1 + k_2c_2^{\alpha}c_3 - k_3(\alpha+1)c_2^{\alpha+1}, x\in\Omega, t>0, \partial_tc_3 - d_3\Delta c_3 = k_1c_1 - k_2c_2^{\alpha}c_3, x\in\Omega, t>0,\end{cases}
|-2 \int_{t}^{t+2(r+1)}ds \lambda''(s)(s-t) \left(\frac{(r^{2}-1-(s-t)^{2})}{\sqrt{\beta}(1+r^{2}+(s-t)^{2}+\sqrt{\beta})}\right)| &\leq C \sup_{x \geq t}\left(|\lambda''(x)|\right) \int_{t}^{t+2(r+1)}ds \frac{|s-t|}{1+r^{2}}&\leq C \sup_{x \geq t}\left(|\lambda''(x)|\right)
\nabla_{\dot{u}}(\chi'(\frac{\dot{u}}{||\dot{u}||_{\chi,u}}))d\mu_{u_t}=\chi''(\frac{\dot{u}}{||\dot{u}||_{\chi,u}})(\frac{\nabla_{\dot{u}}\dot{u}}{||\dot{u}||_{\chi,u}}-\frac{\dot{u}}{||\dot{u}||_{\chi,u}^2}\frac{d}{dt}||\dot{u}||_{\chi,u})d\mu_{u_t}
A_{\mu}d x^{\mu}= \frac{1}{2 g r}\left\{\begin{array}{l}{{\displaystyle{\tau_{\phi}^{n}\left[ H_{1}d r + ( 1 - H_{2}) r d \theta \right]}}}\\{{\displaystyle{- n \left[ \tau_{r}^{n}H_{3}+ \tau_{\theta}^{n}\left( 1 - H_{4}\right) \right] r \sin \theta d \phi}}}\\ \end{array}\right\}
\frac{d^k}{dx^k}p^{n,k}(x)=\left(1-\frac{k}{n}\right)\mu^{n,k}p^{n,k}(x)+\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}p^{n,k}(x), \frac{d^k}{dx^k}\theta_{p}^{n,k}(x)=\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}\theta_{p}^{n,k}(x).
\epsilon_{\mu}^{1}(n)=\frac{1}{\sqrt{2}}(\frac{-in_{1}n_{2}+1+n_{3}-n_{1}^{2}}{1+n_{3}},\frac{-n_{1}n_{2}+in_{1}^{2}+in_{3}^{2}+in_{3}}{1+n_{3}},-n_{1}-in_{2},0),
\left\| e^{\tau \phi}|x|^{-1}w \right\|_{L^2(A_{\rho, r}^+)}+ \left\| e^{\tau \phi}\nabla w \right\|_{L^2(A_{\rho,r}^+)}\leq C(n, \epsilon, \mu) \left\|e^{\tau \phi}|x| f\right\|_{L^2(A_{\rho,r}^+)}.