harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{P(R)-P_{\infty}}{\rho_{L}}=\frac{P_{B}-P_{\infty}}{\rho_{L}}-\frac{4\mu_{L}}{\rho_{L}R}\frac{dR}{dt}-\frac{2\sigma}{\rho_{L}R}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}(\frac{dR}{dt})^{2}
\Gamma_{i j}^{\sf t}= a \left( \frac{d a}{d{\sf t}}\right) \gamma_{i j}\; , \Gamma_{{\sf t}j}^{i}= \frac{1}{a}\left( \frac{d a}{d{\sf t}}\right) \delta_{j}^{i}\; , \Gamma_{j k}^{i}= \frac{\gamma^{i l}}{2}( \partial_{j}\gamma_{l k}+ \partial_{k}\gamma_{l j}- \partial_{l}\gamma_{j k}) \; .
{\rm D}\bar{J}(A_r,B_r,C_r)[(A'_r,B'_r,C'_r)] ={\rm tr}(A_r'^T \nabla_{A_r}\bar{J}(A_r,B_r,C_r)) +{\rm tr}(B_r'^T \nabla_{B_r}\bar{J}(A_r,B_r,C_r)) &+{\rm tr}(C_r'^T \nabla_{C_r}\bar{J}(A_r,B_r,C_r)),
|\tilde{F}(\xi_2' ,\eta_2')| = \Bigl| \frac{3}{2}\, \xi_1' \, \eta_1' + 2 \, \xi_2' \, \eta_2' \Bigr| \geq 2| \xi_2' \, \eta_2'| - \frac{3}{2}|\xi_1' \, \eta_1'| \geq 2^{-54}N_1^2,
{\cal L}_{\mathrm{s p i n}}^{( 2 )}[ B ] ={( 4 \pi )}^{- D}e^{2}\int_{0}^{\infty}{\frac{d T}{T}}e^{- m^{2}T}T^{2 - D}\int_{0}^{\infty}d \hat{T}\int_{0}^{1}d u_{a}\, J ( z , u_{a}, \hat{T}, D ) \, \nonumber \,
\begin{array}{r c l}{{\nabla_{1}F_{0 1}}}&{{=}}&{{- 2 \lambda^{1 / 2}/ R^{7}\left[ Q ( r^{3}- 3 r N^{2}) - P ( 3 r^{2}N - N^{3}) \right] \, ,}}\\{{}}&{{}}&{{}}\\{{\nabla_{1}{}^{\star}F_{0 1}}}&{{=}}&{{- 2 \lambda^{1 / 2}/ R^{7}\left[ P ( r^{3}- 3 r N^{2}) + Q ( 3 r^{2}N - N^{3}) \right] \, ,}}\\ \end{array}
P = \sum_t \left[ \binom{t}{2}\binom{r-2}{t-s}- \binom{r-t}{2}\binom{r-2}{r-t}\right] Q = \sum_t \left[ \binom{t}{2}\binom{r-2}{t}- \binom{r-t}{2}\binom{r-2}{r-t-2}\right].
\left.\frac{\delta\Omega^a}{\delta C^{\alpha b}}\right|_{C=\pi={\cal P}=\lambda=0}=T_\alpha\delta^a_b,\;\;\;\; \left.\frac{\delta\Omega^a}{\delta\pi^\alpha}\right|_{C=\pi=\lambda=0}= \varepsilon^{ab}{\cal P}_{\alpha b},
\mathcal{F}\left\{_{-\infty}D_{x}^{\alpha}f(x)\right\}=\mathcal{F}\left\{_{-\infty}D_{x}^{\alpha-n}f^{(n)}(x)\right\}& =(i\omega)^{\alpha-n}\mathcal{F}\left\{f^{(n)}(x)\right\}& =(i\omega)^{\alpha-n}(i\omega)^{n}F(\omega) & =(i\omega)^{\alpha}F(\omega).
\lim_{m_{\pi}^{2}\rightarrow 0}m_{\pi}^{2}\int d \vec{x}\omega_{\pi}^{( 1 )}= - \lambda \int d \vec{x}( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) \ne 0 .
\partial_{x}\Psi_{n}= \rho_{n}^{0}(x) - \langle \rho_{n}\rangle - \int_{0}^{t}\partial_{x}(\rho_{n}u_{n})(x,s)~ds begin{align*}1ex] = \rho_{n}^{0}(x) - \langle \rho_{n}\rangle + \rho_{n}(x,t) - \rho_{n}(x,0) = \rho_{n}- \langle \rho_{n}\rangle,
A_{j k}( x + y ) ={\frac{\biggl | \begin{matrix}{c_{2}A_{m k}( x )}&{c_{2}A_{m k}( y )}\\{C_{m k}( x )}&{C_{m k}( y )}\\ \end{matrix}\biggr |}{\biggl | \begin{matrix}{G_{m k}( x )}&{G_{m k}( y )}\\{1}&{1}\\ \end{matrix}\biggr |}},
\frac{\partial}{\partial x_i}\xi(x_1,\dots, x_{d-1}, y)=\frac{1}{mq}\Big(Qc_i- \frac{\mu_i\left<\vec{c},\vec{v}\right>x_i}{mr^{-2}}+\left<\vec{c},\vec{v}\right>\frac{\partial \phi}{\partial x_{i}}\Big).
\frac{1}{\sqrt{N}}\sqrt{\frac{\pi}{\pi\!+\!gN}}\Phi^{(1)}_0\; = \;\frac{1}{2\sqrt{\pi}}\left[ \frac{\theta}{N}-\arcsin\left( \frac{e^2}{2mc}\sqrt{\frac{N}{\pi\!+\!gN}}\Phi^{(1)}_0 \right) \right] \; .
\begin{array}{l}\partial_{\tau}X^{2}(0)=\partial_{\sigma}X^{1}(0)=0, \partial_{\tau}X^{1}(\pi)+\partial_{\tau}X^{2}(\pi)\cot(\pi\vartheta)=0, \partial_{\sigma}X^{2}(\pi)-\partial_{\sigma}X^{1}(\pi)\cot(\pi\vartheta)=0,\end{array}
\mathbb{I}^\omega(N\ell)(Z_1,\dots,Z_n) = \prod_i\mathbb{I}^\omega(Nk_i\ell_i)(Z_1,\dots,Z_n) &= \prod_i\mathbb{I}^1(k_i\ell_i)(Z_1^N,\dots,Z_n^N) &= \mathbb{I}^1(\ell)(Z_1^N,\dots,Z_n^N).
\begin{cases}-d_A\frac{d^2}{dy^2}a_0(y) = a_0(y)H_A\big(x^*,a_0(y),b_0(y)\big),\partial_ya_0(y) \leq 0, begin{align*}5pt]-d_B\frac{d^2}{dy^2}b_0(y) = b_0(y)H_B\big(x^*,a_0(y),b_0(y)\big), \partial_yb_0(y) \geq 0, begin{align*}5pt] a_0(0) =b_0(0).\end{cases}
W\left[f_\ell(k,R),\phi_\ell(k^\prime,R)\right] =\left(-k\right)^{\frac{1}{2}-\nu}F_\ell(k)+\left(k^2-k^{\prime2}\right)\int_0^R dr\, f_\ell(k,r)\phi_\ell(k^\prime,r)\,.
L^{\frak{g}}\Bigl(\Bigl(n-\frac{1}{2}\Bigr)\omega_1\Bigr)\Bigl|_{\frak{g}^\sigma}\simeq\bigoplus_{k=0}^{\infty}L^{\frak{so}(2,2l)}\Bigl(\Bigl(n+k-\frac{1}{2}\Bigr)\mu_1\Bigr)\boxtimes F^{\frak{so}(2n-2l+1)}(k\nu_1)
u(x,t) \ge\int_0^t\sum_{n=1}^\infty S(t-s) n^2 s_n \chi_{n^{-2/d}s_n^{-q/d}}(x_n)\,\d s &\ge\int_0^t\sum_{n=1}^\infty c_d n^2 s_n \chi_{n^{-2/d}s_n^{-q/d}}(x_n)\,\d s &= tc_d\sum_{n=1}^\infty n^2 s_n \chi_{n^{-2/d}s_n^{-q/d}}(x_n)
\left( \begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\ \end{array}\right) \longrightarrow \left( \begin{array}{c}{\widetilde \psi_{1}}\\{\widetilde \psi_{2}}\\ \end{array}\right) = \left( \begin{array}{c}{a}\\{c}\\ \end{array}\begin{array}{c c}{b}\\{d}\\ \end{array}\right) \left( \begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\ \end{array}\right) ,
Z_{\Lambda}( \gamma ) = \sum_{R_{1}, R_{2}}\tilde{B}_{R_{1}}^{S_{1}}\tilde{B}_{R_{2}}^{S_{2}}d_{R_{1}}^{1 - 2 r_{1}}d_{R_{2}}^{1 - 2 r_{2}}\{R_{1}, R , R_{2}\}
\mathbb{I}(X;Y,Z)-\mathbb{I}(X,Z) =\mathbb{I}(X,X+N_{eq})-\mathbb{I}(X;X+N_{E}) &=h(X+N_{eq})-h(X+N_{E})+\frac{1}{2}\log(\frac{\sigma_E^2}{\sigma_{DE}^2}).
\begin{cases}\hat{\lambda}^k=\lambda^k-(1-\gamma)(1-\tau^k)\beta^k(A{x}^{k}-b), \hat{x}^k=x^k+\frac{\tau^k(1-\tau^{k-1})}{\tau^{k-1}}(x^k-x^{k-1}), x^{k+1}\in\arg\min\limits_x \{f(x)+x^T\nabla_x\varphi^k(\hat{x}^k,\hat{\lambda}^k)+\frac{\beta^k}{2}\|x-\hat{x}^k\|_D^2+\frac{\sigma(1-\tau^k)}{2\tau^k}\|x-x^k\|^2\}.\end{cases}
\begin{cases}\lfloor r\rfloor:=\max\{k\in\mathbb Z:\k\leq r\}, \lfloor\!\lfloor r\rfloor\!\rfloor:=\max\{k\in\mathbb Z:\k< r\},\end{cases}\begin{cases}\lceil r\rceil:=\min\{k\in\mathbb Z:\k\geq r\}, \lceil\!\lceil r\rceil\!\rceil:=\min\{k\in\mathbb Z:\k>r\},\end{cases}
\left[ \langle A_{m_{0}}\rangle \Phi^{2}\vert_{F}~ + ~ \langle A_{g_{0}}\rangle \Phi^{3}\vert_{F}~ + ~ h . c . \right]{}~ + ~ 2 R e ( \langle F_{m_{0}}\rangle A^{2}~ + ~ \langle F_{g_{0}}\rangle A^{3})
\varphi (g_1 g^{-1}_2) \varphi (g_2 g_3) \overset{{\bf A6}}{=}\varphi (g_1) \varphi (g^{-1}_2) \varphi (g_2) \varphi (g_3) \overset{{\bf A2}}{=}\varphi (g_1) \varphi (g_3) \overset{(\ref{eq:anti3})}{=}\varphi (g_3 g_1) \,,
\beta\sum_{l=1}^{T-1}\theta_l\eta_T\alpha^{T-l-1}+\beta^2\sum_{l=1}^{T-1}\sum_{j=l+1}^T\theta_j^T\eta_T\alpha^{j+T-2l-2}+\eta_T=\frac{X_0}{\sigma^2}\beta\alpha^{T-1}
\sum_{i= 2}^4 U_{h,i}(y) \le M{\min\{|y-ht_1|^{-\left(\frac{N-1}{2}\right)},1\}}e^{(-1+\eta)|y- ht_1|}\sum_{i=2}^4 e^{-\frac{\eta}{2}| ht_i- ht_1|}&= 3 M{\min\{|y-ht_1|^{-\left(\frac{N-1}{2}\right)},1\}}e^{(-1+\eta)|y- ht_1|}e^{-\sqrt{2}\eta h}.
\Delta E_{\mathrm{u p}}= \frac{1}{T^{( 0 )}}\frac{4 e^{S^{( 0 )}}\operatorname{t a n}^{2}\frac{1}{2}( \phi_{\alpha}+ \phi_{\gamma}) + 4 e^{S^{( 0 )}}}{4 e^{2 S^{( 0 )}}\operatorname{t a n}^{2}\frac{1}{2}( \phi_{\alpha}+ \phi_{\gamma}) - 1},
\mathbf{v}_{S}^{S}= \mathbf{h}_{S}^1 + \beta \, \mathbf{P}_{S N}^{1}\, \mathbf{v}^{S}\mathbf{v}_{N \setminus S}^{S}= \mathbf{h}_{N \setminus S}^0 + \beta \, \mathbf{P}_{N \setminus S, N}^{0}\, \mathbf{v}^{S}.
I_2 =\bar{E}[H_0]^{\leq 4+m}+\bar{E}[P]^{\leq 7+m}, I_3 =\bar{E}[H_1,H_2]^{\leq 4}+\bar{E}[Q]^{\leq 7}+\bar{E}[H_0]^{\leq 7+m}+\bar{E}[P]^{\leq 10+m}.
V(x) = \frac{1}{N}\ln (x^2) + V^\prime(x) , \; \;V^\prime(x) = \frac{2}{N}\left[\mbox{K}_0\left(e\sqrt{\frac{N}{\pi}}|x|\right) +\ln\left(\frac{e}{2}\sqrt{\frac{N}{\pi}}\right) + \gamma \right] \; .
b_N'-a_{w(N)}' = b_{N}+1-\frac{n-2N+1}{2}-(a_{w(N)}-\frac{n-2w(N)+1}{2}) = b(f_{t_j})+1-a(j_t,p_{j_t})-\frac{2w(N)-2N}{2}= \mathrm{hk}(\mathfrak{m},t) -\mathrm{ht}(\mathfrak{m},t)+e \mbox{by Lemma \ref{lem count ht}}
\{a ( n_{1}n_{2}n k ; i ) , a ( n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}; i^{\prime})^{*}\}= N o r m ( n_{1}n_{2}n k ; i ) \delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}\delta_{n n^{\prime}}\delta_{k k^{\prime}}\delta_{i i^{\prime}}
R^* (n, D, \epsilon, \delta) &\overset{(a)}{=}R^* (n, D, \epsilon + \delta, 0) &\overset{(b)}{=}R_{X}(D) + \sqrt{\frac{V_{X}(D)}{n}}Q^{-1}(\epsilon + \delta) + O \left( \frac{\log n}{n}\right),
\mathbf{M}^{\pi_{\mathcal{S}(v)}}=\Bigl(\bigotimes_{e\in\mathcal{S}(v|s)}M^{\pi_e^*}\Bigr)\otimes\Bigl(\bigotimes_{e^\prime\in\mathcal{S}(v|t)}M^{\pi_{e^\prime}}\Bigr),
\langle F_{k,m}, F_{k,m}\rangle = \sum_i c_i^2 \langle f_i - \kappa_i f_i|V_2, f_i - \kappa_i f_i|V_2 \rangle + \sum_i d_i^2 2^k \langle f_i|V_2, f_i | V_2 \rangle + \sum_j e_j^2 \langle g_j,g_j \rangle.
G_{\mathrm{i r}}^{( n )}({\bf{r}}_{1}, \ldots ,{\bf{r}}_{n}) = \left. \frac{\delta^{( n )}{\cal{G}}_{\mathrm{i r}}\{\varphi \}}{\delta \varphi ({\bf{r}}_{1}) \ldots \delta \varphi ({\bf{r}}_{n})}\right|_{\varphi = 0}.
\#\{p\in\mathcal{A}{}\}=(1+o(1))\frac{\kappa_{\mathcal{A}}\#\mathcal{A}{}}{\log{X}}+S_1(z_4) &\ge (1+o(1))\frac{\kappa_{\mathcal{A}}\#\mathcal{A}{}}{\log{X}}(1-I_1-I_2-I_3-I_4-I_5-I_6-I_7-I_8-I_9).
\{\; \stackrel{1}{U^{\prime}}( x , \lambda ) \; , \; \stackrel{2}{U^{\prime}}( y , \mu ) \; \}= \, - \frac{1}{4}\frac{\sigma^{a}\otimes \sigma^{b}}{( 1 - \lambda^{2}) ( 1 - \mu^{2})}\, \{\; \lambda J_{0}^{a}( x ) + J_{1}^{a}( x ) \; , \; \mu J_{0}^{b}( y ) + J_{1}^{b}( y ) \; \}=
G_k = \bigcap_{l \,:\, \tilde{X}_{k-1}\leq X_{l-1}< \tilde{X}_k}\bigcap_{j=1}^J G_{j,l}\, , I_{j,l}= \bigcap_{r=1}^3 I_{j,l}^{(r)}\, , I_{k}= \bigcap_{l \, : \, \tilde{X}_{k-1}\leq X_{l-1}< \tilde{X}_k}\bigcap_{j=1}^J I_{j,l}\, .
\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{L}(x)\mapsto\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\prime\mu}}\psi_{L}^{\prime}(x^{\prime})=S\overline{\sigma}^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{L}(x)
\tilde{\mathcal{T}}^4\left\lbrace \begin{array}{l}\lbrack\widehat{x}_{0},\widehat{x}_{i}]=-\frac{i}{\kappa}\widehat{x}_{i},\qquad\lbrack\widehat{x}_{i},\widehat{x}_{j}]=0, \Delta(\widehat{x}_{\mu})=\widehat{x}_{\mu}\otimes1+1\otimes\widehat{x}_{\mu}. \end{array}\right.
I^1_{f,H}=_{\mathbb{C}\{f\}}\{1,\partial_{\lambda_1}F,...,\partial_{\lambda_k}F\}\Leftrightarrow H''_{f,H}=_{\mathbb{C}\{f\}}\{\omega,\partial_{\lambda_1}F\omega,...,\partial_{\lambda_k}F\omega\}
\omega\lVert h_{G_0}\rVert_{2,p}^p+(1-\omega)\lVert h_{H_0}\rVert_{2,p}^p \leq \omega\frac{a^{p/2-1}(1+\delta_{ak})}{1-\delta_{(a+1)k}}\lVert h_{G_0^c}\rVert_{2,p}^p+(1-\omega)\frac{(ak/s)^{p/2-1}(1+\delta_{ak})}{1-\delta_{(a+1)k}}\lVert h_{G_0^c}\rVert_{2,p}^p &=\frac{a^{p/2-1}\gamma(1+\delta_{ak})}{1-\delta_{(a+1)k}}\lVert h_{G_0^c}\rVert_{2,p}^p.
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = 0$ $u(L,t) = 0,$ $u_{x}(L,t) = h_2(t)$, in $(0,T)$, $v(0,t) =0,$ $v(L,t) = 0,$ $v_{x}(L,t) =0$, in $(0,T)$.\end{tabular}\right.
V^{(d)}_M =\prod^{d}_{j=1}\mathbb{Z}/m_j\mathbb{Z}=\left\{(x_1,\ldots,x_d)\,\left|\,x_j\in\{0,1,\ldots,m_j-1\},\j=1,2,\ldots,d\right.\right\}, (V^{(d)}_M)^{*}=\prod^{d}_{j=1}\frac{1}{m_j}\mathbb{Z}\Big/\mathbb{Z}=\left\{(v_1,\ldots,v_d)\,\left|\,v_j\in\left\{0,\frac{1}{m_j},\ldots,\frac{m_j-1}{m_j}\right\},\j=1,2,\ldots,d\right.\right\}.
\mathrm{P}^o_n =Q_2+\mathrm{P}\left(x_k\leq \frac{\epsilon_{1,k}}{\rho}\right) \leq\int^{\frac{\epsilon_{1,k}}{\rho}+\tilde{\epsilon}_{1,n}}_{\frac{\epsilon_{1,k}}{\rho}}f_{{x}_k}(y) dy+\mathrm{P}\left(x_k\leq \frac{\epsilon_{1,k}}{\rho}\right) = F_{{x}_n}\left(\frac{\epsilon_{1,k}}{\rho}+\tilde{\epsilon}_{1,n}\right).
H'(y) \, <\, \log(y+1) \,-\, \frac{1}{2(y+1)}\,-\, \log\left(\frac{y+1}{2}\right) \,-\,\log c -\frac{y}{y+1}\, = \, \frac{y\left[2 \log (2/c) - 2\right] + 2 \log (2/c) - 1}{2(y+1)}.
\lim_{n\rightarrow \infty}r_n=\lim_{n\rightarrow \infty}\frac{p_{n+1}}{p_n(1+\sqrt{1-p_{n+1}})}=\Bigl(\lim_{n\rightarrow\infty}\frac{p_{n+1}}{p_n}\Bigr)\Bigl(\lim_{n\rightarrow\infty}\frac{1}{1+\sqrt{1-p_{n+1}}}\Bigr)=\frac{p}{2}.
D=D_2Y^2+D_1Y+D_0, C_1C_2+C_0C_3=E_2Y^2+E_1Y+E_0, (k+1)(C_1C_2+C_0C_3)^2+(C_1^2+C_0C_2)(C_2^2+C_1C_3)=F_4Y^4+\cdots+F_0,
(-1)^{(N-1)d}\left[q^d\right]\prod_{i\in [N]}\bigg(\frac{\alpha_i-z_{N+1}}{\alpha_i-y}\bigg)^{b_i}\bigg(\frac{z_i}{z_i-y}\bigg)^{-1}\frac{z_i^{d+1}}{P'(z_i)}\prod_{i, j\in [N],\,\,\, i\neq j}(z_i-z_j)\bigg{|}_{\epsilon=0}.
U^T (n_i A_i) U - \epsilon U^T (n_i D_i) = U^T \tilde A U - \epsilon U^T \tilde F - \epsilon \tilde F^T U=\begin{bmatrix}U \epsilon \tilde F\end{bmatrix}^T\begin{bmatrix}\tilde A - I - I 0\end{bmatrix}\begin{bmatrix}U \epsilon \tilde F\end{bmatrix}
B_{k+1}(\boldsymbol{x}^1,\boldsymbol{x}^2,\lambda) = \log \int \exp(\boldsymbol{x}^1 \cdot \boldsymbol{\sigma}^1 + \boldsymbol{x}^2 \cdot \boldsymbol{\sigma}^2 + \lambda \boldsymbol{\sigma}^1 \cdot \boldsymbol{\sigma}^2) d \nu_N^b (\boldsymbol{\sigma}^1) d \nu_N^b (\boldsymbol{\sigma}^2)
\frac{\delta F \left[ f \right]}{\delta f ( k )}= \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon}\left( F \left[ f ( p ) + \varepsilon \delta_{D}^{( 4 )}( p \dot{+}( \dot{-}k ) ) \right] - F \left[ f ( p ) \right] \right) ,
C_{\theta}( \tau ) : = \lim_{\tau \rightarrow \infty}\Big [ \; \langle{\cal A}( \tau ){\cal B}( 0 ) \rangle_{0}^{\theta}\; - \; \langle{\cal A}( 0 ) \rangle_{0}^{\theta}\; \langle{\cal B}( 0 ) \rangle_{0}^{\theta}\; \Big ] \; .
f_{k,n}(z)=\frac{\alpha_{k,n}\beta_{k,n}}{\alpha_{k,n}+\beta_{k,n}-z},\alpha_{k,n}=\alpha_{k,n}(z),\beta_{k,n}=\beta_{k,n}(z),F_{n}(z)=(f_{1,n}\circ\cdot\cdot\cdot\circ f_{n,n})(z)
\|x_1-x_{n+1}\| \leq \|x_1-x_n\|+\|x_n-x_{n+1}\|< \left( 1-\frac{1}{2^n}\right)\delta+\frac{\delta}{2^{n+1}}&= \left( 1-\frac{1}{2^{n+1}}\right)\delta<\delta
\lefteqn{\int_{\{h>0\}}\,\int_0^{h(x)}f(x,z)\,dz\,dx}\overset{\eqref{ms}}=\int_{\{h>0\}}\,\int_0^{h(x)}\big(f(x,z)-f(x,h(x))\big)\,dz\,dx+\int_{\{h>0\}}h\,\kappa\,dx,
DL_{(g,f)}(0,h) = \ 2\Delta^f_g\left(2\Delta_g^f h+h\right)-(4\pi)^{-\frac{n}{2}}\int_{M}h e^{-f}dv_g -2\left\langle\nabla_g h, \nabla_g\left(2\Delta_g f+f-|\nabla f|^2+R_g\right)\right\rangle.
|D^m U_k(x)|\le \sum_{j=0}^k |g_j||D^m u_j(x)|\le C_m\sum_{j=0}^k|g_j|R_0^{-j}(j+|m|)^m &\le C_m \Big(\sum_{j=0}^\infty |g_j|^2\Big)^{1/2}\Big(\sum_{j=0}^\infty R_0^{-2j}(j+|m|)^{2|m|}\Big)^{1/2}\le C_m
F_{k}( \tau , q , \dot{q}, \ddot{q}) = 0 \qquad \dot{q}^{j}= \frac{d q^{j}}{d \tau}\qquad \ddot{q}^{j}= \frac{d^{2}q^{j}}{d \tau^{2}}\qquad j , k = 1 , \cdots , n \ .
\left\|\prod_{i=1}^{\frac{p+1}{2}}\chi(t/T)u_i\right\|_{L^2_{x,t}}\lesssim \|u_1\|_{X^{0,b'}_T}\prod_{i=2}^\frac{p+1}{2}N_i^{\frac{p-5}{2(p-1)}+\varepsilon}\|u_i\|_{X^{0,b'}_T},
D_{\mu\nu}(k)=\frac{-i}{k^{2}-\theta^{2}_{in}}\left(P_{\mu\nu}\,\,-\,\,i\theta_{in}\epsilon_{\mu\nu\rho}\frac{k^{\rho}}{k^{2}}\right)-i\lambda\frac{k_{\mu}k_{\nu}}{k^{4}}f(k^{2}),
\begin{array}{lll}\{x,-\}= P_z \frac{\partial}{\partial y}- P_y\frac{\partial}{\partial z}\,, \{y,-\}= P_x \frac{\partial}{\partial z}- P_z\frac{\partial}{\partial x}\,, \{z,-\}= P_y \frac{\partial}{\partial x}- P_x\frac{\partial}{\partial y}\,. \end{array}
{\tilde{f}}_i^{(h)}=\left({\tilde{g}}_k^{\pm 1}\right)^{(h)}\circ{\tilde{f}}_i^{(ph)}{\tilde{f}}_i^{(t)}={\tilde{f}}_i^{(pt)}\circ \left({\tilde{g}}_k^{\pm 1}\right)^{(t)},{\tilde{f}}_j^{(h)}= \left({\tilde{g}}_l^{\pm 1}\right)^{(h)}\circ{\tilde{f}}_j^{(ph)},{\tilde{f}}_j^{(t)}={\tilde{f}}_j^{(pt)}\circ \left({\tilde{g}}_k^{\pm 1}\right)^{(t)},
\psi_{n}^{\prime}=\psi_{R,n}+\frac{1}{2}\sum_{k\ne n}\sum_{j}\psi_{R,k}V_{kj}^{\prime}V_{jn}^{\prime}\frac{2d_{j}-d_{k}-d_{n}}{(d_{j}-d_{n})(d_{j}-d_{k})(d_{n}-d_{k})}
h_{L}= \sum_{1 \leq s \leq M}\sum_{\beta = 1}^{m}a_{s}^{\beta \dag}a_{s + 1}^{\beta}+ \frac{\lambda}{2}\sum_{1 \leq s \leq M}( n_{s}^{2}- n_{s}) , h_{R}= h_{L}^{\dag}
\left\|\sup_{\mathbb{A},\varphi}\left(\vec f\right)\right\|_{\dot f_{\infty,q}^s}=\left\|\left\{2^{js}g_j\right\}_{j\in\mathbb Z}\right\|_{L\dot F_{\infty,q}^{\tau}}\lesssim\left\|\left\{2^{js}h_j\right\}_{j\in\mathbb Z}\right\|_{L\dot F_{\infty,q}^{\tau}}=\left\|\vec f\right\|_{\dot F_{\infty,q}^s(\mathbb{A},\varphi)}.
\begin{array}{c}{s \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) ,}\\{a \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) .}\\ \end{array}
\sum_{j = 1}^ne^{u_i^{k}+v^{k-1}_j}|A_{ij}|^p + (\gamma/m) e^{u_i^{k}}= n \Leftrightarrow e^{u_i^{k}}= \frac{n}{\sum_{j = 1}^ne^{v^{k-1}_j}|A_{ij}|^p + (\gamma/m)}
\Theta(x,y):=\bigl(\varphi(x)-\varphi(y)\bigr)\left(\xi^2(x)-\xi^2(y)\right)\frac{\varphi(x)+\varphi(y)}{2\varphi(x)\varphi(y)}&\qquad\qquad\qquad\qquad\qquad-\bigl(\varphi(x)-\varphi(y)\bigr)^2\ \frac{\xi^2(x)+\xi^2(y)}{2\varphi(x)\varphi(y)}\leq \bigl(\xi(x)-\xi(y)\bigr)^2.
p^{9}=\left[\begin{array}{cc|c}1 2 1 2 3 2 3 1 3 \end{array}\right],p^{10}=\left[\begin{array}{cc|c}3 3 1 2 1 2 1 2 3 \end{array}\right],p^{11}=\left[\begin{array}{cc|c}3 1 1 2 3 2 1 2 3 \end{array}\right],p^{12}=\left[\begin{array}{cc|c}2 1 1 3 3 2 1 2 3 \end{array}\right],
\begin{aligned}d\in\lbrace 957\,013 , 1\,571\,953 , 1\,734\,184 , 3\,517\,689 , 4\,025\,909 , 4\,785\,845, 4\,945\,973 , 5\,562\,969 , 6\,318\,733 , 7\,762\,296 , 8\,070\,637 \rbrace\end{aligned}
\frac{1}{e(R)}\sum_{f \in E(R)}|\{g \in E(B) : f\cap g \neq \emptyset\}| = \frac{1}{e(R)}\sum_{g \in E(B)}|\{f \in E(R) : f\cap g \neq \emptyset\}| &\le \frac{1}{e(R)}\sum_{g \in E(B)}2 &= \frac{2e(B)}{e(R)}<2.
\mathcal{L}_{N S D_{B \wedge F}}= \mathcal{F}\left( A_{\mu}A^{\mu}, B_{\mu \nu}B^{\mu \nu}\right) - \frac{\chi \theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_{\mu}B_{\nu \rho}A_{\lambda}\; \; ,
d\omega_R(z) =(dR^{-1}(z))\wedge dR(z) &=-\omega_R(z)\wedge \omega_R(z) &=-\left(R^{-1}(z)aR^{-1}(z)t-R^{-1}(z)tR^{-1}(z)a\right)dz_1\wedge dz_2.
\left\langle \sum_{\hat{x}\in \hat{X}}\alpha_{\hat{x}}\hat{x}, \sum_{\hat{y}\in \hat{X}}\beta_{\hat{y}}\hat{y}\right\rangle = \sum_{\hat{x}, \hat{y}\in \hat{X}}\alpha_{\hat{x}}\beta_{\hat{y}}\lambda_{[\hat{x}, \hat{y}]}.
\frac{(\psi_t\psi^*_t)_0 -(\psi \psi^*)_0}{t}= \frac{\psi_t \psi^*_t - \psi \psi^*}{t}-\frac{1}{2}\left(\frac{\psi_t \psi^*_t - \psi \psi^*}{t}\right)1 & = \frac{\psi_t(\psi^*_t - \psi^*)}{t}+ \frac{(\psi_t - \psi) \psi^*}{t}-\frac{1}{2}\Big\{\frac{\psi_t(\psi^*_t -\psi^*)}{t}+ \frac{(\psi_t -\psi)\psi^*}{t}\Big \}1.
\begin{array}{rl}ds^2= (H_1H_5)^{-1/2}(dx_0^2+dx_1^2)+H_1^{1/2}H_5^{-1/2}(dx_2^2+...+dx_5^2) + (H_1H_5)^{1/2}(dx_6^2+...+dx_9^2)\end{array}
\|\nabla(S_{\mathcal W}v)\|_{L^1(\Omega)}= \sum_{i=1}^\infty \|\nabla(S_{\mathcal W}v)\|_{L^1(Q_i)}\le C \sum_{i=1}^\infty\sum_{Q_j \cap Q_i \ne \emptyset}(\|Dv\|(Q_i\cup Q_j)) \leq C \|Dv\|(\Omega).
Z_\Phi(\omega_s,f) =\int\limits_{(P_K\setminus\{0\})^2}|f(x,y)|_K^s\ |dxdy|=\sum_{a=1}^{\infty}\sum_{b=1}^{\infty}\int\limits_{\mathfrak{p}^aR_K^\times\times\mathfrak{p}^bR_K^\times}|f(x,y)|_K^s\ |dxdy| =\frac{\left( 1-q^{-1}\right) q^{-2+us}}{1-q^{-1+us}}.
\bigvee_{i=-n}^0 T_f^i \xi_{-\ell}^{\ell}\subseteq \xi_{-\ell}^{\ell + n j_p}, \bigvee_{i=N}^{N+n}T_f^i \xi_{-\ell}^{\ell}\subseteq \xi_{-\ell - (N + n) j_p}^{\ell - N j_p},
X'_1 = \mathcal{E}_{\pi|_{W'_1}}((T_1,\ldots, T_n)|_{W'_1}), X'_2 = \mathcal{E}_{\pi|_{W'_2}}((T_1,\ldots, T_n)|_{W'_2}), X''_1 = \mathcal{E}_{\pi|_{W''_1}}((T_1,\ldots, T_n)|_{W''_1}), \mbox{and}X''_2 = \mathcal{E}_{\pi|_{W''_2}}((T_1,\ldots, T_n)|_{W''_2}),
\begin{array}{l}{{-{\left( 1 -{\frac{2 M}{r}}\right)}^{- 1}\partial_{t}^{\, 2}{\frac{f}{r}}+{\left( 1 -{\frac{2 M}{r}}\right)}^{- 1}r^{- 2}\partial_{r^{*}}\left( \left( 1 -{\frac{2 M}{r}}\right) r^{2}{\left( 1 -{\frac{2 M}{r}}\right)}^{- 1}\partial_{r^{*}}{\frac{f}{r}}\right)}}\\{{- \frac{l \left( l + 1 \right)}{r^{2}}{\frac{f}{r}}- m^{2}{\frac{f}{r}}= 0 \; ,}}\\ \end{array}
\mu_1 =\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\biggl(\sum_{v_i h}f(v_i)-\sum_{v^j h}f(v^j)\biggr)^2 &=\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\biggl(\sum_{v_i \in h:f(v_i)>0}f(v_i)-\sum_{v^j \in h:f(v^j)<0}f(v^j)\biggr)^2.
m_{ajk}^{b}=2\,d \omega_{a}^{b}(Z_j,Z_k))= Z_j \omega_{a}^{b}(Z_k)- Z_k\omega_{a}^{b}(Z_j)- \omega_{a}^{b}\big{(}[Z_j,Z_k] \big{)}&=R_{jka}^b+\omega_{a}^{\rho}(Z_j) \omega^{b}_{\rho}(Z_k)-\omega_{a}^{\rho}(Z_k) \omega^{b}_{\rho}(Z_j).
\frac d{dt}\sum\limits_{\beta=0}^{1}\sum\limits_{|\alpha|=0}^{3-\beta}(\|\sqrt{B}\partial_\tau^\alpha\partial_y^\beta u\|_{L^2}^2 +\|\sqrt{A}\partial_\tau^\alpha\partial_y^\beta\tilde{h}\|_{L^2}^2)+C_0D(t) &\le CD(t)^{\frac 14}E(t)^{\frac54}+CD(t)^{\frac 12}E(t)+CD(t)E(t)^{\frac14}.
g(0)= f(0)^{\frac{1}{m}},~g(1)=\frac{f(1)}{mg(0)^{m-1}}, g(k)= \frac{1}{m g(0)^{m-1}}\left\{f(k)-\sum_{\sum_{i=1}^{m}k_i=k,~k_i<k}{k\choose k_1,\ldots,k_m}g(k_1)\cdots g(k_m)\right\},~k\geq 2.
\lim_{x^{+}\rightarrow 0}\, \langle 0 | T ( \varphi^{\dagger}( x ) \sigma_{1}\chi ( 0 ) ) | 0 \rangle \neq \langle 0 | \varphi^{\dagger}( x^{+}= 0^{+}, x^{-}, x^{\perp}) \sigma_{1}\chi ( 0 ) | 0 \rangle \ .
\sum_{1\leqslant s<t\leqslant 3}N_{P_{st}^{jl}}(z)\geqslant \sum_{i=j,l}(2\sum_{u=1}^3N^{(n)}_{(f^u,H_i),\leqslant k_i}(r)-3nN^{(1)}_{(f,H_i),\leqslant k_i}(r)) &+\sum_{i\ne j,l}(4N^{(1)}_{(f,H_i),\leqslant k_i}(r)-N(r,\nu_i)).
E(\lambda_2,\lambda_0)-E(\lambda^{\prime}_2,\lambda^{\prime}_0)=-8\pi\sqrt{2}\left(\frac{\pi^2}{12}+\log 2\right)\frac{\kappa^2}{\sqrt{\beta}}\left(\frac{\lambda_0}{\lambda^{\prime}_0}\right)^{\frac{1}{4}}e^{-\frac{8\pi}{\beta}(\lambda_0-{\lambda}^{\prime}_0)}
\sum_{m=0}^{j}\frac{1}{m!}\left. \frac{d^{m}}{d\rho^{m}}w_{k+n}(x_{1},...,x_{k+n}|\rho)\right\vert_{\rho=0}\times\prod_{s=1}^{k}T_{(j-m)+t_{s}}(x_{s})\prod_{s=1+k}^{n+k}U_{(j-m)+t_{s}}(x_{s})=0.
\xi = ([\xi_0^i: \xi_1^i])_{i=1}^N \vec{\xi}= (\xi_i)_{i=1}^N \omega = ([\omega_0^{(k,l)}: \omega_1^{(k,l)}])_{1 \leq k<l\leq N}\vec{\omega}= (\omega_{(k,l)})_{1 \leq k<l\leq N}
\begin{array}{l}{{{\cal O}[ V ] = V^{i_{1}\ldots i_{s}}\sum_{k = 0}^{s}{\frac{( - 1 )^{k}}{k ! \left( k +{\frac{D - 4}{2}}\right) ! ( s - k ) ! \left( s - k +{\frac{D - 4}{2}}\right) !}}\partial_{i_{1}}\cdots \partial_{i_{k}}\phi^{*}\; \partial_{i_{k + 1}}\cdots \partial_{i_{s}}\phi ,}}\\{{g_{i_{1}i_{2}}V^{i_{1}\ldots i_{s}}= 0}}\\ \end{array}
\big\{\sigma^1 > n \big\}\, \subseteq \, \big\{S^1_{K^1_m}- R^1_{K^1_m+1}> d m + z \,:\, m=0, \dots, T^1_n \big\}\,, \big\{\sigma^2 > n \big\}\, \subseteq \, \big\{S^2_{K^2_m}+ R^2_{K^2_m+1}< d m + z \,:\, m=0, \dots, T^2_n \big\}\,.
\begin{array}{rcl}A_e(\zeta) \!\!\! :=\!\!\! \dfrac{1+\Lambda_{1e}^{2}(\zeta)}{(1-\Lambda_{1e}^2(\zeta))^{2}}, A_j(\zeta) := \Lambda_{1j}(\zeta)\dfrac{1+\Lambda_{1j}^2(\zeta)}{(1-\Lambda_{1j}^2(\zeta))^2}, \j=s,s+s_*, B_j(\zeta) \!\!\! :=\!\!\! \dfrac{\Lambda_{1j}(\zeta)}{(1-\Lambda_{1j}^{2}(\zeta))^2}, j=e,s,s+s_*. \end{array}
\mathcal{S}=\Delta+4\mu^2|x+Rt(1-t)e_1|^2+2\mu R(1-2t)(x_1+Rt(1-t)) +(t^2-t+\frac16)R^2-\frac{R^2(1-2t)}{16\mu}, \mathcal{A}=-4\mu(x+Rt(1-t)e_1)\cdot\nabla-2\mu n,
p+q =S_{p}+S_{q}+\overrightarrow{V_{p}}+\overrightarrow{V_{q}}, pq =S_{p}S_{q}+\left\langle \overrightarrow{V_{p}},\overrightarrow{V_{q}}\right\rangle_{\mathbb{L}}+S_{p}\overrightarrow{V_{q}}+S_{q}\overrightarrow{V_{p}}+\overrightarrow{V_{p}}\times_{\mathbb{L}}\overrightarrow{V_{q}},
H(Y_i | Y^{i-1},{\bf S}) \leq \sum_{y^{i-1}}\prod_{j=1}^i \min( \frac{1}{2}, \frac{B_{a,j}}{| \mathcal S(y^{j-1})|})H( \min( \frac{1}{2}, \frac{B_{a,i-1}}{| \mathcal S(y^{i-2})|}) ).
X_0^{(1)}=O,X_n^{(1)}= \bigl(X_{n-1}^{(1)}\bigr)^2 A_{-1}+X_{n-1}^{(1)}A_0+A_1,\n\ge 1, X_0^{(2)}=O,X_n^{(2)}= A_{-1}+A_0 X_{n-1}^{(2)}+A_1 \bigl(X_{n-1}^{(2)}\bigr)^2,\n\ge 1.
= \sum_{i=0}^{k-1}(-1)^i \left[(n-k+1) \binom{n-i}{n-k}+(n-i)\binom{n-i-1}{n-k}\right] \, c_i(\omega) \wedge \omega^{n-i}\wedge \omega_{FS}^{n-k}+ (-1)^k (n-k+1) \, c_k(\omega) \wedge \omega^{n-k}\wedge \omega_{FS}^{n-k}.
dZ_t=\frac 1N\sum\limits_{j=1}^Ne^{-\zeta_t^{(j)}}f^{(j)}(e^{\zeta_t^{(j)}}Z_t)dt+\frac 1N\sum\limits_{i=1}^m\Big(\sum\limits_{j=1}^Nc_i^{(j)}\Big)Z_t\circ dW_t^{(i)},
P_{\pm}( n ) \equiv | \langle n | \widetilde{\alpha_{\pm}}\rangle |^{2}= 2 | \widetilde{N_{\pm}}( | \alpha |^{2}\mathrm{,}\varphi ) |^{2}e^{- | \alpha |^{2}}\frac{| \alpha |^{2 n}}{n !}( 1 \pm \cos ( 2 n \varphi ) ) .
(1-\frac{\epsilon}{3}) \sum\limits_{\substack{I^{c}_{k+l}(n)}}b(i_{1};n)b(i_{2};n)\cdots b(i_{k+l};n) \leq \sum\limits_{\substack{I^{c}_{k+l}(n)}}\tilde{b}(i_{1},i_{2},...,i_{k+l};n) & \leq (1 + \frac{\epsilon}{3}) \sum\limits_{\substack{I^{c}_{k+l}(n)}}b(i_{1};n)b(i_{2};n)\cdots b(i_{k+l};n).
eval \left( e^{\mathcal{B}z}\right) = eval \left( \sum_{n =0}^{\infty}\frac{\mathcal{B}^{n}}{n!}z^{n}\right) = \sum_{n = 0}^{\infty}\frac{eval \left( \mathcal{B}^{n}\right)}{n!}z^{n}= \sum_{n = 0}^{\infty}\frac{B_{n}z^{n}}{n!}= \frac{z}{e^{z}-1}.
\tilde{r}_{a}( \Delta , h ) = \lim_{\stackrel{N \to \infty}{h \mathrm{f i x e d}}}\frac{\tilde{m}_{a}}{\tilde{m}_{F}}= \lim_{\stackrel{N \to \infty}{h \mathrm{f i x e d}}}\frac{\tilde{G}_{a}}{\tilde{G}_{F}}, \qquad a \in \{B 1 , B 2 , C \}